Harmonic Analysis on Reductive Groups

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 G^σ. Assume that G has finite center; then it is known that G has a σ-stable maximal compact subgroup K.

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.

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.

Let E be a Euclidean vector space of dimension n with inner product (·,·). For each α ∈ E with (α, α) = 2 we write for the orthogonal reflection in the hyperplane perpendicular to α.

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 D ∈ D(G/K) is the “ratio” of two µ(Z ₁) and µ (Z ₂) (with Z ₁,Z ₂ ∈ Z(G)). See Theorem 4.1.

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

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.

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 .

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.

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].

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].

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 _ℝ is a semisimple Lie group. The philosophy of coadjoint orbits, as propounded by Kirillov and Kostant, suggests that unitary rep-resentations of G _ℝ are closely related to the orbits of G _ℝ on the dual \( \mathfrak{g}_{\mathbb{R}}^{ * } \) of the Lie algebra g _ℝ of G _ℝ. 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.