LECTURES ON HARMONIC ANALYSIS FOR REDUCTIVE p-ADIC GROUPS

STEPHEN DEBACKER

1. introduction

In his paper The characters of reductive p-adic groups [14] Harish-Chandra outlines his philosophy about harmonic analysis on reductive p-adic groups. According to this philosophy, there are two distinguished classes of distributions on the group: orbital integrals and characters. Similarly, there are two classes of distributions on the Lie algebra which are interesting: orbital integrals and their Fourier transforms. The real meat of his philosophy states that we ought to treat orbital integrals on both the group and its Lie algebra in the “same way” and similarly, we should think of characters and the Fourier transform of orbital integrals in the same way. This philosophy has many manifestations (see, for example, Robert Kottwitz’s excellent article [12]). In this series of lectures, we will examine the various distributions discussed above and discuss one of the deepest connections between them: the Harish-Chandra–Howe local character expansion. Because of requests on the part of participants, I will spend much time reviewing the basics of p-adic fields and discussing some of the uses of Moy-Prasad filtrations in the representation theory of reductive p-adic groups. As such, at a small cost in terms of generality, I will concentrate on those techniques which use this theory. These lectures and the notes are meant as an informal and elementary introduction to the material. For complete, rigorous proofs, please see the references.

Very little of the material in this set of lectures is original. I have borrowed heavily from the work and lectures of Harish-Chandra, Roger Howe, Robert Kottwitz, Allen Moy, Gopal Prasad, Paul Sally, Jr., and J.-L. Waldspurger, among others. I thank the organizers of this conference, in particular, Eng-Chye Tan and Chen-Bo Zhu, for inviting me and allowing me to present this series of tutorials. I thank Jeff Adler, Amritanshu Prasad, Joe Rabinoff, and Loren Spice for their helpful comments on earlier drafts of these notes.

Date: December 20, 2002.

1991 Mathematics Subject Classification. Primary 20E50.

Key words and phrases. Harmonic analysis, reductive p-adic group.

The author was supported by a grant from the National Science Foundation. This paper is based on lectures delivered at the Institute for Mathematical Sciences at the National University of Singapore in 2002. The author thanks the Institute for its support.

2. Basics

2.1. An introduction to the p-adics. The usual introductory mathematical analysis course proceeds roughly as follows: The class agrees to agree that the set of natural numbers

{1, 2, 3, …}

is very natural and therefore a good place to begin the course. In order to form a group with respect to addition, the additive identity and additive inverses are tossed in to the mix to give us the integers

ℤ := {…, −4, −3, −2, −1, 0, 1, 2, 3, …}.

This set does not form a group with respect to multiplication; it is therefore enlarged to form ℚ, the field of rational numbers. Everything so far has been very natural.

At this point, the incompleteness of the rationals is demonstrated by proving that the square root of two is not rational. To compensate for this, the fact that the rationals are ordered (that is, there is a notion of nonpositive and nonnegative) is invoked to define the absolute value, |·|, of any rational number q:

|q| = \begin{cases}q & if q ≥ 0 \\ −q & if q < 0. \end{cases}

As usual, the absolute value gives you a metric with respect to which you complete the rationals. We recall that two Cauchy sequences {q_{n}} and {q′_{n}} of rational numbers are said to be equivalent with respect to |·| provided that {q_{n} − q′_{n}} is a Cauchy sequence which converges to zero. The set of real numbers is then defined to be the set of equivalence classes of Cauchy sequences with respect to |·|, that is, the completion of ℚ with respect to |·|.

It turns out, however, that the normal absolute value provides just one of an infinite number of incompatible ways to complete the rational numbers^{1}, and, from the proper perspective, this completion is a somewhat unnatural object.

Let p be any prime. If q is a nonzero rational number, then there is a unique integer k such that q = p^{k} · a/b with p ∤ a and p ∤ b. We can then define the p-adic absolute value, |·|_{p}, on ℚ by setting |q|_{p} = 0 if q = 0 and |q|_{p} = p^{−k} otherwise. The p-adic absolute value has the following properties.

Exercise 2.1.1. If r_{1} and r_{2} are rational numbers, then (1) |r_{1}|_{p} ≥ 0, and |r_{1}|_{p} = 0 if and only if r_{1} = 0,

(2) |r_{1} · r_{2}|_{p} = |r_{1}|_{p} · |r_{2}|_{p}, and
(3) |r_{1} + r_{2}|_{p} ≤ max(|r_{1}|_{p} , |r_{2}|_{p}).

Exercise 2.1.2. In the last item of the previous exercise, show that if |r_{1}|_{p} ≠ |r_{2}|_{p} then the inequality is an equality. Is the converse of this statement true?

1The field of real numbers is, up to isomorphism, the unique totally ordered completion of the rationals.

From Exercise 2.1.1, it follows that we can define a metric on ℚ with respect to the p-adic absolute value. We define ℚ_{p} to be the completion of ℚ with respect to |·|_{p}. The p-adic absolute value on ℚ extends continuously (and uniquely) to a p-adic absolute value |·|_{p} : ℚ_{p} → {0, p^{k} | k ∈ ℤ}. We define the valuation ν_{p} on ℚ_{p} by |x|_{p} = p^{−ν_{p}(x)} for x ∈ ℚ_{p}^{×} and ν_{p}(0) = ∞. Thus ν_{p}(p^{m}) = m for m ∈ ℤ.

We remark that Ostrowski’s theorem tells us that the only nondiscrete, locally compact fields of characteristic zero are the field of real numbers, the field of complex numbers, and the finite extensions of ℚ_{p}. That is, these are the only fields of characteristic zero on which integration naturally makes sense.

2.2. The structure of ℚ_{p} and additive characters. If we define ℤ_{p} to be the set of x ∈ ℚ_{p} such that |x|_{p} ≤ 1, then ℤ_{p} is a maximal compact open subring in ℚ_{p}. As in [33], it can be identified with the completion of ℤ with respect to |·|_{p}. We call ℤ_{p} the ring of integers in ℚ_{p}. The subset of ℤ_{p} consisting of those elements of ℤ_{p} with p-adic absolute value less than one forms a maximal ideal (p) = pℤ_{p} of ℤ_{p}. The quotient ℤ_{p}/(p) is isomorphic to 𝔽_{p}, the field with p elements. It follows that we can write

ℤ_{p} = ∐_{0≤i≤(p−1)}(i + pℤ_{p}).

As a slight tangent, we remark that Hensel’s lemma (see nearly any book on local fields) tells us that we can lift a generator of 𝔽_{p}^{×} to ℤ_{p}^{×}, the group of units in ℤ_{p}. Thus, the square root of two is an element of, for example, ℚ_{7}.

The set of compact open subrings p^{n}ℤ_{p} for n ≥ 0 forms a neighborhood basis of zero. The sets p^{n}ℤ_{p} also make sense for n < 0, and we note that

ℚ_{p} = ⋃_{n∈ℤ} p^{n}ℤ_{p}.

Since ℤ_{p}^{×} = ℤ_{p} ∖ pℤ_{p}, it follows immediately that

(1)  ℚ_{p} = {0} ∪ ∐_{n∈ℤ} p^{n}ℤ_{p}^{×}.

From this one can show that for each x ∈ ℚ_{p}^{×} if |x|_{p} = p^{−m}, then there exist unique coefficients a_{n} ∈ {0, 1, 2, …, (p − 1)} such that

x = ∑_{n=m}^{∞} a_{n} · p^{n}.

This is called the p-adic expansion of x. For example, the 5-adic expansion of 1/3 is

2 + 3 · (5) + 1 · (5^{2}) + 3 · (5^{3}) + 1 · (5^{4}) + 3 · (5^{5}) + · · · .

If x ∈ ℚ_{p} and |x|_{p} = p^{−m}, then we define the tail of x to be the integer

t(x) = \begin{cases}∑_{n=m}^{0} a_{n} · p^{n} & if |x|_{p} = p^{−m} and m ≤ 0, and \\ 0 & if x ∈ pℤ_{p} . \end{cases}

We then define

Λ(x) = e^{\frac{2·π·i·t(x)}{p}} .

Exercise 2.2.1. Show that Λ defines an additive character of ℚ_{p} and that the restriction to pℤ_{p} of Λ is trivial, yet the restriction of Λ to ℤ_{p} is not trivial.

It is a fact that ℚ_{p} is isomorphic to the Pontrjagin dual ℚ̂_{p} of ℚ_{p} via the map a ↦ (b ↦ Λ(ba)).

2.3. General notation. From now on, let k denote a finite extension of ℚ_{p}, let R denote the ring of integers of k, and let ϖ denote a uniformizer so that ℘ = ϖR where ℘ is the prime ideal. We define ℘^{m} := ϖ^{m} · R for m ∈ ℤ. We let 𝔣 := R/℘ denote the residue field, and we let q = |𝔣|. As before, we fix a valuation ν such that ν(k) = ℤ. We fix an additive character Λ of k that is trivial on ℘ and not trivial on R.

We let G denote the general linear group realized as the set of n × n matrices with entries in k having nonzero determinant. We let 𝔤 denote the Lie algebra of G. We realize 𝔤 as M_{n}(k), the set of n × n matrices with entries in the field k, having the usual bracket operation. Let dX denote a Haar measure on 𝔤 and let dg denote a Haar measure on G.

We let B denote the Borel subgroup of G consisting of upper triangular matrices, and we let T denote the maximal split torus consisting of diagonal matrices.

The map (X, Y) ↦ tr(X · Y) from 𝔤 × 𝔤 to k defines a G-invariant, nondegenerate, symmetric, bilinear form on 𝔤. As such, it allows us to identify 𝔤 with the Pontrjagin dual 𝔤̂ of 𝔤 via the map X ↦ (Y ↦ Λ(tr(X · Y))).

A compact, open, R-submodule of 𝔤 is called a lattice. For example, for each integer i, we can define the standard filtration lattice

𝔨_{i} := ϖ^{i} · M_{n}(R)

of 𝔤. As another example, consider the Iwahori filtration lattices 𝔟_{i/n} defined by

𝔟_{i/n} := \{Y ∈ 𝔤 | Y_{jk} ∈ ϖ^{⌈ \frac{j−k+i}{n} ⌉} · R\}.

Note that for all integers i, j, we have ϖ^{j} · 𝔨_{i} = 𝔨_{i+j} and ϖ^{j} · 𝔟_{i/n} = 𝔟_{j+ \frac{i}{n}}.

More concretely, for n = 2 we have

𝔨_{1} = ( \begin{matrix}℘ & ℘ \\ ℘ & ℘ \end{matrix} )

and

𝔟_{0} = ( \begin{matrix}R & R \\ ℘ & R \end{matrix} ) ⊃ 𝔟_{1/2} = ( ℘ ℘^{℘ R} ) ⊃ 𝔟_{1} = ϖ · 𝔟_{0}.

For n = 3 we have

𝔟_{0} = ( \begin{matrix}R & R & R \\ ℘ & R & R \\ ℘ & ℘ & R \end{matrix} ) ⊃ 𝔟_{1/3} = ( \begin{matrix}℘ & R & R \\ ℘ & ℘ & R \\ ℘ & ℘ & ℘ \end{matrix} ) ⊃ 𝔟_{2/3} = ( \begin{matrix}℘ & ℘ & R \\ ℘ & ℘ & ℘ \\ ^{℘^{2}} & ℘ & ℘ \end{matrix} ) ⊃ 𝔟_{1} = ϖ · 𝔟_{0}.

From these examples, it is easy to see that the image of 𝔟_{0} in 𝔨_{0}/𝔨_{1} = M_{n}(𝔣) is a Borel subalgebra with nilradical equal to the image of 𝔟_{1/n} in 𝔨_{0}/𝔨_{1} = M_{n}(𝔣). (We remark that in all of these examples, the lattice in 𝔤 is a direct sum of copies of lattices of k.)

We have similar filtrations of G by compact open subgroups. The standard filtration subgroups are defined by K_{0} = GL_{n}(R) and K_{i} = 1 + 𝔨_{i} for positive integers i. We note that, up to conjugacy, K_{0} is the unique maximal compact open subgroup of G. We also define the Iwahori subgroup B_{0} = 𝔟_{0}^{×} and, for positive integers i, we define the Iwahori filtration subgroups B_{i/n} = 1 + 𝔟_{i/n}. It is easy to check that if 0 ≤ k ≤ j, then K_{j} is a normal subgroup of K_{k} and the lattices 𝔨_{m} (m ∈ ℤ) are invariant under the action of K_{j}. Similar statements apply to the Iwahori filtration lattices and subgroups.

If 0 < j ≤ k ≤ 2j, then 𝔨_{j}/𝔨_{k} is an abelian group which is isomorphic via the map induced by X ↦ (1 + X) to K_{j}/K_{k}. Similarly, 𝔟_{j/n}/𝔟_{k/n} is isomorphic to B_{j/n}/B_{k/n}.

Given a lattice L in 𝔤 we define the dual lattice L^{∗} of L by

L^{∗} = {Y ∈ 𝔤 | tr(X · Y) ∈ ℘ for all X ∈ L}.

For example, for any integer i, the dual lattice of 𝔨_{i} is 𝔨_{(1−i)}, and the dual lattice of 𝔟_{i/n} is 𝔟_{(1−i)/n}. We let C_{c}^{∞}(𝔤) denote the set of functions on 𝔤 which are compactly supported, complex valued, and locally constant. We recall that a function f on 𝔤 is said to be locally constant if for each X ∈ 𝔤 there exists a lattice L_{X} such that f(X + ℓ) = f(X) for all ℓ in L_{X}. If we assume that our f is compactly supported, then this lattice can be chosen uniformly. We let C_{c}^{∞}(G) denote the functions on G which are compactly supported, complex-valued, and locally constant (that is, translation invariant with respect to some compact open subgroup of G).

For f ∈ C_{c}^{∞}(𝔤) we define f̂, the Fourier transform of f, by the formula

f̂(X) = ∫_{𝔤} f(Y) · Λ(tr(X · Y)) dY

for X in 𝔤. For example, if [L] denotes the characteristic function of a lattice L in 𝔤, then ˆ[L] is [L^{∗}] up to a constant. We normalize the measure dX on 𝔤 so that ˆ f̂(X) = f(−X) for all f ∈ C_{c}^{∞}(𝔤) and X ∈ 𝔤.

Exercise 2.3.1. Show that this normalization of measures implies that for L sufficiently small, the measure of the lattice L with respect to dX is given by one over the square root of the index of L in L^{∗}, that is,

meas_{dX}(L) = [L^{∗} : L]^{−1/2}.

Exercise 2.3.2. Show that the map f ↦ f̂ from C_{c}^{∞}(𝔤) to itself is a bijection.

Finally, we normalize dg so that the map X ↦ (1 + X) takes dX into dg. So, for example, we have that for all j ∈ ℤ_{>0},

meas_{dX}(𝔨_{j}) = meas_{dg}(K_{j}).

3. Moy-Prasad filtrations

In this section we describe the Moy-Prasad filtration lattices of 𝔤 and subgroups of G.

3.1. The apartment of T. Recall that T is the group consisting of diagonal matrices in G. We write t ∈ T as t = (t_{1}, t_{2}, …, t_{n}) with t_{j} ∈ k^{×} for 1 ≤ j ≤ n. Let Φ ⊂ X^{∗}(T) = Hom(T, k^{×}) denote the set of roots of G with respect to T (that is, the nontrivial eigencharacters for the action of T on 𝔤). More explicitly

Φ = {α_{ij} | 1 ≤ i ≠ j ≤ n and α_{ij}(t) = t_{i}/t_{j}}.

With respect to our Borel subgroup B we let Δ denote the set of simple roots; that is, Δ := {α_{i(i+1)} | 1 ≤ i ≤ (n − 1)}. We let Φ^{+} denote the set of positive roots in Φ. We fix a Chevalley basis

{Z, H_{β}, X_{γ} | β ∈ Δ and γ ∈ Φ}.

Here

(Z)_{kl} = δ_{kl}

and

(H_{α_{ij}})_{kl} = \begin{cases}1 & if k = l = i \\ −1 & if k = l = (i + 1) \\ 0 & otherwise, \end{cases}

and

(X_{α_{ij}})_{kl} = δ_{ik}δ_{jl}.

We see that the center of 𝔤 is k · Z and the H_{β} together with Z form a basis for 𝔱, the Lie algebra of T.

For example, for GL_{2}(k) we have

Z = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , H_{α_{12}} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , X_{α_{12}} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , and X_{α_{21}} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) .

Let Z(G) ≤ T denote the center of G. For 1 ≤ k ≤ n define λ_{k} ∈ X_{∗}(T) = Hom(k^{×}, T) by setting

(λ_{k}(s))_{ij} = \begin{cases}s & if i = j = k \\ 1 & if i = j ≠ k \\ 0 & otherwise, \end{cases}

for s ∈ k^{×}.

With respect to our choice of a Chevalley basis, we can identify A = A(T), the apartment^{2} corresponding to T, with the real vector space

(X_{∗}(T) ⊗ ℝ)/(X_{∗}(Z(G)) ⊗ ℝ).

Let x_{0} denote the origin in A. The apartment of T is an (n − 1)-dimensional Euclidean space spanned by the set {x_{0} + λ̄_{k} | 1 ≤ k ≤ n}. Note that these spanning vectors satisfy the relation

0 = ∑_{k} λ̄_{k}.

2In these notes, we are always looking at the reduced building and apartments.

3.2. A simplicial structure for the apartment of T. We let

Ψ := {δ + n | δ ∈ Φ and n ∈ ℤ}

denote the set of affine roots of G with respect to T and ν. For ψ = δ + n ∈ Ψ we let ψ̇ = δ ∈ Φ.

If ψ = δ + n ∈ Ψ and ∑ λ_{i} ⊗ r_{i} ∈ X_{∗}(T) ⊗ ℝ, then we define

ψ(∑ λ_{i} ⊗ r_{i}) = n + ∑ r_{i}⟨δ, λ_{i}⟩.

Here ⟨ , ⟩ denotes the usual pairing between characters and cocharacters. Note that if v_{T} ∈ X_{∗}(T) and v_{Z} ∈ X_{∗}(Z(G)), then ψ(v_{T} + v_{Z}) = ψ(v_{T}). Consequently, we can and shall think of an affine root as a function on A(T). For ψ ∈ Ψ, we define

H_{ψ} = {x ∈ A(T) | ψ(x) = 0}.

The H_{ψ} are hyperplanes in A(T) and they provide us with a simplicial decomposition of A(T).

For example, for GL_{2}(k) the apartment A(T) is one-dimensional and it is spanned by the vectors x_{0} + λ̄_{1} = x_{0} − λ̄_{2}. See Figure 1.

PSfrag replacements

H_{α+0} x_{0}

H_{α−1} λ̄_{1}

Figure 1. The standard apartment for GL_{2}(k).

For GL_{3}(k) the apartment A(T) is two dimensional, and it is spanned by the vectors x_{0} + λ̄_{1}, x_{0} + λ̄_{2}, and x_{0} + λ̄_{3} = x_{0} − (λ̄_{1} + λ̄_{2}). See, for example Figure 2.

The maximal facets in this simplicial decomposition of A(T) are called alcoves or chambers of A(T). For GL_{2}(k), the alcoves are open line segments. For GL_{3}(k), an alcove is the interior of an equilateral triangle. For GL_{4}(k), an alcove is the interior of a regular tetrahedron.

3.3. Some subgroups of G. Note that for all α ∈ Φ, the root group

U_{α} := 1 + k · X_{α}

is naturally isomorphic to k as an additive group. From (1), k, regarded as an additive group, has a natural filtration indexed by ℤ ∪ ∞. Namely,

ϖ^{∞} · R := {0} ⊂ ⋯ ⊂ ϖ^{2} · R ⊂ ϖ · R ⊂ R = ϖ^{0} · R ⊂ ϖ^{−1} · R ⊂ ⋯ ⊂ k.

Similarly, U_{α} has a natural filtration indexed by ℤ ∪ ∞ ≅ {ψ ∈ Ψ | ψ̇ = α} ∪ ∞. The problem is: how do we decide which ψ ∈ {ψ ∈ Ψ | ψ̇ = α} corresponds to which PSfrag replacements

λ̄_{1}

λ̄_{2}
λ̄_{3}
x_{0}

H_{(α+β)+0} H_{1−(α+β)}

H_{β+0} H_{α+0}

Figure 2. The standard apartment for GL_{3}(k)

subgroup of U_{α}? The solution to our problem lies with our choice of a Chevalley basis. The choice of this basis determines the subgroup K_{0} = GL_{n}(R). We define

U_{α+0} := U_{α} ∩ K_{0}.

This, along with the requirement U_{α+1} ⊂ U_{α+0} , completely determines a natural indexing of subgroups in U_{α} by the set {ψ ∈ Ψ | ψ̇ = α}.

For example, for GL_{3}(k), we have

U_{α_{12}+j} = ( \begin{matrix}1 & ϖ^{j}·R & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Similarly, we have a filtration of 𝔤_{α}, the α-eigenspace in 𝔤, by setting

𝔤_{α+0} := M_{n}(R) ∩ 𝔤_{α}.

Note that for all ψ ∈ Ψ we have an isomorphism of the corresponding additive subgroups 𝔤_{ψ} and U_{ψ} via the map X ↦ (1 + X).

The maximal compact open subgroup

T_{0} = {(t_{1}, t_{2}, …, t_{n}) ∈ T | t_{i} ∈ R^{×}}

of T also has a filtration indexed by the integers. Because of future considerations, we define, for r ∈ ℝ_{≥0},

T_{r} := {t ∈ T_{0} | ν(1 − χ(t)) ≥ r for all χ ∈ X^{∗}(T)}.

We also define

T_{r^{+}} := {t ∈ T_{0} | ν(1 − χ(t)) > r for all χ ∈ X^{∗}(T)}.

It is a matter of definition to see that T_{r} = T_{⌈r⌉}, and, for i ∈ ℤ_{≥0}, if i < r ≤ (i + 1), then

T_{r} = {(t_{1}, t_{2}, …, t_{n}) ∈ T_{0} | t_{j} ∈ 1 + ϖ^{(i+1)} · R for 1 ≤ j ≤ n}.

We can define filtrations of 𝔱 = Lie(T) in a similar fashion. For example, for GL_{3}(k), we have

𝔱_{5.2} = ( \begin{matrix}ϖ^{6}·R & 0 & 0 \\ 0 & ϖ^{6}·R & 0 \\ 0 & 0 & ϖ^{6}·R \end{matrix} ) .

3.4. The Moy-Prasad filtrations. Suppose r ∈ ℝ and x ∈ A(T). In [27, 28], Moy and Prasad define filtration lattices of 𝔤 according to the formulae

𝔤_{x,r} := 𝔱_{r} ⊕ ∑_{{ψ∈Ψ | ψ(x)≥r}} 𝔤_{ψ}

and

𝔤_{x,r^{+}} := 𝔱_{r^{+}} ⊕ ∑_{{ψ∈Ψ | ψ(x)>r}} 𝔤_{ψ}.

For some people, it is easier to process these definitions when they are presented in the following form.

𝔤_{x,r} := 𝔱_{r} ⊕ ∑_{α∈Φ} ∑_{\begin{matrix}m∈ℤ \\ (α+m)(x)≥r \end{matrix}} 𝔤_{α+m}

and

𝔤_{x,r^{+}} := 𝔱_{r^{+}} ⊕ ∑_{α∈Φ} ∑_{\begin{matrix}m∈ℤ \\ (α+m)(x)>r \end{matrix}} 𝔤_{α+m}.

Similarly, if r ≥ 0, then they define

G_{x,r} := ⟨T_{r}, U_{ψ}⟩_{{ψ∈Ψ | ψ(x)≥r}}

and

G_{x,r^{+}} := ⟨T_{r^{+}}, U_{ψ}⟩_{{ψ∈Ψ | ψ(x)>r}}.

These lattices (resp., subgroups) are referred to as the Moy-Prasad filtration lattices of 𝔤 (resp., subgroups of G).

It follows immediately from the definitions that if X ∈ 𝔤_{x,r} and Y ∈ 𝔤_{x,s}, then X ·Y ∈ 𝔤_{x,(r+s)}. The following exercise is highly recommended; we have previously considered these statements for the congruence and Iwahori filtrations.

Exercise 3.4.1. Suppose s ∈ ℝ and r ∈ ℝ_{≥0}.

(1) Up to a constant { \atop [𝔤_{x,s}]} = [𝔤_{x,(−s)^{+}}].

(2) If g ∈ G_{x,r}, then g𝔤_{x,s}g^{−1} = 𝔤_{x,s} and g𝔤_{x,s^{+}}g^{−1} = 𝔤_{x,s^{+}}.

(3) If 0 < r ≤ s ≤ 2r, then X ↦ (1 + X) induces an abelian group isomorphism of 𝔤_{x,r}/𝔤_{x,s} and G_{x,r}/G_{x,s}.

3.4.1. The special case r = 0. We first attempt to understand these definitions when r = 0. In this case, if x is a point in A(T), then G_{x,0} is called the parahoric subgroup attached to x and G_{x,0^{+}} is called the pro-unipotent radical of G_{x,0}. Let F be a facet in A and let x, y ∈ F. It follows from the definitions of both the simplicial structure of A and the Moy-Prasad filtrations that G_{x,0} = G_{y,0}, 𝔤_{x,0} = 𝔤_{y,0}, G_{x,0^{+}} = G_{y,0^{+}}, and 𝔤_{x,0^{+}} = 𝔤_{y,0^{+}}. Thus, it is enough to understand these filtrations on a facet by facet basis, and it is natural to label the filtrations using facets rather than points. For GL_{2}(k), Figures 3 and 4 describe the subgroups G_{F,0} and G_{F,0^{+}} for F a facet in the apartment of T.

PSfrag replacements

( \begin{matrix}R & ℘ \\ R & R \end{matrix} )
B_{0}
( \begin{matrix}R & ϖ^{−1}R \\ ϖ^{2}R & R \end{matrix} )
( \begin{matrix}R & ℘ \\ ϖ^{−1}R & R \end{matrix} )
K_{0}
( \begin{matrix}R & ϖ^{−1}R \\ ℘ & R \end{matrix} )
x_{0}

x_{0} + λ̄_{1}

Figure 3. Parahoric subgroups G_{F,0} for GL_{2}(k)

PSfrag replacements

( ℘ ℘_{R ℘} )

B_{1/2}

( \begin{matrix}℘ & ϖ^{−1}R \\ ϖ^{2}R & ℘ \end{matrix} )
( \begin{matrix}℘ & ϖ^{2}R \\ R & ℘ \end{matrix} )
K_{1}
( \begin{matrix}℘ & R \\ ϖ^{2}R & ℘ \end{matrix} )
x_{0}

x_{0} + λ̄_{1}

Figure 4. The subgroups G_{F,0^{+}} for GL_{2}(k)

Note that as we move to the right, that is, in the positive direction as defined by the spherical Weyl chamber corresponding to B, the positive root spaces “expand” and the negative root spaces “shrink”. This simple observation will play a significant role in what is to come.

It is highly recommended that the reader complete the following exercise.

Exercise 3.4.2. Make diagrams similar to those above, but for the filtration lattices 𝔤_{F,0} and 𝔤_{F,0^{+}}. Now, do the same for GL_{3}(k).

The origin x_{0} is the facet in A defined by the intersection of the hyperplanes H_{α+0} for α ∈ Φ. We see from the above examples (and definitions) that G_{x_{0},0} = K_{0} and G_{x_{0},0^{+}} = K_{1}.

For GL_{n}(k), up to conjugation, the facets are indexed by

P(n) := {(μ_{1}, μ_{2}, …, μ_{ℓ}) | μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{ℓ} ≥ 1 and ∑_{i=1}^{ℓ} μ_{i} = n},

the set of ordered partitions of n. We briefly describe how this works. We let C_{0} be the alcove in A with vertices {v_{1} = x_{0} = x_{0} + ∑_{i=1}^{n} λ̄_{i}, v_{2} = v_{1} − λ̄_{1}, v_{3} = v_{2} − λ̄_{2}, …, v_{n} = λ̄_{n}}. To μ ∈ P(n) we attach the facet F_{μ} of dimension ℓ with vertices v_{1}, v_{(μ_{1}+1)}, v_{(μ_{1}+μ_{2}+1)}, …, v_{(μ_{1}+μ_{2}+···+μ_{ℓ−1}+1)}. The parahoric subgroup G_{F_{μ},0} can be described as K_{1} · Q_{μ}(R) where Q_{μ} is the “standard” parabolic subgroup of G containing B which corresponds to the partition μ = (μ_{1}, μ_{2}, …, μ_{ℓ}). The pro-unipotent radical G_{F_{μ},0^{+}} is then the inverse image in G_{F_{μ},0} of the unipotent radical of the image of G_{F_{μ},0} in GL_{n}(𝔣) ≅ K_{0}/K_{1}. Note that if μ, ν ∈ P(n) with μ < ν in the usual partial ordering of ordered partitions, then Q_{μ} ⊃ Q_{ν} and similarly for the associated parahoric subgroups.

More generally, note that if F_{1} and F_{2} are facets in A(T) with F_{1} ⊂ \overline{F_{2}} where \overline{F_{2}} denotes the closure of F_{2}, then it follows that

G_{F_{1},0^{+}} ⊂ G_{F_{2},0^{+}} ⊂ G_{F_{2},0} ⊂ G_{F_{1},0}.

Moreover, as the examples and exercises above show, G_{F_{2},0}/G_{F_{1},0^{+}} is a parabolic subgroup of the connected reductive group G_{F_{1},0}/G_{F_{1},0^{+}}. The unipotent radical of G_{F_{2},0}/G_{F_{1},0^{+}} is G_{F_{2},0^{+}}/G_{F_{1},0^{+}} and G_{F_{2},0}/G_{F_{1},0^{+}} has Levi component isomorphic to G_{F_{2},0}/G_{F_{2},0^{+}}.

As a tangent, we also observe that if W = N_{G}(T)/T ≅ (N_{G}(T) ∩ K_{0})/T_{0} denotes the Weyl group, then there exist |W | alcoves in A(T) which contain x_{0} in their closure. This is because these alcoves correspond to the Borel subgroups in GL_{n}(𝔣) = K_{0}/K_{1} = G_{x_{0},0}/G_{x_{0},0^{+}} which contain “T (𝔣)”.

3.4.2. The Moy-Prasad filtrations for arbitrary r. For arbitrary r, there does not exist such a nice description of what is happening. We first consider GL_{2}(k). In Figure 5 we PSfrag replacements

r

r = (1 − α)(x) r = (2 − α)(x) r = (α + 0)(x)

1 0

( \begin{matrix}R & ℘ \\ R & R \end{matrix} )
( \begin{matrix}R & R \\ R & R \end{matrix} )
( \begin{matrix}R & R \\ ℘ & R \end{matrix} )
( R ^{℘^{−1}}_{℘ R} )
( \begin{matrix}R & ^{℘^{−1}} \\ ^{℘^{2}} & R \end{matrix} )
( ℘ ℘_{R ℘} )
( \begin{matrix}℘ & ℘ \\ ℘ & ℘ \end{matrix} )
( ℘ ℘^{℘ R} )
( ^{℘^{2}} ℘^{℘ R} )
( \begin{matrix}℘ & ^{℘^{−1}} \\ ^{℘^{2}} & ℘ \end{matrix} )
( \begin{matrix}℘ & ^{℘^{2}} \\ ℘ & ℘ \end{matrix} )
( \begin{matrix}℘ & ℘ \\ ^{℘^{2}} & ℘ \end{matrix} )
( ^{℘^{3}} ℘^{℘ R} )

x_{0} x_{0} + λ̄_{1}

Figure 5. (𝔤𝔩_{2})_{x,r}

have identified the apartment of T with the horizontal axis. The vertical axis measures r. Given a pair (x, r) ∈ A(T) × ℝ, we wish to describe the lattice 𝔤_{x,r}. Note that the plane has been divided into open convex polygons. The diagonal dotted lines are the graphs of the equations r = ψ(x) where ψ ∈ Ψ. These measure where the root subgroup filtrations “change”. The horizontal dotted lines are the graphs of the equations r = n where n ∈ ℤ. These measure where the toral subgroup filtrations “change”. Each of the convex polygons is labeled by a lattice: if (x, r) belongs to a convex polygon, then 𝔤_{x,r} is the lattice so identified. For purposes of assigning lattices to every point in the figure, the convex polygons are “closed at the top”. We note that 𝔤_{x,r} = 𝔤_{x,r^{+}} unless the point (x, r) lies on a dotted line. Finally, note that if we fix r and move to the right, then the positive root space “expands” while the negative root space “shrinks”.

Fortunately, even though life is not so nice for arbitrary r, there exists a wonderful result of Moy and Prasad^{3}. Define

O := {x ∈ A(T) | x is the barycenter of a facet}.

An element of O is called an optimal point^{4}.

Lemma 3.4.3 (Moy and Prasad). Suppose z ∈ A(T) and r ∈ ℝ. There exist points x, y ∈ O such that

𝔤_{x,r} ⊂ 𝔤_{z,r} ⊂ 𝔤_{y,r}

and there exist points x′, y′ ∈ O such that

𝔤_{x′,r^{+}} ⊂ 𝔤_{z,r^{+}} ⊂ 𝔤_{y′,r^{+}}.

A similar pair of statements can be made for the Moy-Prasad filtration subgroups.

For example, for GL_{3}(k), the optimal points, up to the action of G (see below), are the points λ̄_{1} + λ̄_{2} + λ̄_{3}, \frac{λ̄_{3}}{2} and \frac{3λ̄_{3}+2λ̄_{2}+λ̄_{1}}{3} .

Exercise 3.4.4. Check that, up to conjugacy, the points listed above are the optimal points for GL_{3}(k). Describe their associated filtration lattices 𝔤_{x,r} and 𝔤_{x,r^{+}}.

4. G-domains

Recall that the ultimate goal of these notes is to relate certain invariant distributions on G with certain invariant distributions on 𝔤. Since the distributions in question are invariant, we will need sets in G and 𝔤 which are invariant. Such a set can never be compact, so the most we can hope for is to have sets which are invariant, open, and closed. A set with these properties is called a G-domain.

As we show in this section, it is possible to use the Moy-Prasad filtrations to define G-domains which have very nice properties. In particular, we will define a family of Gdomains which form a “neighborhood basis” of N, the set of nilpotent elements.

By using the Cartan decomposition of G (see [33]), in [18, Lemma 2.4] Howe demonstrates that, for all integers i, the GL_{n}(k)-orbit of 𝔨_{i} is contained in 𝔨_{i} + N. From, for example, [13, Lemma 12.2], it is also true that for any compact set ω ⊂ 𝔤, there exists a lattice L ⊂ 𝔤 such that ^{G}ω ⊂ L + N. We analyze these statements from the perspective of Moy-Prasad filtration lattices. After doing this, we look at the situation on the group.

3You will not find the result stated exactly as follows, but it is easy to derive this formulation.

4Beware, O plays many roles in these notes. However, there is no possibility of confusion.

4.1. Some comments on the Bruhat-Tits building of G. Before we can continue, we must recall some facts about B, the reduced Bruhat-Tits building of G. To each maximal split torus T′ in G we can attach an apartment A(T′) exactly as we did above. The building of G can then be thought of as the “gluing” together of all these various apartments. In Figure 6 we present a picture of B for GL_{2}(k). An apartment in B is the image of any

Figure 6. A picture of the building of GL_{2}(k)

continuous injective map from ℝ which maps integers to vertices.

Just as any two maximal split tori in G are conjugate, there is a natural action of G on B(G) with respect to which any apartment can be carried into any other. Moreover, a version of the two body problem can be solved: given any two points in the building, there is an apartment which contains both of them. Combining these two facts we have: for any two points x, y in B, there exists a g ∈ G such that gx, gy ∈ A(T). (Here gx denotes the image of x under the action of g on B(G).) Finally, the action of G on B is semisimple; that is, for h ∈ G, either there is a point in B which h fixes, or there is a line (i.e., a one-dimensional subspace of some apartment) on which h acts by nontrivial translation.

Suppose x ∈ B and r ∈ ℝ. We define the Moy-Prasad filtration lattices 𝔤_{x,r} and 𝔤_{x,r^{+}} as follows. Choose g ∈ G so that gx ∈ A(T).

𝔤_{x,r} := g^{−1} · 𝔤_{gx,r} · g and 𝔤_{x,r^{+}} := g^{−1} · 𝔤_{gx,r^{+}} · g.

One can check that these definitions make sense. For r ≥ 0 we define the subgroups G_{x,r} and G_{x,r^{+}} in a similar fashion.

Finally, we note that B is endowed with a nontrivial invariant metric, denoted dist. Moreover, with respect to dist, B has nonpositive sectional curvature.

4.2. G-domains for the Lie algebra. We begin with a generalization of the result of Howe discussed above; the proof of this result (from [1]) nicely illustrates the benefit of working with the Moy-Prasad filtration lattices.

Lemma 4.2.1. Let x, y ∈ B, and let r ∈ ℝ. Then 𝔤_{x,r} ⊂ 𝔤_{y,r} + N.

Proof. Since 𝔤_{x,r} ⊂ 𝔤_{y,r} + N if and only if 𝔤_{gx,r} ⊂ 𝔤_{gy,r} + N for g ∈ G, we may assume that x and y are elements of A(T).

Choose v⃗ ∈ X_{∗}(T) ⊗ ℝ such that x = y + v⃗ (working modulo X_{∗}(Z(G)) ⊗ ℝ). Let B′ be a Borel subgroup determined by v⃗. That is, B′ has a Levi decomposition B′ = T N′ such that for all roots α ∈ Φ_{N′}^{+}, the set of roots which are positive with respect to N′, we have ⟨α, v⃗⟩ ≥ 0.

Let B̄′ = T N̄′ denote the parabolic opposite B = T N′ and let 𝔤 = 𝔫̄′ + 𝔱 + 𝔫′ denote the associated Lie algebras. We have

𝔤_{x,r} = 𝔱_{r} ⊕ ∑_{{ψ∈Ψ | ψ(x)≥r}} 𝔤_{ψ}
= 𝔱_{r} ⊕ ∑_{{ψ∈Ψ | ψ̇∈Φ∖Φ_{N′}^{+} and ψ(x)≥r}} 𝔤_{ψ} ⊕ ∑_{{ψ∈Ψ | ψ̇∈Φ_{N′}^{+} and ψ(x)≥r}} 𝔤_{ψ}
= 𝔱_{r} ⊕ (𝔤_{x,r} ∩ 𝔫̄′) ⊕ (𝔤_{x,r} ∩ 𝔫′)
⊂ 𝔤_{y,r} + 𝔫′ ⊂ 𝔤_{y,r} + N . □

In fact, a kind of converse to the above lemma is true. Namely, if X ∈ 𝔤 belongs to 𝔤_{x,r} + N for all x ∈ B, then there is a point y ∈ B such that X ∈ 𝔤_{y,r}. This result along with the above lemma gives us the following theorem.

Theorem 4.2.2.

⋃_{x∈B} 𝔤_{x,r} = ⋂_{x∈B}(𝔤_{x,r} + N).

In order to simplify our notation, we define

𝔤_{r} := ⋃_{x∈B} 𝔤_{x,r}.

From its definition, 𝔤_{r} is open and invariant. From the above theorem we have that 𝔤_{r} is closed. Consequently, 𝔤_{r} is a G-domain.

We now consider some examples. The set 𝔤_{0} is usually referred to as the set of compact or integral elements of 𝔤. The set 𝔤_{0^{+}} is usually called the set of topologically nilpotent elements in 𝔤; it consists of those X ∈ 𝔤 such that powers of X tend to zero in the p-adic topology. These G-domains have very natural interpretations; namely,

𝔤_{r} = {X ∈ 𝔤 | ν(e) ≥ r for all eigenvalues e of X }.

(If E is a finite extension of k, then there exists a unique extension of ν to E.) For GL_{n}(k) and X ∈ 𝔤, the value of ν(e_{X}) must lie in the set {k/n | k ∈ ℤ}.

Of course, just as we have Moy-Prasad lattices of the form 𝔤_{x,r^{+}}, we also can define 𝔤_{r^{+}} := ⋃_{x∈B} 𝔤_{x,r^{+}}. These G-domains satisfy the obvious analogues of the results discussed above. From the previous paragraph, we have that 𝔤_{r} ≠ 𝔤_{r^{+}} implies that r = k/n for some k ∈ ℤ.

Exercise 4.2.3. Define the subspace D_{r} of C_{c}^{∞}(𝔤) by

D_{r} := ∑_{x∈B} C(𝔤/𝔤_{x,r})

where the sum is interpreted as follows. A function f belongs to D_{r} if and only if f can be written as a finite sum f = ∑_{i} f_{i} with f_{i} ∈ C_{c}(𝔤/𝔤_{x_{i},r}) for some x_{i} ∈ B. We can define D_{r^{+}} in a similar way. Show that the Fourier transform gives us bijective maps from D_{r^{+}} to C_{c}^{∞}(𝔤_{−r}) and from D_{r} to C_{c}^{∞}(𝔤_{(−r)^{+}}).

4.3. G-domains in G. We now turn our attention to the group-side of things. Unlike the Lie algebra, an element of the group can only belong to a Moy-Prasad filtration subgroup if it first belongs to a parahoric. It turns out that the interesting part of the story here rests in proving the equality

⋃_{x∈B} G_{x,0} = ⋂_{x∈B} G_{x,0} · U

where U denotes the set of unipotent elements in G. Once this result is known to be true, it is very easy to establish the equality

G_{r} = ⋂_{x∈B} G_{x,r} · U

where r ≥ 0 and G_{r} = ⋃_{x∈B} G_{x,r}. As above, it follows immediately that G_{r} is a Gdomain.

We show how to prove the equality when r = 0.

Lemma 4.3.1.

G_{0} = ⋂_{x∈B} G_{x} · U.

Proof. We first show that the right-hand side is a subset of the left-hand side. We will argue by contradiction. Suppose that g ∈ ⋂_{x∈B} G_{x,0} · U does not belong to G_{0}.

Since the action of G on B is semisimple, we either have that there is a point x in B which g fixes or a line ℓ in an apartment A′ of B on which g acts by nontrivial translation.

In the first case, we can write g = h · u with h ∈ G_{x,0} and u ∈ U. Since u is unipotent, it must live in G_{y,0} for some y ∈ B. But, from a result of Eugene Kushnirsky this implies that u ∈ G_{x,0} [10, Lemma 4.5.1].

In the latter case, there exists a facet F′ in A′ such that F′ ∩ ℓ is open in ℓ. For all x, y ∈ F′ we have G_{y,0} = G_{x,0}. By hypothesis, there exist elements h ∈ G_{x,0} and u ∈ U such that g = uh. Since u is unipotent, there exists w ∈ B which is fixed by u. We have that for all y ∈ F′ ∩ ℓ

dist(w, y) = dist(w, uy) = dist(w, uhy) = dist(w, gy).

Thus, we have , for x, y ∈ F′ ∩ ℓ a picture something like that described in Figure 7.

However, since B has nonpositive sectional curvature, the line segment from x to w must be shorter than the line segments from y and gy to w. Similarly, the segments from PSfrag replacements

x
y
gx
gy
ℓ
w

Figure 7

gy to w must be shorter than the segments from x and gx to w. Consequently, we have

dist(x, w) < dist(gy, w) < dist(x, w),

a contradiction.

We now show that the left-hand side is a subset of the right-hand side.

We need to show that for x, y ∈ B, we have G_{x} ⊂ U · G_{y}. As in the proof for the Lie algebra, we may assume that x and y both belong to A(T). Let B′ = T N′ be a Borel subgroup of G so that the (spherical) chamber in A(T) determined by N′ is invariant under translation by the vector (y − x).

Let N̄′ be the unipotent radical of the parabolic opposite B = T N′. From [3] we can write G = N′ · N̄′ · N′ · T. With some work, it follows that we can write

G_{x,0} = N′_{x} · N̄′_{x} · N′_{x} · T_{0}

where N′_{x} = N′ ∩G_{x,0} and N̄′_{x} = N̄′ ∩G_{x,0}. Because of the way in which N′ was chosen, we have N′_{x} ⊂ G_{y,0}. Thus, if g ∈ G_{x}, then there exist n_{1}, n_{2} ∈ N_{x}, \overline{n} ∈ \overline{N}_{x}, and t ∈ T_{0} such that

g = n_{1} · \overline{n} · n_{2} · t
= (n_{1} · \overline{n} · n_{1}^{−1}) · (n_{1} · n_{2} · t)
∈ U · G_{y}. □

4.4. A neighborhood basis for the nilpotent cone. We remark that

N = ⋂_{r} 𝔤_{r}

and

U = ⋂_{r} G_{r}.

It follows that the G-domains we have defined provide us with a neighborhood basis of the nilpotent cone (resp., unipotent variety) consisting of open, invariant, closed neighborhoods.

5. Nilpotent orbital integrals as distributions

A distribution on 𝔤 is any element of the linear dual of C_{c}^{∞}(𝔤). The aim of this section is to convince ourselves that integrating against a nilpotent orbit defines a distribution on 𝔤. We follow the argument of Ranga-Rao [29].

5.1. Orbital integrals. Fix an element Y ∈ 𝔤. Let O_{Y} = ^{G}Y := {gY g^{−1} | g ∈ G} denote the G-orbit of Y. We identify the orbit of Y with the homogeneous space G/C_{G}(Y). Thus, the tangent space to the orbit of Y at the point Y is identified with 𝔤/C_{𝔤}(Y).

We define an alternating bilinear form ⟨ , ⟩_{Y} on 𝔤 by ⟨A, B⟩ = tr(Y · [A, B]) for A, B ∈ 𝔤. Fix an element B ∈ 𝔤. A calculation shows that ⟨A, B⟩ = 0 for all A ∈ 𝔤 if and only if B ∈ C_{𝔤}(Y). Since we have a similar statement when we switch the roles of A and B, it follows that ⟨ , ⟩_{Y} induces a nondegenerate alternating form on 𝔤/C_{𝔤}(Y). Thus, the dimension of the orbit is even, say 2m.

Similarly, for each^{g}Y = gY g^{−1} ∈ ^{G}Y, we have a nondegenerate alternating symmetric form on Tan_{^{g}Y} (O_{Y}). Consequently, there exists a nondegenerate, invariant two-form ω for O_{Y} and from this we can form a nonzero, invariant volume form ω ∧ ω ∧ ⋯ ∧ ω (m times) on O_{Y}. Let X_{1}, X_{2}, …, X_{2m} be coordinates for 𝔤/C_{𝔤}(Y). For each 1 ≤ i ≤ 2m, fix a one-form dX_{i} and associated measure |dX_{i}| normalized so that for all f ∈ C_{c}^{∞}(𝔤/C_{𝔤}(Y)), we have ˆ f̂(X) = f(−X), where the Fourier transform is taken with respect to the measure ∏_{i=1}^{2m} |dX_{i}|. There exists a locally convergent power series f(X_{1}, X_{2}, …, X_{2m}) so that (locally)

ω = f(X_{1}, X_{2}, …, X_{2m}) dX_{1} ∧ dX_{2} ∧ ⋯ ∧ dX_{2m}.

Define

|ω| := |f(X_{1}, X_{2}, …, X_{2m})| ∏_{i=1}^{2m} |dX_{i}| .

Thus, we have an invariant measure on O_{Y}. (Since GL_{n}(k) is unimodular, it follows from standard measure theory results that C_{G}(Y) is unimodular as well.)

Exercise 5.1.1. Check that this definition is invariant of the various choices we have made.

5.2. A framing of the problem. If Y is a semisimple element of 𝔤, then O_{Y} is closed in 𝔤. Consequently, if f ∈ C_{c}^{∞}(𝔤), then the restriction of f to O_{Y} is an element of C_{c}^{∞}(O_{Y}). So it makes sense to define the distribution O_{Y} : C_{c}^{∞}(𝔤) → ℂ by setting

O_{Y} (f) := ∫_{G/C_{G}(Y)} f(^{g}Y) dg^{∗}.

for f ∈ C_{c}^{∞}(𝔤). Here dg^{∗} denotes the invariant measure on G/C_{G}(Y) defined above.

What if Y is not semisimple? In particular, what if Y is nilpotent? In this case, we need to do some work to show that for arbitrary f ∈ C_{c}^{∞}(𝔤) the integral

∫_{G/C_{G}(Y)} f(^{g}Y) dg^{∗}

makes sense.

5.3. The basic notation associated to nilpotent orbital integrals. We begin by recalling the familiar parameterization of nilpotent orbits in 𝔤. We then discuss various properties of nilpotent orbits which will be important in the sequel.

From basic linear algebra we know that the nilpotent orbits in 𝔤 can be parameterized by P(n), the set of ordered partitions of n. To μ ∈ P(n) we associate the nilpotent element X_{μ} ∈ N ∩ 𝔤_{x_{0},0} having Jordan canonical form corresponding to μ. That is, if μ = (μ_{1}, μ_{2}, …, μ_{k}), then in the i^{th} block of size μ_{i} × μ_{i} we put the matrix with ones on the superdiagonal and zeroes elsewhere. For example, for μ = (2, 2, 1) ∈ P(5) we have

X_{μ} = (\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) .

Note that the nilpotent orbits in 𝔤_{x_{0},0}/𝔤_{x_{0},0^{+}} ≅ M_{n}(𝔣) are also indexed by P(n) and the map taking X_{μ} to the image of X_{μ} in M_{n}(𝔣) gives a bijective correspondence between O(0), the set of nilpotent orbits in 𝔤, and the set of nilpotent orbits in M_{n}(𝔣).

We let O_{μ} denote the G-orbit of X_{μ}. Note that if μ < μ′ in the usual partial order on P(n), then O_{μ} ⊂ Ō_{μ′}.

Exercise 5.3.1. Show that if O is a nilpotent orbit such that O ∩ (X_{μ} + 𝔤_{x_{0},0^{+}}) ≠ ∅, then O_{μ} is contained in the p-adic closure of O. Moreover, O_{μ} ∩ (X_{μ} + 𝔤_{x_{0},r^{+}}) = ^{G_{x_{0},r^{+}}} X_{μ} for all non-negative r.

Note that if X ∈ N, and t ∈ k^{×}, then tX ∈ N and in fact, tX and X are GL_{n}(k) conjugate. (In general, it follows from Jacobson-Morosov that t^{2}X and X are conjugate, but in GL_{n} we can do better.) So, N and each nilpotent orbit are closed with respect to scaling.

We can associate a parabolic subgroup P (μ) of G to μ as follows: For all positive integers j, the matrix X_{μ}^{j} acts on V = k^{n}, and we define V_{j} = ker(X_{μ}^{j}) ⊂ V. Define P (μ) = {g ∈ G | g · V_{j} ⊂ V_{j} for all positive j }. Then X_{μ} lies in the nilradical 𝔫(μ) of the Lie algebra of P (μ), and the P (μ)-orbit of X_{μ} is dense in 𝔫(μ) and equal to O_{μ} ∩ 𝔫(μ).

We need most of the basic facts about 𝔰𝔩_{2}(k)-triples. For a good reference, see [4]. We can complete X_{μ} to an 𝔰𝔩_{2}(k)-triple (Y_{μ}, H_{μ}, X_{μ}). Explicitly, the i^{th} block of H_{μ} is given by the element diag((μ_{i} − 1), (μ_{i} − 3), …, (1 − μ_{i})) and the i^{th} block of Y_{μ} is given by certain entries on the super-subdiagonal. We let λ_{μ} ∈ X_{∗}(G) denote the associated oneparameter subgroup, that is, the i^{th} block of λ_{μ}(t) looks like diag(t^{(μ_{i}−1)}, t^{(μ_{i}−3)}, …, t^{(1−μ_{i})}).

For i ∈ ℤ, we define

𝔤(i) = {X ∈ 𝔤 |^{λ_{μ}(t)}X = t^{i}X}.

We have 𝔤 = ∑_{i} 𝔤(i). For j ∈ ℤ, define 𝔤(≥ j) := ∑_{i≥j} 𝔤(i). We let 𝔭_{μ} denote the parabolic subalgebra 𝔤(≥0) with Levi subalgebra 𝔪_{μ} = 𝔤(0) and nilradical 𝔫_{μ} (= 𝔤(≥1)). We let P_{μ} denote the corresponding parabolic subgroup with Levi decomposition M_{μ}N_{μ}. The P_{μ}-orbit of X_{μ} is equal to^{M_{μ}}X_{μ} + 𝔤(≥3) and^{M_{μ}}X_{μ} is an open and dense subset of 𝔤(2). Finally, it is true that C_{G}(X_{μ}) ⊂ P_{μ}.

5.4. A sketch of the proof that nilpotent orbital integrals define distributions. Fix μ ∈ P(n). Let f ∈ C_{c}^{∞}(𝔤).

The Iwasawa decomposition (see [33]) tells us that we can write G = G_{x_{0},0}P_{μ}. Fix a Haar measure dk on K and a left Haar measure d_{ℓ}p on P_{μ} so that

∫_{G} h(g) dg = ∫_{P_{μ}} d_{ℓ}p ∫_{G_{x_{0},0}} h(kp) dk

for all h ∈ C_{c}^{∞}(G), the space of complex-valued, compactly supported, locally constant functions on G. Because it will be important, we recall that for all p_{0} = m_{0}n_{0} ∈ P_{μ} = M_{μ}N_{μ} we have d_{ℓ}(p_{0}pp_{0}^{−1}) = |det (m_{0}|_{𝔫_{μ}})|^{−1} d_{ℓ}p.

Define f̄ ∈ C_{c}^{∞}(𝔤) by f̄(X) := ∫_{G_{x_{0},0}} f(^{k}X) dk. Let us assume for the moment that

(*)  ∫_{P/C_{G}(X_{μ})} f̄(^{p}X) d_{ℓ}p^{∗}

converges. (Here d_{ℓ}p^{∗} is the quotient measure.) In this case, we have

(∗) = ∫_{P/C_{G}(X_{μ})} ∫_{K} f(^{kp}X) dk d_{ℓ}p^{∗} = ∫_{G/C_{G}(X_{μ})} f(^{g}X_{μ}) dg^{∗},

and so we can define the invariant distribution O_{μ} by

O_{μ}(f) = ∫_{G/C_{G}(X_{μ})} f(^{g}X_{μ}) dg^{∗}.

for f ∈ C_{c}^{∞}(𝔤).

We now sketch the proof of why (*) makes sense. Let dZ be a Haar measure on 𝔤(≥2). We’d like to define a function F_{0} on 𝔤(≥2) so that

(∗) = ∫_{^{Mμ} X_{μ}+𝔤(≥3)} f̄(Z) |F_{0}(Z)| dZ.

For p_{0} = m_{0}n_{0} as above, we have d(p_{0}Zp_{0}^{−1}) = |det (m_{0}|_{𝔤(≥2)})|^{−1} dZ. Consequently, it is sufficient to produce a nonzero function F_{0} with the property F_{0}(^{p_{0}}Z) = det (p_{0}|_{𝔤(1)}).

The map X ↦ (Y ↦ tr(Y · X)) induces an isomorphism of 𝔤(1) with 𝔤(−1)^{∗}. Just as at the beginning of this section, for Z ∈ 𝔤(2), we can define an alternating bilinear form ⟨ , ⟩_{Z} on 𝔤(−1). When Z = X_{μ}, the form is also nondegenerate. Consequently, both 𝔤(1) and 𝔤(−1) are even dimensional (of dimension 2n′). Let a_{Z} be the matrix which represents ⟨ , ⟩_{Z} with respect to some fixed basis of 𝔤(−1). From [22, Theorem 6.4] there is a polynomial (Pfaffian) Pf of degree n′ on 𝔤(2) so that det(a_{Z}) = (Pf(Z))^{2}. Moreover, since det(a_{(^{m}Z)}) = det (m^{−1}|_{𝔤(−1)})^{2} · det(a_{Z}), the polynomial Pf transforms in the manner we desire. For Z ∈ 𝔤(≥2), set F_{0}(Z) = Pf(Z_{2}) where Z_{2} denotes the image of Z under the projection map from 𝔤(≥2) to 𝔤(2).

Example 5.4.1. For the partition (3, 2) ∈ P(5), we have that the dimension of 𝔤(−1) is four. The space 𝔤(2) is spanned by the Chevalley basis elements X_{α_{12}}, X_{α_{23}} and X_{α_{45}}. If Z = xX_{α_{12}} + yX_{α_{23}} + zX_{α_{45}}, then F_{0}(Z) = xy.

5.5. An important calculation. For j > 0 we calculate O_{μ}([X_{μ} + 𝔨_{j}]).

Proposition 5.5.1.

O_{μ}([X_{μ} + 𝔨_{j}]) = q^{((1−2j)·dim(O_{μ})/2)}

Proof. Suppose that m > j is very large. Let O_{μ}^{m} := ^{K_{m}}X_{μ} = {^{k}X_{μ} | k ∈ K_{m}}. It follows from Exercise 5.3.1 that

O_{μ}([X_{μ} + 𝔨_{j}]) = [K_{j} : K_{m} · C_{K_{j}}(X_{μ})] · O_{μ}([O_{μ}^{m}]).

Now, by definition, we have

O_{μ}([O_{μ}^{m}]) = meas_{dg^{∗}}(O_{μ}^{m})

= meas_{|dX_{1}|·|dX_{2}|···|dX_{2m}|}((𝔨_{m} + C_{𝔤}(X_{μ}))/C_{𝔤}(X_{μ}))

= [𝔨_{1−m} + C_{𝔤}(X_{μ}) : 𝔨_{m} + C_{𝔤}(X_{μ})]^{−1/2}

= \frac{[C_{𝔨_{(1−m)}}(X_{μ}) : C_{𝔨_{m}}(X_{μ})]^{1/2}}{[𝔨_{(1−m)} : 𝔨_{m}]^{1/2}} .

On the other hand,

[K_{j} : K_{m} · C_{K_{j}}(X_{μ})] = \frac{[K_{j} : K_{m}]}{[C_{K_{j}}(X_{μ}) : C_{K_{m}}(X_{μ})]}.

After putting the two pieces together and doing a bit of calculation, we arrive at

O_{μ}([X_{μ} + 𝔨_{j}]) = \frac{[C_{𝔨_{0}}(X_{μ}) : C_{𝔨_{j}}(X_{μ})]}{[𝔨_{0} : 𝔨_{j}]} · \frac{[𝔨_{0} : 𝔨_{1}]^{1/2}}{[C_{𝔨_{0}}(X_{μ}) : C_{𝔨_{1}}(X_{μ})]^{1/2}} .

The proposition follows immediately. □

5.6. Nilpotent orbital integrals are homogeneous. Suppose t ∈ k^{×}. For f ∈ C_{c}^{∞}(𝔤) we define the dilation f_{t} ∈ C_{c}^{∞}(𝔤) of f by t via f_{t}(X) = f(tX). For μ ∈ P(n) we have

O_{μ}(f_{t}) = ∫_{G/C_{G}(X_{μ})} f(^{g}(tX_{μ})) dg^{∗}.

Consequently, since nilpotent orbits are closed with respect to scaling and the invariant measure on a nilpotent orbit is unique up to a constant, O_{μ}(f_{t}) and O_{μ}(f) must differ by a constant. A calculation shows that

O_{μ}(f_{t}) = |t|^{\frac{dim(Oμ)}{2}} O_{μ}(f).

For example, from Proposition 5.5.1 we have

(2)  O_{μ}([ϖ^{(−2j+1)}X_{μ} + 𝔨_{1−j}]) = 1.

6. The Fourier transforms of invariant distributions

6.1. Basics. A distribution on 𝔤 is any element of the linear dual of C_{c}^{∞}(𝔤) (no topological restrictions). We denote the subspace of invariant distributions on 𝔤 by J(𝔤)^{5}.

Suppose T ∈ J(𝔤). We define the Fourier transform T̂ ∈ J(𝔤) of T by

T̂(f) := T (f̂)

for f ∈ C_{c}^{∞}(𝔤).

It is a remarkable fact that T̂ is represented by a locally integrable function; that is, there is a function T̂ ∈ L_{loc}^{1}(𝔤) such that for all f ∈ C_{c}^{∞}(𝔤),

T̂(f) = ∫_{𝔤} T̂(X) · f(X) dX.

Unfortunately, describing this function is beyond our abilities in all but the simplest situations.

Example 6.1.1. Consider the trivial nilpotent orbital integral O_{(1,1,...,1)} ∈ J(𝔤). For f ∈ C_{c}^{∞}(𝔤) we have

Ô_{(1,1,...,1)}(f) = f̂(0) = ∫_{𝔤} f(X) · Λ(tr(0 · X)) dX = ∫_{𝔤} f(X) dX

and we also have

Ô_{(1,1,...,1)}(f) = ∫_{𝔤} f(X) · Ô_{(1,1,...,1)}(X) dX.

We therefore conclude that Ô_{(1,1,...,1)} = 1.

We let 𝔤^{r.s.s.} denote the set of regular semisimple elements in 𝔤; this is a dense open subset of 𝔤. In our situation, 𝔤^{r.s.s.} consists of those elements of 𝔤 having distinct eigenvalues.

The main idea of this section is to present a very elegant description of T̂(X) when T ∈ J(𝔤) and X ∈ 𝔤^{r.s.s.}. (From this description, it will follow that T̂ is represented by a locally constant function on 𝔤^{r.s.s.}.) The existence (of a form) of this expression was conjectured by Paul Sally, Jr. and proved by Reid Huntsinger [21]. The proof follows an argument of Harish-Chandra [13].

Originally, Huntsinger and Sally were only interested in studying the behavior of the Fourier transform of a nilpotent orbital integral. We shall temporarily restrict our attention to this situation. Suppose μ ∈ P(n). For f ∈ C_{c}^{∞}(𝔤) we have

5According to Howe, he chose the letter J because I (for invariant) was already taken, and J follows the letter I in the alphabet.

∫_{𝔤} f(X) · Ô_{μ}(X) dX = Ô_{μ}(f) = O_{μ}(f̂)
= ∫_{G/C_{G}(X_{μ})} f̂(^{g}X_{μ}) dg^{∗}
= ∫_{G/C_{G}(X_{μ})} ∫_{𝔤} f(X) · Λ(tr(^{g}X_{μ} · X)) dX dg^{∗}
“ = ” ∫_{𝔤} f(X) ∫_{G/C_{G}(X_{μ})} Λ(tr(^{g}X_{μ} · X)) dg^{∗} dX.

So, if we could justify the equality in quotation marks, we’d have

\begin{aligned}Ô_{μ}(X) & = ∫_{G/C_{G}(X_{μ})} Λ(tr(^{g}X_{μ} · X)) dg^{∗} \\ & = O_{μ}(Y ↦ Λ(tr(Y · X))). \end{aligned}

This is nearly correct; we now describe what is true.

Theorem 6.1.2 (Reid Huntsinger). Let K be any compact open subgroup of G and let dk denote the normalized Haar measure on K. For all X ∈ 𝔤^{r.s.s.} we have

Ô_{μ}(X) = O_{μ}(Y ↦ ∫_{K} Λ(tr(Y ·^{k}X)) dk).

Remark 6.1.3. For reasons which will become clear later, we remark that the function Ô_{μ} is canonical. On the face of things, it depends on two choices: the additive character Λ and the choice of a measure on O_{μ}. However, our choice of the measure on O_{μ} was not arbitrary; it depended on Λ. As one can verify, this dependence makes Ô_{μ} canonical.

Remark 6.1.4. In our situation, since every nilpotent orbit is Richardson, there is a very nice description, due to Howe, of the function Ô_{μ}. Namely, Ô_{μ} can be related to the character of the representation obtained by inducing the trivial representation on P (μ) up to G.

6.2. A more general statement. More generally, following a suggestion of Bob Kottwtiz, Reid Huntsinger proved the following statement.

Theorem 6.2.1 (Reid Huntsinger). Fix r ∈ ℝ . If T ∈ J(𝔤_{r}), then T̂ is represented on 𝔤^{r.s.s.} by

X ↦ T (η_{X}).

Here, for Y ∈ 𝔤, η_{X}(Y) := ∫_{K}(Λ(tr(Y · ^{k}X))) dk. (As before, K is a compact open subgroup, and dk is the normalized Haar measure on K.)

At the heart of the proof of this theorem lies the statement that the map from 𝔤^{r.s.s.} to C^{∞}(𝔤) sending X to η_{X,r} := η_{X} · [𝔤_{r}] is locally constant. Since the verification of this statement requires some fairly detailed analysis, we shall skip the proof. This statement immediately implies that the function X ↦ T (η_{X,r}) = T (η_{X}) from 𝔤^{r.s.s.} to ℂ is locally constant.

Suppose f ∈ C_{c}^{∞}(𝔤^{r.s.s.}). We need to show

T̂(f) = ∫_{𝔤} f(X) · T (η_{X}) dX.

From the previous paragraph, there exists a finite collection {ω_{i}}_{i=1}^{m} of compact open disjoint subsets of 𝔤^{r.s.s.} such that both X ↦ T (η_{X}) and X ↦ f(X) are constant on ω_{i} and the support of f is contained in ⋃ ω_{i}. Using Fubini’s theorem and the invariance of T, we then have^{6}

∫_{𝔤} f(X) · T (η_{X}) dX = ∑_{i=1}^{m} meas_{dX}(ω_{i}) · f(X_{i}) · T (η_{X_{i}})

= T (∑_{i=1}^{m} meas_{dX}(ω_{i}) · f(X_{i}) · η_{X_{i}})

= T (Y ↦ ∫_{𝔤} f(X) · η_{X}(Y) dX)

= T (Y ↦ ∫_{𝔤} f(X) ∫_{K} Λ(tr(^{k}X · Y)) dk dX)

= T (Y ↦ ∫_{K} f̂(^{k}Y) dk)

= T (f̂).

7. Characters

In this section we define the character of an admissible representation and discuss some properties of characters.

7.1. Admissible representations. Fix an admissible representation (π, V) of G. Recall, from Gordan Savin’s lectures [33], that (π, V) is a representation for which

(1) for all v ∈ V there exists a compact open subgroup K such that

v ∈ V^{K} := {v ∈ V | π(k)v = v for all k ∈ K}

and

(2) for all compact open subgroups K of G we have dim_{ℂ} V^{K} < ∞. The second condition is equivalent to saying that for all compact open subgroups K of G and all irreducible representations σ of K, the multiplicity of σ in π is finite.

Example 7.1.1. Suppose σ̄ is a cuspidal representation of GL_{n}(𝔣) ≅ K_{0}/K_{1}. Inflate σ̄ to a representation of K_{0} and extend this inflation to a representation σ of Z(G) · K_{0} where Z(G) denotes the center of G. The representation (π_{σ}, V_{σ}) obtained by (compact) induction of σ from Z(G) · K_{0} up to G is an (irreducible) admissible representation.

6This part of the proof supports David Vogan’s adage: “The p-adics: if you can add, you can integrate.”

Example 7.1.2. Suppose m > 0 and χ̄ is a character of T/T_{m}. Inflate χ̄ to a character χ of T. Since the unipotent radical N of B is normal in B, we may extend χ to a character of B. The representation (π_{χ}, V_{χ}) obtained by (compact) parabolic induction of χ from B to G is an admissible representation of G.

7.2. The character distribution. We define the character distribution, Θ_{π}, associated to the representation (π, V) as follows.

For f ∈ C_{c}^{∞}(G), we define π(f) ∈ End(V) by

π(f)v = ∫_{G} f(g) · π(g)v dg

for v ∈ V. Since f is compactly supported and locally constant and (π, V) is admissible, this integral is really just a finite sum. In fact, if K is a compact open subgroup of G such that f(k_{1}gk_{2}) = f(g) for all k_{1}, k_{2} ∈ K and g ∈ G, then we have

π(f)v = π([K] ∗ f ∗ [K])v = π([K])π(f)π([K])v.

(Here ∗ denotes the usual convolution operation.) Thus, π(f) is a map from V to V^{K}. If e_{K} := meas_{dg}(K)^{−1} · π([K]), then e_{K} is the projection operator from V to V^{K}, and we have

V = V^{K} ⊕ (1 − e_{K})V.

Consequently, since dim_{ℂ}(V^{K}) < ∞, it follows that π(f) is a finite-rank operator. We can therefore define the character distribution

Θ_{π}: C_{c}^{∞}(G) → ℂ

by sending f to tr(π(f)).

Just as the Fourier transform of an invariant distribution on 𝔤 is represented on 𝔤^{r.s.s.} by a function in C^{∞}(𝔤^{r.s.s.}), so too is the character distribution [15]. We abuse notation and denote this function by Θ_{π} ∈ C^{∞}(G^{r.s.s.}). (G^{r.s.s.} denotes the open, dense subset of G consisting of regular semisimple elements, that is, those elements of G whose eigenvalues are distinct.) We call this function the character of π.

Remark 7.2.1. Fix γ ∈ G^{r.s.s.} and m > 0 large enough so that γK_{m} ⊂ G^{r.s.s.}. Define f_{m} = meas_{dg}(K_{m})^{−1} · [γK_{m}]. We then have

Θ_{π}(f_{m}) = ∫_{G} f_{m}(g) · Θ_{π}(g) dg = meas_{dg}(K_{m})^{−1} ∫_{K_{m}} Θ_{π}(γk) dk.

Since the function Θ_{π} is locally constant on G^{r.s.s.}, the right-hand side becomes Θ_{π}(γ) for all m sufficiently large. Thus, “in the limit” the function Θ_{π} looks like a character.

7.3. A calculation. Fix j ≥ 1.

Suppose that τ̄ is an irreducible representation of K_{j}/K_{2j}. Since K_{j}/K_{2j} is abelian, τ̄ is a character. Let τ denote the corresponding character of K_{j}; we regard τ as an element of C_{c}^{∞}(G) in the obvious way.

Let res_{K_{j}} π denote the restriction of π to K_{j}. Suppose that

res_{K_{j}} π = ⊕_{σ∈K̂_{j}}m(σ, π)σ

where m(σ, π) denotes the multiplicity of σ in res_{K_{j}} π. We have

Θ_{π}(τ) = tr ∫_{G} τ(g) · π(g) dg

= tr ∫_{K_{j}} τ(g) · π(g) dg

= tr ∫_{K_{j}} ∑_{σ∈K̂_{j}} τ(g) · m(σ, π)σ(g) dg

= tr ∫_{K_{j}} ∑_{σ̄∈{ \atop K_{j}/K_{2j}}} τ(g) · m(σ, π)σ(g) dg

= m(τ^{−1}, π) · meas_{dg}(K_{j}).

In other words, the character picks out the multiplicity of τ^{−1} in res_{K_{j}} π (up to a constant).

7.4. Depth. The depth of a representation was introduced in the fundamental papers of Allen Moy and Gopal Prasad [28, 27]. Essentially, the depth of a representation is a rational number which measures the first occurrence of fixed vectors with respect to all filtration subgroups of G that arise naturally from Bruhat-Tits theory.

Recall from §3.4.2 that O denotes the set of optimal points. Up to conjugation, the set of optimal points is finite, and so the subset {r ∈ ℝ | 𝔤_{x,r} ≠ 𝔤_{x,r^{+}} for some x ∈ O} is discrete (and in fact, is a subset of ℚ). For an admissible representation (π, V), we define ρ(π), the depth of (π, V), by

ρ(π) := min{r ∈ ℚ_{≥0} | there is an x ∈ O such that V^{G_{x,r^{+}}} ≠ 0}.

Proposition 7.4.1. ρ(π) is the unique rational number satisfying the following statement. If (x, r) ∈ A(T) × ℝ_{≥0} with V^{G_{x,r^{+}}} ≠ {0}, then r ≥ ρ(π).

Proof. Suppose (x, r) ∈ A(T) × ℝ_{≥0} with V^{G_{x,r^{+}}} ≠ {0}. From the group version of Lemma 3.4.3 there exists y ∈ O such that G_{x,r^{+}} ⊃ G_{y,r^{+}}. Consequently, {0} ≠ V^{G_{x,r^{+}}} ⊂ V^{G_{y,r^{+}}} . □

All the various things you would want to be true about the depth of a representation are true. For example, if σ is an irreducible representation of a parabolic subgroup of G, and π is an irreducible subquotient of the induced representation, then ρ(π) = ρ(σ).

Example 7.4.2. Any representation with Iwahori (that is, B_{0}) fixed vectors has depth zero.

Example 7.4.3. The representation (π_{σ}, V_{σ}) defined in Example 7.1.1 does not have Iwahori fixed vectors, but it does have depth zero.

Example 7.4.4. The representation (π_{χ}, V_{χ}) defined in Example 7.1.2 has depth (m − 1).

7.5. Elementary Kirillov theory. Fix an irreducible representation (π, V).

For nilpotent real groups, Kirillov [24] established that the representations were parameterized by coadjoint orbits in the linear dual of the Lie algebra. In our context, the term Kirillov theory is used to describe the connection between representations of compact open subgroups occurring in π and coadjoint orbits in the linear dual of 𝔤. Via our nondegenerate trace form, we have identified 𝔤 with its dual. The type of result discussed below was first studied, I believe, by Howe [17].

Fix r, s ∈ ℝ_{≥0} and x, y ∈ B.

The map g ↦ (g − 1) induces an isomorphism of the abelian groups G_{x,r^{+}}/G_{x,(2r)^{+}} and 𝔤_{x,r^{+}}/𝔤_{x,(2r)^{+}}. Moreover, every character of 𝔤_{x,r^{+}}/𝔤_{x,(2r)^{+}} is of the form Ȳ ↦ Λ(tr(X·Y) for some X̄ ∈ 𝔤_{x,−2r}/𝔤_{x,−r}. (That is, the Pontrjagin dual of G_{x,r^{+}}/G_{x,(2r)^{+}} ≅ 𝔤_{x,r^{+}}/𝔤_{x,(2r)^{+}} is isomorphic to 𝔤_{x,−2r}/𝔤_{x,−r}.)

Fix a character σ̄ of G_{x,r^{+}}/G_{x,(2r)^{+}} and a character τ̄ of G_{y,s^{+}}/G_{y,(2s)^{+}} Let X_{σ} + 𝔤_{x,−r} ∈ 𝔤_{x,−2r}/𝔤_{x,−r} represent σ̄ and X_{τ} + 𝔤_{y,−s} ∈ 𝔤_{y,−2s}/𝔤_{y,−s} represent τ̄.

Proposition 7.5.1. If σ and τ both occur in π, then there exists a g ∈ G such that

g(X_{τ} + 𝔤_{y,−s}) ∩ (X_{σ} + 𝔤_{x,−r}) ≠ ∅.

Proof. Let V_{σ} ⊂ V (resp. V_{τ} ⊂ V) denote a one-dimensional subspace of V on which res_{G_{x,r^{+}}} π (resp. res_{G_{y,s^{+}}} π) acts by σ (resp. τ). Since (π, V) is irreducible, there exists a g ∈ G such that the image of π(g)V_{σ} under the projection of V onto V_{τ} is nonzero. This implies that for all h ∈ G_{x,r^{+}} ∩^{g}G_{y,s^{+}}, we have

σ(h) = τ(g^{−1}hg).

Thus, for all H ∈ 𝔤_{x,r^{+}} ∩^{g}𝔤_{y,s^{+}}, we have

Λ(tr(X_{σ} · H)) = Λ(tr(^{g}X_{τ} · H)).

This implies that for all H ∈ 𝔤_{x,r^{+}} ∩^{g}𝔤_{y,s^{+}}, we have tr((X_{σ} −^{g}X_{τ}) · H) ∈ ℘. Consequently, (X_{σ} −^{g}X_{τ}) ∈ (𝔤_{x,r^{+}} ∩^{g}𝔤_{y,s^{+}})^{∗} = 𝔤_{x,−r} +^{g}𝔤_{y,−s}. The proposition follows. □

7.6. Understanding the distribution res_{C_{c}^{∞}(G_{ρ(π)^{+}} )} Θ_{π}. Let (π, V) be an irreducible admissible representation.

From our discussion of depth, we know that we can find an x ∈ A(T) such that V^{G_{x,ρ(π)^{+}}} ≠ {0}. Thus, the trivial representation of G_{x,ρ(π)^{+}} occurs in π; the associated coset in 𝔤 is 𝔤_{x,−ρ(π)}.

Now fix s > ρ(π), y ∈ B, and τ̄ ∈ { \atop G_{y,s}/G_{y,s^{+}}} such that the character τ of G_{y,s} occurs in π. Let X_{τ} + 𝔤_{y,(−s)^{+}} ∈ 𝔤_{y,(−s)}/𝔤_{y,(−s)^{+}} be the coset corresponding to τ̄.

From our discussion of Kirillov theory, there exists a g ∈ G such that^{g}𝔤_{x,−ρ(π)} ∩(X_{τ} + 𝔤_{y,(−s)^{+}}) ≠ ∅. However, from Lemma 4.2.1, we have

g𝔤_{x,−ρ(π)} ⊂ 𝔤_{y,−ρ(π)} + N ⊂ 𝔤_{y,(−s)^{+}} + N .

Consequently, we can assume that X_{τ} is nilpotent!

We now use this fact to say something about the distribution res_{C_{c}^{∞}(G_{ρ(π)^{+}} )} Θ_{π}. For a function h ∈ C_{c}^{∞}(𝔤_{ρ(π)^{+}}), we define h̃ ∈ C_{c}^{∞}(G_{ρ(π)^{+}}) by h̃(g) = h(g − 1). We define the distribution Θ_{π,𝔤} on 𝔤 by Θ_{π,𝔤}(f) = Θ_{π}(f̃_{ρ(π)}) where f_{ρ(π)} = f · [𝔤_{ρ(π)^{+}}]. We let Θ̂_{π} denote the Fourier transform of Θ_{π,𝔤}.

From Exercise 4.2.3, we know that the Fourier transform of an element of D_{−ρ(π)} belongs to C_{c}^{∞}(𝔤_{ρ(π)^{+}}). Suppose x ∈ B and s > ρ(π). If f ∈ C(𝔤_{x,−s}/𝔤_{x,−ρ(π)}), then we have Θ̂_{π}(f) = Θ_{π}( ˜f̂). It follows from our discussion above and a few lines of calculation that

Θ̂_{π}(f) ≠ 0 implies supp(f) ∩ (𝔤_{x,(−s)^{+}} + N) ≠ ∅.

This last statement is equivalent to the statement

(3)  Θ̂_{π}(f) ≠ 0 implies supp(f) ∩ 𝔤_{(−s)^{+}} ≠ ∅.

7.7. The Harish-Chandra–Howe local character expansion. Considerations similar to those above led Roger Howe to make his finiteness conjectures [19] (which we will discuss later) and establish the following remarkable connection between the character of an irreducible representation of G and the Fourier transforms of nilpotent orbital integrals.

Theorem 7.7.1 (Harish-Chandra–Howe local character expansion). If π is an irreducible admissible representation of G, then there exist constants c_{μ}(π) indexed by μ ∈ P(n) such that

(4)  Θ_{π}(1 + X) = ∑_{μ∈P(n)} c_{μ}(π) · Ô_{μ}(X)

for all X ∈ 𝔤^{r.s.s.} sufficiently near zero.

Remark 7.7.2. Howe [16] proved this result without any restrictions on the characteristic of k. Later, under the assumption that the characteristic of k is zero, Harish-Chandra [13] generalized the proof to all connected reductive groups. The above theorem and its analogues (local expansions about any semisimple point) play a crucial role in Harish-Chandra’s proof that characters are locally integrable on G (\underline{not} just on G^{r.s.s.}). (The question of integrability is still open for arbitrary groups in positive characteristic. From work of Rodier [30] and Lemaire [25], it is known to be true for GL_{n}(k) when k has positive characteristic.)

One of the difficulties with this theorem is that it provides no indication of where equation (4) ought to hold. The conjecture of Hales, Moy, and Prasad [28] is a precise statement about this issue:

Statement 7.7.3. Equation (4) ought to be valid on 𝔤_{ρ(π)^{+}}^{r.s.s.}.

For integral-depth representations this was proved by Waldspurger [35]. With some restrictions on k, it was proved for arbitrary depth representations in [9].

8. An introduction to Howe’s conjectures and homogeneity

In his paper, Two conjectures about reductive p-adic groups, Howe proposed two remarkable finiteness conjectures that now bear his name. Howe’s conjecture for the Lie algebra looks like:

dim_{ℂ} res_{C_{c}(𝔤/L)} J(ω) < ∞.

Here ω ⊂ 𝔤 is any compactly generated^{7}, invariant, and closed subset of 𝔤, J(ω) denotes the space of invariant distributions supported on ω, and L is any lattice in 𝔤. For T ∈ J(ω),

res_{C_{c}(𝔤/L)} T

denotes the restriction of T to C_{c}(𝔤/L). Howe’s conjecture for the group looks very similar:

dim_{ℂ} res_{C_{c}(G/K)} J(ω) < ∞.

Here K is a compact open subgroup of G, and ω is a closed, compactly generated, invariant subset of G, and J(ω) denotes the space of invariant distributions which are supported on ω.

Of course, although we still refer to these statements as Howe’s conjectures, both are known to be true. Howe’s conjecture for the Lie algebra was proved in the 1970s by Howe [16] and later by Harish-Chandra [13] for general groups (but only in the characteristic zero setting). Waldspurger [37] has also given a proof. Howe’s conjecture for the group was proved in the 1980s by Clozel [6, 7] in the characteristic zero setting. In the 1990s, a characteristic free proof of Howe’s conjecture for the group was given by Barbasch and Moy in their beautiful paper [2].

8.1. Homogeneity. Beginning with Waldspurger’s paper [36], much energy has been spent trying to produce optimal versions of Howe’s conjectures. By optimal, we mean that we’d like to choose ω and L in such a way so that we can not only describe the dimension of res_{C_{c}(𝔤/L)} J(ω), but we can also find a good basis for this space in terms of distributions with which we are very familiar, namely, nilpotent orbital integrals.

These optimal versions of Howe’s conjectures are referred to as homogeneity results. This is an appropriate name, because, according to Webster’s Ninth New Collegiate Dictionary, the word homogeneity means

“the state of having identical distribution functions or values”. We restrict our attention to the Lie algebra. Fix r ∈ ℝ. The role of ω in Howe’s

conjecture will be played by the G-domain 𝔤_{r^{+}} (which is compactly generated). Guided by Harish-Chandra’s philosophy and Theorem 7.7.1, it appears that we want something like the following: For all T ∈ J(𝔤_{r^{+}}) and for all f ∈ C_{c}^{∞}(𝔤_{(−r)})

T̂(f) = ∑_{μ∈P(n)} c_{μ}(T) · Ô_{μ}(f).

7That is, ω can be realized as the closure of^{G}C for some compact subset C of 𝔤.

This last displayed equation is equivalent to requiring

T (f̂) = ∑_{μ∈P(n)} c_{μ}(T) · O_{μ}(f̂).

From Exercise 4.2.3 we have f̂ ∈ D_{r^{+}}. So, the role of C_{c}^{∞}(𝔤/L) will be played by D_{r^{+}}. Putting it all together, we arrive at the homogeneity statement

res_{D_{r^{+}}} J(𝔤_{r^{+}}) = res_{D_{r^{+}}} J(N).

The fact that 𝔤_{r^{+}} ⊂ 𝔤_{x,r^{+}} + N for all x ∈ B gives some feeling as to why this homogeneity statement ought to be true.

8.2. From Howe’s conjectures to the Harish-Chandra–Howe local character expansion. The homogeneity result discussed in the previous section is actually not strong enough to prove Theorem 7.7.1. As in [16, 13, 9, 35] we need something a bit stronger, but, unfortunately, more complicated.

For x ∈ B and s ≤ r, define

J̃_{x,s,r^{+}} := {T ∈ J(𝔤) | for f ∈ C(𝔤_{x,s}/𝔤_{x,r^{+}}), if supp(f) ∩ (𝔤_{s^{+}}) = ∅, then T (f) = 0}.

Note that J(𝔤_{r^{+}}) ⊂ J̃_{x,s,r^{+}}.

Remark 8.2.1. Although this definition seems a bit unnatural, it arises naturally from our understanding of res_{C_{c}^{∞}(G_{ρ(π)^{+}} )} Θ_{π}. In fact, from Equation (3), we have that Θ̂_{π} ∈ J̃_{x,s,r^{+}} if −r > ρ(π).

The following is the stronger homogeneity statement that needs to be proved in order to recover Statement 7.7.3:

res_{D_{r^{+}}} J̃_{r^{+}} = res_{D_{r^{+}}} J(N)

where

J̃_{r^{+}} := ⋂_{x∈B} ⋂_{s≤r} J̃_{x,s,r^{+}}.

9. Proving Homogeneity results

We recall that for r ∈ ℝ we have

𝔤_{r^{+}} = ⋃_{x∈B} 𝔤_{x,r^{+}} = ⋂_{x∈B}(𝔤_{x,r^{+}} + N)

and

D_{r^{+}} = ∑_{x∈B} C_{c}(𝔤/𝔤_{x,r^{+}}) = { \atop C_{c}^{∞}(𝔤_{−r})}.

We want to show

(5)  res_{D_{r^{+}}} J(𝔤_{r^{+}}) = res_{D_{r^{+}}} J(N).

We decided that knowing this, or, in truth, a stronger (but more complicated) statement would tell us that the Harish-Chandra–Howe local character expansion was valid on 𝔤_{ρ(π)^{+}}^{r.s.s.}.

As we discussed before, 𝔤_{r} = 𝔤_{r^{+}} unless r = \frac{k}{n} for some k ∈ ℤ. Consequently, we only need to verify Equation 5 for r ∈ \{\frac{k}{n}\}.

Moreover, by taking advantage of the fact that nilpotent orbital integrals are homogeneous, we can further restrict our attention to r of the form \frac{k}{n} with 0 ≤ k < n. Indeed, pick m ∈ ℤ such that m + r ∈ [0, 1). For T ∈ J(𝔤) and t ∈ k^{×}, define T_{t} by T_{t}(f) = T (f_{t}). Note that

f ∈ D_{r^{+}} if and only if f_{ϖ^{−m}} ∈ D_{(m+r)^{+}}

and

T ∈ J(𝔤_{r^{+}}) if and only if T_{ϖ^{m}} ∈ J(𝔤_{(m+r)^{+}}).

Consequently, if we know

res_{D_{(r+m)^{+}}} J(𝔤_{(r+m)^{+}}) = res_{D_{(r+m)^{+}}} J(N)

then for T ∈ J(𝔤_{r^{+}}) and f ∈ D_{r^{+}} we have

T (f) = T_{ϖ^{m}}(f_{ϖ^{−m}})
= ∑ c_{μ}(T_{ϖ^{m}}) · O_{μ}(f_{ϖ^{−m}})
= ∑ c_{μ}(T_{ϖ^{m}}) · q^{\frac{m·dim(Oμ)}{2}} · O_{μ}(f),

and so we may assume r ∈ [0, 1).

For the remainder of this section we discuss how to prove Equation (5). We first prove it for GL_{1}(k), we next prove it for GL_{2}(k), and finally we discuss a way to go about giving a general proof.

9.1. A proof for GL_{1}(k). From the reasoning above, we need to show

res_{C_{c}(k/℘)} J(℘) = res_{C_{c}(k/℘)} J(N).

The right-hand side is one-dimensional and spanned by the distribution f ↦ f(0) for f ∈ C_{c}(k/℘). Moreover, the right-hand side is a vector subspace of the left-hand side.

Since GL_{1}(k) is abelian, all distributions are invariant. Note that J(℘) consists of linear maps from C_{c}^{∞}(k) to ℂ which are supported on ℘. If f ∈ C_{c}(k/℘), then we can write

f = ∑_{X̄∈k/℘} cX̄ · [X + ℘]

where the cX̄ are complex numbers which are almost always zero. For T ∈ J(℘), we have

T (f) = T (c¯0 · [℘]) = T ([℘]) · f(0).

Consequently, the left-hand side is also one-dimensional and so the equality is established.

9.2. A proof for GL_{2}(k). This is where things begin to become interesting. Thanks to the remarks at the beginning of this section, we only need to verify two statements:

(6)  res_{D_{0^{+}}} J(𝔤_{0^{+}}) = res_{D_{0^{+}}} J(N)

and

(7)  res_{D_{(1/2)^{+}}} J(𝔤_{(1/2)^{+}}) = res_{D_{(1/2)^{+}}} J(N).

We begin by considering the first statement, Equation (6). We note that D_{0^{+}} may be thought of as an invariant version of C_{c}(𝔤/𝔨_{1}), but, for our purposes, this is \underline{not} a good way to think about the space D_{0^{+}}.

9.2.1. Descent and recovery. Fix T ∈ J(𝔤_{0^{+}}). We wish to show that res_{D_{0^{+}}} T is completely determined by res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} T.

Fix f ∈ D_{0^{+}}. We will demonstrate that T (f) is completely determined by res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} T. We write f = ∑_{i} f_{i} with f_{i} ∈ C_{c}(𝔤/𝔤_{x_{i},0^{+}}) for some x_{i} ∈ B. Since T is linear, without loss of generality we may assume that f ∈ C_{c}(𝔤/𝔤_{x,0^{+}}) for some x ∈ B. We can write

f = ∑_{X̄∈𝔤/𝔤_{x,0^{+}}} cX̄ · [X + 𝔤_{x,0^{+}}]

with the cX̄ complex constants which are almost always zero. Again, since T is linear, without loss of generality we may assume that f = [X + 𝔤_{x,0^{+}}].

Now, T (f) = 0 if the support of f does not intersect 𝔤_{0^{+}}. Consequently, since 𝔤_{0^{+}} ⊂ 𝔤_{x,0^{+}} + N, we must have

(X + 𝔤_{x,0^{+}}) ∩ N ≠ ∅.

So, without loss of generality, X ∈ N.

Up to conjugacy, we have two choices for 𝔤_{x,0^{+}}; it is either 𝔨_{1} or 𝔟_{1/2}. In what follows, the reader is encouraged to consult Figure 8 to get a more geometric understanding of what is happening.

We first deal with the 𝔨_{1} case. Since x_{0} is the only point x where 𝔤_{x,0^{+}} = 𝔨_{1}, we may suppose that X ∈ N ∩ (𝔤_{x_{0},−m} ∖ 𝔤_{x_{0},(−m)^{+}}) for some m > 0. In other words, X ∈ N ∩ (𝔨_{−m} ∖ 𝔨_{1−m}). Since we are free to conjugate by G_{x_{0},0} = K_{0}, we may assume that

X = ( \begin{matrix}0 & ϖ^{−m}u \\ 0 & 0 \end{matrix} )

with u ∈ R^{×}. For any point y in the chamber C_{0} (that is, a point between x_{0} and x′ in Figure (8)), we have that X is “closer” to the origin with respect to the y filtration than it is in the x filtration. For example, X ∈ 𝔤_{y_{0},(1/2−m)^{+}} where y_{0} is the barycenter of C_{0}. The problem is: at the point y we require local constancy not with respect to 𝔤_{x_{0},0^{+}} but with respect to 𝔤_{y,0^{+}} = 𝔟_{1/2}. In order to recover the proper type of local constancy, we take advantage of the invariance of T. We write

T ([X + 𝔨_{1}]) = \frac{1}{q^{2}} · ∑_{t̄∈T_{m}/T_{m^{+}}}([^{t}X + 𝔨_{1}])

= \frac{1}{q^{2}} · ∑_{ᾱ,β̄∈R/℘}([X + ( \begin{matrix}0 & u(α−β) \\ 0 & 0 \end{matrix} ) + 𝔨_{1}])

= \frac{1}{q} · T ([X + 𝔟_{1/2}]).

We have succeeded in writing T (f) in terms of T evaluated at a function f′ ∈ D_{0^{+}} which is supported closer to the origin with respect to some other point in the building.

32 STEPHEN DEBACKER

PSfrag replacements r 1 0

−m 1 − m

( \begin{matrix}R & ℘ \\ R & R \end{matrix} )
( \begin{matrix}R & R \\ R & R \end{matrix} )
( \begin{matrix}R & R \\ ℘ & R \end{matrix} )
( R ^{℘^{−1}}_{℘ R} )
( \begin{matrix}R & ^{℘^{−1}} \\ ^{℘^{2}} & R \end{matrix} )
( ℘ ℘_{R ℘} )
( \begin{matrix}℘ & ℘ \\ ℘ & ℘ \end{matrix} )
( ℘ ℘^{℘ R} )
( ^{℘^{2}} ℘^{℘ R} )
( \begin{matrix}℘ & ^{℘^{−1}} \\ ^{℘^{2}} & ℘ \end{matrix} )
( \begin{matrix}℘ & ^{℘^{2}} \\ ℘ & ℘ \end{matrix} )
( \begin{matrix}℘ & ℘ \\ ^{℘^{2}} & ℘ \end{matrix} )
( ^{℘^{3}} ℘^{℘ R} )

ϖ^{−m} · ( \begin{matrix}R & R \\ R & R \end{matrix} )

ϖ^{−m} · ( \begin{matrix}R & R \\ ℘ & R \end{matrix} )

ϖ^{−m} · ( R ^{℘^{−1}}_{℘ R} )

ϖ^{−m} · ( \begin{matrix}℘ & ℘ \\ ℘ & ℘ \end{matrix} )

ϖ^{−m} · ( ℘ ℘^{℘ R} )

ϖ^{−m} · ( ^{℘^{2}} ℘^{℘ R} )

ϖ^{−m} · ( \begin{matrix}℘ & ℘ \\ ^{℘^{2}} & ℘ \end{matrix} )

x_{0}

x′

Figure 8. (𝔤𝔩_{2})_{x,r}

We now examine the 𝔟_{1/2} case. In this case, we are looking at the coset X + 𝔤_{y,0^{+}} where X ∈ N and y is any point in C_{0}. Since we are free to conjugate by G_{y,0} = B_{0}, we may assume that X is either

( \begin{matrix}0 & ϖ^{−m}u \\ 0 & 0 \end{matrix} )

or

( \begin{matrix}0 & 0 \\ ϖ^{(1−m)}u & 0 \end{matrix} )

with m > 0 and u ∈ R^{×}. In the former case, we can write

T ([X + 𝔤_{y,0^{+}}]) = ∑_{ᾱ∈℘/℘^{2}} T ([X + ( \begin{matrix}0 & 0 \\ α & 0 \end{matrix} ) + 𝔤_{x_{1},0^{+}}])

where x_{1} = ^{diag(ϖ^{−1},1)}x_{0} = x′ is the other vertex of C̄_{0}. We have

𝔤_{x_{1},0^{+}} = ( ^{℘^{2}} ℘^{℘ R} ) .

From Figure 8 it is clear that we have expressed T evaluated at [X + 𝔤_{y,0^{+}}] in terms of T evaluated at f′ where f′ ∈ D_{0^{+}} has support closer to the origin with respect to the x_{1} filtration than [X + 𝔤_{y,0^{+}}] had with respect to the y filtration.

Exercise 9.2.1. Do the analogous analysis for the latter case.

To summarize, the point of descent and recovery is as follows. We begin with a simple function f ∈ C((𝔤_{x,s} ∖𝔤_{x,s^{+}})/𝔤_{x,0^{+}}) for some x ∈ B. From this function, we find a point y ∈ B and a function f′ ∈ C(𝔤_{y,s^{+}}/𝔤_{y,0^{+}}) so that T (f) = T (f′). After a finite number of steps, we will have shown that T (f) is completely determined by res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} T.

9.2.2. Counting. We now know the following facts:

(1) Thanks to Harish-Chandra [13], the dimension of the complex vector space res_{D_{0^{+}}} J(N) is equal to the cardinality of P(n), which, in this case, is 2.

(2) From §4.4, we have J(N) ⊂ J(𝔤_{0^{+}}) and so res_{D_{0^{+}}} J(N) ⊂ res_{D_{0^{+}}} J(𝔤_{0^{+}}).

(3) From the previous section, we know that

dim_{ℂ} res_{D_{0^{+}}} J(𝔤_{0^{+}}) = dim_{ℂ} res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} J(𝔤_{0^{+}}).

Consequently, we need only show that

dim_{ℂ} res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} J(𝔤_{0^{+}}) ≤ 2.

Since 𝔤_{0^{+}} ⊂ 𝔤_{x,0^{+}}+N for any x ∈ B, we have that for T ∈ J(𝔤_{0^{+}}), res_{C(𝔨_{0}/𝔨_{1})+C(𝔟_{0}/𝔟_{1/2})} T is completely determined by

T ([( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) + 𝔨_{1}]), T ([𝔨_{1}]), and T ([𝔟_{1/2}]).

Since

T ([𝔟_{1/2}]) = (q − 1) · T ([( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) + 𝔨_{1}]) + T ([𝔨_{1}]),

we are done.

Exercise 9.2.2. Using the above proof as a template, prove Equation (7).

9.3. The general approach. In general, the proof is very much like that produced above, only more complicated.

9.3.1. Descent and recovery. Suppose T ∈ J(𝔤_{r^{+}}). The main point is to show that

res_{D_{r^{+}}} T = 0 if and only if res_{D_{r^{+}}^{r}} T = 0

where

D_{r^{+}}^{r} := ∑_{x∈B} C(𝔤_{x,r}/𝔤_{x,r^{+}}).

That is, we can find a very small space of functions from which we can choose a dual basis for res_{D_{r^{+}}} J(𝔤_{r^{+}}).

Suppose f ∈ D_{r^{+}}. As before, without loss of generality we may assume that f = [X + 𝔤_{x,r^{+}}] for some X ∈ N and some x ∈ B. Under some hypotheses, we can use 𝔰𝔩_{2}(𝔣)-theory to find a direction in the building in which to move so that, for some y ∈ B in that direction, the support of f is nearer the origin with respect to y than it was with respect to x. Moreover, if necessary, we can use the basics of 𝔰𝔩_{2}(𝔣)-representation theory and the invariance of T to “beef-up” the coset 𝔤_{x,r^{+}} so that we arrive at a function f′ ∈ C_{c}(𝔤/𝔤_{y,r^{+}}) with the properties:

(1) T (f′) = T (f) and

(2) the support of f′ is “closer” to the origin with respect to y than the support of f with respect to x.

Remark 9.3.1. Invoking 𝔰𝔩_{2}(𝔣)-theory requires some restrictions. The theory only works well if the highest weights for the representations one wishes to consider are less than (p − 3) where p is the characteristic of 𝔣.

9.3.2. Counting. This is the real key. In the end, you need some type of correspondence between “suitable” elements of D_{r^{+}}^{r} and O(0), the set of nilpotent orbits in 𝔤.

When r is zero, the situation is easy to understand. As discussed before, the conjugacy classes of facets in B are in one-to-one correspondence with the elements of P(n). Similarly, the nilpotent orbits are in one-to-one correspondence with the elements of P(n). For μ ∈ P(n), let F_{μ} be the facet described in §3.4.1. The image of X_{μ} in 𝔤_{F_{μ},0}/𝔤_{F_{μ},0^{+}} is distinguished nilpotent (that is, it does not lie in a proper Levi subalgebra of 𝔤_{F_{μ},0}/𝔤_{F_{μ},0^{+}}). We let [X_{μ} + 𝔤_{F_{μ},0^{+}}] ∈ D_{0^{+}}^{0} denote the characteristic function of the coset X_{μ} + 𝔤_{F_{μ},0^{+}}. This is the correspondence between “suitable” elements of D_{0^{+}}^{0} and O(0) alluded to above. Indeed, for T ∈ J(𝔤_{0^{+}}), the restriction of T to D_{0^{+}}^{0} is zero if and only if

T ([X_{μ} + 𝔤_{F_{μ},0^{+}}])

is zero for all μ ∈ P(n).

When r ≠ 0, life is much more complicated; but something beautiful is true. We refer the reader to [10].

10. A few comments on the c_{μ}(π)s

Let (π, V) denote an irreducible representation of G. We recall that for all regular semisimple X in some neighborhood of zero we have the Harish-Chandra–Howe local character expansion:

Θ_{π}(1 + X) = ∑_{μ∈P(n)} c_{μ}(π) · Ô_{μ}(X).

In his paper [16], Howe proves that for irreducible supercuspidal representations (that is, those representations that do not occur as subrepresentations of parabolically induced representations) the coefficients occurring in the Harish-Chandra–Howe local character expansion are all integers. In fact, given a bit of additional information, the same proof shows that this is true for all smooth irreducible representations. We first recall what we know about the c_{μ}(π)s, and we then prove this result of Howe.

10.1. The coefficient c_{(1,1,...,1)}(π). Suppose that (π, V) is a discrete series representation (that is, the matrix coefficients of (π, V) are square-integrable mod the center of G). In this case, Rogawski [32] extended a result of Harish-Chandra [13] to show that

c_{(1,1,...,1)}(π) = \frac{deg(π)}{(−1)^{ℓ} · deg(St)}.

Here deg(π) denotes the formal degree of π, St is the Steinberg representation (see, for example, [5]), and ℓ denotes the semisimple rank of G.

On the other hand, if (π, V) is a tempered representation (that is, it occurs in the Plancherel formula) which is not in the discrete series, then, using results of Kazhdan [23], Huntsinger [20] showed that c_{(1,1,...,1)}(π) = 0 (see also the paper of Schneider and Stuhler [34]).

10.2. The leading coefficient. According to Mœglin and Waldspurger [26], the set

{μ ∈ P(n) | c_{μ}(π) ≠ 0}

has a unique maximal element; call it μ_{π}. Moreover, according to Mœglin and Waldspurger [26] and Rodier [31], the coefficient c_{μ_{π}}(π) is an integer which is equal to the dimension of the degenerate Whittaker model corresponding to μ_{π}.

Example 10.2.1. Suppose μ = (n) ∈ P(n). Recall that X_{(n)} is regular nilpotent. Let B̄ be the Borel subgroup consisting of lower triangular matrices in G. Let N̄ denote the unipotent radical of B̄; that is, the subgroup of B̄ formed by the set of lower triangular matrices with ones on the diagonal. Define the character δ of N̄ by n̄ ↦ Λ(tr(X_{(n)} · (n̄ − 1)). For an irreducible smooth representation (π, V) of G, let V (δ) denote the subspace of V generated by the set

{(π(n̄) − δ(n̄))v | v ∈ V and n̄ ∈ N̄}.

Define V_{(n)} = V/V (δ). Then according to the results quoted above, we have c_{(n)}(π) = dim_{ℂ}(V_{(n)}) = dim_{ℂ} Hom_{G}(V, IndG ; N̄ δ).

10.3. The remaining coefficients. Not much is known about the remaining coefficients. Following an argument of Roger Howe [16], we show that they are all integers.

We begin by making Remark 6.1.4 precise. If μ′ ∈ P(n), then

Ô_{μ′}(X) = Θ_{Ind_{P (μ′)}^{G} 1}(1 + X)

for all X ∈ 𝔤_{0^{+}}. Here Ind_{P (μ′)}^{G} 1 is the representation of G obtained by inducing the trivial representation of P (μ′) up to G.

Suppose j > 0, μ ∈ P(n), and τ_{μ} is the character of K_{j} represented by the coset ϖ^{(1−2j)}X_{μ} + 𝔨_{(1−j)} ∈ 𝔨_{1−2j}/𝔨_{1−j}. From §7.3 we have

(8)  Θ_{π}(τ_{μ}) = meas_{dg}(K_{j}) · m(τ_{μ}^{−1}, π)

and

(9)  Θ_{Ind_{P (μ′)}^{G} 1}(τ_{μ}) = meas_{dg}(K_{j}) · m(τ_{μ}^{−1}, Ind_{P (μ′)}^{G} 1).

From the discussion above, Exercise 5.3.1, and Equation (2) we have

Ô_{μ′}(τ_{μ}) = O_{μ′}(τ̂_{μ}) = meas_{dX}(𝔨_{j}) · O_{μ′}([−ϖ^{(1−2j)} · X_{μ} + 𝔨_{(1−j)}])

= \begin{cases}meas_{dX}(𝔨_{j}) · m(τ_{μ}^{−1}, Ind_{P (μ′)}^{G} 1) & if μ < μ′, \\ meas_{dX}(𝔨_{j}) & if μ = μ′, and \\ 0 & otherwise. \end{cases}

Thus, if j is sufficiently large, then

m(τ_{μ}^{−1}, π) = ∑_{μ≤μ′} c_{μ′}(π) · m(τ_{μ}^{−1}, Ind_{P (μ′)}^{G} 1).

We now proceed by induction. From Rodier [31] we have that c_{(n)} is always an integer. Suppose μ ∈ P(n). By induction, if μ′ > μ, then c_{μ′}(π) ∈ ℤ. We have

c_{μ}(π) · m(τ_{μ}^{−1}, Ind_{P (μ)}^{G} 1) = m(τ_{μ}^{−1}, π) − ∑_{μ′>μ} c_{μ′}(π) · m(τ_{μ}^{−1}, Ind_{P (μ′)}^{G} 1) ∈ ℤ.

Since m(τ_{μ}^{−1}, Ind_{P (μ)}^{G} 1) = O_{μ}([−ϖ^{(1−2j)} · X_{μ} + 𝔨_{(1−j)}]) = 1, we are done.

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   E-mail address: debacker@math.harvard.edu

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