Trace Paley-Wiener theorem for reductive p-adic groups
type: Item
instance of: scholarly article
title: Trace Paley-Wiener theorem for reductive p-adic groups
zbMATH Open document ID: 0634.22011
DOI: 10.1007/BF02792538
author: Pierre Deligne
author: David Kazhdan
author: Joseph Bernstein
published in: Journal d'Analyse Mathématique
publication date: 1986-01-01T00:00:00Z
review text: Let F be a nonarchimedean local field, \({\mathbb{G}}\) be a reductive F-group, and \(G={\mathbb{G}}(F)\). Let R(G) be the Grothendieck group of G, with basis Irr(G), the set of equivalence classes of smooth irreducible representations of G. Let H(G) be the Hecke algebra \(C_ c^{\infty}(G)\) under convolution. Then each h in H(G) defines a linear form \(f_ h: R(G)\to {\mathbb{C}}\) by \(f_ h(\pi)=trace \pi (h)\). Let M be a standard Levi subgroup of G. The group \(\Psi\) (M) of unramified characters of M has a natural structure of a complex algebraic group. The form \(f=f_ h\) for h in H(G) satisfies: (i) For any standard Levi subgroup M and \(\sigma\) in Irr(M), the function \(\psi \mapsto f(i_{GM}(\psi \sigma))\) is regular on the complex algebraic variety \(\psi\) (M). (ii) There is a compact open subgroup K of G for which f is non-zero only on the G-modules E which have a nontrivial space \(E^ K\) of K-invariant vectors. The Trace Paley-Wiener Theorem is that the conditions (i) and (ii) characterize the trace forms \(\{f_ h:\) \(h\in H(G)\}.\) This theorem describes the image of the morphism tr: H(G)\(\to R^*(G)\). Theorem B of the article reviewed above shows that ker tr\(=[H(G),H(G)]\). Let \(i_{GM}: R(M)\to R(G)\) be the induction functor and let \(R_ I(G)\) be the subgroup of R(G) spanned by \(\{i_{GM}(\sigma)|\) M a proper Levi subgroup of G, \(\sigma\in Irr(M)\}\). An irreducible G-module will be called discrete if its class is not contained in \(R_ I(G)\). An infinitesimal character is called discrete if it is the infinitesimal character of some discrete G-module. A linear form on R(G) is said to be discrete if it vanishes on \(R_ I(G)\). Let \(r_{MG}: R(G)\to R(M)\) be the Jacquet functor. A combinatorial lemma is proved using the operators \(T_ M=i_{GM}\circ r_{MG}\) on R(G), that there exist rational constants \(c_ M\), for M proper, such that for any linear form f on R(G), the form f-\(\sum_ Mc_ Mr^*_{MG}i^*_{GM}(f)\) is discrete. Let \(\Theta\) (G) b…
reviewed by: Charles B. Huijsmans
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 4031986
zbMATH Keywords: nonarchimedean local field
zbMATH Keywords: Grothendieck group
zbMATH Keywords: smooth irreducible representations
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Levi subgroup
zbMATH Keywords: unramified characters
zbMATH Keywords: Paley-Wiener Theorem
zbMATH Keywords: Jacquet functor
zbMATH Keywords: complex algebraic variety of infinitesimal characters
zbMATH Keywords: maximal support property
zbMATH Keywords: order-positive Schauder basis
MaRDI profile type: Publication
Wikidata QID: Q97308020
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full work available at URL: https://doi.org/10.1007/bf02792538
OpenAlex ID: W2057640703
description: scientific article; zbMATH DE number 4031986