Cuspidal geometry of p-adic groups
type: Item
instance of: scholarly article
title: Cuspidal geometry of p-adic groups
zbMATH Open document ID: 0634.22009
DOI: 10.1007/BF02792530
author: David Kazhdan
published in: Journal d'Analyse Mathématique
publication date: 1986-01-01T00:00:00Z
review text: Let F be a nonarchimedean local field of characteristic 0, \({\mathbb{G}}\) be a reductive F-group, and \(G={\mathbb{G}}(F)\). Recall that an irreducible representation of a compact group is necessarily finite-dimensional, and that the character is a finite sum of diagonal matrix coefficients. In general, matrix coefficients of irreducible discrete series representations satisfy standard orthogonality relations. From this, one gets orthogonality relations for the characters of the irreducible representations of compact groups. However, the character of an irreducible infinite-dimensional representation \(\pi\) of a non-compact reductive group G is first a distribution. By theorems of \textit{Harish-Chandra} [Queen's Papers Pure Appl. Math. 48, 281-347 (1978; Zbl 0433.22012)], such a distribution is given by integration against a locally integrable function \(\theta_{\pi}\), which is locally constant (analytic for real groups) on the regular set. One of the main results of this paper is that the \(\theta_{\pi}\) satisfy certain orthogonality relations, for \(\pi\) tempered, involving integration over the elliptic set of G, with respect to a measure arising from the Weyl integration formula. This result is conjectured in Harish- Chandra (notes by van-Dijk) Harmonic analysis on reductive p-adic groups [Lect. Notes Math. 162, (1970; Zbl 0202.411)]. The tools developed are important in their own right. In particular, detailed analysis of various spaces of functions and orbital integrals is developed, which leads to the existence of ``pseudo-coefficients''. The book Représentations des groupes réductifs sur un corps local [Herrmann, 1984; Zbl 0544.00007] edited by \textit{J. Bernstein}, \textit{P. Deligne}, \textit{D. Kazhdan} and \textit{M.-F. Vignéras}, contains the result for GL(n), which is then used in proving a certain correspondence exists between discrete series representations over GL(m,D) and GL(n,F), where D is a central division algebra of F of dimension \(d^ …
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 4031984
zbMATH Keywords: nonarchimedean local field
zbMATH Keywords: reductive F-group
zbMATH Keywords: irreducible discrete series representations
zbMATH Keywords: orthogonality relations
zbMATH Keywords: irreducible representations of compact groups
zbMATH Keywords: Weyl integration formula
zbMATH Keywords: Harish- Chandra
zbMATH Keywords: orbital integrals
zbMATH Keywords: Grothendieck group
zbMATH Keywords: Fourier transform
zbMATH Keywords: Schwartz space
zbMATH Keywords: Selberg principle
zbMATH Keywords: pseudo-coefficients
zbMATH Keywords: tempered representations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4031984