ARITHMETIC QUANTUM CHAOS ON
LOCALLY SYMMETRIC SPACES
LIOR SILBERMAN
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF
MATHEMATICS
NOVEMBER, 2005
c Copyright by Lior Silberman, 2005.
All rights reserved.
Abstract
We report progress on the equidistribution problem of automorphic forms on locally
symmetric spaces. First, generalizing work of Zelditch-Wolpert we construct a representation theoretic analog of the micro-local lift, showing that (under a technical condition of
non-degeneracy) every weak-* limit of the generalized Wigner measures associated to a
sequence of Maass forms with divergent spectral parameters on a locally symmetric space
Γ\G/K can be lifted to a measure on the homogeneous space Γ\G which is invariant by
a maximal split torus A in G. Secondly, we consider the case where G ' PGLd (R) and
Γ < G is a lattice associated to a division algebra over Q of prime degree d. When the
measures are associated to Hecke-Maass eigenforms, we generalize the work of BourgainLindenstrauss to show that every non-trivial a ∈ A acts with positive entropy on each
ergodic component of the lifted measure. Applying recent measure rigidity results of
Einsiedler-Katok we find that the limit measure must be the Haar measure on Γ\G. In particular we prove that a non-degenerate sequence of Hecke-Maass forms becomes equidistributed in Γ\G/K in the semiclassical limit.
These results arise from joint work with Akshay Venkatesh of the Courant Institute of
Mathematical Sciences.
iii
Acknowledgments
I am privileged to have been supervised by Peter Sarnak, who has shown me a lot of
mathematics (and patience), and has directed me toward even more mathematics. Membership in his mathematical community is a great experience.
The community in the department has given me a lot, both mathematically and personally. I have lived and learned here, and among many names would like to mention those of
Adrian Banner, Brent Doran, Harald Helfgott, Jordan Ellenberg and Andreea Nicoara. But
this does not diminish the contribution of everyone else, who are always willing to give of
their time to discuss mathematics. I would also like to thank the assistance of the staff of
the department over the last five years, in particular Scott Kenney and Jill LeClair.
This thesis represents joint work with Akshay Venkatesh, a former fellow student at
Princeton. Collaborating with him has been a pleasure, and I hope to continue our work
in the future. This work started with our conversions at the Clay Mathematics Institute
summer school in Toronto during June 2004, but has its foundations at an ersatz seminar
convened by Prof. Sarnak that May to study the then-recent work of Elon Lindenstrauss on
the arithmetic quantum chaos problem on hyperbolic surfaces. We have also benefited from
conversations with Joseph Bernstein, Manfred Einsiedler, Erez Lapid, Elon Lindenstrauss,
Dima Jakobson and Andre Reznikov.
I have immensely enjoyed the graduate student community at Princeton, a community
that has made my stay here even more memorable. Special thanks to Stefan van der Elst,
Thomas Horine, Peter Keevash, Christopher Mole and Juliet O’Brien, and to Eric Adelizzi,
Morten Kloster and Manish Vachharajani. Also not forgotten are my fellow Officers on
various Graduate College House Committees, and fellow members of the Graduate Student
Government and other student organizations who give much to this community. Last, but
certainly not least, I would like to thank Joanna Karczmarek. You have been wonderful.
To my parents, Giora and Silvia, you have brought me up to where I am today. I
dedicate this thesis to you. I have missed you and my brother Noam, but your support and
encouragement have always been with me.
iv
To my parents
v
Contents
Abstract
iii
Acknowledgments
iv
Chapter 1. Introduction
1.1. General starting point: the semi-classical limit on Riemannian manifolds
1.2. Hyperbolic surfaces and automorphic forms
1.3. Quantum unique ergodicity on locally symmetric spaces
1.4. Arithmetic QUE in the higher-rank case
1.5. The Main Theorem
1
1
3
5
8
10
Chapter 2. Notation and Fundamentals
2.1. Division Algebras
2.2. The Real Group
2.3. The Adelic group and its quotients, Components
2.4. The p-adic groups and Hecke operators
12
12
14
16
19
Chapter 3. The Micro-Local Lift
3.1. Introduction and motivation
3.2. More on Real Lie Groups
3.3. Representation-Theoretic Lift
3.4. Cartan invariance of quantum limits
20
20
22
25
29
Chapter 4.
4.1.
4.2.
4.3.
4.4.
The Method of Hecke Translates I: Tubular Neighbourhood, Translates,
and Diophantine Geometry
Overview
Algebra in tubular neighbourhoods
Diophantine Geometry of Division Algebras
Intersections of Hecke Translates
The Method of Hecke Translates II: Geometry and Harmonic Analysis on
the Building
5.1. The buildings of GLn and PGLn .
5.2. Hecke eigenfunctions – the local contribution
5.3. Split Tori
5.4. The proof of theorem 5.0.1
36
36
37
38
40
Chapter 5.
Bibliography
44
44
50
52
56
58
vi
CHAPTER 1
Introduction
1.1. General starting point: the semi-classical limit on Riemannian manifolds
Let Y be a compact Riemannian manifold, with the associated Laplace operator ∆ and
Riemannian measure dρ. An important problem of harmonic analysis (or mathematical
physics) on Y is understanding the asymptotic behaviour of eigenfunctions of ∆ in the
large eigenvalue limit. The equidistribution problem asks whether for an eigenfunction ψ
with a large eigenvalue λ, |ψ(x)| is approximately constant on Y . This can be approached
“pointwise” and “on average” (bounding kψkL∞ (Y ) and kψkLp (Y ) in terms of λ, respectively), or “weakly”: asking whether as |λ| → ∞, the probability measures defined by
dρ
.
dµ̄ψ (y) = |ψ(y)|2 dρ(y) converge in the weak-* sense to the “uniform” measure vol(Y
)
p
For example, Sogge [29] derives L bounds for 2 ≤ p ≤ ∞, and in the special case of
Hecke eigenfunctions on hyperbolic surfaces, Iwaniec and Sarnak [15] gave a non-trivial
L∞ bound. Here we will consider the weak-* equidistribution problem for a special class
of manifolds and eigenfunctions.
A general approach to the weak-* equidistribution problem was found by Šnirel0 man
[28]. To an eigenfunction ψ he associates a distribution µψ on the unit cotangent bundle
S ∗ Y projecting to µ̄ψ on Y . Generalizing the “Wigner function” formalism of statistical physics (see, e.g. [11, pp. 58–59] or the original account [32]), this construction (the
“microlocal lift”) proceeds using the theory of pseudo-differential operators and has the
2
property that, for any sequence {ψn }∞
n=1 ⊂ L (Y ) with eigenvalues λn tending to infinity,
any weak-* limit of the µn = µψn is a probability measure on the unit tangent bundle S ∗ Y ,
invariant under the geodesic flow. Since any weak-* limit of the µn projects to a weak-*
limit of the µ̄n , it suffices to understand these limits; Liouville’s measure dλ on S ∗ Y plays
here the role of the Riemannian measure on Y .
This construction has a natural interpretation from the point of view of semi-classical
physics. The geodesic flow on Y describes the motion of a free particle (“billiard ball”).
S ∗ Y is (essentially) the phase space of this system, i.e. the state space of the motion. In
this setting one calls a function g ∈ C ∞ (S ∗ Y ) an observable. The state space of the
quantum-mechanical billiard is L2 (Y ), with the infinitesimal generator of time evolution
−∆. “Observables” here are bounded self-adjoint operators B : L2 (Y ) → L2 (Y ). Decomposing a state ψ ∈ L2 (Y ) w.r.t. the spectral measure of B gives a probability measure on
the spectrum of B (which is the set of possible “outcomes” of the measurement). The expectation value of the “measuring B while the system is in the state ψ” is then given by the
matrix element hBψ, ψi. In the particular case where B is a 0th-order pseudo-differential
1
1.1. GENERAL STARTING POINT: THE SEMI-CLASSICAL LIMIT ON RIEMANNIAN MANIFOLDS
2
operator with symbol g ∈ C ∞ (S ∗ Y ), we think of B as a “quantization” of g, and any such
a B will be denoted Op(g).
We can now describe Šnirel0 man construction: it is given by µψ (g) = hOp(g)ψ, ψi.
This indeed lifts µ̄ψ , since for g ∈ C ∞ (Y ) we can take Op(g) to be multiplication by g. If
ψ is taken to be an eigenfunction then, asymptotically, this construction does not depend
on the choice of “quantization scheme,” that is to say, on the choice of the assignment
g 7→ Op(g). Indeed, if B1 , B2 have the same symbol of order 0, and −∆ψ = λψ (i.e. “ψ
is an eigenstate of energy λ”) then one has h(B1 − B2 )ψ, ψi = O(λ−1/2 ).
On a philosophical level we expect our quantum-mechanical description to approach
the classical one at the limit of large energies. We will not formalize this idea (the “correspondence principle”), but depend on it for motivating our main question, whether ergodic
properties of the classical system persist in the semi-classical limit of the “quantized” version:
2
P ROBLEM 1.1.1. (Quantum Ergodicity) Let {ψn }∞
n=1 ⊂ L (Y ) be an orthonormal basis
consisting of eigenfunctions of the Laplacian.
(1) What measures occur as weak-* limits of the {µ̄n }? In particular, when does
wk-*
µ̄n −−−→ dρ hold?
n→∞
(2) What measures occur as weak-* limits of the {µn }? In particular, when does
wk-*
µn −−−→ dλ hold?
n→∞
D EFINITION 1.1.2. Call a measure µ on S ∗ Y a (microlocal) quantum limit if it is a
weak-* limit of a sequence of distributions µψn associated, via the microlocal lift, to a
sequence of eigenfunctions ψn with |λn | → ∞.
In this language, the main problem is classifying the quantum limits of the classical
system, perhaps showing that the Liouville measure is the unique quantum limit. As formalized by Zelditch [35] (for surfaces of constant negative curvature) and Colin de Verdière
[4] (for general Y ), the best general result known is still:
T HEOREM 1.1.3. (Šnirel0 man-Zelditch-Colin de Verdière) Let Y be a compact mani2
fold, {ψn }∞
n=1 ⊂ L (Y ) an orthonormal basis of eigenfunctions of ∆, ordered by increasing eigenvalue. Then:
P
wk-*
−−→ dλ holds with no further assumptions.
(1) N1 N
n=1 µn −
N →∞
(2) Under the additional assumption that the geodesic flow on S ∗ Y is ergodic, there
wk-*
exists a subsequence {nk }∞
−−→ dλ.
k=1 of density 1 s.t. µnk −
k→∞
wk-*
C OROLLARY. For this subsequence, µ̄nk −−−→ dρ.
k→∞
It was proved by Hopf [14] that the geodesic flow on a manifold of contant negative curvature is ergodic. This was generalized to the case of non-constant negative sectional curvature by Anosov [1]. In that situation Rudnick and Sarnak [25] conjecture a simple situation:
1.2. HYPERBOLIC SURFACES AND AUTOMORPHIC FORMS
3
C ONJECTURE 1.1.4. (Quantum unique ergodicity) Let Y be a compact manifold of
strictly negative sectional curvature. Then:
(1) (QUE on Y ) The µ̄n converge weak-* to the Riemannian measure on Y .
(2) (QUE on S ∗ Y ) dλ is the unique quantum limit on Y .
We remark that [25] also give an example of a hyperbolic 3-manifold Y , a point P ∈ Y ,
1/4−
and a sequence of eigenfunctions ψn with eigenvalues λn , such that |ψn (P )| λn
. The
point P is a fixed point of many Hecke operators, and behaves in a similar fashion to the
poles of a surface of revolution. This remarkable phenomenon does not seem to contradict
Conjecture 1.1.4. The scarcity of such points and their higher-dimensional analogues will
play an important role in the analysis of Chapters 4 and 5.
One difficulty associated with this problem is that of multiplicity of the spectrum. For a
negatively curved manifold Y , it is believed that the multiplicities of the Laplacian ∆ acting
on L2 (Y ) are quite small, i.e. the λ-eigenspace has dimension λ . This question seems
extremely difficult even for SL2 (Z)\H, and no better bound is known than the general
O(λ1/2 / log(λ)), valid for all negatively curved manifolds. The freedom associated with
high degeneracy might allow the construction of “scarred” eigenfunctions which become
concentrated on singular subsets of Y .
However, even lacking information on the multiplicities, it transpires that in many natural instances we have a distinguished basis for L2 (Y ). In that context, it is then natural
to ask whether Conjecture 1.1.4 can be resolved with respect to this distinguished basis.
Since it is believed that the ∆-multiplicities are small, this modification is, philosophically, not too far from the original question. However, it is in many natural cases far more
tractable. The main example is that of congruence quotients of symmetric spaces, where
the distinguished basis is that of Hecke eigenforms. This is discussed further below, after
introducing the important work on surfaces of constant negative curvature.
1.2. Hyperbolic surfaces and automorphic forms
The quantum unique ergodicity question for hyperbolic surfaces has been intensely
investigated over the last two decades. We recall some important results.
Zelditch’s work [34, 36] on the case of compact surfaces Y of constant negative curvature provided a representation-theoretic alternative to the original construction of the
microlocal lift via the theory of pseudo-differential operators. It is well-known that the
universal cover of such a surface Y is the upper half-plane H ' PSL2 (R)/SO2 (R), so
Y = Γ\H for a uniform lattice Γ < G = PSL2 (R). Then the SO2 (R) ' S 1 bundle X = Γ\PSL2 (R) Y is isomorphic to the unit cotangent bundle of Y . In this
parametrization,
flow on S ∗ Y is given by the action of the maximal split torus
t/2 the geodesic
e
A=
on X from the right. Zelditch’s explicit microlocal lift starts with
e−t/2
the observation that an eigenfunction ψn (considered as a K-invariant function on X) can
(n)
be thought of as the spherical vector ϕ0 in an irreducible G-subrepresentation of L2 (X).
He then constructs another (“generalized”) vector in this subrepresentation, a distribution
1.2. HYPERBOLIC SURFACES AND AUTOMORPHIC FORMS
(n)
4
δ (n) , and shows that the distribution given by µψn (g) = δ (n) (gϕ0 ) for g ∈ Cc∞ (X) agrees
(up to terms which decay as the λn grow) with the microlocal lift. He then observes that
the distribution µψn is exactly annihilated by a differential operator of the form H + rJn
where H is the infinitesimal generator of the geodesic flow, J a certain (fixed) secondorder differential operator, and λn = − 41 − rn2 . It is then clear that any weak-* limit taken
as |λn | → ∞ will be annihilated (in the sense of distributions) by the differential operator
H, or in other words be invariant under the geodesic flow. Wolpert [33] made Zelditch’s
approach self-contained by showing that the limits are positive measures without using
pseudo-differential calculus. One advantage of this approach is that it is based entirely on
the right action of PSL2 (R) on X, and in particular respects structures on X that commute
with this action.
When Γ < PSL2 (R) is a so-called congruence lattice, there are additional operators
acting on functions on X = Γ\PSL2 (R): for each prime p (except for a finite set of “ramified primes” depending on Γ) there exists an operator Tp : L2 (X) → L2 (X) commuting
with the right action of PSL2 (R). It arises from a PSL2 (R)-equivariant foliation of X into
p + 1-regular graphs (the “Hecke Foliation”; almost all the leafs are trees), and Tp is the
graph Laplacian operator on each leaf. In particular Tp also acts on functions on Y and
commutes with ∆. These are the Hecke operators, and they all commute. The joint eigenfunctions of all the Tp and ∆ are called Hecke-Maass forms. They encode considerable
arithmetic information and are central objects of study in analytic number theory. They are
the prototypical examples of the more general automorphic forms considered below, and
will form our distinguished basis.
Much more results are known on the quantum chaos problem for Hecke-Maass forms.
One example is the Iwaniec-Sarnak result mentioned above. Of interest to us, a quantum
limit µ∞ arising from micro-local lifts of these eigenfunctions is called an arithmetic quantum limit. The arithmetic quantum chaos problem (posed in general below generality) is
the classification of such limits.
The study of arithmetic quantum limits started with the seminal result of Rudnick and
Sarnak [25], that a weak-* limit µ̄∞ coming from µ̄ψn attached to Hecke-Maass eigenforms
cannot be supported on a finite union of closed geodesics. One way to think of this result
is as stating that arithmetic quantum limits cannot be too singular, due to the behaviour of
Hecke eigenfunctions along the Hecke foliation: if a Hecke eigenfunction is too large on
a piece of the geodesic, it must also be somewhat large at translates of this piece by the
Hecke foliation. A clever choice of the prime p (depending on the closed geodesics under
consideration) assured that the translates were all disjoint, and a contradiction was obtained
to the fact that µ̄n (Y ) = 1.
Using many places at once, Bourgain and Lindenstrauss [3] obtained a significantly
stronger result: they showed that the µ∞ -measure of an -neighbourhood of a piece of a
geodesic must decay at least as fast as 2/9 . In the language of ergodic theory, they have
shown that any a ∈ A acts on every ergodic component of an arithmetic quantum limit µ∞
with positive entropy.
1.3. QUANTUM UNIQUE ERGODICITY ON LOCALLY SYMMETRIC SPACES
5
Building on this result, Lindenstrauss [19] proved a theorem classifying A-invariant
measures on X satisfying the positive entropy property as well as a “recurrence” property
easily satisfied by arithmetic quantum limits: such measures must be proportional to the
Haar measure dx. This (almost) answered the arithmetic QUE problem for congruence
surfaces:
T HEOREM 1.2.1. (Lindenstrauss) Let Y = Γ\H be a congruence quotient of the hyperbolic plane, and let µ∞ be an arithmetic quantum limit on X = Γ\PSL2 (R) ' S ∗ Y .
Then µ∞ = c · dx for some c ∈ [0, 1]. If Γ is co-compact (ie arising from a quaternion
algebra) then c = 1.
The main theorem of this thesis is a generalization of this theorem to division algebras
of degree greater than 2. As the basic strategy of the proof remains the same, we shall
record it here:
2
(1) Start with a sequence of Hecke-Maass forms {ψn }∞
n=1 ⊂ L (Y ) and their associated measures {µ̄n }, converging to a limit measure µ̄∞ .
(2) Passing to a subsequence, lift them to measures µn on the bundle X Y converging to a limit µ∞ which is invariant under a subgroup A < PSL2 (R). The lift
is constructed in way which respects the Hecke-eigenform condition.
(3) Using the Hecke correspondence, show that an arithmetic limit µ∞ cannot be too
singular, in that it must have positive entropy w.r.t. the action of elements a ∈ A.
(4) Apply a measure-rigidity theorem to show that µ∞ ∝ µHaar .
We should remark that the special case of congruence surfaces can also be attacked from
a different direction. A beautiful formula of Watson [30] relates the triple-product integral
µ̄n (ψm ) to a special value of an L-function attached to ψn × ψ̄n × ψm . Equidistribution
of the µ̄n would follow from fast enough decay of this special value, which in turn would
follow from an appropriate Generalized Riemann Hypothesis. The rate of decay obtained
this way from the GRH is best possible (that was shown in [22]). Moreover, conditioned on
the GRH the formula permits an evaluation of the normalization of µ∞ in the non-compact
case (the “escape-of-mass” problem) giving µ̄∞ (Y ) = 1 in that case as well. In fact, to
show that c = 1 it suffices to give a sub-convex bound in the eigenvalue aspect for the
Rankin-Selberg L-function L( 21 , ψn × ψ̄n ).
1.3. Quantum unique ergodicity on locally symmetric spaces
Lindenstrauss’s clear exposition [18] of the Zelditch-Wolpert microlocal lift actually
considers the case of Y = Γ\ (H × · · · × H) for an irreducible lattice Γ in PSL2 (R) ×
· · · × PSL2 (R). The natural candidates for ψn there are not eigenfunctions of the Laplacian
alone, but rather of all the “partial” Laplacians associated to each factor separately. Set now
G = PSL2 (R)h , K = SO2 (R)h , X = Γ\G, Y = Γ\G/K, and take ∆i to be the Laplacian
operator associated with the ith factor (so that C [∆1 , . . . , ∆h ] is the ring of K-bi-invariant
differential operators on G). Assume that ∆i ψn + λn,i ψn = 0, where limn→∞ λn,i = ∞ for
each 1 ≤ i ≤ h separately. Generalizing the Zelditch-Wolpert construction, Lindenstrauss
(n)
obtains distributions δ (n) ϕ0 on X, projecting to µ̄ψn on Y , and so that every weak-* limit
1.3. QUANTUM UNIQUE ERGODICITY ON LOCALLY SYMMETRIC SPACES
6
of these (a “quantum limit”) is a finite positive measure invariant under the action of the
full maximal split torus Ah .
It is important to note that the lift is to the bundle X Y which is not the unit
cotangent bundle of Y , in fact much smaller: it is 3h-dimensional whereas S ∗ Y would be
4h − 1-dimensional. Moreover, the limits obtained are invariant by a much larger subgroup
(of dimension h) rather than the 1-dimensional geodesic flow of S ∗ Y . The last fact is
not entirely surprising, in that we have assumed that ψn are eigenfunctions of a family of
h independent commuting differential operators. This phenomenon will repeat with our
general representation-theoretic lift below.
Following the construction, Lindenstrauss proposes the following version of QUE, also
due to Sarnak:
P ROBLEM 1.3.1. (QUE on locally symmetric spaces) Let G be a connected semi-simple
Lie group with finite center. Let K be a maximal compact subgroup of G, Γ < G a lattice,
2
X = Γ\G, Y = Γ\G/K. Let {ψn }∞
n=1 ⊂ L (Y ) be a sequence of normalized eigenfunctions of the ring of G-invariant differential operators on G/K, with the eigenvalues w.r.t.
the Casimir operator tending to ∞ in absolute value. Is it true that µ̄ψn converge weak-* to
the normalized projection of the Haar measure to Y ?
We remark that a central character should certainly play no role in this problem, and it
is possible to consider instead the case where G is a reductive group, and ψn ∈ L2 (Y, ωn )
is a sequence of eigenfunctions which transform under unitary central characters ωn ∈ Ẑ
where Z is the center of G. The measures µ̄ψn are then probability measures on YZ = Z\Y ,
since |ψn (y)|2 is Z-invariant. We take this point of view from now on. We therefore also
will use the notation XZ = Z\X.
Chapter 3 is devoted to showing the first result of this thesis (Theorem 1.3.2 below): the
construction of the microlocal lift in this setting. We will impose a mild non-degeneracy
condition on the sequence of eigenfunctions (see Section 3.3.2; the assumption essentially
amounts to asking that all eigenvalues tend to infinity, at the same rate for operators of the
same order.)
With K and G as in Problem 1.3.1, let A be as in the Iwasawa decomposition G =
N AK, i.e. A = exp(a) where a is a maximal abelian subspace of p. (Full definitions
are given in Section 2.2). For G = GLn (R) and K = On (R), one may take A to be the
subgroup of diagonal matrices with positive entries. Let π : XZ → YZ be the projection.
We denote by dx the G-invariant probability measures on XZ , and by dy the projection of
this measure to YZ .
The content of the Theorem that follows amounts, roughly, to a “G-equivariant microlocal lift” on Y . While our definitions have been specific to GLn (R), the proof will not
make any use of this fact. The theorem holds for any reductive group, with appropriate
generalization of the non-degeneracy assumption.
1.3. QUANTUM UNIQUE ERGODICITY ON LOCALLY SYMMETRIC SPACES
7
T HEOREM 1.3.2. Let ψn ⊂ L2 (Y, ωn ) be a non-degenerate sequence of normalized
eigenfunctions, whose eigenvalues approach ∞. Then, after replacing ψn by an appropriate subsequence, there exist functions ψ̃n ∈ L2 (X, ωn ) and distributions µn on XZ such
that:
(1) The projection of µn to YZ coincides with µ̄n , i.e. π∗ µn = µ̄n .
(2) Let σn be the measure |ψ̃n (x)|2 dx on XZ . Then, for every g ∈ Cc∞ (XZ ), we have
limn→∞ (σn (g) − µn (g)) = 0.
(3) Every weak-* limit σ∞ of the measures σn (necessarily a positive measure of mass
≤ 1) is A-invariant.
(4) (Equivariance). Let E ⊂ EndG (C ∞ (XZ )) be a C-subalgebra of bounded endomorphisms of C ∞ (XZ ), commuting with the G-action. Noting that each e ∈ E
induces an endomorphism of C ∞ (Y ), suppose that ψn is an eigenfunction for E
(i.e. Eψn ⊂ Cψn ). Then we may choose ψ̃n so that ψ̃n is an eigenfunction for E
with the same eigenvalues as ψn , i.e. for all e ∈ E there exists λe ∈ C such that
eψn = λe ψn , eψ̃n = λe ψ̃n .
We first remark that the distributions µn (resp. the measures σn ) generalize the constructions of Zelditch (resp. Wolpert). Although, in view of (2), they carry roughly equivalent information, it is convenient to work with both simultaneously: the distributions µn
are canonically defined and easier to manipulate algebraically, whereas the measures σn
are patently positive and are central to the arguments of Chapter 5.
P ROOF. In Section 3.3.1 we define the distributions µn . (In the language of Definition
3.3.3, we take µn = µψn (ϕ0 , δ)).
Claim (1) is established in Lemma 3.3.6.
In Section 3.3.2 we introduce the non-degeneracy condition. Proposition 3.3.13 defines
ψ̃n and establishes the claims (2) and (4). (Observe that this Proposition establishes (2)
only for K-finite test functions g. Since the extension to general g is not necessary for any
of our applications, we omit the proof.)
Finally, in section 3.4 we establish claim (3) (Corollary 3.4.9) by finding enough differential operators annihilating µn .
R EMARK 1.3.3.
(1) It is important to verify that non-degenerate sequences of eigenfunctions exist. We
mostly consider here the case compact quotients XZ , for which [7, 6] show that a
positive proportion of the unramified spectrum lies in every open subcone of the
Weyl chamber (for definitions see Theorem 3.2.7 and the discussion in Section
3.1). A similar statement for finite-volume arithmetic quotients Y should follow
from the recent techniques of [20]. Earlier, [23, Thm. 5.3] has treated the case of
SL3 (Z)\SL3 (R)/SO3 (R).
(2) We shall use the phrase non-degenerate quantum limit to denote any weak-* limit
of σn , where notations are as in Theorem 1.3.2. Note that if σ∞ is such a limit,
then claim (2) of the Theorem shows that there exists a subsequence (nk ) of the
1.4. ARITHMETIC QUE IN THE HIGHER-RANK CASE
(3)
(4)
(5)
(6)
8
integers such that σ∞ (g) = limnk →∞ µnk (g) for all g ∈ Cc∞ (XZ ). Depending on
the context, we shall therefore use the notation σ∞ or µ∞ for a non-degenerate
quantum limit.
It is not necessary to pass to a subsequence in Theorem 1.3.2. See Remark 3.3.12.
It is likely that the A-invariance aspect of Theorem 1.3.2 could be established by
standard microlocal methods; however, the equivariance property does not follow
readily from these methods and is absolutely crucial for our application. For us
the invariance arises from the action of the ring of invariant differential operators,
which is a polynomial algebra in r generators where r = dim A.
The measures µn all are invariant by the compact group M = ZK (a). In fact,
Theorem 1.3.2 should strictly be interpreted as lifting measures to XZ /M rather
than XZ .
Theorem 1.3.2 admits a natural geometric interpretation. Informally, the bundle
X/M → Y may be regarded as a bundle parameterizing maximal flats in Y , and
the A-action on X/M corresponds to “translation along flats.” We refer to [27,
Sec. 5.3] for a further discussion of this point.
The existence of the microlocal lift already places a restriction on the possible weak-*
limits of the measures {µ̄n } on YZ . For example, the A-invariance of µ∞ shows that the
support of any weak-* limit measure µ̄∞ must be a union of maximal flats.
More importantly, Theorem 1.3.2 allows us to pose a new version of the problem:
P ROBLEM 1.3.4. (QUE on homogeneous spaces) In the setting of Problem 1.3.1, is the
G-invariant measure on XZ the unique non-degenerate quantum limit?
R EMARK 1.3.5. When formulating Conjecture 1.1.4, Rudnick and Sarnak could rely
on part (2) of Theorem 1.1.3 to guarantee that the conjectured unique limit is, in fact, a
quantum limit. As the geodesic flow on the locally symmetric spaces we consider is also
ergodic, this argument extends to the context of Problem 1.3.1 (at least when Y is compact).
While our work on the arithmetic case outlined in the next section implies (in certain special
cases) the analogous fact for Problem 1.3.4, it is likely that a direct proof is possible. This
is especially so in the compact quotient case, when the main problem is of technical nature:
showing that most of the spectrum is non-degenerate in our sense. This should follow from
the results of [7].
1.4. Arithmetic QUE in the higher-rank case
The main result of this thesis is the resolution of Problem 1.3.4 for certain higher rank
symmetric spaces, in the context of arithmetic quantum limits. We first recall their definition and significance.
As in the special case of congruence quotients of the hyperbolic plane, the situation
of having (something close to) a distinguished basis occurs for Y = Γ\G/K and Γ ⊂ G
a congruence lattice. For almost all primes p there exists a commutative algebra Hp of
operators acting on L2 (X) arising from a discrete foliation. These operators commute
with each other and with the G-invariant differential operators. This distinguished basis
1.4. ARITHMETIC QUE IN THE HIGHER-RANK CASE
9
is obtained by simultaneously diagonalizing the action of the Hecke operators. Precise
definitions of the foliation and the Hecke operators in the case under consideration are given
in Section 2.3; in any case, we refer to quantum limits arising via the lift from subsequences
of Hecke-Maass forms as arithmetic quantum limits. A special case of Problem 1.3.4 is
then:
C ONJECTURE 1.4.1. (Arithmetic QUE) Let ψn ∈ L2 (Y, ωn ) be a (non-degenerate ?)
sequence of Hecke-Maass eigenforms. Is dy the unique weak-* limit of the µ̄n ? Is the
G-invariant measure on XZ the unique (non-degenerate ?) arithmetic quantum limit?
In Chapters 4 and 5 we study the properties of arithmetic quantum limits in the case
where Γ arises from the multiplicative group of a division algebra of prime degree d over
Q. The case d = 2 is the theorem of Lindenstrauss discussed above.
For brevity, we state the result in the language of automorphic forms; in particular, A is
the ring of adéles of Q. Detailed discussion of the construction may be found in Chapter 2.
Let D/Q be a division algebra of prime degree d, and let G = D× be the associated
general linear group. Assume that G is split at ∞, ie that G = G(R) ' GLd (R). Let
Kf be an open compact subgroup of G(Af ) such that X = G(Q)\G(A)/Kf contains a
single G-orbit. Then there exists a discrete subgroup Γ < G(R) such that X = Γ\G, and
Section 2.3 develops a Hecke algebra H(R) acting on functions on X via Hecke operators
at almost all primes. There exists an abundance of open compact subgroups Kf satisfying
the condition above. For example, quotients of G by congruence subgroups associated to
Eichler orders are of this type (see Lemma 2.3.7 for details).
The subgroup Γ projects to a co-compact lattice in G/Z ' PGLd (R) where Z is the
center of G. As in the previous section we let XZ = ZΓ\G denote the resulting compact
homogeneous space of PGLd (R), A denote the maximal split torus of diagonal matrices in
G, and ωn denote unitary characters of Z.
The second result of this thesis is:
T HEOREM 1.4.2. Let ψ̃n ∈ L2 (X, ωn ) be a sequence of H(R) eigenforms on X such
that the associated probability measures σn on XZ converge weak-* to an A-invariant
probability measure σ∞ . Then every a ∈ A \ Z acts on every A-ergodic component of σ∞
with positive entropy.
P ROOF. This is essentially a rephrasing of Theorem 5.0.1, where the uniformity of the
estimate means it carries over to weak-* limits. By that theorem we find an η > 0 such
that for any fixed C ⊂ Ma Aa as defined in Section 4.2 and small enough we have for
all x ∈ XZ that σ∞ (xB(C, )) η . For a proof that this bound implies that a acts with
positive entropy see [17, Sec. 8]. While written for the case of quaternion algebras (d = 2),
that discussion readily generalizes to our situation by modifying its “Step 2” to account for
the action of a on the Lie algebra – compare our Section 4.2 and the definitions at the start
of [17, Sec. 7].
1.5. THE MAIN THEOREM
10
R EMARK 1.4.3. The statement of Theorem 5.0.1 gives a direct bound on singular behaviour of the σn . Its proof follows the ideas of Rudnick-Sarnak and Bourgain-Lindenstrauss: translating the set xB(C, ), which is an -neighbourhood of a piece C of a “generalized geodesic” (Levi subgroup) by the Hecke correspondence at many places we show
that it must have small σn -measure.
1.5. The Main Theorem
Following the strategy proposed above, we now state and prove the main result of this
thesis:
T HEOREM 1.5.1. Let YZ ' Γ\PGLd (R)/SOd (R) be a compact locally symmetric
space, where the lattice Γ is associated to an Eichler order in a division algebra of the
2
prime degree d over Q, split over R. Let {ψn }∞
n=1 ⊂ L (YZ ) be a non-degenerate sequence
of Maass forms which are also eigenforms of the Hecke algebra H(R) of Section 2.3. Then
the associated probability measures µ̄n converge weak-* to the normalized Haar measure
on Y , as their lifts µn converge weak-* to the normalized Haar measure dx on XZ =
Γ\PGLd (R).
In other words, then the normalized Haar measure is the unique non-degenerate arithmetic quantum limit in this case.
P ROOF. In fact, the proof generalizes to the case where ψn ∈ L2 (Y, ωn ) for central
characters ωn . The case d = 2 is Lindenstrauss’s Theorem quoted as Theorem 1.2.1 above,
and we will thus assume d ≥ 3. Passing to a subsequence, let ψn ∈ L2 (Y, ωn ) be a nondegenerate sequence of Hecke-Maass forms on Y such that µ̄n → µ̄∞ weakly. Passing to
a subsequence, let ψ̃n and σn be as in Theorem 1.3.2 such that σn → σ∞ weakly and σ∞
lifts µ̄∞ . Then σ∞ is a non-degenerate arithmetic quantum limit on XZ . By Theorem 1.4.2,
σ∞ is an A-invariant probability measure on XZ such that every a ∈ A \ Z acts on every
A-ergodic component of σ∞ with positive entropy. Then [9, Th. 4.1(iv)] shows that σ∞ has
a unique ergodic component, µHaar .
R EMARK 1.5.2.
(1) The assumption that Γ is associated to an Eichler order is of technical nature. The
result certainly holds for Hecke eigenfunctions on an adelic double-coset space
X̃ = G(Q)\G(A)/Kf where G is the group of invertible elements of a Q-division
algebra which is R-split. In general, however, such a space is a disjoint union of
several components of the form X = Γ\G where Γ is a congruence subgroup, and
we would like to consider eigenfunctions on the components themselves. It is not
clear, whoever, whether we can form a sufficiently large explicit Hecke algebra
acting on such a component. For this one is interested in the set of primes p such
that each leaf of the p-Hecke foliation (defined in Section 2.3) is contained in a
single component X of X̃.
(2) In all likelihood it is possible to obtain a version of Theorem 1.3.2 for degenerate sequences as well. The resulting quantum limits µ∞ would be invariant
under subtori A1 < A depending on the degeneracy of the limit parameter ν̃∞ .
1.5. THE MAIN THEOREM
11
A measure rigidity theorem generalizing [19], requiring only invariance under a
one-parameter subgroup, positive entropy and recurrence would then allow us to
drop the non-degeneracy assumption and resolve the AQUE problem for division
algebras of prime degree d ≥ 3.
(3) We expect the techniques developed for the proof of Theorem 1.5.1 will generalize
at least to some other locally symmetric spaces, the case of D being the simplest;
but there are considerable obstacles to obtaining a theorem for any arithmetic locally symmetric space at present. For a general G, the results of Chapter 4 can
be generalized to show that the intersections will be controlled by a proper Qsubgroup. However, this subgroup can be quite large, making the analysis on the
building much more difficult even when G(Qp ) ' GLd (Qp ). Moving from the
building of GLd to buildings of other types might present difficulties of its own.
(4) It is also possible to prove results for the case where G is split, i.e. isomorphic to
GLd over Q. The proof is essentially the same except that the measure rigidity
results of [10] are used instead. Since in that case the quotient is not compact
this does not address the escape-of-mass question. Somewhat surprisingly, however, the normalization of the measure is already controlled by the degenerate
Eisenstein series. Hence a sub-convexity result for the Rankin-Selberg L-function
would control the escape as in the case of GL2 .
CHAPTER 2
Notation and Fundamentals
We define here standard notation and recall basic facts about division algebras over the
rationals and the real, p-adic and adelic Lie groups associated to them.
2.1. Division Algebras
2.1.1. Central simple algebras and their general linear groups. Let K be an infinite
field, D(K)/K a finite-dimensional central simple K-algebra, i.e. a K-algebra with no twodef
sided ideals and center K. Then for any field L/K, D(L) = D(K) ⊗K L is a central simple
L-algebra of the same dimension. It is easy to see ([31, Prop. IX-1-2]) that such an algebra
must be of the form D(K) ' Mn (H) where H is a central division algebra over K. In
particular, dimK D(K) = n2 dimK H.
By the Cayley-Hamilton theorem, every element of D(L) is algebraic over L. In particular, if L/K is algebraically closed then L is the unique division algebra over L, and hence
D(L) ' Md (L) for some d ≥ 1. We then have:
dimK D(K) = dimL D(L) = d2 .
In particular, the number d only depends on D(K) and is called the degree of D(K). It also
follows that the dimension of every central simple K-algebra is a square. An field L/K for
which D(L) ' Md (L) is said to split D(K). Alternatively, we say that D(K) splits over L.
2
Fixing a linear basis {ui }di=1 ⊂ D(K), we note that it is also a basis over L of D(L) for
P2
any L/K. We can then write any x ∈ D(L) uniquely in the form di=1 xi ui . Working in
2
2
this co-ordinate system (x, y) 7→ (x · y)i is then a bilinear map K d × K d → K and hence
there exist aijk ∈ K such that
! d2
!
!
d2
d2
d2
X
X
X
X
xj uj
y k uk =
aijk xi yj ui .
j=1
i=1
k=1
j,k=1
2
N OTATION 2.1.1. We will use the notation {xi }di=1 to denote the co-ordinates of any
2
x ∈ D(L) w.r.t. our basis, {xi (g)}di=1 for the co-ordinates of g ∈ D× (L).
We remark that an L-automorphism of Mn (L) is given by a change of basis, i.e. by
conjugation by an element of GLn (L). It follows that if A is an L-algebra isomorphic to
Mn (L) then the pullback of the map det : Mn (L) → L to A is well-defined independently
of the choice of isomorphism.
12
2.1. DIVISION ALGEBRAS
13
FACT 2.1.2. [31, Prop. XI-2-6]
(1) There exists a map νD(K) : D(K) → K such that for any L/K where D splits we
have det D(K) = νD(K) .
(2) There exists a polynomial νD ∈ K[x1 , . . . , xd2 ] of degree d depending on the
choice of basis such that νD(L) (x) = νD (x1 , . . . , xd2 ) for any L/K (not necessarily split) and any x ∈ D(L). Here νD(L) : D(L) → L is the map constructed in
the first part.
(3) The maps νD(K) and νD are both known as the reduced norm.
Passing to a split extension shows that for any L/K, x ∈ D(L) is invertible iff νD (x) 6=
0, in which case it is possible to compute the co-ordinates of x−1 by polynomial functions
def
of its co-ordinates {xi } and νD (x)−1 . We now identify G = D× with the set of solutions
to νD (x1 , . . . , xd2 ) · x0 = 1 in d2 + 1-dimensional affine space. Since the multiplication
operation in D is also polynomial in the co-ordinates (and νD (xx0 ) = νD (x)νD (x0 )) this
makes G into a linear algebraic group defined over K, with the maps xi : G → A1 all
algebraic and defined over K. We will thus supplement our previous notation by using
x0 (g) ∈ L× to denote the inverse of the reduced norm of g ∈ D× (L).
The center of G is precisely the invertible elements of the center of D, i.e. the invertible
elements of the ground field, and we set Gad = G/ZG . We also set G1 = {g | νD (g) = 1}.
This is a Zariski-closed subgroup.
If K̄ is an algebraic closure of K then we have D(K̄) ' Md (K̄). It is then clear that
G(K̄) ' GLd (K̄), Gad (K̄) ' PGLd (K̄) and G1 (K̄) ' SLd (K̄). In particular, the last
isomorphism shows that G1 is simply connected as an algebraic group.
2.1.2. Algebras over local and global fields. Let Dp be a central simple algebra over
the field Qp . An order Op ⊂ Dp is a finitely-generated Zp -subalgebra which spans Dp .
Equivalently, it is a compact open Zp -subalgebra.
Now let D be a central simple algebra over Q. An order is a finitely-generated Zsubalgebra O ⊂ D(Q) which contains a basis for D(Q) over Q. An order is maximal if
it is not properly contained in another order. These exist (e.g. by Zorn’s Lemma) and we
choose a maximal order O ⊂ D(Q). It is a torsion-free abelian group of rank d2 . We can
2
thus fix a Z-basis {ui }di=1 ⊂ O once and for all. The structure coefficients aijk with respect
to this basis then all lie in Z.
For a place v ∈ |Q|, the local field Qv is an extension of Q and we set Dv = D(Qv ) =
D(Q) ⊗Q Qv and Gv = G(Qv ) = Dv× . In particular we denote G = G∞ . We say that D
splits at v ∈ |Q| if it splits over Qv .
FACT 2.1.3. (for proofs, see[31]) For a finite prime p of Q let Op denote the (topological) closure of O in D(Qp ).
×
×
(1) νDp (Dp× ) = Q×
p and hence νDp (Op ) = Zp .
(2) relatively compact multiplicatively closed subset T ⊂ Dp is contained in a maximal order. In particular K < Gp is maximal compact iff K = Rp× for a maximal
order Rp ⊂ D(Qp ).
2.2. THE REAL GROUP
14
(3) Op is a maximal order in D(Qp ), in particular a maximal compact subring;
the maximal orders of D(Qp ) are all conjugate; Md (Zp ) is a maximal order of
Md (Qp ).
(4) For almost all p we have D(Qp ) ' Md (Qp ). We then have Gp ' GLd (Qp ) can fix
an isomorphism ϕp : Gp → GLd (Qp ) such that ϕp (Op× ) = GLd (Zp ).
2
(5) We have Op = ⊕di=1 Zp ui . In particular xi (g) ∈ Zp for every g ∈ Op× .
Following claim (3) we fix the maximal compact subgroups Kp = Op× of Gp . The first
part of claim (4) is that D splits at almost all places. We let R0 denote the set of finite places
were D does not split.
2.1.3. Division algebras of prime degree. We now make the assumption that D/Q is
a division algebra and that its degree d is a prime and at least 3. If K/Q is a field extension,
then D(K) is a central simple K-algebra, hence a matrix algebra over a central division
H/K. We then have
d2 = dimK D(K) = dimH D(K) · dimK H.
As d is prime and dimH D(K) and dimK H are both squares, there are two possibilities. If
H = D(K) (i.e. D(K) is also a division algebra), we say that D ramifies over K. Otherwise
we have H = K, that is D splits over K.
If D ramifies at v ∈ |Q| then Gad
v is compact. Since R itself and Hamilton’s quaternions
H are the unique central division algebras over R = Q∞ , we see that D can ramify at ∞
only if d = 2, which does not hold by assumption. Denoting G = G∞ , this amounts to
saying that G ' GLd (R).
2.2. The Real Group
We conform to the notation of [16].
We are considering the group G ' GLd (R), obtained as D(R)× where D/Q is a division algebra of prime degree, split at ∞. We choose the Cartan involution Θ(g) = {}t g −1
for G, and let K = Od (R) be the Θ-fixed maximal compact subgroup, Z ' R× the center
of G. Let S = Z\G/K be the symmetric space, with xK ∈ S the point with stabilizer KZ.
We fix a G-invariant metric on S. To normalize it, we observe that the tangent space at the
point xK ∈ S is identified with p/z (see below), and we endow it with the Killing form:
Let g = Lie(G) ' Md (R), and let θ(X) = −X t denote the differential of Θ, giving the
Cartan decomposition g = k ⊕ p with k = Lie(K) (the anti-symmetric matrices) and p the
symmetric matrices. The pairing hX, Y i = Tr(XY t )− d1 Tr(X) Tr(Y ) is Ad G-equivariant
and positive semi-definite (positive definite on gss = [g, g], the subalgebra of matrices of
trace 0). Its isotropic subspace is precisely the center Lie(Z) = ZLie(G) , where Z is the
connected component of the center (in general we would take Z to be the split component
of the torus ZG (R)). We fix a maximal abelian subalgebra a ⊂ p, the subalgebra of diagonal
matrices isomorphic to Rd .
2.2. THE REAL GROUP
15
We denote by aC the complexification a ⊗R C; we shall occasionally write aR for a
for emphasis in some contexts. We denote by a∗ (resp. a∗C ) the real dual (resp. the complex dual) of a; again, we shall occasionally write a∗R for a∗ . For ν ∈ a∗C , we define
Re(ν), Im(ν) ∈ a∗R to be the real and imaginary parts of ν, respectively.
For α ∈ a∗ set gα = {X ∈ g | ∀H ∈ a : ad(H)X = α(H)X},
∆(a : g) = {α ∈ a∗ \ {0} | gα 6= {0}}
and call the latter the (restricted) roots of g w.r.t. a. The subalgebra g0 is θ invariant,
and hence g0 = (g0 ∩ p) ⊕ (g0 ∩ k). By the maximality of a in p, we must then have
g0 = a ⊕ m where m = Zk (a) (here m = {0}). We have ∆(a : g) = {αij }i6=j≤d where
αij (H) = Hii − Hjj . The root subspaces are gij = R · E ij .
The Killing form also induces a natural pairing h·, ·i on a∗ w.r.t. which ∆(a : g) ⊂ a∗
is a root system. The associated Weyl group, generated by the root reflections sα , will
be denoted W (a : g). sij acts on Rd by exchanging the ith and jth co-ordinate, so that
W (a : g) ' Sd . This group is also canonically isomorphic to the analytic Weyl groups
NG (A)/ZG (A) and NK (A)/ZK (A), where a set of representatives is given by the permutation matrices. The fixed-point set of any sα is a hyperplane in a∗ , called a wall. The
connected components of the complement of the union of the walls are cones, called the
(open) Weyl chambers. A subset Π ⊂ ∆(a : g) will be called a system of simple roots (by
abuse of notation a “simple system”) if every root can be uniquely expressed as an integral
combination of elements of Π with either all coefficients non-negative or all coefficients
non-positive. For a simple system Π, the open cone CΠ = {ν ∈ a∗ | ∀α ∈ Π : hν, αi > 0}
is an open Weyl chamber, and the map Π 7→ CΠ is a 1-1 correspondence between simple systems and chambers. The Weyl group acts simply transitively on the chambers and
simple systems. The closure of an open chamber will be called a closed chamber. The
action of W (a : g) on a∗ extends in the complex-linear way to an action on a∗C preserving
ia∗ ⊂ a∗C , and we call an element ν ∈ a∗C regular if it is fixed by no w ∈ W (a : g). We use
P
ρ = 21 α>0 (dim gα )α ∈ a∗ to denote half the sum of the positive (restricted) roots.
Fixing the simple system Π = {ei,i+1 }d−1
i=1 we get a notion of positivity. For n =
⊕α>0 gα (strictly upper-triangular matrices) and n̄ = Θn we have g = n ⊕ a ⊕ m ⊕ n̄ and
(Iwasawa decomposition) g = n ⊕ a ⊕ k. By means of the Iwasawa decomposition, we may
uniquely write every X ∈ g in the form X = Xn + Xa + Xk . We sometimes also write
H0 (X) for Xa .
Let N , A be the subgroups of G corresponding to the subalgebras n, a ⊂ g respectively
(upper-triangular unipotent matrices and diagonal matrices with positive entries, respectively), and let M = ZK (a) (diagonal matrices with entries in {±1}). Then A is a maximal
split torus in G, and m = Lie(M ), though M is not necessarily connected. Moreover
P0 = N AM is a minimal parabolic subgroup of G, with the map N × A × M → P0 being
a diffeomorphism. The map N × A × K → G is a (surjective) diffeomorphism (Iwasawa
decomposition), so for g ∈ G there exists a unique H0 (g) ∈ a such that g = n exp(H0 (g))k
for some n ∈ N , k ∈ K. The map H0 : G → a is continuous; restricted to A it is the
inverse of the exponential map.
2.3. THE ADELIC GROUP AND ITS QUOTIENTS, COMPONENTS
16
Let gC = g ⊗R C denote the complexification of g. It is a complex semi-simple Lie
algebra. Let θC denote the complex-linear extension of θ to gC . It is not a Cartan involution
of gC . We fix a maximal abelian subalgebra b ⊂ m and set h = a ⊕ b. Then hC = h ⊗ C ⊂
gC is a Cartan subalgebra, with the associated root system ∆(hC : gC ) satisfying ∆(a : g) =
{α a }α∈∆(hC : gC ) \ {0}. Moreover, we can find a system of simple roots ΠC ⊂ ∆(hC : gC )
and a system of simple roots Π ⊂ ∆(a : g) such that the positive roots w.r.t. Π are precisely
the nonzero restrictions of the positive roots w.r.t. ΠC . We fix such a compatible pair of
simple systems, and let ρh denote half the sum of the roots in ∆(hC : gC ), positive w.r.t. ΠC .
Let F0 ⊂ ∆(hC : gC ) consist of the roots that restrict to 0 on a, F0+ ⊂ F0 those positive
w.r.t. ΠC . Let nM = ⊕α∈F0+ (gC )α , n̄M = ⊕α∈F0+ (gC )−α . Then mC = nM ⊕ bC ⊕ n̄M and
gC = nC ⊕ nM ⊕ hC ⊕ n̄M ⊕ n̄C .
For ν ∈ a∗C , set kνk2 = hRe(ν), Re(ν)i + hIm(ν), Im(ν)i (with the inner products taken
in a∗R ).
If lC is a complex Lie algebra, then we denote by U (lC ) its universal enveloping algebra,
and by Z(lC ) its center. In particular we set Z = Z(gC ).
2.3. The Adelic group and its quotients, Components
Q
Let G(Af ) denote the subgroup of the Cartesian product p Gp consisting of those seQ
quences g such that gp ∈ Kp for almost all p. Declaring K = p Kp (in the product
topology) to be an open compact subgroup of G(Af ) makes G(Af ) into a totally disconnected locally compact topological group. Finally, we set G(A) = G∞ · G(Af ). This is a
locally compact group. This construction is called a restricted direct product and denoted:
Y
0
(Gp , Kp ).
G(A) = G∞ ·
p
For g ∈ G(A) (or g ∈ G(Af )) we denote gv (resp. gp ) its components at specific places.
R EMARK 2.3.1. In the alternate one can define G(A) as the A-points of the variety
2
νD (g) · x0 (g) = 1, with the topology as a subset of Ad +1 (in both senses). This shows that
the group obtained by the restricted direct product procedure above is independent of the
choice of the maximal order O.
L EMMA 2.3.2. Let Kf be an open compact subgroup of G(Af ). Then there exists a
Q
finite set R1 of finite places and an open compact subgroup KR1 < p∈R1 Gp such that
Q
Kf = KR1 × p∈R
/ 1 Kp .
P ROOF. A set of basic neighbourhoods of the identity in G(Af ) is given by the sets
Q
of the form p Up where Up ⊂ Gp are open and Up = Kp for almost all p. Since Kf
Q
is open we conclude that there exists a finite set R1 of finite places such that p∈R
/ 1 Kp
is contained in Kf . For any p ∈
/ R1 the projection map G(Af ) → Gp is continuous and
the image of Kf under this map is a compact subgroup containing the maximal compact
Q
Q
subgroup Kp . It follows that Kf is an open compact subgroup of p∈R1 Gp × p∈R
/ 1 Kp
Q
which contains the second factor. It is hence obviously of the form KR1 × p∈R
/ 1 Kp for a
Q
subgroup KR1 < p∈R1 Gp . This subgroup equals the image of Kf under a quotient map
and in particular is compact and open.
2.3. THE ADELIC GROUP AND ITS QUOTIENTS, COMPONENTS
17
Q
Q
Letting R = R0 ∪ R1 we set KR = KR1 × p∈R0 \R1 Kp so that Kf = KR × p∈R
/ Kp .
In that case we have for every p ∈
/ R an isomorphism ϕp : Gp → GLd (Qp ) such that
ϕ(Kp ) = GLd (Zp ).
FACT 2.3.3. Let G be a linear algebraic group. We identify G(Q) with its image in the
Q
Cartesian product v∈|Q| Gv under the diagonal embedding, and let Kf < G(Af ) be an
open compact subgroup.
(1) G(Q) lies in G(A). It is discrete in the adelic topology of G(A).
(2) (finiteness of class number) The space X̃ = G(Q)\G(A)/Kf has finitely many
connected components.
In certain cases we can say more about the space described in claim (2):
FACT 2.3.4. Returning to our previous notation, let G be the group of invertible elements of a division algebra defined over Q, G1 the group of elements of norm 1. Then:
(1) The quotient G(Q)ZG (A)\G(A) is compact.
(2) G1 (Q) is dense in G1 (Af ). Equivalently (see below), G1 (Q)\G1 (A)/Kf1 is connected for any open compact Kf < G1 (Af ).
D EFINITION 2.3.5. A component of X̃ is a G-orbit X = x̃G for some x̃ ∈ X̃.
Since G is not connected the ‘components’ we have just defined do not coincide with
the topological connected components of X̃. These are closely related concepts though:
L EMMA 2.3.6. X̃ is the disjoint union of its components, of which there is a finite
number. Furthermore, let X be a component of X̃. Then:
(1) X is a union of connected components of X̃.
(2) There exists a discrete subgroup Γ < G such that X ' Γ\G.
(3) XZ = Z\X is compact. Hence ΓZ/Z is a co-compact lattice in Gad = G/Z.
P ROOF. As Kf is open in G(Af ), the quotient G(Af )/Kf is discrete. The space of
components
X̃/G = G(Q)\G(A)/G · Kf = G(Q)\G(Af )/Kf
is a quotient of this and hence discrete as well. By the second claim of the previous Lemma
it has finitely many connected components, i.e. it is finite. Now every component X is
closed and open in X̃ (being the inverse image under the quotient map of a point of X̃/G),
that is a union of connected components. Since the components are closed and open, X̃ is
their disjoint union.
Now let X be a component of X̃, and let gf ∈ G(Af ) be a representative for the class
of X in G(Q)\G(Af )/Kf . We can then set:
Γ = γ∞ | γ ∈ G(Q) : (γp )p<∞ ∈ gf Kf gf−1 .
We first verify that this is a discrete subgroup of G. For this let U ⊂ G be a relatively
compact open neighbourhood of the identity. Then Ũ = U × gf Kf gf−1 is a relatively
compact open neighbourhood of the identity in G(A), and it follows that Γ∩U ' G(Q)∩ Ũ
is finite.
2.3. THE ADELIC GROUP AND ITS QUOTIENTS, COMPONENTS
18
To see that Γ\G ' X we start with the surjective map ϕ : G → X given by ϕ(g∞ ) =
0
G(Q)g∞ gf Kf . By definition we have ϕ(g∞ ) = ϕ(g∞
) iff there exist γ ∈ G(Q), kf ∈ Kf
such that γg∞ gf kf = g∞ gf . At the finite places this reads γ = gf kf gf−1 , at the infinite
0
0
place γg∞ = g∞
and we conclude that ϕ(g∞ ) = ϕ(g∞
) iff there exists γ ∈ Γ such that
0
γg∞ = g∞ . Since Γ is closed in G this means our map ϕ induces a bijective continuous
map Γ\G → X. It is open since the map G → G(A)/Kf given by g∞ 7→ g∞ gf Kf is open
(the space G(Af )/Kf is discrete).
Finally we note that XZ is a closed subset of X̃Z = Z\X̃. It thus suffices to verify the compactness of the latter space. A direct computation shows: ZG (Q)\ZG (A) =
Q ×
Q
+
Z + × p Z×
p where Z is the connected component of Z. It follows that X̃Z /
p Zp =
×
G(Q)ZG (A)\G(A)/Kf . In fact, for p ∈
/ R1 we have Z×
⊂
O
=
K
while
the
fact
that
p
p
p
Q
×
KR1 < GR1 is open implies that is contains a subgroup of finite index of p∈R1 Zp . We
conclude that X̃Z is a finite cover of the compact space G(Q)ZG (A)\G(A)/Kf .
L EMMA 2.3.7. Let Kf < G(Af ) be an open compact subgroup.
Q
point.
(1) Assume νD (Kf ) = p Z×
p . Then G(Q)\G(Af )/Kf reduces to a single
Q ×
(2) For any maximal order O, the maximal compact subgroup Kf = p Op satisfies
the above condition. The same holds for the intersection of two such subgroups,
in which case we say that Kf is associated to an “Eichler order”.
Q
P ROOF. For the first part, let gf ∈ G(Af ). We then have νD (gf ) = p pkp up ∈ A×
f for
×
some kp ∈ Z (almost all of which are zero) and up ∈ Zp .
By a Eichler’s Theorem (see [31, Prop. XI-3-3] and note that our D is R-split) there
Q
exists γ ∈ G(Q) such that r = νD (γ) = p p−kp . Moreover, for any p the number rpkp is
p-integral, so that rpkp up ∈ Z×
p . By assumption there exists kf ∈ Kf such that νD (kf ) =
−1
Q
kp
. It follows that the element hf = γgf kf of G(Af ) has νDp ((hf )p )) = 1 at
p rp up
every prime p, i.e. hf ∈ G1 (Af ). Now since G1 (Af ) is a topological subgroup of G1 (Af ),
Kf1 = Kf ∩ G1 (Af ) is an open subgroup there. By part (2) of Fact 2.3.4 we can thus find
γ 0 ∈ G1 (Q) and kf0 ∈ Kf1 such that γ 1 kf1 = hf . We thus have:
0
0
1 ≡ γ kf ≡ γgf kf ≡ gf
as desired, where equivalence is read in the double coset space G(Q)\G(Af )/Kf .
The second part follows immediately from part (1) of Fact 2.1.3. If Kf , Kf0 are both
associated to maximal orders O, O0 it suffices to show that νDp (Kp ∩ Kp0 ) = Z×
p at every
place separately. At every place p where D ramifies Dp has a unique maximal order Rp .
We then have Kp = Kp0 = Rp× and are in the same situation as before. At a place where
p splits we use the isomorphism with GLd (Qp ) and the computation (up to conjugation) in
Lemma 5.1.16 of the joint stabilizer of two vertices in the associated building to obtain an
explicit form (up to conjugation) for Kp ∩ Kp0 . One sees that the intersection must contain a
conjugate of the subgroup diag(u, 1, . . . , 1) | u ∈ Z×
of GLd (Qp ) and hence an element
p
×
with reduced norm u for any u ∈ Zp . In this case Kf ∩ Kf0 is the open compact subgroup
associated to the order O ∩ O0 , which in the case d = 2 is the class of orders constructed
in [8, pp. 48–55].
2.4. THE p-ADIC GROUPS AND HECKE OPERATORS
19
We henceforth assume that Kf satisfies the condition of the Lemma, so that we can
identify X̃ and its single component X. The quotient Y \G/K is then a locally symmetric
space of non-positive curvature (this appellation is sometimes reserved for YZ = Z\Y ).
We also remark that (unlike G) the symmetric space G/K is always connected. Hence Y
and YZ are connected manifolds even when X isn’t.
We normalize the Haar measures dx on XZ , dk on K and dy on YZ to have total mass
1 (here dy is the pushforward of dx under the the projection from XZ to YZ given by
averaging w.r.t. dk).
For any unitary character ω ∈ Ẑ consider the space of functions ψ : X → C such that
ψ(xz) = ω(z)ψ(x) for all z ∈ Z, x ∈ X. Since ω is unitary the map x 7→ |ψ(x)|2 is
Z-invariant, and hence a function on the compact
space XZ . We let L2 (X, ω) denote the
R
space of the functions ψ as above such that XZ |ψ(x)|2 dx < ∞. We also let L2 (Y, ω)
be the subspace consisting of K-invariant functions. We note that L2 (X, ω) is a unitary
representation of G under right translation and that as a G-representation L2 (X) is the
direct integral of the L2 (X, ω).
2.4. The p-adic groups and Hecke operators
For a prime p ∈
/ R0 we have Gp ' GLd (Qp ), Kp ' GLd (Zp ). Let Hp denote the
convolution algebra of the bi-Kp -invariant functions of compact support on Gp . It is commutative, and generated by the elements Kp ap Kp with ap ∈ Ap , the subgroup of diagonal
matrices of GLd (Qp ) (see Fact 5.1.15).
By assumption we can think of functions on X as functions on G(Q)\G(A) which are
invariant by Kf on the right. For almost all primes p (except for p ∈ R1 ), Kp is a direct
factor of Kf and hence if p ∈
/ R = R0 ∪ R1 , Hp acts on the space of functions on X by
convolution on the right. Moreover, the actions of Hp and Hp0 for p 6= p0 commute, and
they both commute with the right regular action of G = G∞ .
We call an operator Th on functions of X associated to an element h ∈ Hp a Hecke
operator, and think of it as arising from a discrete foliation of the manifold X where the leaf
of G(Q)g∞ gf Kf is given by {G(Q)g∞ gf xp Kf }xp ∈Gp : each leaf is of the form Hp \Gp /Kp
for some (generically trivial) subgroup Hp < Gp and the action of Th on f : X → C is given
by restricting f to each leaf and convolving on the right with h. This action is analyzed in
detail in Chapter 5, where the control of subgroups Hp causing the Hecke correspondence
to return is achieved using the results of Chapter 4.
Together the Hecke operators at all places p ∈
/ R generate the (commutative) Hecke
algebra H(R) . This is the algebra we have in mind when we apply Theorem 1.3.2.
D EFINITION 2.4.1. We call ψ ∈ L2 (X, ω) a Hecke eigenfunction if it is a joint eigenfunction of the Hecke algebra H(R) .
CHAPTER 3
The Micro-Local Lift
3.1. Introduction and motivation
Let ψ ∈ L2 (Y, ω) be normalized as well as an eigenfunction of Z. The aim of the
present section is to construct a distribution µψ on XZ that lifts the measure µ̄ψ on YZ , and
establish some basic properties of µψ . We will of course take ψ = ψn , and the corresponding distribution will be the distribution µn discussed in the proof of Theorem 1.3.2. The
functions ψ̃n will then be chosen so that the measures |ψ̃n (x)|2 dx approximate µn ; finally,
both |ψ˜n (x)|2 and µn will become A-invariant as n → ∞.
We begin by fixing notation and providing some motivation for the relatively formal
definitions that follow.
Setting ψ(x) = ψ(xK) for any x ∈ X, we can think of ψ as a function in L2 (X, ω).
By the uniqueness of spherical functions [13, Th. 4.3 & 4.5], ψ generates an irreducible
spherical G-subrepresentation of L2 (X). As discussed in Section 3.2.1 below, we can then
find ν ∈ a∗C such that this representation is isomorphic to a principal series representation
πν (in particular, πν is unitarizable). We will assume for the rest of this section that Re(ν) =
0, i.e. πν is tempered, and that ν is regular. This will eventually be the only case of interest
to us in view of the non-degeneracy assumption (Definition 3.3.8). In this case the induced
representation (VK , Iν ) (living on a space of functions on K defined below, including the
constant function ϕ0 ) is irreducible and isomorphic to πν .
It follows that there is a unique G-homomorphism Rψ : (V, Iν ) → L2 (X) such that
Rψ (ϕ0 ) = ψ. The normalization kψkL2 (X,ω) = 1 now implies kRψ (f )kL2 (X) = kf kL2 (K)
for any f ∈ VK , i.e. that Rψ is an isometry.
We now give the rough idea of the construction that follows in the language of Wolpert
and Lindenstrauss; the language we shall use later is slightly different, so the discussion
here also provides a translation. The strategy of proof is similar to theirs; in a sense, the
main difficulty is finding the “correct” definitions in higher rank. For instance, the proofs of
Wolpert and Lindenstrauss use heavily the fact that K-types for PSL2 (R) have multiplicity
one, and the explicit action of the Lie algebra by raising and lowering operators. We shall
need a more intrinsic approach to handle the general
case.
R
The measure µ̄ψ on YZ is defined by g 7→ YZ g(y)|ψ(y)|2 dy. More generally, suppose
that ψ 0 ∈ L2 (X) belongs to the G-subrepresentation generated by ψ, i.e. ψ 0 ∈ Rψ (V ).
Then ψ(x)ψ 0 (x) is Z-invariant and we can consider the (signed) measure on XZ given by:
Z
(3.1.1)
σ : g 7→
ψ(x)ψ 0 (x)g(x)dx.
XZ
20
3.1. INTRODUCTION AND MOTIVATION
21
If g(x) is K-invariant, then so is the product ψ(x)g(x), and it follows that the right-hand
side of (3.1.1) depends only on the projection of ψ 0 onto Rψ (V )K . The space Rψ (V )K is
one-dimensional, spanned by ψ, and it follows that if ψ 0 − ψ ⊥ ψ then the measure σ on X
projects to the measure µ̄ψ on Y .
The distribution µψ we shall construct will be in the spirit of (3.1.1), but with ψ 0 a
“generalized vector” in Rψ (V ). Suppose, in fact, that ψ10 , ψ20 , . . . ψn0 , . . . are an infinite
sequence of elements of Rψ (V ) that transform under different K-types, and suppose further
R
that g ∈ Cc∞ (XZ )K . Then, by considering K-types, the integral X ψ(x)ψj0 (x)g(x)dx
vanishes for all sufficiently large j. It follows that, if one sets ψ 0 to be the formal sum
P∞ 0
j=1 ψj , one can make sense of (3.1.1) by interpreting it as:
∞ Z
X
σ(g) =
ψ(x)ψj0 (x)g(x)dx
j=1
X
In other words, if g ∈ Cc∞ (XZ )K , we may make sense of (3.1.1) while allowing ψ 0 to
belong to the space Vc
K of “infinite formal sums of K-types.” Our definition of µψ will,
indeed, be of the form (3.1.1) but with ψ 0 an “infinite formal sum” of this kind.
For a certain choice of ψ 0 (denoted Φ∞ in [18]), we will wish to show that (3.1.1) is “approximately a positive measure” and “approximately A-invariant,” where both statements
become true in the large eigenvalue limit in an appropriate sense. For the “approximate
positivity,” we shall integrate
(3.1.1) by parts to show
that there exists another unit vector
R
R
00
00
0
ψ ∈ Rψ (V ) such that X ψ(x)ψ (x)g(x)dx ≈ X |ψ (x)|2 g(x)dx, where the right-hand
side is evidently a positive measure. For the “approximate A-invariance,” we will construct
differential operators that annihilate ψ(x)ψ 0 (x); this reduces to a purely algebraic question of constructing elements in U (g) that annihilate a vector in a certain tensor product
representation.
0
The space Vc
K is very closely linked to the dual VK of the K-finite vectors: the conjugate linear isomorphism T : V → V 0 (3.2.1) extends to a conjugate-linear isomorphism
0
0
c
T : Vc
K → VK . For formal reasons, it is simpler to work with VK than VK ; this is the viewpoint we shall take in Definition 3.3.1. To motivate this viewpoint, let us rewrite (3.1.1)
in a different fashion. Let v 0 ∈ V be chosen so that ψ 0 = Rψ (v 0 ), and let P be the orthogonal projection of L2 (X) onto Rψ (V ). We may rewrite (3.1.1) – using the notations of
Definition 3.2.3 – as follows:
σ(g) = hψ(x)g(x), ψ 0 (x)iL2 (X) = hP (ψ(x)g(x)), ψ 0 (x)iL2 (X)
(3.1.2)
= hRψ−1 ◦ P (ψ(x)g(x)), v 0 iV = T (v 0 ) ◦ Rψ−1 ◦ P (ψ(x)g(x))
Now, if g ∈ Cc∞ (XZ )K , then the quantity Rψ ◦ P (ψ(x)g(x)) is K-finite, i.e. belongs to
VK . It follows that, if g ∈ Cc∞ (XZ )K , the last expression of (3.1.2) makes formal sense if
we replace T (v 0 ) by any functional Φ ∈ VK0 .
3.2. MORE ON REAL LIE GROUPS
22
3.2. More on Real Lie Groups
3.2.1. Spherical Representations and the model (VK , Iν ). We recall some facts from
the representation theory of compact and reductive groups. At the end of this section we
analyze a model (the “compact picture”) for the spherical dual of G.
T HEOREM 3.2.1. [16, Th. 1.12] Let K be a compact topological group and let K̂fin be
the set of equivalence classes of irreducible finite-dimensional unitary representations of
K.
(1) (Peter-Weyl) Every ρ ∈ K̂fin occurs discretely in L2 (K) with multiplicity equal to
its dimension d(ρ). Moreover, L2 (K) is isomorphic to the Hilbert direct sum of its
isotypical components {L2 (K)ρ }ρ∈K̂fin .
(2) Let π : K → GL(W ) be a representation of K on the Frêchet space W . Then
⊕ρ∈K̂ Wρ is dense in W , where Wρ is the ρ-isotypical subspace.
(3) Every irreducible representation of on a Frêchet space is finite-dimensional and
hence unitarizable. In particular, K̂fin is the unitary dual of K.
(4) For K as in Section 2.2, K̂ is countable.
Note that while [16, Th. 1.12(c-e)] are only claimed for unitary representations on
Hilbert spaces, their proofs only rely on the action of the convolution algebra C(K) on
representations of K, and hence carry over with little modification to the more general
context needed here. The last conclusion follows from the separability of L2 (K), which in
turn follows from the separability of K.
N OTATION 3.2.2. Let π : K → GL(W ) be as above. The algebraic direct sum
def
WK = ⊕ρ∈K̂ Wρ
consists precisely of these w ∈ W which generate a finite-dimensional K-subrepresentation. We refer to WK as the space of K-finite vectors. We will use W K to denote these
vectors of W fixed by K.
D EFINITION 3.2.3. Set V = L2 (M \K), and set VK ⊂ V to be the space of K-finite
vectors. Let C ∞ (M \K) be the smooth subspace, C ∞ (M \K)0 the space of distributions
on M \K. Let VK0 (resp. V 0 ) be the dual to VK (resp. V ). Then we have natural inclusions VK ⊂ C ∞ (M \K) ⊂ V and VK0 ⊃ C ∞ (M \K)0 ⊃ V 0 ; further, we have (Riesz
representation) a conjugate-linear isomorphism
T
(3.2.1)
V ,→ V 0
R
where the map T : V → V 0 is defined via the rule T (f )(g) = hg, f iV = M \K gf dk.
Fix an increasing exhaustive sequence of finite dimensional K-stable subspaces of VK ,
i.e. a sequence V1 ⊂ V2 ⊂ · · · ⊂ VN ⊂ VN +1 ⊂ . . . of subspaces such that ∪∞
i=1 Vi = VK
and each Vi is a K-subrepresentation.
For Φ ∈ VK0 and 1 ≤ N ∈ Z, define the N -truncation of Φ as the unique element
ΦN ∈ VN such that T (ΦN ) − Φ annihilates VN .
Finally let ϕ0 ∈ VK be the function that is identically 1.
3.2. MORE ON REAL LIE GROUPS
23
D EFINITION 3.2.4. Let µ be a regular Borel measure on a space X. Call aRsequence of
non-negative functions {fj } ∈ L1 (µ) a δ-sequence
at x ∈ X if, for every j, fj dµ = 1,
R
and moreover if, for every g ∈ C(X), limj→∞ fj · gdµ = g(x).
2
L EMMA 3.2.5. There exists a sequence {fj }∞
j=1 ⊂ VK such that |fj | is a δ-sequence
on M \K.
P ROOF. Let {hj }∞
j=1 ⊂ C(M \K) be a δ-sequence. By the Peter-Weyl theorem VK is
dense
in
C(M
\K),
so
that for every j we can choose fj0 ∈ VK such that the difference
p
f0
hj (k) − fj0 (k) ≤ 21j . Then one may take fj = f 0j .
k j k2
∞
Secondly, we recall the construction of the spherical principal series representations
of a reductive Lie group. An irreducible representation of G is spherical if it contains a
K-fixed vector. Such a vector is necessarily unique up to scaling.
To any ν ∈ a∗C we associate the character χν (p) = exp(ν(H0 (p)) of P0 and the unitarily
induced representation with (g, K)-module
∞
hν+ρ,H0 (p)i
(3.2.2)
IndG
f (g) .
P0 χν = f ∈ C (G)K | ∀p ∈ P, g ∈ G : f (pg) = e
By the Iwasawa decomposition, every f ∈ IndG
P0 χν is determined by its restriction to K;
this restriction defines an element of the space VK . Conversely, every f ∈ VK extends
uniquely to a member of IndG
P0 χν .
D EFINITION 3.2.6. For ν ∈ a∗C , we denote by (Iν , VK ) the representation of g on VK
fixed by the discussion above; we shall also use Iν to denote the corresponding action of g
on C ∞ (M \K) and of G on V . We shall denote by Iν0 the dual action of g on either VK0 or
C ∞ (M \K)0 .
Note also that ϕ0 ∈ VK (see Definition 3.2.3) is a spherical vector for the representation
(Iν , VK ).
T HEOREM 3.2.7. (The unitary spherical dual; references are drawn from [16])
(1) For any ν ∈ a∗C , IndG
P0 χν has a unique spherical irreducible subquotient, to be
denoted πν . [Th. 8.37] Any spherical irreducible unitary representation of G is
isomorphic to πν for some ν. [Th. 8.38] We have πν1 ' πν2 iff there exists w ∈
W (a : g) such that ν2 = wν1 .
(2) [§7.1-3] If Re(ν) = 0 then IndG
P0 χν is unitarizable, with the invariant Hermitian
R
form given by hf, gi = M \K f (k)g(k)dk. This representation has a unique spherical summand (necessarily isomorphic to πν ), and we let jν : VK → πν denote the
orthogonal projection map. [Th. 7.2] If ν is regular then IndG
P0 χν is irreducible.
(3) [§16.5(7) & Th. 16.6] If πν is unitarizable then Re(ν) belongs to the convex hull
of {wρ}{w∈W(a : g)} ⊂ a∗ , a compact set. Moreover, there exists w ∈ W (a : g) such
that w2 = 1 and wν = −ν̄. In particular if Re(ν) 6= 0, then w 6= 1, and since
Im(ν) is w-fixed it is not regular.
Note that the norm on πν is only unique up to scaling. If Re(ν) = 0 and Im(ν) is
regular (the main case under consideration), we choose kϕ0 kπν = 1.
3.2. MORE ON REAL LIE GROUPS
24
For future reference we compute the action of g on VK via Iν . First, note that the action
of K on V = L2 (M \K) is given by right translation, and the action of k ⊂ g on VK is then
given by right differentiation.
Secondly, recall that if U ⊂ Rn is open, a differential operator D on U is an expression
P
α1
αn
of the form K
i=1 fi ∂1 . . . ∂n , where the fi are smooth and αj ≥ 0. If M is a smooth
n-manifold, we say a map D : C ∞ (M ) → C ∞ (M ) is a differential operator if it is defined
by a differential operator in each coordinate chart.
L EMMA 3.2.8. Let f ∈ VK and let X ∈ g. Then there exists a differential operator DX
on M \K (depending linearly on X and independent of ν) such that for every k ∈ K,
(Iν (X)f )(k) = hν + ρ, H0 (Ad(k)X)i f (k) + (DX f )(k).
P ROOF. Let t ∈ R be small, and consider f (k exp(tX)) = f (exp(t Ad(k)X) · k). We
write the Iwasawa decomposition of Ad(k)X ∈ g as Ad(k)X = Xn (k) + Xa (k) + Xk (k)
where Xa (k) = H0 (Ad(k)X). By the Baker-Campbell-Hausdorff formula, exp(t Ad(k)X)
has the form exp(tXn (k)) · exp(tXa (k)) · exp(tXk (k)) + O(t2 ), so that:
d
d
f (exp(tXn (k)) · exp(tXa (k)) · k) t=0 + f (exp(tXk (k))k) t=0 .
dt
dt
To conclude, observe that f 7→ dtd f (exp(tXk (k))k) t=0 defines a differential operator DX
on M \K.
(Iν (X)f )(k) =
X
Lemma 3.2.8 will be used in the following way: as kνk → ∞, the operator Iν ( kνk
) acts
−1
on VK in a very simple fashion, modulo certain error terms of order kνk . The simplicity
of this “rescaled” action as kνk → ∞ will be of importance in our analysis.
3.2.2. Some Functional Analysis. We collect here some simple functional analysis
facts that we shall have need of.
Let Cc∞ (XZ ) denote the space of smooth functions of compact support on X. It is
endowed with the usual “direct-limit” topology: fix a sequence of K-invariant compact sets
C1 ⊂ C2 ⊂ . . . such that their interiors exhaust X. Then the Cc∞ (Ci ) exhaust Cc∞ (XZ ).
Cc∞ (Ci ) is endowed as usual with a family of seminorms, viz. for any D ∈ U (gC ) we define
kf kCi ,D = supx∈Ci |Df |. These seminorms induce a topology on each Cc∞ (Ci ). We give
Cc∞ (XZ ) the topology of the union of Cc∞ (Ci ), i.e. a map from Cc∞ (XZ ) is continuous if
and only if its restriction to each Cc∞ (Ci ) is continuous.
In other words: a sequence of functions converges in Cc∞ (XZ ) if their supports are
all contained in a fixed compact set, and all their derivatives converge uniformly on that
compact set.
Cc∞ (XZ )is a locally convex complete space in this topology. In particular, its subspace
Cc∞ (XZ )K of K-finite vectors is dense. We denote by Cc∞ (XZ )0 (resp. Cc∞ (XZ )0K ) the
topological dual to Cc∞ (XZ ) (resp. the algebraic dual to Cc∞ (XZ )K ). Both spaces will
be endowed with the weak-* topology. We shall refer to an element of Cc∞ (XZ )0 as a
distribution on X.
Let C0 (X) be the Banach space of continuous functions on X decaying at infinity,
endowed with the supremum norm. Let C0 (X)0 be the continuous dual of C0 (X); the
3.3. REPRESENTATION-THEORETIC LIFT
25
Riesz representation theorem identifies it with the space of finite (signed) Borel measures
on X. We endow C0 (X)0 with the weak-* topology.
It is easy to see that Cc∞ (XZ )K is dense in C0 (X). In particular any (algebraic) linear
functional on Cc∞ (XZ )K which is bounded w.r.t. the sup-norm extends to a finite signed
measure on X, with total variation equal to the norm of the functional. Moreover, if this
functional is non-negative on the non-negative members of Cc∞ (XZ )K then the associated
measure is a positive measure.
3.3. Representation-Theoretic Lift
3.3.1. Lifting a single (non-degenerate) eigenfunction.
D EFINITION 3.3.1. Let Φ ∈ VK0 be an (algebraic) functional, and f ∈ VK . Let µψ (f, Φ)
be the functional on Cc∞ (XZ )K defined by the rule:
(3.3.1)
µψ (f, Φ)(g) = Φ ◦ Rψ−1 ◦ P (Rψ (f ) · g)
where g ∈ Cc∞ (XZ )K , P : L2 (X, ω) → Rψ (V ) is the orthogonal projection, and Rψ (f ) · g
denotes pointwise multiplication of functions on X.
R EMARK 3.3.2. In fact, if Φ ∈ C ∞ (M \K)0 (see equation (3.2.1)) then µψ (f, Φ) extends to an element of Cc∞ (XZ )0 , i.e. defines a distribution on X: µψ is the composite
Cc∞ (XZ )
g7→Rψ (f )g
−→
−1
Rψ
P
Φ
Cc∞ (XZ ) −→ C ∞ (M \K) → C,
and it is easy to verify that each of these maps is continuous. This is never used in our
arguments: we use this observation only to refer to certain µψ as “distributions”.
def
D EFINITION 3.3.3. Let δ ∈ VK0 be the distribution δ(f ) = f (1), and call µψ =
µψ (ϕ0 , δ) the (non-degenerate) microlocal lift of µ̄ψ .
The rest of the section will exhibit basic formal properties of this definition. We will
establish most of the formal properties of µψ by restricting Φ to be of the form T (f2 ), where
the conjugate-linear mapping T is as defined in (3.2.1). This situation will occur sufficiently
often that, for typographical ease, it will be worth making the following definition:
D EFINITION 3.3.4. Let f1 , f2 ∈ VK . We then set µTψ (f1 , f2 ) = µψ (f1 , T (f2 )).
L EMMA 3.3.5. Suppose f1 , f2 ∈ VK . Then
Z
T
Rψ (f1 )(x)Rψ (f2 )(x)g(x)dx.
(3.3.2)
µψ (f1 , f2 )(g) =
XZ
and µTψ defines a signed measure on XZ of total variation at most ||f1 ||L2 (K) ||f2 ||L2 (K) . If
f1 = f2 , then µTψ (f1 , f1 ) is a positive measure of mass ||f1 ||2L2 (K) .
P ROOF. (3.3.2) is a consequence of the definition of µ. The Cauchy-Schwarz inequality
implies that |µTψ (f1 , f2 )(g)| ≤ ||f1 ||L2 (K) ||f2 ||L2 (K) ||g||L∞ (X) , whence the second conclusion. The last assertion is immediate.
In fact, it may be helpful to think of µψ as being given by a distributional extension of
the formula (3.3.2); see the discussion of Section 3.1.
3.3. REPRESENTATION-THEORETIC LIFT
26
L EMMA 3.3.6. The distribution µψ (ϕ0 , δ) on X projects to the measure |ψ|2 dy on Y .
P ROOF. In view of the previous Lemma, it will suffice to show that the distribution
µψ (ϕ0 , δ) − µTψ (ϕ0 , ϕ0 ) on X projects to 0 on YZ . This amounts to showing that µψ (ϕ0 , δ −
T (ϕ0 )) annihilates any K-invariant function g ∈ Cc∞ (XZ )K . Taking into account that the
functional δ − T (ϕ0 ) on VK annihilates any K-invariant vector, the claim follows from the
definition of µψ .
L EMMA 3.3.7. The map µψ : VK ⊗ VK0 → Cc∞ (XZ )0K is equivariant for the natural
g-actions on both sides.
P ROOF. This follows directly from the definition of µψ .
Concretely speaking, this says that for f ∈ VK , Φ ∈ VK0 , g ∈ Cc∞ (XZ )K , X ∈ g we
have
(3.3.3)
µψ (Xf1 , Φ)(g) + µψ (f1 , XΦ)(g) + µψ (f1 , Φ)(Xg) = 0
where X acts on VK via Iν and on VK0 via Iν0 . In particular, if f1 , f2 ∈ VK we have
(3.3.4)
µTψ (Xf1 , f2 )(g) + µTψ (f1 , Xf2 )(g) + µTψ (f1 , f2 )(Xg) = 0
3.3.2. Sequences of eigenfunctions and quantum limits. In what follows we shall
consider ψn ∈ L2 (Y, ωn ), a sequence of eigenfunctions with parameters {νn }∞
n=1 diverging
νn −dωn
to ∞ (i.e. with kνn k → ∞). Set ν̃n = kνn k (i.e. remove the central character part of the
parameter). For f1 , f2 ∈ VK and Φ ∈ VK0 , we abbreviate µTψn (f1 , f2 ) (resp. µψn (f, Φ)) to
µTn (f1 , f2 ) (resp. µn (f, Φ)), and we abbreviate the microlocal lift µψn (:= µn (ϕ0 , δ)) to µn .
D EFINITION 3.3.8. (Gad = G/Z simple) We say a sequence ψn is non-degenerate if
every limit point of the sequence ν̃n is regular.
We say that it is conveniently arranged if it is non-degenerate, Re(νn ) = 0 for all n,
the limit in limn→∞ ν̃n exists, the νn are all regular, and for all f1 , f2 ∈ VK the measures
µTn (f1 , f2 ) converge in C0 (XZ )0 as n → ∞. In this situation we denote limn→∞ ν̃n by ν̃∞ .
The existence of non-degenerate sequences of eigenfunctions was discussed in Remark
1.3.3. This follows from strong versions of Weyl’s Law on Y . By Theorem 3.2.7, the
non-degeneracy of a sequence ψn as in the Definition implies Re(νn ) = 0 for all large
enough n. For fixed f1 , f2 ∈ VK the total variation of the measures µTn (f1 , f2 ) is bounded
independently of n (Lemma 3.3.5); in view of the (weak-*) compactness of the unit ball in
C0 (X)0 it follows that this sequence of measures has a convergent subsequence. Combining
this remark with the fact that VK has a countable basis, a diagonal argument shows that
every non-degenerate sequence of eigenfunctions has a conveniently arranged subsequence.
Now suppose {ψn } is a conveniently arranged sequence and fix f1 ∈ VK , Φ ∈ VK0 , g ∈
Cc∞ (XZ )K . Let ΦN be the N -truncation of Φ (see Definition 3.2.3). In view of (3.3.1),
if we choose N := N (f1 , g) sufficiently large, then µn (f1 , Φ)(g) = µTn (f1 , ΦN )(g). It
follows that the limit limn→∞ µn (f1 , Φ)(g) exists.
3.3. REPRESENTATION-THEORETIC LIFT
27
We may consequently define µ∞ : VK × VK0 → Cc∞ (XZ )0K and µT∞ : VK × VK →
Cc∞ (XZ )0K by the rules:
µ∞ (f, Φ)(g) = lim µn (f1 , Φ)(g),
(3.3.5)
VK0
n→∞
(g ∈ Cc∞ (XZ )K )
µT∞ (f1 , f2 ) = µ∞ (f1 , T (f2 ))
L EMMA 3.3.9. For fixed f1 ∈ VK , the map Φ → µ∞ (f1 , Φ) is continuous as a map
→ Cc∞ (XZ )0K , both spaces being endowed with the weak topology.
P ROOF. This is an easy consequence of the definitions.
It is natural to ask whether µ∞ (f1 , Φ) extend to an element of Cc∞ (XZ )0 , at least when
Φ ∈ C ∞ (M \K)0 . Indeed a uniform bound on the distributions µn (f1 , Φ) follows from
making the argument of Remark 3.3.2 quantitative. This is not needed for our choice of
(f1 , Φ), however, when we can address this directly.
Henceforth {ψn }∞
n=1 will be a conveniently arranged sequence. We will show that
µ∞ (ϕ0 , δ) is positive and bounded w.r.t. the L∞ norm on Cc∞ (XZ )K . It hence extends to a
finite positive measure.
The key to the positivity of the limits is the following lemma (cf. [33, Prop. 3.3], [18,
Th. 3.1]).
L EMMA 3.3.10. (Integration by parts) Let {ψn } be conveniently arranged. Then, for
any f, f1 , f2 ∈ VK we have:
(3.3.6)
µT∞ (f1 , f · f2 ) = µT∞ (f¯ · f1 , f2 ).
Here e.g. f · f2 denotes pointwise multiplication of functions on M \K.
P ROOF. We start by exhibiting explicit functions f for which (3.3.6) is valid.
Extend every ν ∈ a∗C to g∗C via the Iwasawa decomposition g = n ⊕ a ⊕ k. For any
X ∈ gss , let pX (k) = 1i hν̃∞ , Ad(k)Xi. For fixed X, k 7→ pX (k) defines a K-finite element
of L2 (M \K).
By (3.3.4), for every X, f1 , f2 , g, and n, we have
(3.3.7)
µTn (Xf1 , f2 )(g) + µTn (f1 , X f2 )(g) + µTn (f1 , f2 )(Xg) = 0.
Divide by kνn k and apply Lemma 3.2.8 (as well as hdωn , gss i = 0) to see:
µTn (ipn · f1 , f2 )(g) + µTn (f1 , ipn · f2 )(g)
µT (DX f1 , f2 )(g) + µTn (f1 , DX f2 )(g) + µTn (f1 , f2 )(Xg)
,
=− n
kνn k
D
E
where pn (k) = 1i ν̃n + kνρn k , Ad(k)X .
As n → ∞, the right-hand side of (3.3.8) tends to zero by Lemma 3.3.5. On the
other hand, pn fi (considered as continuous functions on K) converge uniformly to pX fi .
Another application of Lemma 3.3.5 shows that the left-hand side of (3.3.8) converges to
iµT∞ (pX f1 , f2 )−iµT∞ (f1 , pX ·f2 ). Since pX = pX this shows that (3.3.6) holds with f = pX .
Now let F ⊂ C(M \K) be the C-subalgebra generated by the pX and the constant
function 1. Clearly (3.3.6) holds for all f ∈ F. This subalgebra is K-stable since
(3.3.8)
3.3. REPRESENTATION-THEORETIC LIFT
28
pX (kk1 ) = pAd(k1 )X (k) and hence F ∩ Vρ ⊂ F for all ρ ∈ K̂. Showing F is dense in
L2 (M \K) suffices to conclude that F = VK .
We will prove the stronger assertion that F is dense in C(M \K) using the StoneWeierstrass theorem. Note that 1 ∈ F, and F is closed under complex conjugation since
pX = pX . It therefore suffices to show that F separates the points of M \K. To this
end, let k1 , k2 ∈ K be such that pX (k1 ) = pX (k2 ) for all X ∈ g. Then hν̃∞ , Ad(k1 )Xi =
hν̃∞ , Ad(k2 )Xi for all X ∈ g, i.e. hAd(k1 )−1 ν̃∞ −Ad(k2 )−1 ν̃∞ , Xi = 0 for all X ∈ g. This
implies that Ad(k1−1 )ν̃∞ = Ad(k2 )−1 ν̃∞ ; by the non-degeneracy assumption, ZK (ν̃∞ ) =
ZK (A) = M , so M k1 = M k2 , i.e. k1 and k2 represent the same point of M \K.
Lemma 3.3.10 shows easily that µ∞ (ϕ0 , δ) extends to a positive measure. Indeed,
choosing fj as in Lemma 3.2.5, we see that
(3.3.9)
µ∞ (ϕ0 , δ) = lim µT∞ (ϕ0 , |fj |2 ) = lim µT∞ (fj , fj ).
j→∞
j→∞
Here we have invoked Lemma 3.3.9 for the first equality. It is clear that µT∞ (fj , fj ) defines
a positive measure on X; thus µ∞ (ϕ0 , δ), initially defined as an (algebraic) functional on
Cc∞ (XZ )K , extends to a positive measure on X. To obtain the slightly stronger conclusion
implicit in (2) of Theorem 1.3.2, we will analyze this argument more closely.
C OROLLARY 3.3.11. Notations as in Lemma 3.3.10, there exist a constant Cf1 ,f2 ,f and
a seminorm || · || on Cc∞ (XZ ) such that
(3.3.10) µTn (f1 , f · f2 )(g) − µTn (f · f1 , f2 )(g) ≤ Cf1 ,f2 ,f kgk kν̃∞ − ν̃n k + kνn k−1
P ROOF. We keep track of the error term in the proof of of Lemma 3.3.10.
Fix a basis {Xi } for g = gss ⊕ Zg , and define a seminorm on Cc∞ (XZ ) by kgk =
P
kgkL∞ (XZ ) + i kXi gkL∞ (XZ ) . With this seminorm, (3.3.10) holds for f1 , f2 ∈ VK and f =
pX . This follows from (3.3.8), utilizing Lemma 3.3.5 and the fact that kpX −pn kL∞ (M \K) kν̃∞ − ν̃n k.
Next suppose f1 , f2 , f, f 0 ∈ VK and α, α0 ∈ C. Then, if (3.3.10) is valid for (f1 , f2 , f )
and (f1 , f2 , f 0 ), it is also valid for (f1 , f2 , αf + α0 f 0 ). Further, if (3.3.10) is valid for
(f1 , f 0 · f2 , f ) and for (f f1 , f2 , f 0 ), then it is also valid for (f1 , f2 , f · f 0 ).
Consider now the set of f ∈ VK for which (3.3.10) holds for all f1 , f2 ∈ VK . The
remarks above show that this is a subalgebra of VK that contains each pX . The Corollary
then follows from the equality F = L2 (M \K)K established in the Lemma.
R EMARK 3.3.12. It is possible to obtain a bound of the form Cf1 ,f2 ,f,ν̃n kgkkνn k−1 , with
the constant uniformly bounded if the ν̃n are uniformly bounded away from the walls. This
result can be used to avoid passing to a subsequence in Theorem 1.3.2 or the following
Proposition; this is unnecessary for our applications, however.
P ROPOSITION 3.3.13. (Positivity and equivariance: (2) and (4) of Theorem 1.3.2).
Let {ψn } be non-degenerate. After replacing {ψn } by an appropriate subsequence,
there exist functions ψ̃n on X with the following properties:
R
(1) Define the measure σn via the rule σn (g) = X g(x)|ψ˜n (x)|2 dx. Then, for each
g ∈ Cc∞ (XZ )K we have limn→∞ (σn (g) − µn (g)) = 0.
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
29
(2) Let E ⊂ EndG (C ∞ (XZ )) be a C-subalgebra of endomorphisms of C ∞ (XZ ),
commuting with the G-action. Note that each e ∈ E induces an endomorphism
of C ∞ (Y ). Assume in addition that ψn is an eigenfunction for E. Then we may
choose ψ̃n so that each ψ̃n is an eigenfunction for E with the same eigenvalues as
ψn .
P ROOF. Without loss of generality we may assume that {ψn } are conveniently arranged.
Let {fj }∞
j=1 ⊂ VK be the sequence of functions provided by Lemma 3.2.5, so that
2
T (|fj | ) approximates δ. The main idea is, as in (3.3.9), to approximate µn = µn (ϕ0 , δ)
using µTn (fj , fj ).
For any g ∈ Cc∞ (XZ )K we have:
µn (g) − µTn (fj , fj )(g) ≤ µn (ϕ0 , δ)(g) − µn (ϕ0 , |fj |2 )(g)
(3.3.11)
+ µn (ϕ0 , |fj |2 )(g) − µn (fj , fj )(g) .
Corollary (3.3.11) provides a seminorm k·k on Cc∞ (XZ ) and a constant Cj such that
µn (ϕ0 , |fj |2 )(g) − µn (fj , fj )(g) ≤ Cj ||g|| · kν̃n − ν̃∞ k + kνn k−1 .
Choose a sequence of integers {jn }∞
n=1 such that jn → ∞ and:
Cjn · kν̃n − ν̃∞ k + kνn k−1 −−−→ 0
n→∞
We now estimate the other term on the right-hand side of (3.3.11). Choosing N = N (g)
large enough so that µn (ϕ0 , δ)(g) = µn (ϕ0 , δN )(g), we have
µn (ϕ0 , δ)(g) − µn (ϕ0 , |fj |2 )(g) ≤ |fj |2N − δN 2
kgk∞ .
L (M \K)
As j → ∞ (in particular, if j = jn ), |fj |2N → δN in VN , so this term tends to zero. It
follows that
(3.3.12)
lim µn (g) − µTn (fjn , fjn )(g) = 0.
n→∞
Setting ψ̃n = Rψn (fjn ), we deduce that
Z
(3.3.13)
lim µn (g) −
n→∞
2
|ψ̃n | g(x)dx
=0
XZ
holds for every g ∈ Cc∞ (XZ )K . In particular, we obtain (1) of the Proposition.
To obtain the equivariance property note that the representation Iνn is irreducible as a
(g, K)-module. By [16, Corollary 8.11], there exists un ∈ U (g) such that Iνn (un )ϕ0 = fjn .
Thus ψ̃n = un ψn . Now every e ∈ E commutes with the right G-action; in particular,
eun = un e. It follows that ψ̃n transforms under the same character of E as ψn .
3.4. Cartan invariance of quantum limits
In this section we show that a non-degenerate quantum limit µ∞ is invariant under
the action of A < G. This invariance follows from differential equations satisfied by the
intermediate distributions µn . The construction of these differential equations is a purely
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
30
algebraic problem: construct elements in the U (gC )-annihilator of ϕ0 ⊗ δ ∈ VK ⊗ VK0 ,
where the U (gC )-action is by Iν ⊗ Iν0 .
Ultimately, these differential equations are derived from the fact that each z ∈ Z =
Z(gC ) acts by a scalar on the representation (VK , Iνn ). To motivate the method and provide
an example, we first work out the simplest case, that of PSL2 (R), in detail. In this case the
resulting operator is due to Zelditch.
This section is written without reference to the central character – assume it to be trivial.
Allowing the central character to vary would amount to writing Z(g) = Z(gss ) ⊗ Z(Zg ) and
only working with the first part.
3.4.1. Example of G = PSL2 (R). Set G = PSL2 (R), Γ ≤ G a lattice, and A the subgroup of diagonal matrices. Let H (explicitly given below) be the infinitesimal generator
of A, thought of first as a differential operator acting on X = Γ\G via the differential of
the regular representation. If {ψn } is a conveniently arranged sequence of eigenfunctions
on Γ\G/K, and µn the corresponding distributions (Definition 3.3.3), we will exhibit a
second-order differential operator J such that for all g ∈ Cc∞ (XZ )K ,
(3.4.1)
µn ((H −
J
)g) = 0,
rn
where rn ∼ |λn |1/2 . Since the µn (Jg) are bounded (they converge to µ∞ (Jg)), we will conclude that µ∞ (Hg) = 0, in other words that µ∞ is A-invariant. This operator in equation
(3.4.1) is given in [36]. Its discovery was motivated by the proof (via Egorov’s theorem)
of the invariance of the usual microlocal lift under the geodesic flow. We show here how it
arises naturally in the representation-theoretic approach.
By Lemma 3.3.7, it will suffice to find an operator annihilating the element ϕ0 ⊗ δ ∈
VK ⊗ VK0 , where
acts via Iν⊗ Iν0 . U (gC ) 1
0 1
0
Let H =
, X+ =
, X− =
be the standard generators
−1
0
1 0
of SL2 , with the commutation relations [H, X± ] = ±2X± , [X+ , X− ] = H. The roots w.r.t.
the maximal split torus a = R · H are given by ±α(H) = ±2. We also set W = X+ − X− ,
so that R·W = k. Letting +α be the positive root, n = R·X+ , we have ρ(H) = 12 α(H) = 1.
Set exp a = A as in the introduction.
The Casimir element C ∈ Z(SL2 C) is given by 4C = H 2 + 2X+ X− + 2X− X+ . For
the parameter ν ∈ ia∗ given by ν(H) = 2ir (r ∈ R), C acts on πν with the eigenvalue
λ = − 14 − r2 . The Weyl element acts by mapping ν 7→ −ν. On S = G/K with the
metric normalized to have constant curvature −1, C reduces to the hyperbolic Laplacian.
In particular, every eigenfunction ψ ∈ L2 (Γ\G/K) with eigenvalue λ < − 41 generates a
unitary principal series subrepresentation. Definition 3.3.3 associates to ψ a distribution
µψ (ϕ0 , δ) on Γ\G.
As in Definition 3.2.6, we have an action Iν of G on V and of g on VK . Note that
for g ∈ N A, f ∈ VK , (Iν (g)f ) (1) = f (g) = ehν+ρ,H0 (g)i f (1). Since δ(f ) = f (1) and
the pairing between VK and VK0 is G-invariant, it follows that for X ∈ a ⊕ n, Iν0 (X)δ =
− hν + ρ, H0 (X)i δ.
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
31
Suppressing Iν from now on this means that X·(f ⊗δ) = (Xf )⊗δ−hν + ρ, H0 (X)i f ⊗
δ. Extend ν + ρ trivially on n to obtain a functional on a ⊕ n. Then
(X + (ν + ρ)(X)) · (f ⊗ δ) = (Xf ) ⊗ δ.
(3.4.2)
Now since a normalizes n and ν + ρ is trivial on n, the map X 7→ X + (ν + ρ) (X) is a Lie
algebra homomorphism a ⊕ n → a ⊕ n, and hence extends to an algebra homomorphism
τν+ρ : U (aC ⊕ nC ) → U (aC ⊕ nC ). (3.4.2) shows that, for u ∈ U (aC ⊕ nC ),
τν+ρ (u) · (f ⊗ δ) = (uf ) ⊗ δ
(3.4.3)
In view of (3.4.3) any operator u ∈ U (aC ⊕ nC ) annihilating ϕ0 gives rise to an operator
annihilating ϕ0 ⊗ δ.
The natural starting point is the eigenvalue equation (4C +1+4r2 )ϕ0 = 0. Of course, C
is not an element of U (nC ⊕aC ). Fortunately, it “nearly” is: there exists an C 0 ∈ U (nC ⊕aC )
such that C − C 0 annihilates ϕ0 .
In detail, we use the commutation relations and the fact that X− = X+ − W to write
4C = H 2 − 2H + 4X+2 − 4X+ W . Since φ0 is spherical, it follows that W φ0 = 0. Thus
(3.4.4)
H 2 − 2H + 4X+2 + 1 + 4r2 ϕ0 = 0
Since (ν + ρ)(H) = 2ir + 1, we conclude from (3.4.3) that:
(H + 2ir + 1)2 − 2(H + 2ir + 1) + 4X+2 + 1 + 4r2 · ϕ0 ⊗ δ = 0.
Collecting terms in powers of r we see that this may be written as:
(2H)(2ir) + (H 2 + 4X+2 ) ϕ0 ⊗ δ = 0
H 2 +4X 2
+
and dividing by 4ir we see that the operator H + Jr annihilates ϕ0 ⊗ δ,
Setting J =
4i
and so also the distribution µn . One then deduces the A-invariance of µ∞ as discussed in
the start of this section.
Notice that the terms involving r2 in (3.4.4) canceled. This is a general feature which
will be of importance.
3.4.2. The general proof. We now generalize these steps in order. Notations being as
in Sections 2.2,3.2 and in Definition 3.3.3, we first compute the action of U (mC ⊕ aC ⊕ nC )
on δ (Lemma 3.4.1) and then on ϕ0 ⊗ δ (Corollary 3.4.2). Secondly we find an appropriate
form for the elements of Z(gC ) (Corollary 3.4.5), which gives us the exact differential
equation (3.4.6). We then show that the elements we constructed annihilating µψ are (up to
J
scaling) of an appropriate form H + kνk
∗ (Lemma 3.4.6), and “take the limit as ν → ∞”
(Corollary 3.4.7) to see that µ∞ is invariant under a sub-torus of A.
A final step (not so apparent in the PSL2 (R) case) is to verify that we have constructed
enough differential operators to obtain invariance under the full split torus (Lemma 3.4.8).
In fact, even in the rank-1 case one needs to verify that the “H” part is non-zero.
Given λ ∈ a∗C , we extend it to a linear map mC ⊕ aC ⊕ nC → C. Since mC ⊕ nC is an
ideal of this Lie algebra, λ is a Lie algebra homomorphism; thus it extends to an algebra
homomorphism λ : U (mC ⊕ aC ⊕ nC ) → C. We denote by τλ the translation automorphism
of U (mC ⊕aC ⊕nC ) given by X 7→ X +λ(X) on mC ⊕aC ⊕nC . Similarly, given χ ∈ h∗C , we
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
32
define τχ : U (hC ) → U (hC ). We shall write U (gC )≤d for the elements of U (gC ) of degree
≤ d, and similarly for other enveloping algebras and Z = Z(gC ) (e.g. Z≤d = Z ∩ U (gC )≤d ).
Let ν ∈ a∗C . Let χν : Z → C be the infinitesimal character corresponding to Iν (that
is, the scalar by which Z acts in (Iν , VK ).) Recall that ρh denotes the half-sum of positive
roots for (hC : gC ), ρ the half-sum for (a : g).
L EMMA 3.4.1. For X ∈ m ⊕ a ⊕ n, Iν (X)δ = − hν + ρ, Xi δ.
P ROOF. This follows from the definitions.
C OROLLARY 3.4.2. For any u ∈ U (mC ⊕ aC ⊕ nC ) and f ∈ VK ,
Iν ⊗ Iν0 (τν+ρ (u)) · (f ⊗ δ) = (Iν (u)f ) ⊗ δ.
P ROOF. This follows from the previous Lemma.
R EMARK 3.4.3. Denote by DG (G/K) the ring of G-invariant differential operators
on S = G/K. There is an evident homomorphism U (gC )K → DG (G/K) with kernel
U (gC )K ∩ kU (gC ), see [12, 2.6] or [5, 9.2]. We recall that “projection to U (aC )” under
the Poincaré-Birkhoff-Witt isomorphism U (gC ) = U (nC ) ⊗ U (aC ) ⊗ U (kC ) descends (after composing with an appropriate translation on U (aC )) to an isomorphism of DG (G/K)
with U (aC )W (g:a) . We shall need some very slightly refined information about this decomposition. There is an evident map Z → DG (G/K) and it will suffice for our purpose to
understand the decomposition on the image of Z(gC ). (Although we do not need this, the
map from Z → DG (G/K) is in most cases nearly surjective; in all cases the quotient field
of the image coincides with the quotient field of DG (G/K), c.f. [13, 3.16].)
D EFINITION . Let pr : U (gC ) → U (hC ) be the projection corresponding to the decomposition U (gC ) = U (hC ) ⊕ [(nC ⊕ nM )U (gC ) + U (gC )(n̄C ⊕ n̄M )] (arising from the
decomposition gC = nC ⊕ nM ⊕ hC ⊕ n̄C ⊕ n̄M by the Poincaré-Birkhoff-Witt Theorem).
L EMMA 3.4.4. For z ∈ Z≤d , we have
z − pr(z) ∈ U (nC )U (aC )≤d−2 U (kC ).
P ROOF. It suffices to show that z − pr(z) ∈ U (nC )U (gC )≤d−2 U (kC ), since gC = nC ⊕
aC ⊕ kC .
Let B(nC ), B(n̄C ), B(nM ) and B(n̄M ) be bases for nC , n̄C , nM and n̄M , respectively,
consisting of hC -eigenvectors. Let B(aC ) and B(bC ) be bases for aC and bC , respectively.
By Poincaré-Birkhoff-Witt, one may uniquely express z as a linear combination of
terms of the form:
D = X1 . . . Xn Y1 . . . Ym A1 . . . At B1 . . . Br X 1 . . . X k Y 1 . . . Y l
where X∗ ∈ B(nC ), Y∗ ∈ B(nM ), A∗ ∈ B(aC ), B∗ ∈ B(bC ),X ∗ ∈ B(n̄C ) and Y ∗ ∈ B(n̄M ).
Then z − pr(z) consists of the sum of all terms D for which n + m + k + l 6= 0. We show
that each such term satisfies D ∈ U (nC )U (gC )≤d−2 U (kC ).
In view of the fact that z − pr(z) commutes with aC , one has n = 0 iff k = 0. Further,
if n = k = 0, then the fact that z − pr(z) commutes with bC implies m = 0 iff l = 0. Also
one has n + m + t + r + k + l ≤ d.
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
33
We now proceed in a case-by-case basis, using either the inclusion nM ⊕ bC ⊕ n̄M =
mC ⊂ kC , or the observation that for X ∈ n̄C we have θC X ∈ nC , while X + θC X ∈ kC (it
is θC -stable!).
(1) k = l = 0 is impossible, for this would force n = m = 0.
(2) k ≥ 1 and l ≥ 1. Then n ≥ 1 so that X1 . . . Xn ∈ U (nC ), Y 1 . . . Y l ∈ U (kC ), and
m + t + r + k ≤ d − 2.
(3) k = 0 and l ≥ 1. Then n = 0 and m ≥ 1, so t ≤ d − 2. Since [a, m] = 0 we may
commute the A-terms past the Y -terms, so that D is the product of the A-terms (at
most d − 2 of them) and Y1 . . . Ym B1 . . . Br Y 1 . . . Y l ∈ U (kC ).
(4) k ≥ 1 and l = 0. Then n ≥ 1. Set s = Y1 . . . Ym A1 . . . At B1 . . . Br X 1 . . . X k−1
so that D = X1 . . . Xn · s · X k . Since m + t + r + (k − 1) ≤ d − 1 − n ≤ d − 2,
we have s ∈ U (gC )≤d−2 . Then (recall θC is the complex-linear extension of the
Cartan involution θ to gC ),
(3.4.5)
D = X1 . . . Xn sX k = X1 . . . Xn · s · (X k − θC (X k ))
+ X1 . . . Xn θC (X k )s
+ X1 . . . Xn (sθC (X k ) − θC (X k )s).
From the observation above, the first
on the right clearly belong to
two terms
U (nC )U (gC )≤d−2 U (kC ). Moreover, s, θC (X k ) ∈ U (gC )≤d−2 (the general fact
that [p, q] ∈ U (gC )dp +dq −1 whenever p ∈ U (gC )≤dp , q ∈ U (gC )≤dq follows by
induction on the degrees from the formula [ab, c] = a[b, c] + [a, c]b). Thus the
third term of (3.4.5) belongs to U (nC )U (gC )≤d−2 U (kC ) also.
C OROLLARY 3.4.5. Let z ∈ Z≤d . Then there exists b = b(z) ∈ U (nC )U (aC )≤d−2 such
that z − pr(z) + b(z) ∈ U (gC ) · kC .
Since Iν (kC ) annihilates ϕ0 and z·ϕ0 = χν (z)ϕ0 , we have Iν (χν (z)−pr(z)+b(z))·ϕ0 =
0. In view of Corollary 3.4.2inf} we obtain:
(3.4.6)
Iν ⊗ Iν0 (τν+ρ pr(z) − τν+ρ b(z) − χν (z))(ϕ0 ⊗ δ) = 0
In what follows, we shall freely identify the algebra U (hC )W(hC : gC ) with the Weylinvariant polynomial functions on h∗C .
Given P ∈ U (hC )W(hC : gC ) , we denote by P 0 : h∗C → hC its differential. In other words,
we identify P with a polynomial function on h∗C , and P 0 denotes the derivative of this
function; it takes values in the cotangent space of h∗C , which is canonically identified at
every point with hC .
We shall use the notation U (gC )[aC ]≤r to denote polynomials of degree ≤ r on a∗C ,
valued in the vector space U (gC ). Note that given J ∈ U (gC )[aC ]≤r and ν ∈ a∗C we can
speak of the “value of J at ν.” We denote it by J(ν) and it belongs to U (gC ).
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
L EMMA 3.4.6. Let P ∈ U (hC )W(hC : gC ) have degree ≤ d. Set H =
there exists J ∈ U (gC )[aC ]≤d−2 such that
!
J(ν)
Iν ⊗ Iν0 H +
· ϕ0 ⊗ δ = 0.
kνkd−1
34
P 0 (ν)
||ν||d−1
∈ hC . Then
(As defined in Section 2.2, kνk denotes the norm of ν ∈ a∗C w.r.t. the Killing form.)
P ROOF. The map γHC : Z → U (hC )W(hC : gC ) given by γHC (z) = τρh pr(z) is an isomorphism of algebras, the Harish-Chandra homomorphism. With the above identification,
the infinitesimal character of (VK , Iν ) corresponds to “evaluation at ν + ρ − ρh ,” i.e. for
P ∈ U (hC )W(hC : gC ) :
(3.4.7)
−1
χν (γHC
(P)) = P(ν + ρa − ρh )
(See [16, Prop 8.22]; w.r.t. the maximal torus bC ⊂ mC , the infinitesimal character of the
trivial representation of mC is (the Weyl-group orbit of) ρ − ρh ).
−1
Given P ∈ U (hC )W(hC : gC ) of degree d, we set z = γHC
(P) in (3.4.6), writing b(P)
≤d
for the element b(z). Note that z ∈ Z(gC ) , as the Harish-Chandra homomorphism “preserves degree” (see [5, 7.4.5(c)]), and hence b(P) ∈ U (nC )U (aC )≤d−2 .
Combining (3.4.6) and (3.4.7): ϕ0 ⊗ δ is then annihilated by the operator
(3.4.8)
τν+ρ−ρh P − P(ν + ρa − ρh ) − τν+ρ b(P) ϕ0 ⊗ δ = 0
Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ). If a polynomial p ∈ C[x] has degree d,
p(x + y) − p(y) = p0 (y)(x) + q(x, y) where q ∈ (C[x]) [y] has degree at most d − 2 in y,
and the derivative p0 (y) is understood to act as a linear functional on x.
Applying this to p = P, y = ν + ρ − ρh we see that there exists J1 ∈ U (gC )[aC ]≤d−2
with deg(J) ≤ d − 2 and
(3.4.9)
τν+ρ−ρh P − P(ν + ρ − ρh ) = P 0 (ν + ρ − ρh ) + J1 (ν)
Now b(P) ∈ U (nC ) · U (aC )≤d−2 , so the map ν 7→ τν+ρ b(P) can be regarded as an
element J2 ∈ U (gC )[aC ]≤d−2 . Similarly ν 7→ P 0 (ν + ρ − ρh ) − P 0 (ν) defines an element
J3 ∈ U (gC )[aC ]≤d−2 .
Combining these remarks with (3.4.8) and (3.4.9), we see that
(P 0 (ν) + J1 (ν) + J2 (ν) + J3 (ν))ϕ0 ⊗ δ = 0
Set J ≡ J1 + J2 + J3 and divide by kνkd−1 to conclude.
C OROLLARY 3.4.7. Let P ∈ U (hC )W(hC : gC ) . Notations being as in Definition 3.3.8 and
Lemma 3.3.9, suppose {ψn } is conveniently arranged. Then µ∞ (ϕ0 , δ) is P 0 (ν̃∞ )-invariant.
P ROOF. It suffices to verify this for P homogeneous, say of degree d. Combining
Lemma 3.4.6 and Lemma 3.3.7, and using the homogeneity of P, we see that there exists
J ∈ U (gC )[aC ]≤d−2 so that
J(νn )
0
P (ν̃n ) +
· µn (ϕ0 ⊗ δ) = 0
||νn ||d−1
3.4. CARTAN INVARIANCE OF QUANTUM LIMITS
35
Here (P 0 + . . . ) acts on µn (ϕ0 ⊗ δ) according to the natural action of U (gC ) on
Cc∞ (XZ )0K . Now fix g ∈ Cc∞ (XZ )K . Let u → ut be the unique C-linear anti-involution of
U (gC ) such that X t = −X for X ∈ gC ⊂ U (gC ). Then we have for each d
! !
t
J
(ν
)
n
(3.4.10)
µn (ϕ0 ⊗ δ)
P 0 (ν̃n ) −
g = 0.
d−1
kνn k
t
n)
Note that, as n varies, the quantity P 0 (ν̃n ) − kνJ (ν
g remains in a fixed finite did−1
nk
mensional subspace of C ∞ (X)K . Further, it converges in that subspace to P 0 (ν̃∞ )g.
With these remarks in mind, we can pass to the limit n → ∞ in (3.4.10) to obtain
µ∞ (ϕ0 ⊗ δ)(P 0 (ν∞ )g) = 0, i.e. P 0 (ν∞ ) annihilates µ∞ as required.
It remains to show that the subspace
(3.4.11)
S = P 0 (ν̃∞ ) | P ∈ U (hC )W(hC : gC ) ⊂ hC
contains aC . By the Corollary this will show that a annihilates any limit measure, or that
this measure is A-invariant.
L EMMA 3.4.8. Let W0 ⊂ W (hC : gC ) be the stabilizer of ν̃∞ ∈ a∗C , and define S as in
0
(3.4.11). Then S = hW
C . In particular, if ν̃∞ is regular, then S contains aC .
P ROOF. This can be seen either from the fact that S is the image of the map on cotangent spaces induced by the quotient map h∗C → h∗C /W0 , or more explicitly: first construct
many elements in U (hC )W(hC : gC ) by averaging over W (hC : gC ), and then directly compute
derivatives to obtain the claimed equality.
W0 is generated by the reflections in W (hC : gC ) fixing ν̃∞ . In the case where ν̃∞
is regular as an element in ia∗R , the corresponding roots must be trivial on all of a∗C . In
particular, any element of W0 fixes all of aC .
C OROLLARY 3.4.9. Let notations be as in Proposition 3.3.13. Then any weak-* limit
σ∞ of the measures σn is A-invariant.
P ROOF. After passing to an appropriate subsequence, we may assume that {ψn } is conveniently arranged. Proposition 3.3.13, (1), shows that σ∞ (g) = µ∞ (ϕ0 , δ)(g) whenever
g ∈ Cc∞ (XZ )K . Corollary 3.4.7 and Lemma 3.4.8, together with the fact that Cc∞ (XZ )K is
dense in C0 (X), show that σ∞ is A-invariant.
CHAPTER 4
The Method of Hecke Translates I: Tubular Neighbourhood,
Translates, and Diophantine Geometry
4.1. Overview
In the previous chapter we have seen that a Maass form cannot be too concentrated –
its associated measure can be approximately lifted to a measure which is approximately
A-invariant (in the non-degenerate case). This chapter and the next one are devoted to
showing a similar conclusion for Hecke eigenforms. We hence fix a unitary Hecke character
ω ∈ Ẑ andR a normalized Hecke eigenfunction ψ ∈ L2 (X, ω), and let µ denote the measure
µψ (f ) = XZ f |ψ|2 dx on XZ . We will of course take ψ to be one of the functions ψ̃n of
Theorem 1.3.2. As before, π will denote any of the quotient maps G X XZ .
In Section 4.2 we will describe certain relatively compact open sets Ba (C, ) ⊂ G to
be called “tubular neighbourhoods”, depending on a parameter , their “width” , which will
tend to zero. We will also use the term for sets of the form xBa (C, ) ⊂ XZ where x ∈ XZ .
Using the isomorphism XZ = ΓZ\G ' G(Q)Z\G × G(Af )/Kf , any pair g∞ ∈ G and
gf ∈ G(Af ) gives rise to a relatively compact open subset
def
g∞ Ba (C, )gf = {ZG(Q)g∞ bgf Kf | b ∈ Ba (C, )} ⊂ XZ ,
which we will call a Hecke translate of the tubular neighbourhood xBa (C, ) of x = ZΓg∞ .
Writing gf = γkf for some γ ∈ Γ, kf ∈ Kf , we see that g∞ Ba (C, ) = x0 Ba (C, ) where
−1
x0 = ZΓ(γ∞
g∞ ). In other words, a Hecke translate of a tubular neighbourhood is again a
tubular neighbourhood of the same type.
In the next chapter we will analyze the behaviour of ψ on a large number of such translates to bound the measures µψ (xBa (C, )) and show they must decay as a power of (“positive entropy for the action of a”). We would like to choose a disjoint set of translates,
and hence we need to understand the intersection pattern of such a set. The main result of
this chapter (Theorem 4.4.4 of Section 4.4) is the first step in that direction, showing that
under certain conditions on S ⊂ G(Af ) there exists a proper Q-subalgebra of our division
algebra which controls the intersection pattern of the Hecke translates {xBa (C, )gf }gf ∈S .
Since we assume D to be of prime degree d, this proper subalgebra will then be commutative, i.e. a number field of dimension d contained in D(Q).
The key idea is the analysis in section 4.3 of the polynomial nature of the condition
“the subalgebra of D(Q) generated by the subset U ⊂ D(Q) is proper”.
For a very concrete version the ideas of this chapter (in the case d = 2) the see Lemmata
3.1, 3.2 and 3.3 of [3].
36
4.2. ALGEBRA IN TUBULAR NEIGHBOURHOODS
37
4.2. Algebra in tubular neighbourhoods
Let a ∈ A \ Z. We then set na ⊂ n be the Lie subalgebra spanned by the root spaces
nα associated to roots α such that α(a) > 0. This is the unipotent radical of a parabolic
subalgebra associated to that set of roots. The Levi factor is Ma Aa = ZG (a), where Aa =
Z(Ma Aa ) = ∩α(a)=0 Ker(α) is its center. We then set n̄a = na , the unipotent radical of the
opposite parabolic subgroup.
For any > 0 let na = {X ∈ na | kXk < } (note that n̄a = θna ), and set Na () =
exp na , N̄a () = ΘNa () = exp n̄a . For any relatively compact symmetric neighbourhood
of the identity C ⊂ Aa Ma and every > 0, we set Ba (C, ) = N̄a () · C · Na () ⊂ G.
We call C̃ ⊂ G a neighbourhood of C if C̃ is neighbourhood of every c ∈ C̄ (topological
closure).
D EFINITION 4.2.1. We call a set of the form Ba (C, ) a tubular neighbourhood of the
piece C ⊂ Aa Ma .
Thinking of as small (C will be fixed), the elements of Ba (C, ) are very close to lying
on Aa Ma . Since Aa Ma is a subgroup, one expects that a set of the form Ba (C, )Ba (C 0 , 0 )
also consists of elements close to a neighbourhood of the identity of Aa Ma .
L EMMA 4.2.2. (c.f. [3, Lemma 3.2]) For any C, C 0 , any relatively compact symmetric
neighbourhoods C ⊂ C̃ resp. CC 0 ⊂ C̃, and any , 0 small enough w.r.t. the choice of
C, C 0 , C̃ we have:
Ba (C, )−1 ⊂ B(C̃, OC ())
resp.
Ba (C, )Ba (C 0 , 0 ) ⊂ Ba (C̃, OC,C 0 ( + 0 )).
P ROOF. This is a direct computation. We only prove the first assertion.
Since the adjoint action is differentiable and B(C, 1) is a relatively compact subset of
G, there exists a constant rC such that kAd(m)Xk ≤ rC kXkfor any m ∈ B(C, 1) and
X ∈ g. Secondly, for any (vector-space) direct sum decomposition g = ⊕i Vi the map
Q
(Xi )i 7→ i exp Xi is a local diffeomorphism. Thus there exists δ, r0 > 0 such that if
X1 , X2 , Y ∈ g are all of norms≤ δ then there exists X 0 ∈ g and (Y1 , Y2 , Y3 ) ∈ n̄a ⊕ ma ⊕
na ' g such that kX 0 k ≤ r0 (kX1 k + kX2 k), kYi k ≤ r0 kY k and exp X 0 = exp X1 exp X2
and exp Y = exp Y1 exp Y2 exp Y3 (the existence of X 0 follows from the smoothness of
the multiplication operation in the co-ordinate system exp : g → G, the existence of Yi
from the equivalence of that co-ordinate system with the one induced from the direct sum
decomposition). We also observe that if C1 ⊂ G is any relatively compact subset, and C2 ⊃
C1 is a neighbourhood then for small enough we have C1 exp {X ∈ g | kXk ≤ } ⊂ C2 .
Assume that rC ≤ 1, rC ≤ δ, r0 rC ≤ δ. As the sets na , n̄a are symmetric, so are
Na () and N̄a (), so for the first assertion it suffices to show that if b = n̄mn ∈ Ba (C, )
then nmn̄ ∈ Ba (C̃, OC ()). For this write n = exp X, n̄ = exp Y with kXk , kY k ≤ and
write n0 = m−1 nm so that:
nmn̄ = mn0 n̄ = mn0 n̄n0−1 n0 .
4.3. DIOPHANTINE GEOMETRY OF DIVISION ALGEBRAS
38
We next note that n0 = exp(Ad(m−1 )X) ∈ B(C, 1) since kAd(m)Xk ≤ rC kXk ≤ 1.
Since n0 n̄n0−1 = exp(Ad(n0 )Y ) we conclude kAd(n0 )Y k ≤ rC kY k ≤ rC . We can thus
write exp(Ad(n0 )Y ) = exp(Y1 ) exp(Y2 ) exp(Y3 ) with Y1 ∈ n̄a , Y2 ∈ ma and Y3 ∈ na and
kYi k ≤ r0 rC . It follows that:
nmn̄ = exp(Ad(m)Y1 ) (m exp(Y2 )) (exp(Y3 ) exp(Ad(m)X)) .
Finally, choosing small enough will insure that m exp(Y2 ) ∈ C̃, while kAd(m)Y2 k ≤
r0 rC2 and exp(Y3 ) exp(Ad(m)X) = exp X 0 for some X 0 ∈ na such that kX 0 k ≤ r0 (1 +
r0 )rC .
The next Lemma formalizes the notion that the xBa (C, ) are “tubular neighbourhoods”
of the pieces xC, uniformly over compacta in G:
L EMMA 4.2.3. Let Ω∞ ⊂ G be compact. Then there exists a constant c > 0 (depending
on Ω∞ , C) such that for every x∞ ∈ Ω∞ and small enough > 0, if g ∈ x∞ Ba (C, )x−1
∞
−1
then there exists g∞ ∈ x∞ Cx−1
⊂
x
A
M
x
with
|x
(g
)
−
x
(g)|
≤
c
for
every
i.
∞
a
a
i
∞
i
∞
∞
P ROOF. We know that g = x∞ ng̃n̄x−1
∞ with g̃ ∈ C, n ∈ Na () and n̄ ∈ N̄a (). Now
−1
set g∞ = x∞ g̃x∞ and consider the maps ti : na × n̄a × G → R given by
(X, X̄, x∞ ) 7→ xi exp (Ad(x∞ )X) g∞ exp Ad(x∞ )X̄ .
Being the composition of smooth maps this is one as well; in particular it is continuously differentiable on a relatively compact open neighbourhood
of n1a × n̄1a × Ω∞ . We
1
1 can hence
find c > 0 such that for X ∈ na , X̄ ∈ n̄a ti (X, X̄, x∞ ) − ti (0, 0, x∞ ) ≤
c max kXk , X̄ . Since we have g∞ = t(0, 0, x∞ ) and g = t(X, X̄, x∞ ) for some
X ∈ na , X̄ ∈ n̄a we have the desired result.
4.3. Diophantine Geometry of Division Algebras
We will show that if I ⊂ D(Q) are close to a proper R-subalgebra DR0 ⊂ D(R) and
have small denominators, they generate a proper Q-subalgebra of D of dimension at most
dimR DR0 . Along the way we will make extensive use the co-ordinates xi on D introduced
as Notation 2.1.1.
L EMMA 4.3.1. Let
n
o
Vr,s (K) = x = x(1) , . . . , x(s) ∈ D(K)s dimK SpK x ≤ r ,
where SpK x is the K-subspace of D(K) = D(Q) ⊗ K spanned by x. Then Vr,s (K) =
∩p∈F {p = 0} where F is a finite family of homogeneous polynomials in the co-ordinates
of the x(i) , with coefficients in Q and degrees bounded as a function of d.
P ROOF. We need to verify that the statement “the rank of the s×d2 matrix M is at most
r” is equivalent to the joint vanishing of some polynomials in the entries of M . Indeed, M
having rank≤ r is equivalent to the vanishing of the determinant of every (r + 1) × (r + 1)
minor of M , and the coefficients of the polynomials are in fact ±1.
4.3. DIOPHANTINE GEOMETRY OF DIVISION ALGEBRAS
39
L EMMA 4.3.2. For each 1 ≤ k ≤ d2 and any field K extending Q, let
n
o
d2 Vr (K) = x ∈ D(K) dimK K[x] ≤ k ,
where K[x] is the K-subalgebra of D(K) = D(Q) ⊗ K generated by x. Then Vr is an
algebraic variety defined over Q, when defined in terms of the co-ordinates of the elements
of x in the basis. Moreover, it is defined by a finite set of homogeneous polynomials of
a-priori bounded degrees.
2
P ROOF. For x ∈ D(K)d let Wt (x) denote the subspace of D(K) spanned by all prod2
ucts of length at most t in the elements x(1) , . . . , x(d ) . This is a non-decreasing sequence of
subspace of the d2 -dimensional K vector space D(K). Hence there must exist a 1 ≤ t ≤ d2
such that Wt+1 = Wt . This means that the subspace Wt , spanned by products of the x(i) ,
is closed under left and right multiplication by them. In other words, Wt = K[x], the
subalgebra generated by the x. Since the Wt cannot increase further we conclude that
Wd2 = K[x] in all cases. Now, the set of products of at most d2 of the x is an element
Pd2
2l
of (D(K)) l=0 d which depends polynomially on x (fix an ordering). Since the previous
Lemma showed that Vr,Pd2 d2l is defined by homogeneous polynomial equations, we are
l=0
done. Note that the “structure constants” aijk enter into the coefficients of these polynomials.
D EFINITION 4.3.3. Let x ∈ Q. We define its denominator d(x) and height h(x) by:
Y
d(x) =
p−vp (x) ,
p:vp (x)<0
h(x) =
Y
p|vp (x)| .
p<∞
For a sequence x ∈ Qr we set d(x) = gcd {d(xi )}.
L EMMA 4.3.4. (Properties of denominators and heights) Let x, x0 ∈ Q.
(1) d(xx0 ) ≤ d(x)d(x0 ) and d(x + x0 ) ≤ d(x)d(x0 ).
(2) h(xx0 ) ≤ h(x)h(x0 ). If x ∈ Q× then h(x−1 ) = h(x).
(3) Let P ∈ Q[x] be a multivariate polynomial in r variables. Then there exist CP > 0
and an integer tP such that for all x ∈ Qr :
d(P (x)) ≤ CP d(x)tP .
P ROOF. Direct calculation and induction.
D EFINITION 4.3.5. For xp ∈ Qp we set dp (xp ) = 1 if xp ∈ Zp , dp (xp ) = p−vp (xp )
Q
Q
otherwise and hp (xp ) = p|vp (xp )| . If x ∈ Af we set d(x) = p dp (xp ), h(x) = p hp (xp ).
L EMMA 4.3.6. d : Af → Z is uniformly continuous in the adelic topology; Our definitions are compatible with the standard embeddings Qp ,→ Af and Q ,→ Af .
Q
P ROOF. Let x ∈ Af , x0 ∈ p Zp . Then d(x + x0 ) = d(x): at places where x is integral,
so is x + x0 . At places where x is not, we have vp (x + x0 ) = vp (x) by the ultrametric
behaviour of vp . The second assertion is obvious.
4.4. INTERSECTIONS OF HECKE TRANSLATES
40
We now extend the notion of denominator to elements of G:
2
D EFINITION 4.3.7. For γ ∈ D(Q)× set d(γ) = gcd {d(xi (γ))}di=1 . As above we
extend this notion to maps dp : Gp → Z and d : G(Af ) → Z. For g ∈ G(Af ) we will also
Q
write hp (gp ) = hp (νD (gp )) and h(g) = p hp (gp ) = h(νD (g)).
C OROLLARY 4.3.8. d(gg 0 ) ≤ d(g)d(g 0 ) and h(g −1 ) = h(g) for any g ∈ G(Af ). The
map d : G(Af ) → Z is continuous.
P
P ROOF. We have xi (gg 0 ) = j,k aijk xj (g)xk (g 0 ) with all the aijk ∈ Z. It is also clear
that the map is locally constant.
The continuity of the denominator map implies, in particular, its boundedness on the
compact subgroup KR , and we will assume that d(k) ≤ c1 for any k ∈ KR . If p ∈
/ R and
2
kp ∈ Kp and by Fact 2.1.3(4) we have xi (kp ) ∈ Zp for all 1 ≤ i ≤ d and hence dp (kp ) = 1.
In fact, if gp ∈ D(Qp ) then xi (gp kp ) and xi (kp gp ) are all linear combinations with integral
coefficients of the xi (gp ). This means dp (kp gp ), dp (gp kp ) ≤ dp (gp ). Multiplying by kp−1
(also an element of Kp ) we get:
L EMMA 4.3.9. Let p ∈
/ R, kp ∈ Kp and gp ∈ Gp . Then dp (gp kp ) = dp (kp gp ) = dp (gp ).
If g ∈ G(Af ), k ∈ Kf and kp = 1 for p ∈ R then d(kg) = d(gk) = d(g).
4.4. Intersections of Hecke Translates
Let a ∈ A \ Z, C ⊂ Aa Ma . Let Ω∞ ⊂ G be compact such that π(Ω∞ ) = XZ , and
choose a neighbourhood C̃ ⊂ Aa Ma so that for small enough , Ba (C, )Ba (C, )−1 is
contained in B̃ = Ba (C̃, O()) as in Lemma 4.2.2. We will also shorten B = Ba (C, ).
Fixing some x ∈ XZ we are ready to analyze intersections of Hecke translates of xB.
Writing x = ΓZx∞ for some x∞ ∈ Ω∞ , let g, g 0 ∈ G(Af ) be such that x∞ Bg ∩ x∞ Bg 0 is
nonempty. This entails the existence of γ ∈ G(Q), z∞ ∈ Z, k ∈ Kf and b, b0 ∈ B such that
(4.4.1)
γz∞ x∞ bg = x∞ b0 g 0 k
holds as an equality of adéles, where γ is embedded diagonally in G(A). We will say that
such γ cause an intersection. At the infinite place this reads γz∞ x∞ b = x∞ b0 or:
(4.4.2)
−1 −1
−1
γz∞ = x∞ b0 bx−1
∞ ∈ x∞ BB x∞ ⊂ x∞ B̃x∞ ,
where the last inclusion follows from the choice of C̃. Now lemma 4.2.3 shows that γz∞ is
0
O()-close to x∞ Aa Ma x−1
∞ (independently of g or g !). Recalling that Aa Ma is contained
in a proper subalgebra of D(R), we see that these γ are very close to satisfying the polynomial equations we have just constructed, except for the annoyance of the factor z∞ : γ
is O(/z∞ )-close to an element of the subalgebra, and |z∞ | might be very small. Lemma
4.4.3 addresses this problem.
Noting that the sets B̃ only decrease with , we also see that (assuming < 1) if
γ causes an intersection there exists z∞ ∈ R× such that γz∞ belongs to the relatively
compact set:
(4.4.3)
Ω∞ Ba (C̃, O(1))Ω−1
∞ ⊂ G∞ .
4.4. INTERSECTIONS OF HECKE TRANSLATES
41
Given S ⊂ G(Af ), we denote I(a, x∞ , B̃, S) the set of γ ∈ G(Q) that cause an intersection for translates of x∞ B by some g, g 0 ∈ S, where γz∞ lies in x∞ B̃x−1
∞.
D EFINITION 4.4.1. Let T > 0. We will say that S ⊂ G(Af ) is T -bounded if every
g ∈ S satisfies:
(1) d(g), d(g −1 ) ≤ −T ;
(2) h(νD (g)) ≤ −T (where νD : G(Af ) → Af is the reduced norm);
(3) (gp )p∈R ∈ KR .
L EMMA 4.4.2. Let S ⊂ G(Af ) be T -bounded, and let x∞ ∈ Ω∞ , γ ∈ I(a, x∞ , B̃, S).
Then d(γ), |νD (γ)|∞ −2T where the implied constant depends only on the choice of
co-ordinates and on the compact subgroup Kf .
P ROOF. By definition we have g 6= g 0 ∈ S and k ∈ Kf such that γ = g 0 kg −1 holds as
Q
an equality of finite adéles. We now use this place-by-place. Projecting to GR = p∈R Gp ,
we have gR , gR0 ∈ KR and hence γ ∈ KR under the diagonal embedding. It follows that
Q
/ R we use
p∈R dp (γ) is uniformly bounded by the number c we fixed above. If p ∈
Corollary 4.3.8 and Lemma 4.3.9 to get:
dp (γp ) ≤ dp (gp0 kp )dp (gp−1 ) = dp (gp0 )dp (gp−1 ).
Multiplying over all primes we conclude:
d(γ) e−2T .
Q
Next, by the product formula we have |νD (γ)|∞ = p |νD (γp )|−1
p . The absolute value of the
reduced norm is a positive multiplicative quasi-character. In particular it is the constant 1 on
any compact subgroup such as Kf . Also, we have defined the height so that νD (gp−1 )p ≤
hp (gp ). It follows again that:
Y
|νD (γ)|∞ ≤
hp (gp0 )hp (gp ) ≤ h(g)h(g 0 ) ≤ −2T .
p
L EMMA 4.4.3. There exists a constant c > 0 (depending on Ω∞ , C̃) such that for
every x∞ ∈ Ω∞ , small enough > 0, S ⊂ G(Af ) and γ ∈ I(a, x∞ , B̃, S) there ex−1/d
−1
ist z∞ ∈ R×
and g∞ ∈ x∞ C̃x−1
>0 with |z∞ | |ν(γ)|∞
∞ ⊂ x∞ Aa Ma x∞ such that
|xi (g∞ ) − z∞ xi (γ)| ≤ c for every 1 ≤ i ≤ d2 .
P ROOF. Let γ ∈ I(a, x∞ , B̃, S), diag(z∞ ) ∈ Z such that γ diag(z∞ ) ∈ x∞ B̃x−1
∞ (note
that xi (γ diag(z∞ )) = z∞ xi (γ) by definition). Assuming, as we may, that < 1, we have
observed above that γz∞ must belong to a specific compact subset of G. The continuity
of the map det : G → R× shows that det(γ diag(z∞ )) is uniformly bounded below. As
d
ν(γ) = det(γ) when on the right we embed γ in D(R), and det(diag(z∞ )) = z∞
we are
done.
4.4. INTERSECTIONS OF HECKE TRANSLATES
42
T HEOREM 4.4.4. (c.f. [3, Lemma 3.3]) There exists T > 0 such that, given a T bounded S and a point x∞ ∈ Ω∞ we can find a number field F ⊂ D(Q) of dimension d such that I(a, x∞ , B̃, S) ⊂ F × ⊂ G(Q), whenever is small enough. Moreover,
d
0
there exist γ 0(j) j=1 ⊂ O, each of height at most O(−T ) for some known T 0 , such that
F = Q(γ 0(1) , . . . , γ 0(d) ).
d2
P ROOF. Choose γ (j) j=1 ⊂ I(a, x∞ , B̃, S) which have the same Q-span as all of
I(a, x∞ , B̃, S), and let D0 denote the Q-subalgebra of D generated by the γ (j) , necessarily a division algebra. It suffices to show that this is a proper subalgebra of D, since in that
case dimQ D0 (Q) will be a proper divisor of dimQ D(Q) = d2 . With d assumed prime, the
possibilities are dimQ D0 (Q) = 1 where D0 (Q) = Q and dimQ D0 (Q) = d, where D0 (Q)
must be monogenic and hence commutative, i.e. a number field.
By Lemma 4.4.2 we have d(γ (j) ) 2T for all j. Applying Lemma 4.4.3 as well we
can find for each j
(j)
and z∞
(j)
−1
−1
g∞
∈ x∞ C̃x−1
∞ ⊂ x ∞ A a M a x ∞ ⊂ Ω ∞ Aa M a Ω ∞
(j) ∈ ZG (R) such that z∞ 2T /d and such that for each 1 ≤ i ≤ d2 we have:
(j)
(j)
(j) xi (g∞
) − z∞
xi (γ∞
) .
Now let r = dimR Aa Ma , and let {fm }M
m=1 be a set of polynomials with integer coefficients
n od2
(j)
∈ Vr (R) (they
defining Vr , each homogeneous of degree hm in the x(j) . That g∞
j=1
n o (j)
g∞
= 0.
generate an R-subalgebra of at most that dimension!) can be written as fm
j
Since we can uniformly bound the gradient of fm in a c-neighbourhood of the relatively
d2
−1
compact set Ω∞ Ba (C̃, O(1))Ω∞
for small enough, we have:
(j) (j) fm z∞
γ∞ ,
and hence:
(j) fm γ∞
1−2hm T /d .
n o (j)
is at most
Now since the coefficients of fm are integers, the denominator of fm
γ∞
(j)
xi (γ∞ ),
j
−2hm T
the hm -th power of the maximal denominator
i.e. at most . The
n o of an
(j)
6= 0, it is then at least 2hm T . Choosing T
crucial observation is then that if fm
γ∞
j
so that for all m
T <
2d
hm ,
d+1
we can make sure that for small enough we will have fm
dim D0 ≤ r.
n o
(j)
γ∞
= 0 for all m, i.e. that
4.4. INTERSECTIONS OF HECKE TRANSLATES
43
d
Finally, since [F : Q] = d, we can assume w.l.g. that γ (j) j=1 generate F , and the
same will hold if we replace γ (j) with γ 0(j) = d(γ (j) )γ (j) ∈ O. Now the discriminant of γ 0 is
at most that of the characteristic polynomial of the automorphism x 7→ γ 0 x of D(Q). Since
the discriminant of a polynomial is a polynomial expression in its coefficients, and since in
our case these coefficients are polynomials in the xi (γ 0 ), which are bounded by the height
of γ 0 . From this one can recover an exponent T 0 as in the statement of the Theorem.
00
R EMARK 4.4.5.
(1) The discriminant of F as above is also O(−T ), since it is at
most the product of the discriminants of the γ 0(j) .
(2) We have used the observation that every γ causing an intersection must satisfy
γ∞ ∈ Ω∞ Ba (C̃, O(1))Ω−1
∞ , and the latter is a compact set independent of .
CHAPTER 5
The Method of Hecke Translates II: Geometry and Harmonic Analysis
on the Building
As discussed in the introduction, this method was initiated by [25, 15]. Our analysis
stems from the very concrete Lemmata 3.3 and 3.4 of [3].
The following is the second main ingredient of the main theorem:
T HEOREM 5.0.1. Let a ∈ A \ Z. Then there exists η > 0 such that for every Hecke
Eigenfunction ψ ∈ L2 (X, ω) and every relatively compact neighbourhood of the identity
C < Ma ,
µψ (xBa (C, )) a η
holds for small enough and any x ∈ XZ . In particular, the implied constant is independent of f .
In the previous chapter we saw that given a T -bounded S ⊂ G(Af ), intersections between sets of the type x∞ Bg and x∞ Bg 0 for g, g 0 ∈ S could occur only if γgKf = g 0 Kf
for some γ ∈ F × , where F ,→ D(Q) is a number field of degree d and bounded discriminant. The proof hinges on choosing an S that will on the one hand be small enough to
(almost) fail to admit such intersections, while on the other be large enough to have µψ (xB)
P
bounded by a small multiple of g∈S µψ (xBg). The resulting disjointness of the translates
implies that the latter sum is bounded by 1, hence that µψ (xB) is small. The proof will be
broken up in several stages, alternating geometrical considerations and harmonic analysis
estimates.
We first analyze the intersection pattern at a single place, in other words the action
of the torus F × on the quotient Gp /Kp . We will do this by embedding the discrete set
Gp /Kp in a geometric structure, the building of Gp . In Section 5.1 we give a summary
of the properties of the building and use its geometry to construct a subset Sp ⊂ Gp of
translates for which we understand the (local) intersection pattern. In Section 5.2 we then
P
bound gp ∈Sp |ψ(x∞ bgp )|2 from below by a not-too-small multiple of |ψ(x∞ b)|2 for any
x∞ ∈ Ω∞ and b ∈ B. We use there bounds toward the Ramanujan conjecture due to
Luo-Rudnick-Sarnak.
The next step is to combine the information from many places. Section 5.3 shows that
by taking the union of the Sp over many places we still have no intersections, allowing us
to complete the proof of Theorem 5.0.1 in Section 5.4 .
5.1. The buildings of GLn and PGLn .
Let p be a finite rational prime, v the p-adic valuation on Q, with completion Qp ,
valuation ring Zp and maximal ideal p = pZp C Zp . The residue field Zp /p ' Fp is
44
5.1. THE BUILDINGS OF GLn AND PGLn .
45
then finite, and we will denote its cardinality q. We choose a uniformizer $ ∈ p \ p2 (e.g.
$ = p) and normalize the absolute value on Qp so that |$| = q −1 .
R EMARK 5.1.1. The distinction between p and q appears silly. It amounts to distinguishing between a finite cardinal and the associated integer, thought of as an object of
arithmetic. However, the discussion below would remain unchanged if Qp was replaced by
a field complete w.r.t. a discrete valuation, Zp by its valuation ring O, and p by the maximal
ideal of O. In that case read q for the cardinality of the residue field, p for its characteristic.
The buildings (defined below) are locally finite exactly when q is a finite cardinal.
N OTATION 5.1.2. For this Section (5.1) and the next only, we drop the subscript ’p’ we
use elsewhere, writing G = GLn (Qp ), K = GLn (Zp ). Thus A will denote the subgroup of
invertible diagonal matrices and Z = ZG ' Q×
p the center, i.e. the subgroup of non-zero
scalar matrices.
D EFINITION 5.1.3. Let B 0 = PGLn (Qp )/PGLn (Zp ), B̃ 0 = B 0 × Z. We let G act on
B 0 via the quotient PGLn (Qp ), on B̃ 0 by:
g · (x, n) = (gx, n + v(det(g))) .
R EMARK 5.1.4. One can identify G/K with the space of Zp -lattices in Qnp , B 0 with the
space of homothety classes of lattices.
Noting that det(k) ∈ O× for any k ∈ K, for x̃ = gK ∈ G/K the integer ν(det g)
is independent of the choice of representative g ∈ x̃. We will denote it c(x̃). Then the
map ϕ : G/K → B̃ 0 given by ϕ(x̃) = (x̃Z, c(x̃)) is a G-equivariant embedding. However,
it is not surjective: for x̃ ∈ G/K and z ∈ Q×
p thought of as an element of Z we have
0
c(z x̃) = nv(z) + c(x̃). Hence, to each x ∈ B we can associate a residue class ax ∈ Z/nZ
so that the image of ϕ is precisely {(x, t) | x ∈ B 0 , t ∈ ax }.
D EFINITION 5.1.5. Call a sequence {xi }di=0 ⊂ B 0 an oriented d-simplex if there exist
representative lattices Λi ∈ xi (also let Λd+1 = pΛ0 ) such that Λi ⊃ Λi+1 for all 0 ≤
i ≤ d. Denote by B d the set of d-simplexes (forgetting the orientation for the moment).
n-dimensional simplexes are called chambers.
n
FACT 5.1.6. B = B d d=0 is a chamber complex:
(1) It is a simplicial complex, i.e. the intersection of two simplexes is again a simplex.
(2) Every simplex is contained in a chamber.
Moreover, the G action on B 0 is simplicial.
D EFINITION 5.1.7. The complex B is called the simplicial building of PGLn (Qp ).
Endowing Z with its standard 1-dimensional simplicial complex structure, the simplicial
complex B̃ = B × Z (whose set of vertices is precisely B̃ 0 ) is called the (poly-)simplicial
building of GLn (Qp ). The elements of B 0 and B̃ 0 will be called the vertices of the respective buildings. In particular, we will thing of the cosets of G/K as vertices of B̃.
Now let x0 ∈ B0 be the vertex stabilized by ZK (i.e. the identity coset), and
let A00 ⊂
n
n
B 0 be the orbit Ax0 . Identifying A ' Q×
, we have StabA (x0 ) = Z · Z×
(with the
p
p
5.1. THE BUILDINGS OF GLn AND PGLn .
46
center acting diagonally). We then identify A00 with A/ StabA (x0 ) ' Zn /Z(1, . . . , 1). The
vertex corresponding to r + Z(1, . . . , 1) is the homothety class of the lattice generated by
{$ri ei }ni=1 where {ei }ni=1 is the standard basis of Qnp .
FACT 5.1.8. The subcomplex A0 ⊂ B consisting of those simplexes supported on A00 is
a chamber subcomplex; for every two simplexes ∆1 , ∆2 ∈ B there exists g ∈ G such that
g∆1 , g∆2 ∈ A0 .
We have {g ∈ G | gA0 = A0 } = NG (A), and NG (A) = W n A where W < K is the
subgroup of permutation matrices (henceforth called the Weyl group of G w.r.t. A).
D EFINITION 5.1.9. A subcomplex A of the form gA0 for some g ∈ G is called an
apartment of B. We have just seen that any two simplexes are contained in an apartment.
In particular, any two points x, y of the geometric realization |B| lie in the geometric realization |A| of an apartment A.
In order to give a canonical metric on B, we start with the positive semidefinite bilinear
form
! n !
n
n
X
X
1 X
ui vi −
ui
vi
hu, vi =
n i=1
i=1
i=1
on Rn . Its isotropic subspace is one-dimensional and spanned by the vector (1, . . . , 1).
In particular, this pairing descends to a positive-definite bilinear form on Rn /R(1, . . . , 1).
This defines a norm and hence a metric on this space.
FACT 5.1.10. The identification of the vertices of the standard apartment with the quotient Zn /Z(1, . . . , 1) extends uniquely to a piecewise-linear isomorphism of the geometric
realization|A0 | and Rn /R(1, . . . , 1).
Pulling back the norm we have defined gives a metric on |A0 |. We have remarked
before that the elements of G that preserve A0 are generated by NG (A) = W A. Since W
acts by permuting the co-ordinates, and A by affine translations, the metric we have defined
is NG (A)-invariant.
FACT 5.1.11. This metric extends uniquely to a G-invariant metric d(·, ·) on |B|, in
particular on the set of vertices B 0 . |B| is a simply connected complete CAT(0) metric
space. The geometric realization of an apartment is a (flat) geodesic subspace.
The standard metric on Z extends to a metric on its realization
R as a simplicial com plex. We extend our metric d (with the same notation) to B̃ ' |B| × R by taking the
Euclidean product (d2 ((x, s), (y, t)) = d2 (x, y) + |s − t|2 ).
D EFINITION 5.1.12. We call the metric space (|B| , d)together with the isometric G action the building of PGLn (Qp ), the metric space B̃ , d the building of G. Both spaces
are simply connected complete CAT(0) spaces.
FACT 5.1.13. Let (X, d) be a CAT(0) metric space. Then:
(1) (X, d) is uniquely geodesic. We will use [x, y] to denote the unique geodesic segment between x, y.
5.1. THE BUILDINGS OF GLn AND PGLn .
47
(2) Let Y ⊂ X be convex (x, y ∈ Y ⇒ [x, y] ⊂ Y ) and closed. Then for any x ∈ X
the function y 7→ d(x, y) on Y is strictly convex and has a unique minimum, say
πY (x).
(3) The map x 7→ πY (x) is a retraction of X to Y . It does not increase distances.
C OROLLARY 5.1.14. Let x, y ∈ |B| satisfy pr|A0 | (x) = pr|A0 | (y) = x0 . Then we have
{a ∈ A | ax = y} ⊂ A ∩ KZ.
P ROOF. Since the (isometric!) action of a on |B| preserves |A0 |, and since the orthogonal projection operator pr is defined via the metric, we have pr|A0 | (ax) = a pr|A0 | (x) for all
x ∈ |B|, a ∈ A. In particular, if ax = y we have ax0 = x0 , i.e. a ∈ StabG (x0 ) = KZ. Finally, we introduce a co-ordinate system of sorts on G, in terms of the set A+ =
{diag($r1 , · · · , $rn ) | 0 = r1 ≤ r2 ≤ · · · ≤ rn } ⊂ A. It is easy to see that A+ x0 ⊂ A00 is
a set of representatives for the orbits of W . From this one gets:
FACT 5.1.15. (Cartan decomposition) Let x, y ∈ B 0 be vertices. Then there exists a
unique a ∈ A+ such for some g ∈ G we have gx = x0 , gy = ax0 . In particular (“KAK
decomposition”), for any y ∈ B 0 there exists a unique a ∈ A+ and some k ∈ StabG (x0 ) =
K such that ky = ax0 .
We call a the relative position of y w.r.t. x (in that order!). Clearly this is a G-equivariant
notion. It is also clear that the distance d(x, y) only depends on the relative position of x
and y.
L EMMA 5.1.16. Let r ∈ A+ and let N (x0 , r) be the set of vertices of relative position
r to x0 . Then
#N (x0 , r) ∼ q 2δ(r) ,
P
asymptotically as q → ∞ where r is fixed. Here δ(r) = − ni=1 n+1
− i ri .
2
P ROOF. The KAK decomposition shows N (x0 , r) = {kax0 | k ∈ K} where a =
diag($r ). Now it suffices to compute the index of StabK (ax0 ) in K. A direct computation shows that
StabK (ax0 ) = {k ∈ K | ∀i, j : v(kij ) ≥ ri − rj } .
Since v(kij ) ≥ 0for all i, j, the condition
only has meaning for i > j. We also let N = rn
N
and set KN = k ∈ K | k ≡ In (p ) , the kernel of the quotient map Q : GLn (Zp ) →
GLn (Z/pN Z). By the choice of N we see that pri −rj | pN for all i > j, and hence that
KN ⊂ Kr . Setting K̄r = Q(Kr ), Ḡ = GLn (Z/pN Z) and letting B̄ < Ḡ denote the
subgroup of upper-triangular matrices, we then have:
#Ḡ
.
#N (x0 , r) = [K : Kr ] = Ḡ : K̄r =
#B̄ K̄r : B̄
Next, let Ū < Ḡ be the subgroup of lower-triangular unipotent matrices, let W̄ < Ḡ
denote the subgroup of permutation matrices, and set Ūr = Ū ∩ K̄r . We will show that
5.1. THE BUILDINGS OF GLn AND PGLn .
48
#Ūr ≤ K̄r : B̄ ≤ #W · #Ūr , and hence that:
#N (x0 , r) ∼
#Ḡ
.
#B · #Ūr
It now suffices to compute the orders of the finite groups Ḡ, B, Ūr . Since Ūr is the set of
ḡ ∈ Ḡ such that ḡij = δij for i ≤ j and ḡij ∈ pri −rj Z/pN Z for i > j, we have
Y
n(n−1)/2 −2δ(r)
#Ūr ∼
p
pN −(ri −rj ) = pN
.
i>j
It is also clear that for N fixed and p → ∞ (recall that N only depends on r), we have
n2
#Ḡ ∼ pN
and
n(n+1)/2
.
#B̄ = ϕ(pN )n · (pN )n(n−1)/2 ∼ pN
It remains to estimate K̄r : B̄ . Since both Ūr , B̄ are subgroups of K̄r , we have Ūr B̄ ⊂
K̄r . Standard linear algebra (Gaussian elimination) shows that any element of Ḡ has at
most
a unique
representation in the form ūb̄ where ū ∈ Ū , b̄ ∈ B̄. This shows that #Ūr ≤
K̄r : B̄ . On the other side, we will use Gaussian elimination show W̄ Ūr B̄ = K̄r : Start
with k ∈ Kr . If r2 > r1 , we must have k11 ∈ Z×
p (every other entry in the first column is
×
in pZp ). If r1 = r2 · · · = rs then the top-left s × s minor of k must be invertible for the
same reason (all entries below it are divisible by p) and we thus can permute the first s rows
to ensure the pivots of the image of this minor in GLn (Z/pN Z) are the diagonal elements.
Moreover, the permuted matrix is still in Kr . Multiplying on the right by an element of B
(the set of upper-triangular matrices in K) we can now assume that the first s rows of k are
the first s standard unit vectors. Next, we assume rs+1 = · · · = rs+t . Permuting these rows
(an operation which essentially commutes with what we have done so far), we can assume
that the pivots for the t × t minor on the diagonal at position s + 1, . . . , s + t has its pivots
on the diagonal, and continue the elimination by induction.
L EMMA 5.1.17. Let x ∈ N (x0 , en ) (we think of en as a representative of a coset modulu
Z(1, . . . , 1)). Then either # (N (x) ∩ A00 ) ≥ 2 or pr|A0 | (x) = x0 .
P ROOF. We may assume x ∈
/ A0 , z0 = pr|A0 | (x) 6= x0 . Being strictly convex, the
function y 7→ d(x, y) is strictly decreasing on the geodesic segment [x0 , z0 ]. Let x0 ∈
∆ ∈ A0 be a chamber such that [x0 , z] = |∆| ∩ [x0 , z0 ] has positive length. Let A be an
apartment containing the simplexes ∆ and {x0 , x}. Then x, x0 ∈ |A| and pr|∆2 | (x) 6= x0 ,
since z ∈ |∆2 | is closer to x.
Without loss of generality we can identify the vertices of A with Zn /Z as before, with
P
x0 = 0, ∆2 being the standard simplex with vertices xi = j≥n−i ej , and x = ek for some
k (there are all the neighbours with the correct relative position). The assumptions on the
existence of z amount to saying that x − x0 has a positive projection on xi − x0 for some
1 ≤ i ≤ n (otherwise x − x0 would have non-positive projection on the direction z − x0 ,
5.1. THE BUILDINGS OF GLn AND PGLn .
49
contradicting d(x, z) < d(x, x0 )). But the inner product is:
*
+
X
i
ek ,
ej = −
n
j≥n−i
where = 0 if k < n − i, = 1 otherwise. To make this positive, we must have = 1, i.e.
k ≥ n − i. But then x is in fact a neighbour of xi ∈ ∆02 ⊂ A00 .
Conversely, if x is a neighbour of two vertices of A0 then (since all edges in B 1 have
the same length) the projection of x to |A0 | is closer to x than any of them, and in particular
cannot be a vertex.
L EMMA 5.1.18. Let x ∈ N (x0 , en ) be such that pr|A0 | (x) = x0 and let x0 ∈ N (x, −en ).
Then pr|A0 | (x0 ) = x0 .
P ROOF. Assume x0 6= x0 , z0 = pr|A0 | (x) 6= x0 . As before let x0 ∈ ∆ be a simplex
containing an initial segment [x, z] of [x0 , z0 ] and let A be an apartment containing ∆ and
{x, x0 }. As before we can choose co-ordinates such that ∆ = {xi }n−1
i=0 and x = e1 (the
last by the proof of the previous lemma). We then have x0 = e1 − ek for some k, and the
assumption x0 6= x0 amount to assuming k 6= 1. We now show that x0 is the point of |∆|
nearest to x0 by computing the inner products hx0 − x0 , xi − x0 i and showing they are all
non-positive. Indeed, with depending on i, k as in the previous lemma, we have:
he1 − ek , xi i = − ≤ 0.
We now set S1 = x1 ∈ N (x0 , a) | pr|A0 | (x1 ) = x0 . Continuing outward we set
Nx1 = N (x1 , −en ) \ {x0 } for x1 ∈ S1 .
L EMMA 5.1.19. The union S2 = ∪x1 ∈S1 Nx1 is disjoint. S1 and S2 are disjoint.
P ROOF. The second assertion is immediate (the elements of S1 and S2 clearly have
different relative positions to x0 ). For the first, let x1 ∈ S1 , x2 ∈ Nx1 , and let x0 ∈ S1 be
distinct from x1 . Let A be an apartment containing the simplexes {x0 , x1 } and {x01 , x2 }.
We can choose the co-ordinates in such a way that x1 = e1 , x01 = e2 , and x2 = e1 − ek for
some k 6= 1. Assuming k 6= 2, the distance squared between x01 and x2 is:
1
he1 − e2 − ek , e1 − e2 − ek i = 9 − ,
n
while the squared distance between adjoining vertices is easily seen to be 1 − n1 . In the case
k = 2 the squared distance is 5 − n1 .
C OROLLARY 5.1.20. Let S = S1 ∪ S2 , let x, y ∈ S, and let a ∈ A satisfy ax = y. Then
a ∈ KZ.
L EMMA 5.1.21. #S1 ∼ q n−1 , #S2 ∼ q 2(n−1) .
P ROOF. By Lemma 5.1.16, we have #N (x, en ), #N (x, −en ) ∼ q n−1 for any vertex
x. Since #Nx1 = #N (x1 , −en ) − 1 for any x1 ∈ S1 and since these are all disjoint,
we see that #S2 ∼ q n−1 #S1 and that it suffices to show #N (x0 , en ) − #S1 q n−2 .
5.2. HECKE EIGENFUNCTIONS – THE LOCAL CONTRIBUTION
50
Since N (x0 , en ) \ S1 consists of those elements of N (x0 , en ) that are also neighbours of
another vertex of A0 , its cardinality is at most the size of the link of x0 in A0 times a bound
for the number of 2-simplexes in B 2 containing a fixed 1-simplex. Fixing a neighbour
x ∈ Lk0 (x0 ) is equivalent to giving a lattice pΛ0 < Λ < Λ0 . In this language we need to
enumerate lattices Λ0 of index p in Λ0 containing Λ. Reducing mod pΛ0 this is equivalent to
enumerating the subspaces of Fnq of codimension 1 containing a fixed non-trivial subspace.
Dualizing, we need to bound the number of 1-dimensional subspaces of Fnq contained in a
fixed proper subspace, a number which is easily verified to be ∼ q d−1 when d ≤ n − 1 is
the dimension of the subspace.
Since the structure of the apartment A0 is independent of q, the size of the link is
uniformly bounded (in fact by (n − 1) · n!) and we are done.
D EFINITION 5.1.22. Let S̃1 = {(x, 1) | x ∈ S1 }, S̃2 = {(x0 , 0) | x0 ∈ S2 }, S̃ = S̃1 ∪S̃2 .
P ROPOSITION 5.1.23. (transversals) S̃ ⊂ B̃ 0 is contained in the G-orbit of (x0 , 0), i.e.
in the image of G/K. Moreover it does not intersect the A-orbit of (x0 , 0) and if a ∈ A
carries x ∈ S̃ to y ∈ S̃ then a ∈ K.
P ROOF. The first assertion is clear by construction. The second assertion follows from
noting that every x ∈ S satisfies pr|A| (x) = x0 while every x0 ∈ Ax0 is fixed by this
projection, and that x0 ∈
/ S. For the last claim assume a(x, ) = (y, δ) with , δ ∈ {0, 1}.
From ax = y we conclude a = kz for some k ∈ K, z ∈ Z. From v(det a) + = δ we
conclude that |v(det(z))| ≤ 1, and since it is a multiple of n we must have v(det(z)) = 0.
This actually implies z ∈ K and we are done.
5.2. Hecke eigenfunctions – the local contribution
We keep here the notation of the previous section. However, we assume p ∈
/ R and
×
2
identify Gp ' GLd (Qp ) with D (Qp ). Let ψ ∈ L (X, ω) be our Hecke eigenfunction.
To any x∞ ∈ G∞ we have associated its p-Hecke orbit {G(Q)x∞ gp Kf }gp ∈Gp ⊂ X. This
is isomorphic to a quotient Gp /Kp , and by assumption the restriction f of ψ to this orbit
is a Hecke eigenfunction on Gp /Kp . The following Proposition can best be described by
saying f cannot be too concentrated on apartments: if f (1) is large, then f must also be
large on the transversal S̃ which lies away from the apartment. It demonstration relies on
a bound toward the Generalized Ramanujan Conjecture, the proof of which is reproduced
below.
FACT 5.2.1. There exists δ > 0 such that the Hecke eigenvalue λ considered below
satisfies:
1
1
|λ| (#S1 ) 2 · q 2 −δ .
(5.2.1)
P ROPOSITION 5.2.2. (“part of the tree” method) Let f be obtained as above. Then
X
1
|f (x̃)|2 1−2δ |f (1)|2 .
q
x̃∈S̃
5.2. HECKE EIGENFUNCTIONS – THE LOCAL CONTRIBUTION
51
P
P
P ROOF. Let L1 = x̃∈S̃1 f (x̃), L2 = x̃∈S̃2 f (x̃). There exist λ = λf (en , 1) depending only on f and the relative position (en , 1) such that for each x ∈ S1 we have:
X
f (x0 , 0) + f (x0 , 0) = λ · f (x, 1).
x0 ∈Nx
Summing over S1 and using the disjointness of the union defining S2 we have:
L2 + #S1 · f (1) = λ · L1 .
Therefore, at least one of the following holds:
#S1
|L1 | |f (1)|
|λ|
|L2 | #S1 |f (1)| .
Squaring, and using Cauchy-Schwartz, we get one of:
X
#S1
2
|f (x̃)|2 2 |f (1)|
|λ|
x̃∈S̃1
X
x̃∈S̃2
2
|f (x̃)|
(#S1 )2
|f (1)|2 .
#S2
Using Lemma 5.1.21 and the bound (5.2.1) complete the proof.
Digression: proof of the estimate 5.2.1. Our eigenfunction ψ ∈ L2 (X, ω) generates
an subrepresentation π̃ ⊂ L2 (G(Q)\G(A), ω) of G(A). Since ψ is invariant under right
translations by the maximal compact subgroup Kp , there must exist an irreducible π ⊂ π̃
containing a non-zero Kp -invariant vector. It is easy to see that any Kp -invariant vector
in π̃ must have the same eigenvalue λ w.r.t. the Hecke operator under consideration as ψ,
and we may thus switch to the case where ψ is a Kp -spherical vector in an irreducible
subrepresentation π ⊂ L2 (G(Q)\G(A), ω).
The component πp of this representation at the place p is then a spherical representation
of Gp ' GLd (Qp ) and hence isomorphic to the spherical constituent of the representation
Q
of GLd (Qp ) induced from the character diag(a1 , . . . , ad ) 7→ j |aj |µp j +it/d of Ap (where
P
t ∈ R and j µj = 0).
The eigenvalue λ is (up to normalization) the eigenvalue of the convolution operator
πp (1N (x0 ,en ) ) acting on the spherical vector of πp , where 1N (x0 ,en ) is the characteristic function of the subset ∪x∈N (x0 ,en ) xKp of Gp . This can be computed explicitly in terms of the
parameters µj :
T HEOREM 5.2.3. (the Satake Isomorphism; see [26]) To the convolution operator associated to the characteristic function of KaK with a ∈ A it is possible to associate
a permutation-invariant polynomial P (x1 , . . . , xn ) such that its eigenvalue acting on the
spherical function of πp is given by P (q µ1 +it/d , . . . , q µd +it/d ). If a = diag(q r1 , . . . , q rd )
Q r
with 0 ≤ r1 ≤ · · · ≤ rd , a monomial of maximal degree in P is j xj j .
5.3. SPLIT TORI
52
In our case we get a symmetric polynomial P of degree 1 in d variables such that
λ = P (q µ1 , . . . , q µn )q it
for all choices of {µj } , t. Since there is a unique such polynomial up to rescaling, we have:
X
λf = cq it (
q µj ).
j
We evaluate the constant c by considering the special case t = 0, µj = d+1
− j, where πp is
2
the trivial representation. In that case f (the restriction of ψ to an orbit Gp /Kp ) is constant
and we have the explicit evaluation λf = #N (x0 , ed ) ∼ q d−1 . We conclude that as q → ∞
d−1
#N (x0 , e )
c = P d+1 d ∼ q 2 .
−j
2
jq
As #S1 ∼ q d−1 this means:
1/2
|λf | ∼ (#S1 )
X q µj .
j
We now obtain a bound on the parameters µj in three steps.
First, we construct an automorphic representation of GLd (A) which also has πp as its
local component at p:
T HEOREM 5.2.4. (Arthur-Clozel; see [2]) Let π be an automorphic representation on
D× (A). Then there exists an automorphic representation Π on GLd (A) in the discrete
spectrum such that for every finite place v where D splits we have πv ' Πv .
Secondly, we argue that in the case where d is prime Π is in fact a cuspidal representation: the classification of the residual spectrum due to Mœglin-Waldspurger [24] implies
that for d prime the discrete non-cuspidal spectrum of GLd (A) consists of 1-dimensional
representations. Π is not a character since πp isn’t.
Thirdly, the cuspidality implies a bound on the spectral parameters of Πp ' πp :
T HEOREM 5.2.5. (Luo-Rudnick-Sarnak; see [21]) Let Π be a cuspidal automorphic
representation of GLd (A). At every place v where Πv is unramified, let it be the unitary
spherical constituent of the representation induced from the character diag(a1 , . . . , ad ) 7→
Q
P
µj +it/d
of Av (where t ∈ R and j µj = 0). Then
j |aj |v
|<µj | ≤
1
1
1
− 2
.
2 d +1
1
This gives |λ| (#S1 ) 2 · q 2 −δ where δ = d21+1 . We also note that our estimate of c
above shows that the implied constant is independent of q.
5.3. Split Tori
From here on we return to the usual notations: G = G(R) ' GLd (R) etc. In order
to apply the Diophantine results of the previous chapter, we need to fix C̃ ⊂ Aa Ma and
5.3. SPLIT TORI
53
Ω∞ ⊂ G as in the beginning of Section 4.4. We retain the notations B = Ba (C, ) and B̃
as defined there.
We first estimate the denominator of an element of G(Af ) in terms of the geometry of
the building. Recall that for each p ∈
/ R we fixed an algebra isomorphism ϕp : D(Qp ) →
Md (Qp ) such that ϕp (Op ) = Md (Zp ), and let T(Qp ) ⊂ G(Qp ) be the inverse image under
ϕ of the torus of diagonal matrices Ap < GLd (Qp ). Pulling back the Cartan decomposition
(Fact 5.1.15) we see that for every gp ∈ Gp there exists a unique ap ∈ A+
p and some
0
−1
×
kp , kp0 ∈ Op = Kp such that gp = kp ϕp (ap )kp . If ap has co-ordinates r = (r1 ≤ · · · ≤ rn )
we write1 rp (gp ) = max {−r1 , rn }. Necessarily a non-negative number, we call it the
radius of gp . It is immediate that rp (gp−1 ) = r(gp ).
L EMMA 5.3.1. dp (gp ) ≤ prp (gp ) , hence also dp (gp−1 ) ≤ prp (gp ) .
P ROOF. Let ap be defined as above, and let z ∈ ZGLd (Qp ) be the scalar matrix p−r1 so
that zap ∈ Md (Zp ). Since ϕp is an algebra homomorphism, we conclude that p−r1 gp ∈ Op ,
and hence that xi (gp ) ∈ pr1 Zp for all i.
Now let T be as in Theorem 4.4.4. For each prime p ∈
/ R, let S̄p denote all elements of
Gp of radius at most 1 such that hp (gp ) ≤ p (we set S̄p = ∅ if p ∈ R) also identify every
h ∈ S̄p with the element g ∈ G(Af ) such that gp = h and gp0 = 1 for all primes p0 6= p.
Finally, given > 0 let:
S̄ = ∪p2 ≤−T S̄p ⊂ G(Af ).
This family of sets is T -bounded by construction. It follows that for small enough, we can
d
associate to each x ∈ Ω a commuting subset γ (j) j=1 ⊂ O, of discriminants bounded by
0
O(−T ), such that every γ ∈ G(Q) causing an intersection
for xB w.r.t. Hecke translation
by S̄ lies in the subalgebra F = Q γ (1) , . . . , γ (d) ⊂ D(Q), which is isomorphic to a
number field also to be denoted F . Given this data we let E = Z[γ (1) , . . . , γ (d) ]denote
0
the subring of O generated by the γ (j) , and let D = O(−dT ) denote the product of their
discriminants, a multiple of the discriminant ofE. Since O is a Z-algebra of finite type all
its elements are integral over Z. In particular we have E ⊆ OF and hence the discriminant
of F divides D. Reflecting this we set
RD = R ∪ {p | p|D} ,
P = p ≤ −T /2 \ RD .
Let TF ⊂ G be the (maximal) Q-torus such that TF (Q) = F × . We will be interested in
the Qp -points of this torus, a subtorus of Gp . Clearly TF (Qp ) = (F ⊗Q Qp )× ⊂ D(Qp )× .
As is well-known, F ⊗Q Qp ' ⊕v|p Fv where the direct sum is over the places of F lying
over p. We thus have:
Y
(F ⊗Q Qp )× '
Fv× .
v|p
We now assume p ∈
/ RD . Then every v ∈ |F | lying above p is unramified, and hence
×
×
× 1
1
p is still a uniformizer of Fv so that Fv× = Q×
p OFv . In fact, Fv = Qp Fv , where Fv =
1
This definition is independent of the choice of isomorphism ϕp , but we shall not need this fact.
5.3. SPLIT TORI
54
x ∈ Fv | NQFvv (x) = 1 ⊂ OFv . We thus have:

 

Y
Y
 ×  Fv1  .
TF (Qp ) '  Q×
p
v|p
Tsp
F (Qp )
v|p
1
an
Setting
= v|p Q×
p , TF (Qp ) =
v|p Fv we note that these are the Qp -points,
respectively, of the maximal split and anisotropic Qp -subtori of TF . In other words, we
have just written our torus as an almost-direct product of its split and anisotropic parts.
Q
Q
L EMMA 5.3.2. (torus orbit contained in an apartment) Assume p ∈
/ RD . Then there
an
−1 sp
exist kp ∈ Kp for which kp TF (Qp )kp ⊂ T(Qp ). In addition, TF (Qp ) ⊂ Kp , so that
TF (Qp ) ⊂ kp T(Qp )Kp .
P ROOF. We first note that since O is a free Z-module of finite rank, every element of
O is integral over Z. In particular, the elements of the ring E ⊂ F (defined above) are
algebraic integers of F . Since the γ (j) generate F , we see that E is an order of F , and its
discriminant divides D. Since p does not divide D this implies that E is dense in OFv for
any place v ∈ |F | lying above p. For the remainder of this proof v will denote such a place,
and sums or products will be over the set of places of F lying above p.
The proof that F is dense in ⊕v Fv can be extended to show that E ⊗ Zp is dense in
⊕v OFv and hence that
⊕v OFv = E ⊗ Zp ⊂ Op .
Restricting our attention to the invertible elements we conclude that
Y
OF×v ⊂ Kp .
Tan
F (Qp ) ⊂
v
In order to diagonalize the split part we let xv ∈ F ⊗ Qp denote the idempotent given
by the identity element of Fv under the isomorphism of F ⊗ Qp with ⊕v Fv . Since
(
)
X
sp
TF (Qp ) =
av xv | av ∈ Q×
,
p
v
it suffices to simultaneously diagonalize the {xv }. Since xv ∈ OFv , the previous discussion
shows that xv ∈ Op . Applying the isomorphism ϕp it now suffices to show that a family
{xv } of commuting idempotents in ϕ(Op ) = Md (Zp ) can be diagonalized by an element
of GLd (Zp ). Equivalently we need to find a minimal generating set of the standard lattice
Λ = ⊕i Zp ei ⊂ Qdp which consists of joint eigenvectors of the xv .
Q
For each choice of eigenvalues εv ∈ {0, 1}, we set P (ε) = v (−1)εv (xv − εv ) ∈
Md (Zp ). Since the xv commute, this is a collection of commuting idempotents as well.
P
Q
Furthermore, it is easy to check that ε P (ε) = v (xv + (1 − xv )) = 1. This way we
obtain a direct sum decomposition Λ = ⊕ε Λε where Λε = P (ε)Λ. It is clear that each Λε
is a torsion-free Zp -module consisting of those elements t ∈ Λ for which xv t = εv t for
every v. In particular, we can choose a Zp -basis for each of them. Combining these bases
we obtain the desired basis for Λ, and hence an element of GLd (Zp ) conjugating the {xv }
to diagonal 0 − 1 matrices.
5.3. SPLIT TORI
55
For each prime p ∈ P now fix kp as in the lemma and let S̃ be the transversal constructed in Definition 5.1.22. The first claim of Proposition 5.1.23 assures us we can choose
a set of representatives gs ∈ GLd (Qp ) such that {gs (x0 , 0)} = S̃.
−1
C OROLLARY 5.3.3. Let Sp = kp ϕ−1
. Then:
p (gs )kp
s∈S̃
(1) Sp ⊂ S̄p . In other words, every gp ∈ Sp is of radius at most 1 and height at most
p.
(2) We have Sp ⊂ S̄p \ TF (Qp )Kp .
(3) For x 6= y ∈ Sp and g ∈ TF (Qp ) gx = y implies g ∈ kp−1 Kp kp = Kp .
(4) For every x∞ ∈ G we have
X
1
|ψ(ZΓx∞ gp )|2 1−δ |ψ(ZΓx∞ )|2 .
p
g ∈S
p
p
P ROOF. (1) The elements of S̃ are of relative positions en and −e1 + en to the identity
coset, respectively. Their pull-backs by ϕ−1
p are thus of radius 1 and reduced norm of absolute value either p or 1. Since multiplication by an element of Kp on the left or right does
not change the radius or the height of an element of Gp , the same holds for the elements of
Sp .
(2) and (3) follow directly from the corresponding claims of Proposition 5.1.23 via the
Lemma. Part (4) follows from 5.2.2.
Again constructing our set of translates place-by-place, we set:
[
Sp .
S =
p2 ≤−T
L EMMA 5.3.4. (intersections only occur place-by-place) Let γ cause an intersection
for Hecke translates of x∞ B by S . Then there exists a prime p such that γp0 ∈ Kp0 for all
p0 6= p.
P ROOF. Recall the basic observation from the previous chapter: if g, g 0 ∈ S are distinct
and γ causes an intersection between x∞ Bg and x∞ Bg 0 then (the finite part of equation
(4.4.1)):
γ ∈ g 0 Kg −1 .
Let g ∈ Sp , g 0 ∈ Sp0 . If p00 6= p, p0 then for gp00 , gp0 00 ∈ Kp00 so γp00 ∈ Kp00 . If p = p0 we
are done. In the case p0 6= p the p0 -component of g, is an element gp0 ∈ Kp0 . We then read
off γp0 ∈ gp0 0 Kp0 . Since γ ∈ TF (Qp ) this means gp0 0 ∈ TF (Qp )Kp , which contradicts the
construction of Sp0 as interpreted in the first part of Corollary 5.3.3.
The main geometric property of our construction is now clear:
P ROPOSITION 5.3.5. There exists finite subset I ⊂ Γ such that for small enough, the
set of γ ∈ G(Q) that cause intersections for {x∞ Bg}g∈S (x∞ ∈ Ω∞ fixed) is contained in
S
I. In particular, any point of the union x∞ B ∪g∈S x∞ Bg ⊂ XZ is contained in at most
|I| of the translates forming the union.
5.4. THE PROOF OF THEOREM 5.0.1
56
P ROOF. Recalling the observation leading to equation (4.4.3), we set
n
o
1
×
−1
Q = g ∈ G∞ | |det(g)| = 1 ∧ ∃z∞ ∈ R : gz∞ ∈ Ω∞ Ba (C̃, O(1))Ω∞ .
For g ∈ Q and z∞ as in the definition, we have |det(z∞ )| = |det(gz∞ )| belonging to
a compact subset of R× . It follows that Q is relatively compact, and we will see that
I = Γ ∩ Q works as claimed.
Let γ ∈ F × = TF (Q) cause an intersection between x∞ Bg and x∞ Bg 0 . By the Lemma
we have g, g 0 ∈ Sp for some p, so that γp0 ∈ Kp0 for all p0 6= p. At the place p itself the
second claim of Corollary 5.3.3 now gives γp ∈ Kp , so that γ ∈ Kf , i.e. γ ∈ Kf ∩G(Q) = Γ.
Q
Next, γ ∈ Kf implies |ν(γ)|∞ = p |ν(γp )|−1 = 1 so that |det(γ)| = 1 and γ ∈ Q.
Finally, let y ∈ x∞ Bg with g ∈ Sp . Then, if y ∈ x∞ Bg 0 for some other g 0 ∈ S we
must have g 0 ∈ Sp and some γ ∈ I causing that intersection. As usual we write this in the
form:
γp gp Kp = gp0 Kp .
In particular, the coset gp0 Kp ∈ Gp /Kp can be recovered from γ and g. Now since Sp
was chosen to be a system of representatives for a set of such cosets, it follows that g 0 is
uniquely determined by γ, so that there can be at most |I| such g 0 .
5.4. The proof of theorem 5.0.1
In summary, we fixed an open compact subgroup Kf < G(Af ), an element a ∈ A \ Z,
a compact fundamental domain Ω∞ ⊂ G, and relatively compact neighbourhood C, C̃ ⊂
M a Aa .
Then, for > 0 small enough, we have found in order a number field F with discriminant bound D controlling the intersections, a set of primes P avoiding ramification, and
finally a set of Hecke translates S satisfying both geometric and spectral properties. We
now show that for any central character ω unramified at R, Hecke eigenfunction ψ and any
x ∈ XZ , µψ (xB) decays polynomially with .
P ROOF. Choose some x∞ ∈ Ω∞ projecting to x ∈ XZ , and let 1x∞ Bg denote the
characteristic function of the translate x∞ Bg ⊂ XZ , Proposition 5.3.5 can be interpreted
to read:
X
1x∞ Bg (y) ≤ |I| .
g∈S
2
Multiplying by |ψ(y)| and integrating over XZ we conclude:
X
µψ (x∞ Bg) ≤ |I| .
g∈S
Recall the construction of S as ∪p∈P Sp . Changing the order of summation and integration, we obtain:
Z X
X
µψ (x∞ Bg) =
|ψ(x∞ bg)|2 dm(b)
g∈Sp
B g∈S
p
5.4. THE PROOF OF THEOREM 5.0.1
57
Where dm is the Haar measure on XZ . We now apply part (4) of Corollary 5.3.3 and
conclude
Z
X
1
1
µψ (x∞ Bg) 1−δ
|ψ(x∞ b)|2 dm(b) = 1−δ µψ (xB).
p
p
B
g∈S
p
Summing over p ∈ P we get:
X
µψ (x∞ Bg) g∈S
Since
X
p∈RD
!
X 1
p1−δ
p∈P
µψ (xB).
1
p1−δ
≤
X 1
+ log D log −1 ,
1−δ
p
p∈R
while
X
p2 ≤−T
1
p1−δ
−T δ/3 ,
the latter expression also bounds the asymptotics of
P
p∈P
p−1+δ . We thus have:
µψ (xBa (C, )) T δ/3 .
We remark that the implicit constant indeed only depends on a, on properties of D and
Kf such that the set R of ramified places and the structure constants aijk , and finally on
the choices of Ω∞ , C̃. The exponent η = T δ/3, furthermore, only depends on the degree
d of the division algebra (since T depends on that and on the dimension r < d2 of the
subalgebra spanned by Ma Aa ).
Bibliography
1. D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst.
Steklov. 90 (1967), 209. MR MR0224110 (36 #7157)
2. James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace
formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR
MR1007299 (90m:22041)
3. Jean Bourgain and Elon Lindenstrauss, Entropy of quantum limits, Comm. Math. Phys. 233 (2003), no. 1,
153–171.
4. Yves Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985),
no. 3, 497–502. MR 87d:58145
5. Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996, Revised reprint of the 1977 translation. MR 97c:17010
6. J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Erratum: “Spectra of compact locally symmetric
manifolds of negative curvature”, Invent. Math. 54 (1979), no. 1, 101. MR 82a:58050b
7.
, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979),
no. 1, 27–93. MR 82a:58050a
8. Martin Eichler, Lectures on modular correspondences, Lectures on Mathematics, vol. 9, Tata Institue of
Fundamental Research, Bombay, 1957.
9. Manfred Einsiedler and Anatole Katok, Invariant measures on G/Γ for split simple Lie groups G, Comm.
Pure Appl. Math. 56 (2003), no. 8, 1184–1221, Dedicated to the memory of Jürgen K. Moser. MR
2004e:37042
10. Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions
to littlewood’s conjecture, http://www.math.princeton.edu/ẽlonl/Publications/EKLslnr.pdf.
11. Richard P. Feynman, Statistical mechanics: a set of lectures, Frontiers in Physics, W. A. Benjamin, Inc.,
Reading, Massachusetts, 1972.
12. Ramesh Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive
groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas],
vol. 101, Springer-Verlag, Berlin, 1988. MR 89m:22015
13. Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83,
American Mathematical Society, Providence, RI, 2000, Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original. MR 2001h:22001
14. Eberhard Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh.
Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304. MR 1,243a
15. H. Iwaniec and P. Sarnak, L∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141
(1995), no. 2, 301–320. MR 96d:11060
16. Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36,
Princeton University Press, Princeton, NJ, 1986, An overview based on examples. MR 87j:22022
17. Elon Lindenstrauss, Adelic dynamics and arithmetic quantum unique ergodicity, to be published in the
Proceedings of the 2005 Current Developments in Mathematics Conference.
, On quantum unique ergodicity for Γ\H × H, Internat. Math. Res. Notices (2001), no. 17, 913–
18.
933. MR 2002k:11076
19.
, Invariant measures and arithmetic quantum unique ergodicity, preprint (2003), (54 pages).
58
BIBLIOGRAPHY
59
20. Elon Lindenstrauss and Akshay Venkatesh, Existence and weyl’s law for spherical cusp forms,
math.NT/0503724.
21. Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for GL(n),
Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos.
Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310. MR 2000e:11072
22. Wenzhi Luo and Peter Sarnak, Quantum variance for Hecke eigenforms, Ann. Sci. École Norm. Sup. (4)
37 (2004), no. 5, 769–799. MR MR2103474
23. Stephen D. Miller, On the existence and temperedness of cusp forms for SL3 (Z), J. Reine Angew. Math.
533 (2001), 127–169. MR 2002b:11070
24. C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in
Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995, Une paraphrase de l’Écriture [A
paraphrase of Scripture]. MR MR1361168 (97d:11083)
25. Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm.
Math. Phys. 161 (1994), no. 1, 195–213. MR 95m:11052
26. Ichirô Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Inst. Hautes
Études Sci. Publ. Math. (1963), no. 18, 5–69. MR MR0195863 (33 #4059)
27. Lior Silberman and Akshay Venkatesh, On quantum unique ergodicity for locally symmetric spaces I,
preprint available at http://arxiv.org/abs/math/407413.
28. A. I. Šnirel0 man, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk 29 (1974), no. 6(180), 181–
182. MR 53 #6648
29. Christopher D. Sogge, Concerning the Lp norm of spectral clusters for second-order elliptic operators
on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–138. MR 89d:35131
30. Thomas Watson, Rankin triple products and quantum chaos, Ph.D. thesis, Princeton University, 2003.
31. André Weil, Basic number theory, third ed., Springer-Verlag, New York, 1974, Die Grundlehren der
Mathematischen Wissenschaften, Band 144. MR MR0427267 (55 #302)
32. Eugene Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), no. 5,
749–759.
33. Scott A. Wolpert, Semiclassical limits for the hyperbolic plane, Duke Math. J. 108 (2001), no. 3, 449–
509. MR 2003b:11051
34. Steven Zelditch, Pseudodifferential analysis on hyperbolic surfaces, J. Funct. Anal. 68 (1986), no. 1,
72–105. MR 87j:58092
35.
, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55
(1987), no. 4, 919–941. MR 89d:58129
36.
, The averaging method and ergodic theory for pseudo-differential operators on compact hyperbolic surfaces, J. Funct. Anal. 82 (1989), no. 1, 38–68. MR 91e:58194