COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA
GERMS
CHENG-CHIANG TSAI
Abstract. For a reductive group G over a non-archimedean local field, with some
assumptions on (residue) characteristic we give an method to compute certain orbital
integrals using a method close to that of [8] but in a different language. These orbital
integrals allow us to compute the Shalika germs at some “very elliptic” elements
in terms of number of rational points on some quasi-finite covers of the Hessenberg
varieties in [8] which are subvarieties of (partial) flag varieties. Such values of Shalika
germs determine the Harish-Chandra local character expansions of the so-called very
supercuspidal representations.
Contents
1. Introduction
Acknowledgments
2. A computation for orbital integrals
2.1. Very elliptic elements
2.2. Cartan decomposition
2.3. Organization of double cosets Wy wWx
3. Shalika germs
3.1. Homogeneity of nilpotent orbits
3.2. Relatedness and strong-relatedness
3.3. A vanishing result
Appendix A. Assumptions on p
References
1
5
5
5
10
12
14
14
16
20
21
22
1. Introduction
Throughout the article, let F be a non-archimedean local field with residue field k
of order q. Let G be a connected reductive group over F . We’ll actually work with
the assumption that G is semi-simple and simply connected (this makes no harm;
see beginning of Sec. 2.1). The F -points of G will be denoted by G = G(F ), and
the F -points of the Lie algebra of G by g. Similar conventions will be applied for
other algebraic groups, which will be defined either over F or k. We will assume that
char(k) is very good for G, G is tamely ramified and char(F ) is sufficiently large (see
Appendix).
For any element g ∈ G, let CG (g) denote the centralizer of g in G and Ad(G)(g) ∼
=
G/CG (g) the orbit of g. Ad(G)(g) is a p-adic manifold equipped with a G-invariant
Date: June 29, 2015.
1
2
CHENG-CHIANG TSAI
R
measure [19, III.3.27]. One may consider the orbital integral µg (f ) = Ad(G)(g) f (x)dx =
R
f (Ad(g)x). Orbital integrals on the Lie algebra are defined in the same manG/CG (g)
ner. For regular semisimple g, such orbital integrals appear on the geometric side of
the trace formula and are fundamental objects of study in the Langlands program.
For representations of G we have the local character expansion (By Howe [11] for
GLn over a p-adic field, Harish-Chandra [9] for p-adic fields and Adler-Korman [2] in
general) for representations of G. Let π be any irreducible admissible representation
of G and γ ∈ G be semisimple. Let M = (CG (γ))o be its connected centralizer,
M = M(F ) and m = Lie M . Then there exists a neighborhood U of 0 ∈ m such that
(1.1)
Θπ (γ · e(X)) =
X
cO (π)µ̂O (X), X ∈ U
O∈O(γ)
where Θπ is the character of π, e is a suitable generalization of the exponential map
(or just the exponential map when char(F ) = 0), and O(γ) is the set of nilpotent orbits
in m, with cO ∈ C and µ̂O the Fourier transform of µO . Both side of the equality may
be viewed either as distributions on U , or as locally constant functions defined almost
everywhere that represent the distributions.
In fact, a general philosophy dated back to Harish-Chandra is that characters should
resemble Fourier transforms of orbital integral. Another important result of this philosophy is the work of Kim and Murnaghan [12]. Roughly speaking, Kim and Murnaghan
showed that whenever an irreducible admissible representation has certain K-type, the
local character near the identity can be written as a linear combination of Fourier transforms of orbital integrals (not necessarily semisimple or nilpotent). Adler and Spice [4]
generalize their results in the case of very supercuspidal representations (see below) to
give full a character formula, where the character is locally given as Fourier transforms
of elliptic orbital integrals.
We’d like to interpret their results in the case of very supercuspidal representations.
The definition relies on the construction of supercuspidal representations by J.-K.
Yu [22]. The construction begins with a datum consists of a sequence of subgroups
G0 ( G1 ( ... ( Gd = G where each Gi is a Levi subgroup of G after base change to
a tame extension of F , a depth-zero supercuspidal representation π0 of G0 = G0 (F ),
and a sequence (φi )di=0 of characters on (Gi )di=0 . A supercuspidal representation is then
constructed as the compact induction, from a compact open subgroup determined by
(Gi )di=0 and depths of (φi )di=0 , of some finite dimensional representation determined by
π0 and (φi )di=0 .
Such supercuspidal representations were shown to exhaust all supercuspidal representations when char(F ) = 0 and char(k) is large enough. If Gd−1 is anisotropic
(i.e. Gd−1 is compact, possibly modulo center), the supercuspidal representations
constructed are called very supercuspidal representations. The result of Kim and
Murnaghan, when restricted to a very supercuspidal representation π, says that there
exists T ∈ g satisfying (CG (T ))o = Gd−1 such that locally near the identity, Θπ (−) =
deg(π)µ̂T (−), where deg(π) ∈ Q is the formal degree of π.
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
3
On the other hand, the Lie algebra version of a theorem of Shalika [18] states that
for any T ∈ g, there exists a lattice Λ ⊂ g such that
X
(1.2)
µT (f ) =
ΓO (T )µO (f ), ∀f ∈ Cc∞ (g/Λ).
O∈O(0)
where O(0) is the set of nilpotent orbits, and ΓO (X) are constants that depends on
T ∈ g. These ΓO (X) are called the Shalika germs. When Θπ (−) = deg(π)µ̂T (−),
comparing (1.1) and (1.2) gives us cO (π) = deg(π)ΓO (T ).
Our main goal in this article is to present an algorithm to compute these numbers
ΓO (T ). This gives the local character expansion at the identity for very supercuspidal
representations as explained above. Our method begins with using test functions
constructed via DeBacker’s parametrization of nilpotent orbits [7]. We first take f to
be the characteristic function of some nice set. The set has the property that it meets a
nilpotent orbits O and does not meet any other nilpotent orbits O0 of smaller or equal
dimension. The test functions are then fπ2n (X) = f (π 2n X). Nilpotent orbital integrals
satisfy the property µO (fπ2n ) = q dim O µO (f ). This gives the term ΓO (T )µO (fπ2n ) a
unique scaling behavior with respect to n. To compute ΓO (T ), the essential thing
remained is then to compute the asymptotic of µT (fπ2n ).
To do so, we compute the orbital integral using the Cartan decomposition. We’ll
choose (based on T and the test functions) specific points x, y ∈ B(G, F ) be two
points in the Bruhat-Tits building of G and Gx , Gy associated parahoric subgroups.
Let S ⊂ G be a maximal split torus whose corresponding apartment A = A(S, F )
contains x and y. Denote by W̃ be the affine Weyl group of G (associated to S)
and Wx , Wy ⊂ W̃ the finite subgroups stabilizing xFand y, respectively. The Cartan
decomposition is the following decomposition: G = w∈Wy \W̃ /Wx Gy wGx .
Since we are doing orbital integral of T who has a compact stabilizer, up to a
normalizing constant we can write
Z Z
Z
X
µT (f ) =
f (Ad(g)T )dg =
f (Ad(gy )Ad(w)Ad(gx )T )dgx dgy .
G
w∈Wy \W̃ /Wx
Gy
Gx
For all but finitely many w the integral will be zero (Theorem 2.1). We thus obtain
µT (f ) as a somewhat large combinatorial sum. We then prove some transversality
result (Theorem 2.3), in a spirit very close to that of the work of Goresky, Kottwitz
and MacPherson [8, Theorem 0.2 and Sec. 3.7]. The transversality result reduces each
term in the sum into counting points on a quasi-finite cover of Hessenberg varieties
(see [8] for definition of Hessenberg varieties) defined over the residue field k.
The combinatorics of the sum is a priori very complicated. Note the terms in the sum
are indexed by w ∈ Wy \W̃ /Wx . Let V = X∗ (S)⊗R the real span of the cocharacters of
S. We have a map ev : Wy \W̃ /Wx → V/Wy by w 7→ x − w−1 y. Here Wy acts linearly
on V (i.e. fixing the origin). For a certain real number r related to our orbital integral
problem, we’ll draw walls (affine hyperplanes) of the form α(v) = 0 and α(v) = r on V,
where α ∈ Φ(G, S) runs over roots of G with respect to S. Since the set of these walls
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CHENG-CHIANG TSAI
are invariant under Wy -action, they decompose V/Wy into finitely many polyhedrons.
Call this set of polyhedrons P̄. We then have induced map ev new : Wy \W̃ /Wx → P̄.
These walls are where the behavior of the term indexed by w changes. It turns out
that we should group terms indexed by elements in Wy \W̃ /Wx via their image in P̄.
In Theorem 2.1 we prove that if ev new (w) is an unbounded polyhedron then the term
indexed by such w vanishes. Let P̄ bd ⊂ P̄ be the subset of bounded polyhedrons. We
end up writing the orbital integral into
X X
µT (test function) =
(the term indexed by w).
Π∈P̄ bd w∈Π
Intuitively, what we then arrive is that there exists combinatorial constants pΠ and
geometric constant J(Π) such that
X
µT (test function) =
pΠ · J(Π).
Π∈P̄ bd
and the corresponding result on Shalika germs
X
ΓO (T ) =
cpΠ ,O · JO (Π).
Π∈P̄ bd
See (3.2) and Theorem 3.9 for a precise statement. As mentioned, the numbers in the
sum consist of the combinatorial part cpΠ ,O and the geometric part JO (Π). The latter
part counts number of rational points on certain varieties over k and is in general
a mystery. Nevertheless, in Theorem 3.7 and Theorem 3.10 we give conditions for
cpΠ ,O and JO (Π) to vanish, respectively. This greatly reduces the complexity of the
combinatorics. Roughly speaking, the complexity now is exponential in the difference
in dimension of O and that of a regular orbit. This makes it feasible to compute ΓO (T )
for the top orbits for a given type of groups.
Our method gives for each T and nilpotent orbit O an explicit list of quasi-finite covers of Hessenberg varieties, for which the point-counting gives the geometric numbers
JO (Π). These varieties are described in terms to the root system. In [21, Sec. 3] the
author describes some examples of such varieties. For example, when O comes from
the top four nilpotent geometric orbits of a ramified unitary group and T is certain
nice half-integral depth elements, the author shows that JO (Π) and thus ΓO (T ) appear
to be numbers of points on varieties over k whose `-adic cohomologies are generated
by that of specific hyperelliptic curves.
We now briefly describe the structure of this article. In Section 2, we begin by
introducing the notion of very elliptic elements. They are elements in the Lie algebra
that arise from local characters of very supercuspidal representations. After proving
such elements have anisotropic stabilizer (Theorem 2.1) and the transversality result
(Theorem 2.3) in 2.1, we begin with a general discussion of orbital integrals of a very
elliptic element φ̃x . In 2.2 we apply the Cartan decomposition and eventually in 2.3
write the orbital integral into a sum of products of combinatorial number and geometric
numbers in (2.6).
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
5
In Section 3 we turn to the discussion of Shalika germs. In 3.1 we describe how to
find a sequence of suitable test functions using DeBacker’s parametrization of nilpotent
orbit, and explain how we use the homogeneity property of nilpotent orbital integrals.
We then in 3.2 describe how the notions in 2.3 can be used to derive our main result
for Shalika germ, which is Theorem 3.9. Lastly, in 3.3, we prove a vanishing result that
greatly simplifies the formula in practice.
Acknowledgments. The author would like to thank his Ph.D. advisor Benedict Gross
for all inspiration and encouragement. He would like to thank Zhiwei Yun for many
beneficial discussions, from which in particular the method in the first half of this
article was improved. He would also like to thank Thomas Hales, Tasho Kaletha,
Bao Le Hung, Loren Spice, Jack Thorne, Pei-Yu Tsai and Jerry Wang for helpful
suggestions and discussions.
2. A computation for orbital integrals
2.1. Very elliptic elements. Our purpose is to compute some orbital integral on the
g. Let Gsc be the simply connected cover of the derived group Gder of G. Gsc is also
the simply connected cover of the adjoint quotient Gad of G. Since we assume char(F )
is very good for G, in the exact sequence Gsc (F ) → Gad (F ) → H 1 (F, Z(Gsc )) the last
term is finite.
The map Gsc → Gad factors through Gsc → G → Gad . Therefore the image of
Gsc (F ) in Gad (F ) is a finite index subgroup of the image of G = G(F ). When char(F )
is very good for G, the maps Gsc → G → Gad induce full-rank differentials on the
Lie algebras. Computation of orbital integrals on the Lie algebra can thus be reduced
from G to Gsc . We thus assume from now on, as mentioned in the introduction, that
G = Gsc is semi-simple and simply connected.
In this and the following section, beside the group G over the local field F , x, y
will be two points on the building B(G, F ) of G over F . We will assume that G
is tamely ramified, i.e. G splits over a tamely ramified extension. We also assume
that the coordinate of x has denominator coprime to p, i.e. x becomes a hyperspecial
point after base change to a tame extension. Take dx > dy two real numbers. For our
computation we’ll make use of a maximal split torus S whose corresponding apartment
A = A(S, F ) contains x and y. We fix such an S from now on. Let Gx and Gy be the
parahorics stabilizing x and y respectively in G.
Let R̃ be the totally ordered set of symbols R̃ = {r, r + | r ∈ R} and likewise for
R̃≥0 . We have Moy-Prasad filtrations [14, Sec. 2] {Gx,r }r∈R̃≥0 and {gx,r }r∈R̃ , where
Gx,r+ := lims→r+ Gx,s and gx,r+ := lims→r+ gx,s . We’ll denote by gx,r:s = gx,r /gx,s for
any r < s in R̃. Finally define Lx = Gx /Gx,0+ and Vx = gx,dx :dx + . The same notations
will be used when x is replaced by y or other points on the building.
Since we assume G to be simply connected, the group Lx is the k-points of a connected reductive group Lx defined over k. Lx acts on the k-vector space Vx by adjoint
6
CHENG-CHIANG TSAI
action Ad, and this action also arises from an algebraic representation of Lx . We consider an element φx ∈ Vx with the following assumptions: (1) the Lx orbit of φx is
(Zariski) closed and (2) The stabilizer of φx in Lx is an anisotropic torus.
Let φ̃x ∈ gx,dx be any lift1 of φx , and φy ∈ Vy be any element. The main goal of this
section is to compute the following orbital integral:
(2.1)
µφ̃x (1φy +gy,dy + ) = ?
Here 1φy +gy,dy + denote the characteristic function of φy + gy,dy + and µφ̃x the orbital
integral of φ̃x .
We shall explain more about this setting, especially about the point x, the group
Lx and its representation Vx . For the result in this subsection we have to assume G is
tamely ramified. We first recall some results in [17, Sec. 4]. Let’s say x has denominator
m. Then mdx must be an integer, since Vx = 0 otherwise. Fix ζm a m-th root of unity
in k̄. By [17, Theorem 4.1], there is an algebraic group G defined over k together with
an order m automorphism
θ : G → G, which is only defined over k(ζm ), but such that
L
i
},
the grading Lie G = i∈Z/m (Lie G)(i), where (Lie G)(i) := {X ∈ Lie G | θ(X) = ζm
θ
is defined over k. In particular the fixed subgroup G is also a k-subgroup of G.
The statement is that the automorphism θ is such that we have compatible kisomorphisms Lx ∼
= (Gθ )o and Vx ∼
= (Lie G)(mdx ), that is, the action of Lx on Vx
agrees with the adjoint action of (Gθ )o on (Lie G)(mdx ). We can therefore think of φx
as an element in (Lie G)(mdx ) with closed orbit and anisotropic stabilizer under the
action of (Gθ )o . In addition, that φx has closed orbit is equivalent to φx ∈ Lie G is
semisimple [13, Lemma 2.12 and Cor. 2.13]. In this subsection we prove two crucial
results for the orbital integral of φ̃x , that is Theorem 2.1 and Theorem 2.3:
Theorem 2.1. φ̃x has anisotropic stabilizer in G.
Proof. For any g ∈ G which centralizes φ̃x , we shall prove that g ∈ Gx . Supoose
otherwise that g.x = x0 6= x. Then since g centralizes φ̃x we have φ̃x ∈ gx0 ,dx as well.
Without loss of generality we may assume x0 ∈ A = A(S, F ). Let Ψ(G, S) be the
set of affine roots of S in G. One has
M
Vx =
gx,dx :dx +,ψ̇ , and
ψ∈Ψ(G,S), ψ(x)=dx
φx ∈ Im(gx,dx ∩ gx0 ,dx → Vx ) =
M
gx,dx :dx +,ψ̇ .
ψ∈Ψ(G,S), ψ(x0 )≥ψ(x)=dx
where ψ̇ denotes the gradient of ψ, and gx,dx :dx +,ψ̇ is the graded piece in the root
space gψ̇ of depth dx . Since we assume x has rational coordinates with denominator
m, m(x0 − x) ∈ X∗ (S) corresponds to a cocharacter λ : Gm → Lx with image in
1This
is what we meant by a very elliptic element in the title of this subsection.
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
7
the maximal k-split torus of Lx that corresponds to S. The previous discussion says
φx ∈ Vx lies in the non-negative weight space of λ.
Identify Lx as (Gθ )o and Vx with (Lie G)(mdx ). Since λ has image in Gθ , the Zgrading given by λ is compatible with the grading given by θ, i.e. we can write
M
Lie G =
(Lie G)(i)j
i∈Z/m, j∈Z
where (Lie G)(i)j is the subspace of (Lie G)(i) on which λ acts by z j . We have φx ∈
(Lie G)(mdx )≥0 is semisimple. G has a parabolic subgroup P with Lie algebra Lie P =
(Lie G)≥0 of non-negative weights of λ, and P has a Levi subgroup L with Lie L =
(Lie G)0 . Denote by φx,0 the image of φx to Lie L which is just given by projection to
the weight-0 subspace.
By the action of λ(z) with z → 0, we see φx,0 is in the (Zariski) closure of the
(Gθ )o -orbit of φx and thus φx is conjugate to φx,0 by elements in (Gθ )o (k̄). One can
possibly compute Galois cohomology (especially as groups are now defined over the
finite field k) to show that φx is conjugate to φx,0 by elements in (Gθ )o (k). If so, then
φx,0 should have anisotropic stabilizer in (Gθ )o (k) but λ(Gm ) centralizes φx,0 , and we
derive the contradiction we want. However we are not able to carry out a general
Galois cohomology argument. Instead, we prove the following lemma:
Lemma 2.2. Let U be the unipotent radical of P with Lie U = (Lie G)>0 . Then for
k-rational X ∈ (Lie G)0 and Y ∈ (Lie G)>0 such that X and X + Y are semisimple,
there exists u ∈ U(k) satisfying Ad(u)(X + Y ) = X.
Moreover, if X, Y ∈ (Lie G)(mdx ), we may take u ∈ (Gθ )o .
Proof. We may assume Y ∈ (Lie G)≥n \(Lie G)>n for some integer n > 0. It suffices
to find u ∈ U(k) such that Ad(u)(X + Y ) ≡ X modulo (Lie G)>n then use induction
on m, as (Lie G)>n = 0 for m 0. Write Y = Yn + Y>n with Ym ∈ (Lie G)n (k)
and Y>n ∈ (Lie G)>n (k) (note (Lie G)n (k) means k-points of Lie G, same for all other
places of this proof). If we can find V ∈ (Lie G)n (k) such that [V, X] = −Yn , then
Ad(e(V ))(X + Y ) − X ∈ (Lie G)>n and we are done. Note here if char(k) is not large
enough we cannot have e as the naive exponential map e; we’ll not deal with this issue
until the end. Now it suffices to prove that the image of ad(X) : (Lie G)n → (Lie G)n
contains Yn . This statement is independent of the base field and we work over an
algebraic closure k̄.
Take a maximal torus T of G that contains X and is contained in CG (λ(Gm )). We
claim that if X + Y is semisimple, then for every root α ∈ Φ(G, T) with hλ, αi = n, we
have (dα)(X) 6= 0 or Yn,α = 0. Assuming the claim, we have
X
ad(X)( [(dα)(X)]−1 Yn,α ) = Yn .
α
The claim is a statement only on a particular root space, and can be checked directly
by embedding G into GLN and T into the diagonal torus. This proves the lemma except
for the last sentence. For the last sentence, note that when we take V ∈ (Lie G)n such
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CHENG-CHIANG TSAI
that [V, X] = −Yn , since Yn ∈ (Lie G)(mdx )n , it makes no harm to project V to its
component in (Lie G)(0)n . One then has e(V ) ∈ (Gθ )o .
Lastly, as promised we justify the exponential map e on (Lie G)n (k) ⊂ (Lie G)>0 (k)
when char(k) is small by finding an approximation. Let Φ+ = {α ∈ Φ(G,
L T), hα, λi >
0} be the subset with positive weight with respect to λ, so Lie U = α∈Φ+ (Lie U)α
(this decomposition might not be defined over k). On each (Lie U)α , α ∈ Φ+ we
have the exponential map eα : (Lie U)α ∼
= Uα , which is an isomorphism to the root
subgroup
U
.
Taking
product
in
any
order
gives an isomorphism en : (Lie U)n ∼
= Un :=
α
Q
Q
hα,λi≥n Uα /
hα,λi>n Uα defined over k (which does not depend on the order for the
product).
Now we can define a map on k-points of Lie U to U compatible with all en following
P
[1, Sec. 1.3] . This is done by writing an element in V ∈ (Lie U)(k) as V = i≥1 Vi ,
Vi ∈
Q(Lie U)i (k) and take e(V ) = e(V1 ) · e(V2 ) · ... where e(Vi ) are chosen representatives
in hα,λi≥i Uα that are defined over k. We can also choose θ-fixed representatives
whenever Vi ∈ (Lie G)(0), which is needed for the last sentence of the lemma.
Lemma 2.2 gives explicit rational element u ∈ (Gθ )o (k) such that Ad(u)(φx ) = φx,0 ,
and thus φx,0 has anisotropic stabilizer and we have a contradiction unless λ is trivial,
that is x0 = x. This finishes the proof of Theorem 2.1.
With Theorem 2.1, up to a normalizing constant we have
Z
µφ̃x (1φy +gy,dy + ) =
1φy +gy,dy + (Ad(g)φ̃x )dg.
Gx
Theorem 2.3. The value of
the lift φ̃x chosen).
R
Gx
1φy +gy,dy + (Ad(g)φ̃x )dg depends only on φx (but not on
Proof. We may decompose Gx =
to prove that for every ġ
F
ġGx,0+
ġGx,0+ into left Gx,0+ -cosets, and it suffices
Z
(2.2)
Gx,0+
1Ad(ġ−1 )(φy +gy,dy + ) (Ad(h)φ̃x )dh depends only on φx .
By letting y = ġ −1 y and φy = Ad(ġ −1 )(φy ) we may drop ġ in (2.2). We claim the
following general statement: Let r ∈ R̃>0 and φ̃x ∈ φx + gx,dx + be any element in the
coset. Then
Z
|Gx,r | · |(φ̃x + gx,dx +r ) ∩ (φy + gy,dy + )|
(2.3)
1φy +gy,dy + (Ad(h)φ̃x ) =
.
|(φ̃x + gx,dx +r )|
Gx,r
Note the measures |(φ̃x + gx,dx +r )| = |gx,dx +r | and |(φ̃x + gx,dx +r ) ∩ (φy + gy,dy + )| =
|gx,dx +r ∩gy,dy + | if the first intersection is non-empty and zero otherwise. Taking r = 0+
in (2.3) gives (2.2). To prove (2.3) we use induction on r in the way that we assume
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
9
it works for all r0 > r and prove the statement for r. This works because Gx,r jumps
semicontinuously, and that when r is large enough, gx,dx +r ⊂ gy,dy + in which case the
conclusion is immediate.
(2.3) is clear when r is not a jump for the Moy-Prasad filtration of Gx or when
(φ̃x + gx,dx +r ) ∩ (φy + gy,dy + ) = ∅. Suppose otherwise (φ̃x + gx,dx +r ) ∩ (φy + gy,dy + ) 6= ∅
and denote by r0 is the next jump of the filtration, by comparing integration over Gx,r
and over Gx,r0 , the induction is exactly to show that
#{ḣ ∈ Gx,r /Gx,r0 | (Ad(ḣ)φ̃x + gx,dx +r0 ) ∩ (φy + gy,dy + ) 6= ∅}
(2.4)
=
|Gx,r | · |gx,dx +r0 | · |gx,dx +r ∩ gy,dy + |
.
|Gx,r0 | · |gx,dx +r | · |gx,dx +r0 ∩ gy,dy + |
We may decompose
gx,dx +r:dx +r0 =
M
α(x−y)<dx +r−dy
gx,dx +r:dx +r0 ,α ⊕
M
gx,dx +r:dx +r0 ,α .
α(x−y)≥dx +r−dy
Let’s now in the remaining proof of this lemma denote the first summand by W1
and the second by W2 . They are subspaces of gx,dx +r:dx +r0 as k-vector spaces. In the
decomposition, W1 is the image of gx,dx +r ∩ gy,dy + , consequently the RHS of (2.4) is
equal to
#(Gx,r /Gx,r0 )
.
#W2
Given the assumption (φ̃x + gx,dx +r ) ∩ (φy + gy,dy + ) 6= ∅, that whether (Ad(ḣ)φ̃x +
gx,dx +r0 ) ∩ (φy + gy,dy + ) 6= ∅ depends only on the image of Ad(ḣ)φ̃x − φ̃x in gx,dx +r:dx +r0 ,
and actually only on its projection to W2 . In fact, let πW2 be the projection to from
gx,dx +r:dx +r0 or gx,dx +r to W2 . Then
(Ad(ḣ)φ̃x + gx,dx +r0 ) ∩ (φy + gy,dy + ) 6= ∅ and (Ad(ḣ0 )φ̃x + gx,dx +r0 ) ∩ (φy + gy,dy + ) 6= ∅
⇒ Ad(ḣ)φ̃x − Ad(ḣ0 )φ̃x ∈ gx,dx +r0 + gy,dy + ⇒ πW2 (Ad(ḣ)φ̃x − Ad(ḣ0 )φ̃x ) = 0,
that is, the intersection is non-empty exactly when πW2 (Ad(ḣ)φ̃x − φ̃x ) hits a specific
element in W2 .
We make use of the mock-exponential map e(·) : gx,0+ → Gx,0+ defined in [4,
Appendix A], which is a bijection satisfying properties as if the exponential map
∼
was defined. In particular we identify e : gx,r /gx,r0 −
→ Gx,r /Gx,r0 . Observe that
Ad(ḣ)φ̃x + gx,dx +r0 = φ̃x + [e−1 (ḣ), φx ] + gx,dx +r0 .
Claim 2.4. The linear transformation (over k) from gx,r:r0 to W2 given by X 7→
πW2 ([X, φx ]) is surjective.
Suppose the validity of the claim. Then πW2 (Ad(ḣ)φ̃x − φ̃x ) runs over all elements in
W2 . Thus (Ad(ḣ)φ̃x + gx,dx +r0 ) ∩ (φy + gy,dy + ) 6= ∅ exactly (#W2 )−1 of the time. This
is exactly (2.4).
10
CHENG-CHIANG TSAI
It remains to prove Claim 2.4. By [3, Proposition 4.1], as we assume char(k) is very
good, there is a G-invariant non-degenerate symmetric bilinear form κ(·, ·) on g whose
reduction κ̄ identifies gx,dx +r:dx +r0 with the dual of gx,−dx −r:(−dx −r)+ . For every root
α ∈ Φ(G, S), it identifies gx,dx +r:dx +r0 ,α with gx,−dx −r:(−dx −r)+,−α . In particular, we can
identify the dual of W2 :
M
W2∗ =
gx,−(dx +r):(−dx −r)+,α ⊂ gx,−dx −r:(−dx −r)+
α(x−y)≤dx +r−dy
Suppose the map in Claim 2.4 is not surjective. Then there exists a non-zero element
N ∈ W2∗ ⊂ gx,−dx −r:(−dx −r)+ such that κ̄([X, φx ], N ) = 0 ∈ k for all X ∈ gx,r:r+ .
However κ̄([X, φx ], N ) = κ̄(X, [φx , N ]). Hence we have [φx , N ] = 0.
Recall we have L
algebraic group G with an order m automorphism θ which gives Z/mgrading Lie G = i∈Z/m (Lie G)(i) such that we may realize φx ∈ (Lie G)(mdx ), N ∈
(Lie G)(m(−dx − r)) with [φx , N ] = 0. Note N is a nilpotent element, and we shall
derive a contradiction with the assumption that φx has anisotropic stabilizer in (Gθ )o .
Let H := CG (φx )o be the connected centralizer of φx . By assumption (Hθ )o is an
anisotropic torus. Since θ acts on φx by a constant, H is invariant under θ and we
have likewise a Z/m-grading on Lie H. In particular N ∈ (Lie H)(m(−dx − r)). The
proof in [13, Lemma 2.11] gives a unique one-parameter subgroup in (Hθ )o which acts
non-trivially on the line generated by N . But this contradicts with the assumption
(Hθ )o is anisotropic.
2.2. Cartan decomposition. We are now ready to evaluate (2.1). Because of Theorem 2.1 and that φy + gy,dy + is invariant under Gy,0+ , up to a normalizing constant,
one can write
Z
µφ̃x (1φy +gy,dy + ) =
Gy,0+ \G
1φy +gy,dy + (Ad(g)φ̃x )dg.
We have the Cartan decomposition Gy \G/Gx ∼
= Wy \W̃ /Wx , where W̃ is as in [20,
1.2] and Wx and Wy are the stabilizers of x and y in W̃ , respectively. We shall
choose an arbitrary set of representatives for Wy \W̃ /Wx in G; whenever we write
w ∈ Wy \W̃ /Wx , w can be understood as an element in W̃ or G (while whatever
follows should be essentially independent of this choice). Note that since we assume
G to be semi-simple and simply connected, W̃ agrees with the affine Weyl group.
Suppose we normalize our measure so that Gy,0+ has measure 1. With the Cartan
decomposition we can rewrite the integral:
Z
Gy,0+ \G
1φy +gy,dy + (Ad(g)φ̃x )dg =
X
X
1φy +gy,dy + (Ad(g)φ̃x )
w∈Wy \W̃ /Wx g∈Gy,0+ \Gy wGx
=
X
X
X
w∈Wy \W̃ /Wx h∈Ly ġ∈Gy \Gy wGx
1φy +gy,dy + (Ad(h)Ad(ġ)φ̃x )
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
X
=
X
X
11
1Ad(w−1 h−1 )(φy )+gw−1 y,dy + (Ad(γ̇)φ̃x ).
w∈Wy \W̃ /Wx h∈Ly γ̇∈(Gx ∩Gw−1 y )\Gx
Here the point w−1 y ∈ A on the building is characterized by Gw−1 y = Ad(w−1 )Gy ,
and Ad(w−1 ) sends Ad(h−1 )(φy ) ∈ gy,dy :dy + to Ad(w−1 h−1 )(φy ) ∈ gw−1 y,dy :dy + .
We’ll show that there are only finitely many w ∈ Wy \W̃ /Wx such that Ad(g)φ̃x
is in the compact lattice gy,dy , i.e. only finitely many w in the sum contributes (see
Lemma 2.6). The order of sum and integral thus don’t matter. For convenience write
φw−1 y = Ad(w−1 )φy ∈ gw−1 y,dy :dy + . By Theorem 2.3, we can, by assigning gx,dx + to
have Haar measure 1 in the following integral, rewrite
µφ̃x (1φy +gy,dy + )
=
X
X
Z
X
w∈Wy \W̃ /Wx h∈Lw−1 y γ̇∈(Gx ∩Gw−1 y )\Gx
=
X
0
X
q −mw
φx +gx,dx +
X
1Ad(h−1 )(φw−1 y )+gw−1 y,dy + (Ad(γ̇)φ̃x )dφ̃x
1(Ad(h−1 )(φw−1 y )+gw−1 y,dy + +gx,dx + )/gx,dx + (Ad(γ̇)φx ).
h∈Lw−1 y γ̇∈(Gx ∩Gw−1 y )\Gx
w∈Wy \W̃ /Wx
0
where m0w is the non-negative number such that q mw = [gx,dx + : gx,dx + ∩ gw−1 y:dy + ].
Now one note that the sum over (Gx ∩ Gw−1 y )\Gx is [(Gx ∩ Gw−1 y )Gx,0+ : Gx ∩ Gw−1 y ]
times the sum over (Gx ∩ Gw−1 y )Gx,0+ \Gx , which is a quotient of Lx by a parabolic
subgroup.
More precisely, one knows [20, 3.5] (Gx ∩Gw−1 y )Gx,0+ \Gx = Px,w (k)\Lx (k), for some
algebraic subgroup Px,w ⊂ Lx as given in the following lemma. Note that by Lang’s
theorem Px,w (k)\Lx (k) = (Px,w \Lx )(k).
Lemma 2.5. Let S̄ be the maximal k-split torus of Lx that correspond to S as in [20,
3.5]. In particular, the character lattice X ∗ (S) can be identified to X ∗ (S̄). The group
Px,w is then the parabolic subgroup of Lx generated by ZLx (S̄) and the root subgroups
of those roots α ∈ Φ(Lx , S̄) satisfying α(x − w−1 y) ≤ 0, where x − w−1 y ∈ X∗ (S) ⊗ R
is evaluated by α via the identification X ∗ (S) ∼
= X ∗ (S̄).
We now rewrite the integral:
µφ̃x (1φy +gy,dy + )
=
X
w∈Wy \W̃ /Wx
0
q mw −mw
X
X
1(Ad(h−1 )(φw−1 y )+gw−1 y,dy + +gx,dx + )/gx,dx + (Ad(γ̇)φx ),
h∈Lw−1 y γ̇∈(Px,w \Lx )(k)
where mw is a non-negative integer such that q mw = [(Gx ∩ Gw−1 y )Gx,0+ : Gx ∩
Gw−1 y ] = [Gx,0+ : Gx,0+ ∩ Gw−1 y ] = [gx,0+ : gx,0+ ∩ gw−1 y ].
12
CHENG-CHIANG TSAI
2.3. Organization of double cosets Wy wWx . We shall rewrite the last equation
in the previous subsection. Note that Ad(γ̇)(φx ) ∈ gx,dx :dx + . Thus we may replace
(Ad(h−1 )(φw−1 y )+gw−1 y,dy + +gx,dx + )/gx,dx + by ((Ad(h−1 )(φw−1 y )+gw−1 y,dy + +gx,dx + )∩
gx,dx )/gx,dx + . The latter is contained in (2.5) of the following:
(2.5)
((gw−1 y,dy + gx,dx + ) ∩ gx,dx )/gx,dx + =
X
α∈Φ(G,S), α(x−w−1 y)≤d
gx,dx :dx +,α ,
x −dy
where for any r, s ∈ R̃ we define gx,r,α to be the intersection gx,r ∩ uα (uα being the
root space of α), and gx,r:s,α = gx,r,α /gx,s,α ⊂ gx,r:s .
Consider the vector space V := X∗ (S) ⊗ R. We draw affine hyperplanes {v | α(v) =
0} and {v | α(v) = dx − dy }. These hyperplane cut the whole space into polyhedrons
of various dimensions. We have
Lemma 2.6. If w ∈ Wy \W̃ is such that x−w−1 y lies in an unbounded polyhedron, then
((gw−1 y,dy + gx,dx + ) ∩ gx,dx )/gx,dx + ⊂ gx,dx :dx + contains no element with closed Lx -orbit
and anisotropic stabilizer.
This implies that such w, or say their image in Wy \W̃ /Wx (which are all but finitely
many), does not contribute to the formula for the orbital integral.
Proof. Recall dx − dy > 0. Let Φ = Φ(G, S) denote the roots in the root system. For an unbounded polyhedron described above, let Φ[ := {α ∈ Φ | α(v) ≤
dx − dy on that polyhedron}. The positive span of α ∈ Φ[ should not be the whole
space, for the polyhedron has to be bounded if so. Consequently, there is a hyperplane on X ∗ (S) ⊗ R such that all roots lies on either the boundary or one side of the
hyperplane.
We may take such a hyperplane to have integral coefficient. In other words, we
may find a cocharacter λ : Gm → S such that hλ, αi ≥ 0, ∀α ∈ Φ[ . In other words,
α(x − w−1 y) > dx − dy for all α with hλ, αi < 0. We may identify the cocharacter
λ as λ̄ : Gm → S̄ just like in Lemma 2.5. By (2.5), an element in ((gw−1 y,dy +
gx,dx + ) ∩ gx,dx )/gx,dx + lies on the non-negative root space of λ̄. However, the proof of
Theorem 2.1 exactly shows that this cannot happen for elements with closed Lx -orbit
and anisotropic stabilizer.
Consider W̃ = Z̃ o W where Z̃ is the quotient of CG (S) by the stabilizer of any
point on A = A(S, F ) in CG (S) (this Z̃ is the group denoted by Λ in [20, 1.2]).
W = NG (S)/CG (S) is the finite Weyl group of G over F . We have Wx and Wy maps
injectively into W . There is a “derivative” map D : Wy \W̃ → Wy \W . We’ll need
another definition:
Definition 2.7. Elements w, w0 ∈ Wy \W̃ with the same image D(w) = D(w0 ) in
Wy \W is called linked if x − w−1 y and x − (w0 )−1 y lies in the same polyhedron in V.
They are called strongly linked 2 if they are linked, and there exists a lift z ∈ CG (S)
2the
reader might want to skip this notion on reading; see Remark 2.9
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
13
of the element z̄ ∈ Z̃ with wz̄ = w0 such that the action of Ad(z) on
X
gx,dx :dx +,α
α∈Φ(G,S), α(x−w−1 y)=dx −dy
is the identity.
We say elements w, w0 ∈ Wy \W̃ /Wx are linked (resp. strongly linked) if some lifts
of them in Wy \W̃ are linked (resp. strongly linked). We write w ∼ w0 if they are linked
and w ≈ w0 if they are strongly linked. A linked class (resp. strongly linked class)
is an equivalence class for ∼ (resp. ≈) in Wy \W̃ or Wy \W̃ /Wx whose image under
ev lies in a bounded polyhedron.
Note that in the first part of the definition, the action of Ad(z) on the space is always
well-defined because w ∼ P
w0 implies α(x − w−1 y) = dx − dy ⇔ α(x − w0−1 y) = dx − dy
and that Ad(z) preserves such α gx,dx ,α . The following lemma shows that there are at
most a bounded number of strongly linked classes among a linked class.
Lemma 2.8. Let Z = S(F )/S(OF ), identified as a finite index subgroup of Z̃. If for
w, w0 ∈ Wy \W̃ we have w ∼ w0 and z̄ ∈ Z with wz̄ = w0 , then w ≈ w0 .
Proof. Choose a uniformizer π for our local field F . One has a natural identification
Z ∼
= X∗ (S) by identifying the image of the discrete valuation of F to Z. This gives
a lift Z ∼
= X∗ (S) → S = S(F ) by χ 7→ χ(π). That w ∼ w0 implies (z̄, α) = 0 for all
α ∈ Φ(G, S) with α(x − w−1 y) = dx − dy . But then for all such α, z = z̄(π) acts on uα
by π (z̄,α) = 1.
Remark 2.9. The lemma implies that there can only be a bounded amount of strongly
linked class inside a linked class. In fact, the author does not know a single example
of a linked class in which there are more than one strongly linked class, and it can be
proved that for a quasi-split group (i.e. when CG (S) is a torus) a linked class is always
a strongly linked class. We don’t know if this is true in general or not.
Now we can finally rewrite the last equation of the last subsection. Denote by C the
set of strongly linked classes in Wy \W̃ /Wx .
Theorem 2.10. We have
! 
(2.6)
µφ̃x (1φy +gy,dy + ) =
X
X
Π∈C
w∈Π
0
q mw −mw
·

X
jΠ,φy (Ad(γ̇)φx ) .
γ̇∈Hx,Π
The notations are as follows: firstly Fx,Π := Px,w \Lx for any w ∈ Π by Lemma 2.5,
and Fx,Π = Fx,Π (k). And
Hx,Π = {γ̇ ∈ Fx,Π | Ad(γ̇)φx ∈
X
α∈Φ(G,S),α(x−w−1 y)≤dx −dy
gx,dx :dx +,α }.
14
CHENG-CHIANG TSAI
Lastly, jΠ,φy : gx,dx :dx + → Z≥0 is given by
(2.7)
jΠ,φy (a) = #StabLw−1 y (φw−1 y ) ·#{φ0 ∈ Ad(Lw−1 y )(φw−1 y ) | φ0 ∈ (gw−1 y,dy + +gx,dx )/gw−1 y,dy +
and its reduction in
X
gx,dx :dx +,α agrees with a}.
α∈Φ(G,S),α(x−w−1 y)=dx −dy
0
Recall the notations φw−1 y = Ad(w−1 )φy and q mw −mw = [gx,0+ : gx,0+ ∩gw−1 y ]·[gx,dx + :
gx,dx + ∩ gw−1 y,dy + ]−1 . Note that for w ∈ Π the definitions of jΠ,φy are independent of
the choice of w, since wz = w0 ⇒ φw−1 y = Ad(z)φw0−1 y ; this is where we use the notion
of strongly linked.
The set Hx,Π is actually the k-points of a subvariety Hx,Π of the flag variety Fx,Π
which are called Hessenberg varieties in [8]. In fact, when y is a hyperspecial vertex,
dy = 0 and if we replace 1φy +gy,dy + in (2.6) by 1gy,dy , then we are computing orbital
integral on the indicator function of (the Lie algebra of) a hyperspecial parahoric, and
what we are doing should be thought as a p-adic integral analogue of the work of [8].
In general, we should be counting points on quasi-finite covers of Hessenberg varieties.
The author studies some examples in [21, Sec. 3].
At this stage, the combinatorics of the set C of strongly linked classes can be very
complicated, since the set of linked classes, or equivalently, the set of polyhedrons cut
out on V is already very complicated. In the next section we’ll see a situation where
we simplify this combinatorics, and use the result to compute Shalika germs.
3. Shalika germs
3.1. Homogeneity of nilpotent orbits. Let x, dx , φx and φ̃x any lift of φx be as in
the previous section. Fix in this section N a nilpotent element in g. The (G-conjugacy)
orbit of N will be denoted G N . We shall assume in this section char(F ) is large enough
so that there are only finitely many such orbits, see Appendix. We shall compute the
value of the Shalika germ ΓN (φ̃x ) in terms of counting points on subvarieties of flag
varieties and those combinatorics same as in (2.6).
Hypothesis 3.1. There exists a point y on the building, a real number d∗y and a coset
φ∗y ∈ gy,d∗y /gy,d∗y + such that N ∈ φ∗y + gy,d∗y + , and for any nilpotent orbit other than G N
that meets φ∗y + gy,d∗y + , the dimension of that orbit is strictly larger than the dimension
of G N .
By work of DeBacker [7, Corollary 5.2.4] the hypothesis always holds when char(k)
is large enough; In his construction, all other orbits that meets the coset contains G N
in its (p-adic) closure.
We now suppose the hypothesis holds for N , and we fix a choice of y, d∗y and φ∗y
that satisfy the condition in the hypothesis. Write µO for the nilpotent orbital integral
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
15
on a nilpotent orbit O and µN := µ(G N ) . As a consequence, µN (1φ∗y +gy,d∗y + ) > 0, and
µO (1φ∗y +gy,d∗y + ) = 0 for a different orbit O of the same or less dimension as that of G N .
For any nilpotent element N 0 , by [15, Proposition 1], there is always a 1-dimensional
torus in G that acts on the line generated by N 0 but non-trivially. Such torus must
split, and thus we have a torus j : Gm ,→ G which acts on N 0 by Ad(j(t))N 0 = t`N 0
for some `N 0 ∈ Z − {0}. When F is a p-adic field one can integrate an sl2 -triple and
take `N 0 = 2. We take ` to be the least common multiple for a choice of such `N 0 for
each nilpotent orbit, so that for all nilpotent element N 0 , t` N 0 ∈ G N 0 for any t ∈ F ∗ .
Fix also a choice of the uniformizer π ∈ F . We in particular have π ` N 0 ∈ G N 0 .
The assumption that char(F ) is very good for G guarantees that the centralizer of N 0
in g is the Lie algebra of the centralizer of N 0 in G [5, Theorem 1.2] (this implies [6,
Proposition 6.7] that the nilpotent orbit is smooth with the expected tangent space).
Recall by [3, Proposition 4.1] we have a G-invariant non-degenerate symmetric bilinear form κ(·, ·) on g. One can then consider the symplectic form on G N 0 (as an
F -analytic manifold over F ), for which this form at the point N 0 ∈ G N 0 is given by
hX, Y iN 0 = κ(Ad(N 0 )X, Y ) for X, Y ∈ g/gN 0 , where gN 0 is the Lie algebra of the
centralizer of N 0 .
The top exterior power of this symplectic form give rises to an G-invariant measure
on G N 0 . Obviously h·, ·it` N 0 = t` h·, ·iN 0 for t ∈ F ∗ . This implies that the measure on
the nilpotent orbit satisfies the property that for any subset S ⊂ g, the measure of
dim G N 0
G 0
N ∩ π ` S is q − 2 ·` times the measure of G N 0 ∩ S.
For any function f ∈ Cc∞ (g), t ∈ F ∗ , denote by ft the function ft (X) = f (t−1 X).
Then for any nilpotent orbit O
µO (fπ`n ) = q −
dim O
·`n
2
µO (f ), ∀n ∈ Z.
The theorem of Shalika germs [18] states that under certain assumptions on char(F )
(see Appendix), there exists smooth functions ΓO on grs the dense open subset of
regular semisimple elements of g, such that for any X ∈ grs , there exists a lattice
ΛX ⊂ g such that for any compactly supported function f ∈ Cc∞ (g/ΛX ) invariant
under translation by ΛX , we have
X
µX (f ) =
ΓO (X)µO (f ).
O
Here O runs over nilpotent orbits, and µX and µO are orbital integrals for the orbit
of X and O, respectively, with suitable normalization for their Haar measures. For
nilpotent element N we also write ΓN := Γ(G N ) .
Let y, d∗y and φ∗y be as in Hypothesis 3.1. Let n be a positive integer. In the rest of
this section let dy = d∗y +`n and φy = π −`n φ∗y depend on n. We consider µφ̃x (1φy +gy,dy + ).
Since multiplying by π −`n preserves all nilpotent orbits, the new pair of dy and φy still
satisfy Hypothesis 3.1. Together with the theorem of Shalika germ, we have
16
CHENG-CHIANG TSAI
(3.1) µφ˜x (1φy +gy,dy + ) =
X
ΓO (φ̃x )µO (1φy +gy,dy + ) =
O
X
q
dim O
·`n
2
ΓO (φ̃x )µO (1φy +gy,dy + ).
O
G
When O = G N , dim2 O = dim(2 N ) . Hypothesis 3.1 says that for all other orbits O0
G
0
that appears in this sum, dim2 O > dim(2 N ) . We shall compare (3.1) with (2.6) in the
next section.
3.2. Relatedness and strong-relatedness. Let n0 be a positive integer and d0y =
0
d∗y + `n0 , φ0y = π −`n φ∗y .
Definition 3.2. Let Wd∼y denote the set of linked classes in Wy \W̃ (while the definition
depends on dy and n) and Wd∼0y instead be the set of linked classes defined with dy
replaced by d0y . We say Π ∈ Wd∼y and Π0 ∈ Wd∼0y are related if any (hence all) w ∈ Π,
w0 ∈ Π0 we have D(w) = D(w0 ), and for any α ∈ Φ(G, S) one has α(x − w−1 y) and
α(x−w0−1 y) have the same sign and α(x−w−1 y)−(dx −dy ) and α(x−w0−1 y)−(dx −d0y )
have the same sign.
∼
∼
Similarly, let Wd=y and Wd=0y denote the set of strongly linked classes in Wy \W̃ , defined
∼
∼
with dy and d0y , respectively. We say Π ∈ Wd=y and Π0 ∈ Wd=0y are strongly related if
they belong to related linked classes, and there exists z̄ ∈ Z̃ and its lift z ∈ CG (S) such
that for some w ∈ Π, w0 ∈ Π0 one has wz̄ = w0 and that Ad(z) acts on
M
M
gx,dx :dx +,α =
gx,dx :dx +,α
α∈Φ(G,S), α(x−w−1 y)=dx −dy
α∈Φ(G,S), α(x−w0−1 y)=dx −d0y
∼
∼
by π kn . Lastly, we write Cd∼y := Wd∼y /Wx and Cd=y := Wd=y /Wx , the same for d0y and
induce the notion of relatedness and strong relatedness on them. That is, elements in
∼
∼
Cd∼y and Cd∼0y (resp. Cd=y and Cd=0y ) are related (resp. strongly related) if some of their
preimages are.
Exactly the same argument as in Lemma 2.8 shows that if wz̄ = w0 for related
w ∈ Π, w0 ∈ Π0 and z̄ ∈ Z, then Π and Π0 are strongly related. This notion of
strong relatedness is for the reason that, just like the notion for strongly linkedness,
the geometrical sum (i.e. the sum in the last round bracket) in (2.6) depends only up
to strong relatedness.
Recall in the paragraph before Lemma 2.6 we took V = X∗ (S) ⊗ R. Consider this
time affine hyperplanes α(v) = 0 and α(v) = 2 for α ∈ Φ(G, S). The affine hyperplanes
cut V into numerous polyhedrons (of different dimensions). Call P the set of these
polyhedrons. The group Wx acts on the space V by fixing the origin. Denote by P̄
the set of Wx -orbits in P. By the very definition
of relatedness
we have injective maps
i
h
2
−1
∼
∼
ev : Wdy → P and ev : Cdy → P̄ by w 7→ dx −d0 (x − w y) .
y
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
17
By Lemma 2.6, only those (strongly) linked class that maps to bounded polyhedrons
⊂ Wd∼y the subset of those which map to bounded polyhematter. Denote by Wd∼,bd
y
⊂ Cd∼y those which map to orbits of bounded polyhedrons. As we have
drons and Cd∼,bd
y
∼
∼
∼
∼
, and similarly
and Cd=,bd
natural map Wd=y → Wd∼y and Cd=y → Cd∼y , we can define Wd=,bd
y
y
for d0y .
Recall that in (2.7), we have
!
µφ˜x (1φy +gy,dy + ) =
(3.2)
X
X
∼
Π∈Cd=,bd
y
w∈Π
q
mw −m0w
J(Π).
where J(Π), being the second inner sum in the RHS of (2.6), is some “geometric
number” that only depends on Π up to strongly relatedness. We now have
Theorem 3.3. After possibly replacing ` by a multiple of it, we have
∼
(1) There is an integer N such that for all n, n0 ≥ N , we have bijections Cd∼,bd
−
→ Cd∼,bd
0
y
y
∼
∼
∼
−
→ Cd=,bd
given by relatedness and strong relatedness.
and Cd=,bd
0
y
y
∼
and similarly by C =bd the set identified
(2) Let C ∼,bd be the set identified to all Cd∼bd
y
∼
∼
to all Cd=y as in (1). Then we have for any strongly related class Π ∈ C =,bd , the sum
P
mw −m0w
is a polynomial pΠ (q n , n) in q n and n.
w∈Π q
Proof. For (1), since for any σ ∈ Wx , w ∈ Wd∼,bd
and w0 ∈ Wd∼,bd
are related if and
0
y
y
0
only if wσ and w σ are related (and the same for strongly related), it suffices to find
∼
∼
∼
=,bd
=,bd ∼
a multiple of ` so that Wd∼,bd
−
→ Wd∼,bd
−
→
W
under relatedness and
and
W
0
0
d
d
y
y
y
y
strong relatedness. We need an extension of Lemma 2.8, whose proof is the same:
∼
∼
Lemma 3.4. Let Z ⊂ Z̃ be as in Lemma 2.8. If for Π ∈ Wd=y , Π0 ∈ Wd=0y , Π and Π0
are related and for some (hence all) w ∈ Π, w0 ∈ Π0 there exists z̄ ∈ Z with wz̄ = w0 ,
then Π and Π0 are strongly related.
Let Wd≡y be a further partition of Wd∼y by the property in Lemma 2.8. The above
∼
lemma stated that Wd≡y is also a partition of Wd=y . Let’s invent a new term that Π ∈ Wd≡y
and Π0 ∈ Wd≡0y are called equivalently related if for some w ∈ Π, w0 ∈ Π0 , the property
∼
in the lemma holds. Then it suffices to prove the bijection Wd≡,bd
−
→ Wd≡,bd
under
0
y
y
equivalent relatedness.
Note that w 7→ x−w−1 y gives an injection from Wy \W̃ to A. For elements w, w0 ∈ Π
for some Π ∈ Wd≡,bd
, we have x − w−1 y and x − w0−1 y satisfy that α(x − w−1 y) and
y
α(x − w0−1 y) shares the same sign, that α(x − w−1 y) − (dx − dy ) and α(x − w0−1 y) −
(dx − d0y ) shares the same sign, and that x − w−1 y and x − w0−1 y differs by a translation
from Z, where Z acts on A by translation through a lattice. The assertion then follows
from the following combinatorial lemma:
18
CHENG-CHIANG TSAI
Lemma 3.5. We use notations independent from the rest of the article. Consider
lattice Zm ⊂ Rm and a finite set of linear equations and inequalities of the form a1 x1 +
... + an xm = c(n) and a1 x1 + ... + am xm < c(n), where ai are rational numbers and
c(n) is a linear function in integer variable n with rational coefficients. Suppose the
equations and inequalities satisfy that the region of solutions in Rm is bounded. Let q
be a formal variable and λ(x) be a linear function on Zm with integral coefficients. Let
Sλ (n) ∈ Z[q, q −1 ] be the sum of q λ(x) over all x ∈ Zm in the region, i.e. that satisfy the
equations and inequalities. Then there exists positive integers c and N such that for
all n > N , Sλ (cn) is a Laurent polynomial in q n and n.
One takes the conditions regarding the signs of α(x−w−1 y) and α(x−w−1 y)−(dx −dy )
to be the equations and the inequalities. The lemma implies that the number of lattice
points inside such a set of equations and inequalities is either always non-zero or always
zero (one sees this by subtituting q = 1). Hence the result for (1).
For (2) of the theorem, the lemma almost provide the result except that we are summing over w in subsets of Wy \W̃ /Wx instead of Wy \W̃ (as in the setting of Definition
3.2, Lemma 3.4 and the above lemma. This difference can be resolved by adding extra
linear equations to “mark” the fixed points of various subgroups of Wx , so that in
each subset Π of Wy \W̃ considered, Wx acts on the right with the order of the stabilizer being constant, and the sum over Π ⊂ Wy \W̃ /Wx becomes just the sum over its
preimage in Wy \W̃ divided by this order. This finishes the proof of Theorem 3.3. ∼
Let’s now fix Π ∈ C =,bd . We analyze the polynomial pΠ (q n , n) in terms of ev(Π) ∈ P̄.
0
Recall that in (2.6) q mw −mw was given by
0
q mw −mw = [gx,0+ : gx,0+ ∩ gw−1 y ] · [gx,dx + : gx,dx + ∩ gw−1 y,dy + ]−1
=
Y
α∈Φ(G,S)
|gx,0+,α | · |gx,dx +,α ∩ gw−1 y,dy +,α |
,
|gx,0+,α ∩ gw−1 y,0,α | · |gx,dx +,α |
where | · | was used to denote any Haar measure on the F -vector space gα . We have
following easy estimate
Lemma 3.6. For any point z in the Bruhat-Tits building, any real numbers d1 , d2 and
any α ∈ Φ(G, S). Let ρα = dimF (gα ). Suppose [gz,d1 ,α : gz,d1 ,α ∩ gz,d2 ,α ] = q m . Then
|m − ρα · max(d2 − d1 , 0)| < ρα ,
Applying the estimate, we have
X
mw −m0w =
ρα · max{α(x − w−1 y), 0} − max{α(x − w−1 y) − (dx − dy ), 0} +,
α∈Φ(G,S)
P
with the error term || < α ρα = dim G − dim CG (S). Recall that dy = d∗y − `n
2
−1
depends on n. Suppose n is large and w ∈ Wy \W̃ is such that dx −d
y) is very
0 (x − w
y
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
close to a vertexPv, i.e. 0-dimensional polyhedron in P, then one has mw −m0w ∼
where τ (v) := α∈Φ(G,S) ρα τα (v), with

if α(v) ≥ 2.
 2
if 0 ≤ α(v) ≤ 2.
τα (u) = α(v)
 0
otherwise.
19
`n
·τ (v),
2
More precisely, we have
Theorem 3.7. For each monomial in the polynomial pΠ (q n , n), the degree in q n is equal
to `·τ2(v) , for some vertex v ∈ P̄ contained in the closure of ev(Π) (see the definition of
ev(Π) before Theorem 3.3.)
Proof. To prove the theorem, one simply use the following extension of the combinatorial lemma 3.5:
Lemma 3.8. We use notations in Lemma 3.5 and independent from all other parts
of the article. Suppose d is the degree in q n of a term in Sλ (cn) in Lemma 3.5. Then
possibly after replacing c by a multiple of it, there exist vectors v, w ∈ Rm and ∈ R,
such that d · cn = λ(v + cnw) − , and satisfy:
(1) By changing some of the inequalities into equations (by turning the < sign into
= sign) and throwing away other inequalities, one has that for every n, λ(v + nw) is
the unique solution in Rm to the resulting set of equations.
(2) There exists an N 0 such that for all n > N 0 that are divided by c, λ(v + nw) is
contained in the closure of the region cut out by the equations and inequalities.
The theorem is then proved by using Lemma 3.6 and substituting λ(x − w−1 y) =
mw − m0w as when we used Lemma 3.5.
It’s now time to compare (3.1) with (3.2). They are
X dim O
µφ˜x (1φy +gy,dy + ) =
q 2 ·`n ΓO (φ̃x )µO (1φy +gy,dy + ).
O
µφ˜x (1φy +gy,dy + ) =
X
pΠ (q n , n)J(Π).
∼
Π∈Cd=,bd
y
Recall our goal is to compute ΓN (φ̃x ), and Hypothesis 3.1 says µO (1φy +gy,dy + ) 6= 0
only when O = G N or dim O > dim G N . Comparing two equations gives
Theorem 3.9. Let φx ∈ gx,dx :dx + be with closed orbit and anisotropic stabilizer under
Lx -action, φ̃x ∈ gx,dx any lift of φx , and y, dy , φy and N as in Hypothesis 3.1. Then
X
ΓN (φ̃x ) = (µN (1φy +gy,dy + ))−1 ·
c `·dim O (pΠ )J(Π),
=,bd
Π∈C ∼
where c `·dim O (pΠ ) denote the coefficient of the q
2
is the second inner sum in the RHS of (2.6).
2
`·dim O
n
2
-term in pΠ (q n , n), and J(Π)
20
CHENG-CHIANG TSAI
Logically Theorem 3.9 is independent from Theorem 3.7. However, Theorem 3.7
gives a necessary condition for c `·dim O (pΠ ) to be non-zero; it can be non-zero only
2
when a vertex v on the boundary of ev(Π) has τ (v) = dim O. In the next subsection
we give another strong result along this line.
3.3. A vanishing result. This subsection is devoted to the following result.
∼
Theorem 3.10. Let Π ∈ C =,bd . Suppose there is a vertex v on the boundary of ev(Π)
such that τ (v) < dim O. Then J(Π) = 0.
For our convenience, we copy from (2.6), (2.7) that
J(Π) =
X
jΠ,φy (Ad(γ̇)φx )
γ̇∈Fx,Π
where


X
jΠ,φy (a) = #StabLw−1 y (φw−1 y ) · id a ∈
gx,dx :dx +,α  ·
α∈Φ(G,S),α(x−w−1 y)≤d
x −dy
#{φ0 ∈ Ad(Lw−1 y )(φw−1 y ) | φ0 ∈ (gw−1 y,dy + + gx,dx )/gw−1 y,dy + and its reduction in
X
gx,dx :dx +,α agrees with a}.
α∈Φ(G,S),α(x−w−1 y)=dx −dy
In particular, we see that in order to have J(Π) 6= 0, we need to have
X
φ0 ∈ ((gw−1 y,dy + + gx,dx ) ∩ gw−1 y,dy )/gw−1 y,dy + =
gw−1 y,dy :dy +,α .
α∈Φ(G,S),α(x−w−1 y)≥dx −dy
where φ0 is some element in Ad(Lw−1 y )(φw−1 y ). We can then take a lift N ∈ gw−1 y,dy
of φ0 such that
(3.3)
N∈
X
gα .
α∈Φ(G,S),α(x−w−1 y)≥dx −dy
Since dx − dy > 0, N is nilpotent. φ0 ∈ Ad(Lw−1 y )(φw−1 y ) implies that for some
g ∈ Gy , π −`n Ad(g)Ad(w)N ∈ φ∗y + gy,d∗y + . By Hypothesis 3.1, the dimension of
the orbit of π −`n Ad(g)Ad(w)N is larger than or equal to dim O. By definition of
`, π −`n Ad(g)Ad(w)N and N have the same orbit. To prove the theorem, it suffices to
show that for every vertex v on the boundary of ev(Π) we have τ (v) ≥ dim G N .
Since the polyhedron ev(Π) (in fact a Wx -orbit of polyhedrons, while the Wx -action
does not matter here) is cut out by several affine walls of the form α(v) = 0 or α(v) = 2,
2
(x − w−1 y) ∈ ev(Π), we see that for every
and w and Π is related by ev(w) = dx −d
y
point in the closure of the polyhedron, in particular for every vertex v on the boundary
of ev(Π).
COMPUTATIONS OF ORBITAL INTEGRALS AND SHALIKA GERMS
21
While v is a Wx -orbit of points on V = X∗ (S) ⊗ Q, we can think of it as L
a point on
V and thus a rational cocharacter, giving a Q-grading on g. We write g = i∈Q g(i).
By (3.3), N ∈ g(≥ 2). We have
L
Lemma 3.11. Let H = P
H(i) be an R-grading of a Lie algebra H (over an arbitrary
field) with X ∈ H(≥ 2) := i≥2 H(i). Then for every r ∈ R,
X
dim{Y ∈ H | [Y, X] = 0} ≥ dim
H(i).
r≤i<r+2
Proof. The LHS is greater than or equal to the dimension of the kernel of ad(X) on
H(≥ r), and the projection of ad(X)|H(≥r) to the space on the RHS is the zero map. On the other hand, we have by definition
Z
X
2τ (v) = τ (v) + τ (−v) =
min(|i|, 2) · dim g(i) = 2 dim g −
i∈Q
0
−2
dim
X
g(i)dr.
r≤i<r+2
With the above lemma for H = g, this says τ (v) ≥ dim G N as asserted.
Appendix A. Assumptions on p
In this appendix, we summary assumptions on either char(F ) or char(k) at various
places of this article.
To begin with, char(F ) is assumed to be very good for a lot of purposes, including
even the well-definedness of orbital integrals [19, III.3.27]. Recall we say a prime p is
good (0 is always very good) for G if p does not divide the coefficient of any simple root
in the highest roots, for the absolute root system of any simple factor of (the isogeny
type) of G. p is very good for G if p is good and it does not divide n + 1 whenever G
(up to isogeny) has a simple factor of type An .
For the proof of Theorem 2.1 and Claim 2.4 we require that G is tamely ramified,
that is, G splits over a tamely ramified extension. We also assume that the rational
coordinate of our point x on the Bruhat-Tits building has denominator coprime to
char(k). However, that gx,dx has semisimple Lx -orbit implies that x must be a (0dimensional) intersection of walls defined by affine roots. Both tameness condition
(on G and on x) will thus be automatic if char(k) is larger than a certain number
determined by the absolute root system of G.
In the proof of Claim 2.4 we need char(k) to be very good to identify g with its dual
in a way compatible with the Moy-Prasad filtration. It might be that in some cases
the method in [8, Sec. 5] will allow us to drop this assumption.
In Section 3, if char(F ) > 0 we need to verify the assumptions (U1)∼(U4) in [18].
(U1) and (U2) are implied by that char(F ) is very good [5, Theorem 1.2], [6, Proposition 6.7] and [19, III.3.27]. (U3) is proved by Ranga Rao’s [16], which was stated for
22
CHENG-CHIANG TSAI
char(F ) = 0 but in fact only uses (U1), (U2) and [19, III.4.14]. The last was proved
under the assumption char(F ) ≥ 4m + 3 where m is the sum of coefficients for the
highest root in the absolute root system (for each simple factor). For (U4), the number
of geometric (stable) nilpotent orbits is always finite [10]. For the number of orbits
to be finite, we need the first Galois cohomology of the centralizer of any nilpotent
element to be finite. If char(F ) is larger than rkF̄ (G) + 1 where rkF̄ (G) denotes the
absolute rank of G, then the Levi factor of the centralizer is always tamely ramified
and consequently the H 1 is finite.
Lastly, for the computation of Shalika germs of a chosen nilpotent orbit we uses
Hypothesis 3.1. The hypothesis is valid if DeBacker’s result [7] is available, which is
good when char(k) is large enough; we refer the reader to [7, Sec. 4.2]. For applications
in not-so-large residue characteristic one can usually check Hypothesis 3.1 directly.
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