Last revised 12:09 p.m. February 20, 2016
Introduction to admissible representations of p-adic groups
Bill Casselman
University of British Columbia
cass@math.ubc.ca
Chapter IV. Harmonic analysis on the tree of SL(2)
In this chapter I shall discuss at length a family of representations of PGL2 (k) constructed by geometric
means. This will provide both interesting examples of admissible representations and a preview of what is
to come concerning the asymptotic behaviour of matrix coefficients.
The motivation for the construction is classical. The group GL2 (R) acts on the space of all symmetric 2 × 2
real matrices:
X: S 7−→ XS tX ,
and preserves the open cone C of positive definite matrices. The quotient PGL2 (R) therefore acts on the
space P(C), which is the quotient of such matrices modulo positive scalars. The isotropy subgroup of I is the
image O(2) = O(2)/{±I} in PGL2 (R), so that P(C) may be identified with PGL2 (R)/O(2). The embedding
of SL2 into GL2 identifies this with SL2 (R)/SO(2). There exists a Riemannian metric on P(C), invariant with
respect to PGL2 (R) and unique with this property, up to a positive scalar multiple.
The hollow cone ∂ C of non-negative symmetric matrices of rank one that borders C is also stable under
GL2 (R). To each point of ∂ C corresponds the null line of the corresponding quadratic form, and P(∂ C) may
be identified with P1 (R), the space of lines in R2 . This space compactifies P(C).
If we choose coordinates
z+y x
x z−y
for symmetric matrices, the space C is where z > 0, z 2 > y 2 + x2 . The intersection of this and the plane z = 1
is the open disc x2 + y 2 < 1, which may be identified with P(C). Interesting representations of SL2 (R) are
obtained on eigenspaces of the non-Euclidean Laplacian. Among these, for example, is the space of harmonic
functions that are smooth on {z | |z| ≤ 1}. As is well known, restriction of such functions to the unit circle S
is an isomorphism of this space with C ∞ (S).
Among many remarkable parallels between the structures of real and p-adic groups, one of the most remarkable is that there exist analogues of real symmetric spaces. These are the buildings constructed by Bruhat and
Tits. Among them is the tree on which PGL2 (k) acts. In this chapter I shall define it, prove some elementary
properties, and show how it can be used in harmonic analysis.
For groups of higher rank, buildings are important in understanding the structure of such groups, but play a
very small role in analysis. Doing analysis on the tree offers a unique opportunity to perceive many features
of representations of p-adic groups intuitively.
At the beginning,I assume only that k is a discrete valuation field, but eventually I shall assume it to be
complete. I’ll write ∗ ̟n for an element equal to a unit times ̟n . Throughout, let
G = GL2 (k)
K = GL2 (o)
G = PGL2 (k)
K = PGL2 (o) .
Chapter IV. Harmonic analysis on the tree of SL(2)
2
If λ = (m, n) then
λ
̟ =
̟m
◦
◦
̟n
.
The standard reference for the geometric material to be presented is Chapitre II of [Serre:1977]. and references
for analysis on the tree are [Cartier:1971/2] and [Cartier:1973]. For buildings [Macdonald:1972], particularly
Chapter III, is concise and very readable.
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Lattices
The tree
The action of K
Apartments
Orbits of N
The tree and the Hecke algebra
The representations
Relations with representations induced from parabolic subgroups
References
1. Lattices
A lattice in k 2 is any finitely generated o-submodule that spans k 2 as a vector space, for example o2 .
IV.1.1. Proposition. Every lattice in k 2 is free over o of rank 2.
Proof. Suppose given m generators of the o-submodule L, and suppose ML to be the 2 × m matrix whose
columns are those generators. We shall apply what I call integral column operations to reduce ML to a
matrix with only two non-zero columns, which are linearly independent and hence form an o-basis of L.
There are three types of integral column (or, for that matter, row) operations:
(a) permuting columns (rows);
(b) multiplying one column (row) by a unit of o;
(c) adding to any column (row) an integral multiple of another.
These column operations may be effected through multiplication on the right by elementary matrices in
GLm (o). Any multiplication on the right by a matrix in GLm (o), hence one of these operations, does not
change the lattice generated by the columns. The row operations can be carried out through multiplication
on the left by matrices in GL2 (o), and amount to a change of basis in k 2 .
The Proposition is therefore a consequence of:
IV.1.2. Lemma. Every 2 × m matrix of rank 2 with coefficients in k may be reduced through elementary
column operations to one that vanishes except in the first two columns, and in those columns is of the form
̟m x
◦
̟n
.
The integers m, n are unique, and the entry x is unique modulo pm .
Proof. Because the lattice contains a basis, there exists at least one non-zero entry in the second row. One
among them will have maximal norm, and we may assume it to be in the lower left-hand corner (position
(2, 1)). By an operation of type (b), we may make it ̟n for some n, and then we may apply operations of
type (c) to reduce the rest of the second row to 0.
The matrix now looks something like
x ∗
̟n ◦
∗
◦
.
Chapter IV. Harmonic analysis on the tree of SL(2)
3
Let’s look at the first row. Beginning with a swap if necessary, possibly followed by a unit column multiplication, we may get an entry in position (1, 2) of the form ̟m and of maximal norm in columns c ≥ 2. We
may then apply operations of type (c) to make the first row vanish in columns c ≥ 3. The only non-zero
entries are now in the first two columns. After a swap, it will be of the right form.
As for uniqueness: ω n is the least common multiple of the second row, and the elementary column operations
do not change this. The determinant fixes ω m+n . The only column operations leaving the matrix in the given
form do not change x modulo pm .
The group G acts transitively on bases of k 2 , hence also on the set of lattices. The stabilizer of o2 is K , so with
that choice of base lattice the set of lattices may be identified with G/K .
Another way to phrase the Lemma:
IV.1.3. Corollary. Every matrix in GL2 (k) can be expressed as
1 x
◦
1
̟m
◦
◦
̟n
k
with k in K , unique m, n, and x in k unique modulo pm−n .
Later, we shall see how this can be interpreted in terms of the geometry of the tree.
IV.1.4. Proposition. Given an invertible 2 × 2 matrix g with coefficients in k, there exist matrices k1 , k2 in K
and λ = (m, n) with m ≤ n such that
g = k1 ̟λ k2 .
The pair λ is unique.
Proof. The proof is a variation on that of the previous Proposition. By column and row permutations, we
may assume that the left corner entry is that of maximal norm in the entire matrix, and by a unit column
operation we may assume it to be ̟m . Row and column operations of type (3), followed by a unit column
multiplication, make it of the right form.
As for uniqueness, the greatest common divisor of the entries of the matrix is ∗̟m , and ∗̟m+n is its
determinant.
IV.1.5. Corollary. (Principal divisor theorem) If L and M are two lattices, there exists a basis (e, f ) of L and
integers m ≤ n such that (̟m e, ̟n f ) is a basis of M .
In these circumstances I call [̟m : ̟n ] the matrix index of the pair (L, M ) and q m+n the index. If m, n ≥ 0
this last is indeed the index, the size of L/M . If L = o2 and (e, f ) form an o-basis of M , this is also
det [ e f ] −1 .
Proof. I suppose L and M to be given as 2 × 2 matrices λ and µ of rank 2. Use a coordinate system in which
L = o2 . This means replacing λ by I and µ by λ−1 µ. Apply the previous Proposition to it. The columns of
k1 form a basis of L, and those of µk2−1 = k1 a form one of M .
IV.1.6. Proposition. The stabilizer of any lattice in G is a compact open subgroup. Conversely, any compact
subgroup stabilizes some lattice, and is therefore contained in a conjugate of K .
Proof. If L = go2 then its stabilizer is g K g −1 .
Suppose K∗ to be a compact subgroup, and let L = o2 . The intersection of two, hence of any finite number
of, lattices
S is again
S a lattice. IfTH = K∗T∩ K , then K∗ /H is in bijection with K∗ K/K , hence finite. If
K∗ = ki H = k∈K kH then ki L = k∈K kL is a lattice stable under it.
Chapter IV. Harmonic analysis on the tree of SL(2)
4
2. The tree
The Bruhat-Tits tree of the group G is a graph X on which it acts. The geometry of this graph encodes neatly
much of its structure.
• The nodes of the tree are the lattices in k2 modulo similarity.
These are the analogues of the points of the open unit disk. One point (so to speak) of similarity is that
whereas a point of the real symmetric space corresponds to a Euclidean metric (modulo similarity) on R2 ,
the choice of a lattice L in k 2 determines a norm on k 2 :
kvkL = inf v∈cL |c| .
In effect, the choice of L here is roughly the same as specifying a unit disk in the Euclidean case.
For each lattice L let hhLii be the corresponding node of the tree, or in other words its equivalence class, the
set of lattices {̟n L}.
If L and M are lattices, the principal divisor theorem asserts that we may find a basis (e, f ) of L such that
(̟m e, ̟n f ) is a basis of M , for some integers m ≤ n. The difference n − m is an invariant of the similarity
class of M , so that the definition inv(hhLii: hhM ii) = n − m makes sense. This invariant is 1 if and only if the
two nodes possess representatives L and M with L/M ∼
= o/p, or equivalently
̟L ⊂ M ⊂ L .
In this case, I’ll call them neighbours.
• There is an edge between two nodes of the tree if and only if they are neighbours.
The nodes linked by an edge to hhLii thus correspond to lines of L/̟L ∼
= (Fq )2 , and there are q + 1 of them.
If u and v form a basis of k2 , let [u, v] be the lattice they span and hhu, vii the corresponding node. Fix basis
vectors and particular nodes
u0 = (1, 0)
v0 = (0, 1)
νm = hh̟m u0 , v0 ii = hhu0 , ̟−m v0 ii (m ∈ Z) ,
so that ν0 is the equivalence class of o2 .
If g is in GL2 (k), it takes a lattice [u, v] to the lattice [gu, gv]. The group GL2 (k) preserves equivalence of
lattices, and it also preserves the lattice pair invariant. Hence it transforms edges to edges, and therefore
acts on the graph X. By definition, this action factors through PGL2 (k). The group PGL2 (k) acts transitively
on nodes of the tree. The stabilizer in PGL2 (k) of the node ν0 is the maximal compact subgroup PGL2 (o),
which is therefore the analogue in PGL2 (k) of the image of O(2) in PGL2 (R).
If
α=
̟
◦
◦
1
then α(νn ) = νn+1 for all n.
The principal divisor theorem gives us the Cartan decompositions
GL2 (k) = K̟λ K
with m ≤ n and hence a bijection of K\G/K with −N.
(λ = (m, n))
Chapter IV. Harmonic analysis on the tree of SL(2)
5
Suppose that L = o2 and that the matrix index of [L: M ] is [̟m : ̟n ]. I call hhM ii even or odd depending on
the parity of n − m. The action of SL2 (k) preserves this parity, and in fact there are exactly two orbits of the
group SL2 (k) among the nodes of the tree, each one corresponding to lattices of a given parity.
A path in the tree X is a finite or half-infinite sequence of nodes linked by edges. Every path may be
represented by a sequence of lattices
L0 ⊃ L1 ⊃ . . . ⊃ Ln ⊃ Ln+1 . . .
with
Ln ⊃ Ln+1 ⊃ ̟Ln
for all n. I’ll call a path a chain if it does not fold back on itself. For example, the path
hhu0 , v0 ii — hhu0 , ̟v0 ii — hhu0 , v0 ii
is not a chain. A standard chain is one of the form
ν0 — ν−1 — ν−2 — · · · ,
whether finite or infinite. It is represented by the sequence of lattices
Lm = [u0 , ̟m v0 ]∞
m=0 .
The two chains
ν0 — ν−1 — ν−2 — · · ·
and
ν0 — ν1 — ν2 — · · · ,
are swapped by the action of
◦
1
1
◦
.
This is a special case of:
IV.2.1. Proposition. Every finite simple chain in the building may be transformed to a standard one by an
element of GL2 (k). If k is complete, this remains true for all half-infinite simple chains.
Proof. The proof is by induction on the length of the associated chain of lattices
L0 ⊃ L1 ⊃ · · · ⊃ Ln ,
in which we may assume Lk ⊃ Lk+1 ⊃ ̟Lk for all k . Since GL2 (k) acts transitively on nodes, we may
assume L0 = o2 .
If n = 1, the image of L1 in L0 /̟L0 is a line. We can find a matrix g in GL2 (Fq ) transforming it to the line
through (1, 0), and if g in GL2 (o) has image g , then gL1 is [1, ̟], corresponding to the node ν−1 .
The first part of the Proposition will now follow by induction from this:
IV.2.2. Lemma. Suppose given a chain (Li ) ( for 0 ≤ i ≤ n + 1) with Li = [1, ̟i ] for 1 ≤ i ≤ n. There exists
x ∈ pn such that
1 ◦
x 1
takes every Li to [1, ̟i ].
Implicit in this statement is that when x lies in pn this matrix takes [1, ̟i ] to itself for every i ≤ n. This is
easy to verify.
Chapter IV. Harmonic analysis on the tree of SL(2)
6
Proof of the Lemma. The lattices linked to ν−m are those with normal forms
1 ◦
,
x ̟m+1
1
◦
̟m−1
◦
,
x in pm /pm+1 . The last one amounts to a step backwards. For the first set, the Lemma is clear.
To conclude the proof of Proposition IV.2.1 in case k is complete, note that under this assumption the product
of the matrices
Y
n≥0
1
xn
◦
1
found inductively will then converge.
IV.2.3. Corollary. The distance |x:y| between two nodes x and y is the pair invariant inv(x: y).
Only a short additional argument is necessary to prove:
IV.2.4. Corollary. The graph X is a connected tree.
Proof. If M is any lattice, we may find a basis (e, f ) of L = o2 such that some (̟m e, ̟n f ) is a basis of M .
Replacing M by some multiple of itself, then we may assume m = 0, n ≥ 0. That means that there exists
a chain of lattices [u0 , ̟k v0 ] from L to M . This proves that the graph is connected. That it is a tree follows
from the preceding Proposition, since no standard chain has a loop.
The node ν0 may be chosen as root. The structure of X is completely determined by the properties: (a) it is
connected; (b) it is a tree; (c) every node has q + 1 neighbours. For example, when q = 2 it looks like this:
ν2
ν1
ν0
ν−1
ν−2
Chapter IV. Harmonic analysis on the tree of SL(2)
7
3. The action of K
The group K = GL2 (o) fixes the point ν0 . How does it act on the rest of the tree?
The distance between ν0 and νm is |m|. The Cartan decomposition implies that K = GL2 (o) acts transitively
on the q m−1 (q + 1) nodes at distance m from ν0 . This is true for SL2 (o) as well.
The nodes at distance 1 from ν0 may be identified with the points of P1 (o/p). There is a similar description of
those at distance m. The space P1 (o/pm ) is that of all pairs λ = (x, y) with x, y in o/pm , at least one of them
a unit, modulo scalar multiplication by units of o. To such a pair λ corresponds the lattice Lλ = oλ + ̟m o2 ,
and then in turn the node hhLλ ii.
IV.3.1. Proposition. The map taking λ to hhLλ ii is a K -equivariant bijection of P1 (o/pm ) with the nodes of X
at distance m from ν0 .
Left as exercise. This is related to Lemma IV.2.2.
Define the congruence subgroup
Km = g ∈ GL2 (o) g ≡ I (mod pm ) .
As a consequence of this Proposition:
IV.3.2. Corollary. The fixed points of Km are are those at distance ≤ m from ν0 . For m < n, the Km -orbit of
νn is the set of all nodes at distance n from ν0 and n − m from νm .
That is to say, at distance n − m from νm and at the end of a geodesic that passes from ν0 through νm .
The group K = SL2 (o) fixes ν0 , representing the lattice o2 , while its twin K∗ = αKα−1 , with
α=
1 0
0 ̟
,
fixes its neighbour α(ν0 ) = ν−1 . Since every compact subgroup fixes some lattice, these two subgroups of
SL2 (k) are maximal compact. They are not conjugate to each other.
4. Apartments
A branch from a node is an infinite chain starting at that node. One branch is the chain B0 made up of the
nodes νm for m ≥ 0, and another is the chain B∞ of the νm with m ≤ 0. Proposition IV.2.1 says that any
branch can be transformed to B∞ by an element of GL2 (k) if k is complete, and hence that GL2 (k) then acts
transitively on branches.
An apartment is the union of two branches from one node with no common edge. One apartment is
A = B0 ∪ B∞ = {νm | m ∈ Z} .
THE ACTION OF G .
Elements of G take apartments to apartments.
IV.4.1. Proposition. If k is complete, the group GL2 (k) acts transitively on apartments.
Proof. It suffices to prove this when one of the apartments is A. Suppose given some other apartment χ,
say with two branches χ0 and χ∞ running out in opposite directions from the same node. Since GL2 (k) acts
transitively on branches, we may transform χ0 to the branch B0 . In effect, we may assume χ0 = B0 . By
Lemma IV.2.2 we may now find a matrix
1 ◦
x 1
with x in o that transforms the other branch χ∞ of X into the other branch B∞ of A. But these matrices fix
the all the nodes on B0 , so X is taken to A.
Chapter IV. Harmonic analysis on the tree of SL(2)
8
The limit of the lattices [u0 , ̟n v0 ] as n → ∞ is the line in k2 through u0 . It is called the end of the chain
B∞ . The notation for (say) B∞ is motivated by this observation, since by convention this line is expressed
as ∞ in P1 (k). Every point of P1 (k) is the end of some branch, and if k is complete every branch ends at a
point of P1 (k). In effect, adding P1 (k) is a compactifies the tree. The parallel with what happens for SL2 (R)
is striking.
Since GL2 (k) acts transitively on apartments, every apartment is stabilized by a single split torus. If its ends
are λ and µ in P1 (k), these lines are the eigenspaces of that torus. The apartment can be characterized as
containing all the nodes corresponding to lattices that split compatibly with the direct sum λ ⊕ µ.
Here is an indication of a graphical rendering of the apartment A:
ν4
ν3
ν2
ν1
ν0
ν−1
ν−2
ν−3
ν−4
If [a, b] is an interval in Z, let A[a, b] be
νa — . . . — νb .
It is a matter of convention which infinite geodesic I choose to be standard, since all are equivalent. The
choice I have made is the conventional one, and is convenient for visualization.
Proposition IV.4.1 also implies:
IV.4.2. Proposition. Suppose k to be complete. Given two apartments and an oriented edge in each, there
exists g in GL2 (k) inducing an isometry of one with the other mapping one oriented edge to the other.
The stabilizer in K of the node g ν0 is gKg −1 ∩ K . This is the same as
a
b
̟n c d
if
g=
1 0
0 ̟n
(n ≥ 0)
with a, b, c, d in o. As n gets larger and larger, this has as limit the group K ∩ P , and this brings out again that
asymptotically the building is isomorphic to K/K ∩ P or G/P . More precisely, we can see that the points at
Chapter IV. Harmonic analysis on the tree of SL(2)
9
distance m from ν0 correspond naturally to the points of Km \P1 (k), with Km equal to the congruence group
of level pm .
THE STRUCTURE OF AN APARTMENT .
Let
A = the group of diagonal matrices in GL2 (k) .
Elements of A act as translations on A. The compact subgroup A(o) acts trivially on it, so the action factors
through A/A(o). The matrix
1
◦
◦
̟
translates νm to νm−1 . Since
̟m
◦
◦
̟−m
≡
1
◦
◦
̟−2m
modulo scalar matrices ,
the subgroup A ∩ SL2 (k) shifts by an even number of nodes. The element
σ=
◦
1
−1
◦
also takes A to itself, reflecting νm to ν−m . The group generated by A and σ is precisely the stabilizer of A.
We shall find the following useful later on:
IV.4.3. Proposition. Any two branches running from ν0 but not containing any edges in A are taken into each
other by some element of A(o).
Proof. The nodes at distance n from ν0 and at distance n from A correspond to points (x, 1) in o/pn with x
not in p. Given this, the proof becomes obvious.
The analogue for SL2 (k) is not true, as is already easy to see for nodes at distance 1 from A when p is odd.
One feature of the apartment A that becomes more significant for groups of higher rank is that its structure
mirrors that of the unipotent subgroup of upper triangular matrices. This group is filtered by subgroups
1 pn
◦
1
,
and the set of points on A fixed by this subgroup consists of all those on the branch
νn — νn−1 — νn−2 — · · · .
5. Orbits of N
As m → ∞ the lattice [1, ̟m ] passes off to the line through (1, 0) in P1 (k). The group N of all upper
triangular unipotent matrices fixes the end of the branch {νm | m ≤ 0}, which amounts to ∞ in P1 (k). There
is a finite approximation of this phenomenon. Let N (pm ) be the subgroup of
1 x
0 1
with x ∈ pm . The following is elementary, but useful to refer to.
IV.5.1. Proposition. (a) Elements in N (pm ) fix all nodes νk with k ≤ m. (b) The N (pm )-orbit of νn for n > m
are the points x at distance m − n from νm other than those on a path starting back to νm−1 .
Chapter IV. Harmonic analysis on the tree of SL(2)
10
6. The tree and the Hecke algebra
The integral Hecke algebra HZ (G//K) of G = PGL2 (k) is the ring made up of linear combinations of
functions on K\G/K with values in Z. Multiplication is convolution. But it can also be interpreted as a
ring of ‘algebraic correspondances’ on the tree X. The definitions in these terms mimic Hecke’s original
definitions of the classical operators Tn on automorphic forms.
Let C(X) be the space of functions on the nodes of X. The group G acts on it by the left regular representation:
[Lg F ](x) = F g −1 (x) .
There is one operator Tm (analogous to Hecke’s Tpm ) for each m ≥ 0. According to this, to each node of the
building corresponds the set of nodes at distance m from it. Let T = T1 . The operator T /(q + 1) − I is the
p-adic analogue of a Laplacian—I recall, for example, that the value of a harmonic function at a point P is
the average of its values on the unit circle around P , and the same thing holds here for ‘harmonic’ functions.
Let RZ be the ring generated by the Tm . Its unit is T0 . This ring may be identified with a ring of operators
on the space C(X) through the formula:
[Tm f ](x) =
X
f (y) .
y
|x:y|=m
Since distances are preserved by G, the Hecke algebra commutes with the left regular representation of G on
C[X].
What is the relationship between this ring and the Hecke algebra HZ (G//K)? If f is a Z-valued function
on K\G/K , convolution by f on the right takes C(G/K) into itself. The Hecke algebra has as basis the
characteristic functions
char(K̟λ K) with (λ = (m, 0), m ≥ 0) .
It is precisely the right action of this function that amounts to the operator Tm . To match this with classical
computations on automorphic forms, we can use:
IV.6.1. Lemma. We have
T ◦ T = T2 + (q + 1)I
T ◦ Tm = Tm+1 + qTm−1 (m ≥ 2) .
m+1
m+2
m
m−1
ν0
m−2
Proof. As the figure above shows, every node has q + 1 neighbours. If y is at distance m ≥ 1 from ν0 it has q
neighbours at distance m + 1 from ν0 and 1 at distance m − 1. Thus, for example:
XX
[T ◦ T ](x) =
z = T2 (x) + (q + 1)I(x) .
y∼x z∼y
Chapter IV. Harmonic analysis on the tree of SL(2)
11
7. The representations
I recall that an admissible representation (π, V ) of a p-adic group is one with these two properties: (1) every
vector in V is fixed by some open subgroup (i.e. is smooth); (2) for any open subgroup, the subspace of
vectors fixed by all elements in that subgroup is finite-dimensional.
The action of G on C(X) commutes with all Hecke operators, and in consequence it acts on the space of
eigenfunctions of a Hecke operator.
For λ in C, let Vλ be the space of functions f on the nodes of X such that Tf = λf .
IV.7.1. Proposition. The representation of G on Vλ is admissible. The dimension of VλK is 1.
The condition on an eigenfunction ϕ is that
λϕ(x) =
X
ϕ(y) .
y∼x
Proof. In several steps.
Step 1. For each n ≥ 0 let Kn be the congruence subgroup PGL2 (pn ). Thus K = K0 . Suppose the
eigenfunction ϕ to be fixed by Kn . Let Bn be the ‘ball’ of nodes at distance ≤ n from ν0 , which are all fixed
by Kn . I claim that the values of ϕ at all nodes at distance > n from ν0 are determined by its values on Bn .
Suppose x to be one of the nodes at distance exactly n from ν0 . Call a node y external to x if the geodesic to
ν0 passes through x. This includes x itself. Then Kn fixes x and, if m ≥ n, acts transitively on all nodes at
distance m from ν0 and external to x. Hence ϕ takes the same value, say ϕx,m , at all those nodes. My earlier
claim will follow from the new claim that all these ϕx,m are determined by the values of ϕ at the neighbours
of x, if any, inside Bn .
Step 2. First we look at the value of ϕ at the external neighbours of x at distance 1. There are two cases,
according to whether x = ν0 or not.
Suppose x = ν0 . This happens only when n = 0 and ϕ is fixed by K itself. The function ϕ takes the same
value ϕm at all nodes at distance m from ν0 . There are q + 1 neighbours of ν0 at distance 1, so
(IV.7.2)
λϕ0 = (q + 1) ϕ1 ,
ϕ1 =
λ
· ϕ0 .
q+1
If n ≥ 1, there is one neighbour y in the interior of Bn , q outside, and we must have
qϕx,n+1 = λϕx,n − ϕ(y) .
In either case, the values ϕx,n+1 are determined by the values of ϕ inside Bn .
Step 3. Suppose now that m ≥ n + 1. Then
λϕx,m = ϕx,m−1 + qϕx,m+1 .
In other words, the function Fn = ϕx,n satisfies this difference equation of second order:
(IV.7.3)
qFm+1 − λFm−1 + Fm−2 = 0 . .
A solution is determined by the ‘initial values’ Fn−1 and Fn .
Step 4. There is a well known recipe for the solution of a difference equation, setting ϕm = rm . Plugging
into the equation, we see that r must be a root of the ‘indicial equation’
r2 − (λ/q)r + 1/q = 0 .
Chapter IV. Harmonic analysis on the tree of SL(2)
√
12
√
Set r = z/ q where now we require that z + z −1 = λ/ q .
If this equation has separate roots the solution is
Fm = q −m/2 (cz m + dz −m )
for constants c, d satisfying the given initial conditions near the boundary of Bn .
If it has a single root r, the solutions are of the form
Fm = q −m/2 z m (c + dm) .
√
This is the case when λ = ±2 q , z 2 = 1.
Step 5. If n = 0 and x = ν0 , there is a unique function ϕm satisfying the difference equation with a given
value of ϕ0 , and proportional to ϕ0 . Therefore VλK has dimension one.
SUMMARY . In all cases, ϕ has a well defined asymptotic behaviour on every branch running out from Bn .
(a) If the polynomial
√
r2 − (λ/ q)r + 1 = 0
has distinct roots z ±1 then there exist constants c, d such that
ϕ(y) = q −m/2 (cz m + dz −m )
if y lies at distance m from Bn . (b) If it has one root z then
ϕy = q −m/2 z m (c + dm) .
The constants c, d depend on the branch.
If n = 0 we can find a completely explicit formula for ϕ with a bit more work. We know in this case that
ϕ0 = 1
λ
q+1
= λϕn − ϕn−1
ϕ1 =
qϕn+1
Suppose at first that z 2 6= 1, so that
(n ≥ 1) .
ϕn = q −n/2 (az n + bz −n )
√
in which z + 1/z = λ/ q , and we must solve the initial conditions for a, b:
a+b=1
√
λ q
az + b/z =
.
q+1
After some messy calculation, we deduce the first half of:
IV.7.4. Proposition. Suppose z + 1/z = λ/(q + 1).
Chapter IV. Harmonic analysis on the tree of SL(2)
13
√
(a) Suppose λ 6= ±2 q . Then z 2 6= 1 and the unique solution of the difference equation for ϕ is
ϕn =
q −n/2
1 + 1/q
1 − q −1 z −2 z
1 − q −1 z 2 −n
.
z
+
z
1 − z −2
1 − z2
√
(b) Suppose λ = ±2 q , so z 2 = 1. Then the solution is
q
−m/2 m
z
q−1
q+1
m+1
.
This is a special case of a much more general formula first found in [Macdonald:1972].
Proof. Assertionb must be proved. It follows from l’Hôpital’s rule, but there is a more interesting method.
The formula for ϕm can be put into a curious form. It becomes for m ≥ 1
m−1
m+1
q −m/2
− z −(m−1)
z
− z −(m+1)
−1 z
−
q
·
1 + 1/q
z − z −1
z − z −1
1
=
· q −m/2 (z m + z m−2 + · · · + z −m ) − q −(m−2)/2 (z m−2 + · · · + z −(m−2) )
1 + 1/q
ϕm =
Setting z = ±1 in this causes no problem.
The expression z m + · · · + z −m is the same as the character of the irreducible representation of SL2 (C) of
dimension m + 1, evaluated at
z
◦
◦
1/z
.
This is not at all an accident. In general, Macdonald’s formula expresses unramified matrix coefficients in
terms of Weyl characters for Langlands’ dual group, which is a reductive group defined over C. Macdonald’s
formula evaluated at singular parameters is in general related to Weyl’s formula for the dimension of
irreducible representations of semi-simple groups.
8. Relations with representations induced from parabolic subgroups
Recall that ∆ = T /(q + 1) − I is the analogue of the non-Euclidean Laplacian on the open unit disk. The
space ∆ = 0 is a representation of G, the eigenspace V = Vλ of T for λ = q + 1. This space contains the
constants, and the terms in the expansion of eigenfunctions along branches are of the form c + dq −m for
constants c, d depending on the branch. As m → ∞ every matrix coefficient ϕ has as limit a locally constant
function Φ on P1 (k). (This is very similar to what happens with the Euclidean unit disk!) The map taking
ϕ to Φ is a G-equivariant homomorphism from V = C(X) to C ∞ (P1 (k)). As we’ll see in a moment, it is an
isomorphism (again, just as with the unit disk).
Now C ∞ (P1 (k)) is the smooth representation induced from the trivial representation of the Borel subgroup
P of upper triangular matrices in G, and according to Frobenius reciprocity a homomorphism from V to it
is equivalent to a P -equivariant homomorphism from V to the trivial representation. Here, this takes ϕ to
lim ϕ(ν−n )
n→∞
since P is the isotropy subgroup of ∞.
What happens for other eigenspaces? For general λ, the boundary values of eigenfunctions are functions in
other representation induced from P . This is best explained by Frobenius reciprocity. If χ is an unramified
character of A:
χ:
̟n
◦
◦
1
7−→ αn ,
Chapter IV. Harmonic analysis on the tree of SL(2)
let
14
Iχ = {f ∈ C ∞ (G) | f (ang) = δ 1/2 (a)χ(a)f (g)} .
I want to find G-equivariant maps from V = Vλ to some Iχ . In effect, we are looking for P -equivariant—in
particular N -invariant—maps from V to δ 1/2 χ. If f lies in V , then there exists m ≪ 0 with the property that
f restricted to
A[−m, −∞) = νm — νm−1 — νm−2 — · · ·
lies in the space U of solutions of the difference equation (IV.7.3). If n lies in N , then nνk = νk for k ≪ 0.
Therefore f and Ln f have the same restriction to A[m, ∞) for m ≪ 0. he A-stable space spanned by it has
dimension two. The map from f to U is therefore N -invariant and also A-equivariant.
√
If λ 6= ±2 q , the A-module U is the direct sum of two one-dimensional representations χ±1 , and by
√
Frobenius reciprocity we get G-embeddings into the two principal series I(χ−1 ). If λ = ±2 q , let z = ±1.
The A-module U is the space spanned by δ −1/2 z m , δ −1/2 mz m and we get just one P -equivariant character
as quotient.
The exact form of these embeddings is related to intertwining operators between representations induced
from P , which we’ll examine later.
9. References
1. Pierre Cartier, ‘Géométrie et analyse sur les arbres’, Exposé 407 of Séminaire Bourbaki, 1971/1972.
2. ——, ‘Harmonic analysis on trees’, pp. 419–424 in [Moore:1973].
3. Ian G. Macdonald, Spherical functions on a group of p-adic type, University of Madras, 1972.
4. Calvin Moore (editor), Harmonic analysis on homogeneous spaces, volume XXVI in the series Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1973.
5. Jean-Pierre Serre, Arbres, amalgames, et SL(2), Astérisque 46 (1977). Translated into English as Trees,
Springer-Verlag, 1980.