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## Über dieses Buch

A conference on Harmonic Analysis on Reductive Groups was held at Bowdoin College in Brunswick, Maine from July 31 to August 11, 1989. The stated goal of the conference was to explore recent advances in harmonic analysis on both real and p-adic groups. It was the first conference since the AMS Summer Sym­ posium on Harmonic Analysis on Homogeneous Spaces, held at Williamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer (three weeks) and somewhat broader (nilpotent groups, solvable groups, as well as semisimple and reductive groups), the structure and timeliness of the two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec­ ond, there was a series of single lectures in which the speakers presented an overview of their latest research.

## Inhaltsverzeichnis

### Lifting of Characters

Abstract
In Lifting of Characters and Harish-Chandra’s Method of Descent [1] we discussed lifting of characters from endoscopic groups in terms which we need for Arthur's conjectures [4]. This paper is an expository version of [1], written with some important special cases in mind and illustrated by numerous examples. We refer the reader to the introduction to [1] for motivation; here we limit ourselves to a summary of some essential points and a discussion of how this paper differs from [1].

### Handling the Inverse Spherical Fourier Transform

Abstract
We use the standard notation and refer to [GV], [H] for more details. Let X = G/K be a Riemannian symmetric space of the noncompact type.
Jean-Philippe Anker

### Some Problems in Local Harmonic Analysis

Abstract
The purpose of this article is to discuss some questions in the harmonic analysis of real and p-adic groups. We shall be particularly concerned with the properties of a certain family of invariant distributions. These distributions arose naturally in a global context, as the terms on the geometric side of the trace formula. However, they are purely local objects, which include the ordinary invariant orbital integrals. One of our aims is to describe how the distributions also arise in a local context. They appear as the terms on the geometric side of a new trace formula, which is simpler than the original one, and is the solution of a natural question in local harmonic analysis. The local trace formula seems to be a promising tool. It might have implications for the difficult local problems which are holding up progress in automorphic forms.
James Arthur

### Asymptotic Expansions on Symmetric Spaces

Abstract
Let G/H be a semisimple symmetric space, where G is a connected semisimple real Lie group with an involution σ, and H is an open subgroup of the fix point group Gσ. Assume that G has finite center; then it is known that G has a σ-stable maximal compact subgroup K.
Erik van den Ban, Henrik Schlichtkrull

### The Admissible Dual of GL N Via Restriction to Compact Opent Subgroups

Abstract
Let G be a reductive group over a p-adic field F. Then, as with reductive groups over any field, it is natural to cast the representation theory of G in terms of parabolic induction. This leads to the notion of supercuspidal representation and, in the case of GLn, to the classification of irreducible (admissible) representations given in the work of Bernstein-Zelevinski [BZ], [Z]. On the other hand, the fact that G is a totally disconnected, locally compact group accounts for the existence of open, compact modulo center subgroups of G which in turn has a strong influence on its representation theory. In particular, one is led to consider the possibility that supercuspidal representations may be constructed by induction from such subgroups (see [Ku2] for historical background) and, more generally, to inquire into the possibility of classifying admissible representations of G by considering the subrepresentations they may have when restricted to such subgroups (the possibility of classifying the admissible dual in this fashion was first raised in [H].) In what follows, we report on recent progress in this direction in the case G = GLn(F)\ we begin with some general background.
Colin J. Bushnell, Philip C. Kutzko

### Invariant Harmonic Analysis on the Schwartz Space of a Reductive p-ADIC Group

Abstract
At the Williastown conference on harmonic analysis, R. Howe publicized two conjectures concerning the orbital integrals of functions on a reductive p-adic group and on its Lie algebra. He proved the Lie algebra case of the conjecture himself; recently we have proved the conjecture on the group, at least in zero characteristic [4c,4d].
Laurent Clozel

### Constructing the Supercuspidal Representation of GL n (F), F p—ADIC

Abstract
In this paper I give a description of a construction of the supercuspidal representations of GL n (F). This construction, the result of work done in late 1988 and early 1989, was first given in [6], and the description here follows the same lines.
Lawrence Corwin

### A Remark on the Dunkl Differential—Difference Operators

Abstract
Let E be a Euclidean vector space of dimension n with inner product (·,·). For each αE with (α, α) = 2 we write
$${r_{\alpha }}(\lambda ) = \lambda - (\alpha, \lambda )\alpha, \lambda \in E$$
(1.1)
for the orthogonal reflection in the hyperplane perpendicular to α.
Gerrit J. Heckman

### Invariant Differential Operators and Weyl Group Invariants

Abstract
In this research announcement we describe the relationship between the algebra Z(G) of bi-invariant differential operators on a simple noncompact Lie group G and the algebra D(G/K )of invariant differential operators on the symmetric space G/K associated with G (cf. §4). The natural map µ: Z(G) → D(G/K) turns out to be surjective except in exactly four cases. These cases involve the exceptional groups and for them the relationship between Z(G) and D(G/K) is quite complicated. However for all cases of G/K, each DD(G/K) is the “ratio” of two µ(Z 1) and µ (Z 2) (with Z 1,Z 2Z(G)). See Theorem 4.1.
Sigurdur Helgason

### The Schwartz Space of a General Semisimple Lie Group

Abstract
Let G be a connected semisimple Lie group. The tempered spectrum of G consists of families of representations unitarily induced from cuspidal parabolic subgroups. Each family is parameterized by unitary characters of a θ-stable Cartan subgroup. The Schwartz space C(G)is a space of smooth functions decreasing rapidly at infinity and satisfying the inclusions $$C_c^{\infty }(G) \subset C(G) \subset {L^2}(G)$$. The Plancherel theorem expands Schwartz class functions in terms of the distribution characters of the tempered representations. Very roughly, we can write $$f(x) = \sum\nolimits_H {{f_H}(x)}$$,$$f \in C(G)$$,$$x \in G$$, where
$${f_H}(x) = \int\limits_{{{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{H}}}} {\Theta (H:x)(R(x)f)m(H:x){d_X}}$$
Rebecca A. Herb

### Intertwining Functors and Irreducibility of Standard Harish—Chandra Sheaves

Abstract
Let g be a complex semisimple Lie algebra and σ an involution of g. Denote by t the fixed point set of this involution. Let K be a connected algebraic group and ϕ a morphism of K into the group G = Int(g) of inner automorphisms of g such that its differential is injective and identifies the Lie algebra of K with t. Let X be the flag variety of g, i.e. the variety of all Borel subalgebras in g. Then K acts algebraically on X, and it has finitely many orbits which are locally closed smooth sub varieties. The typical situation is the following: g is the complexification of the Lie algebra of a connected real semisimple Lie group G 0 with finite center, K is the complexification of a maximal compact subgroup of G 0, and σ the corresponding Cartan involution.
Dragan Miličić

### Fundametal G—Strata

Abstract
Let k be a non-archimedean local field, with valuation ring O and prime ideal P. Then O/P is a finite field with q elements which we denote byF q .
Lawrence Morris

### Construction and Classification of Irreducible Harish—Chandra Modules

Abstract
Typically, irreducible Harish-Chandra modules are constructed not directly, but as unique irreducible submodules, or unique irreducible quotients, of so-called standard modules. As in some other contexts, standard modules are obtained by cohomological constructions which tend to be “easy” on the level of Euler characteristic. For certain values of the parameters in these constructions, there is a vanishing theorem; standard modules arise when the vanishing theorem applies.
Wilfried Schmid

### Langlands’ Conjecture on Plancherel Measures for p-Adic Groups

Abstract
One of the major achievements of Harish-Chandra was a derivation of the Plancherel formula for real and p-adic groups [9,10]. To have an explicit formula, one will have to compute the measures appearing in the formula; the so called Plancherel measures and formal degrees [12]. (For reasons stemming from L-indistinguishability, we would like to distinguish between the formal degrees for discrete series and the Plancherel measures for non-discrete tempered representations, cf. Proposition 9.3 of [29].) While for real groups the Plancherel measures are completely understood [1, 9, 22], until recently little was known in any generality for p-adic groups [29] (except for their rationality and general form due to Silberger [39]). On the other hand any systematic study of the non-discrete tempered spectrum of a p-adic group would very likely have to follow the path of Knapp and Stein [20, 21] and their theory of 72-groups. Since the basic reducibility theorems for p-adic groups are available [40, 41], it is the knowledge of Plancherel measures which would be necessary to determine the R-groups. This is particularly evident from the important and the fundamental work of Keys [16, 17, 18] and the work of the author [29, 30, 31].
Freydoon Shahidi

### Transfer and Descent: Some Recent Results

Abstract
A basic tool for studying the transfer of representations is the dual transfer of orbital integrals. In this paper we report on some recent results for orbital integrals and, in particular, on a descent theorem [LS3].

### On Jacquet Modules of Induced Representations of p—Adic Symplectic Groups

Abstract
We fix a reductive p-adic group G. One very useful tool in the representation theory of reductive p-adic groups is the Jacquet module. Let us recall the definition of the Jacquet module. Let (π, V) be a smooth representation of G and let P be a parabolic subgroup of G with a Levi decomposition P = MN. The Jacquet module of V with respect to N is $${V_N} = V/spa{n_{\mathbb{C}}}\left\{ {\pi (n)v - v;n \in N,v \in V} \right\}$$.
Suppose G is a semisimple Lie group. The philosophy of coadjoint orbits, as propounded by Kirillov and Kostant, suggests that unitary rep-resentations of G are closely related to the orbits of G on the dual $$\mathfrak{g}_{\mathbb{R}}^{ * }$$ of the Lie algebra g of G . One knows how to attach representations to semisimple orbits, but the methods used (which rely on the existence of nice “polarizing subalgebras” of g) cannot be applied to most nilpotent orbits.