On the Harish-Chandra homomorphism for the Chevalley groups over \(p\)-adic field
type: Item
instance of: scholarly article
title: On the Harish-Chandra homomorphism for the Chevalley groups over \(p\)-adic field
zbMATH Open document ID: 0830.22009
published in: Chinese Annals of Mathematics. Series B
publication date: 1995-04-17T00:00:00Z
review text: Given a Chevalley group \(G\) (not of type \(G_2\)) defined over a \(p\)- adic field and a finite-dimensional Hilbert space \(V\), let \(C_c(G : \text{End }V)\) be the algebra of the continuous functions \(G \to \text{End }V\) with compact support. For an open compact subgroup \(J\) of \(G\) and an irreducible unitary representation \(\sigma\) of \(J\) in \(V\) define the \(\sigma\)-spherical Hecke algebra \(H(G // J, \sigma)\) consisting of the functions in \(C_c (G : \text{End } V)\) such that \[ f(k_1 gk_2) = \sigma(k_1) f(g) \sigma(k_2),\qquad k_1, k_2 \in J,\;g \in G. \] The author is interested in the case when \(J = K_q\) is the \(q\)-congruence maximal compact subgroup of \(G\). For some \(\sigma\) he constructs the Harish-Chandra homomorphism \(H(G' // J', \sigma') \to H(G // J,\sigma)\) related to the Levi component of a parabolic subgroup of \(G\) and proves that it is an algebra isomorphism and an isometry of \(L^2\)-spaces. It generalizes the results obtained for \(G = GL_n\) by \textit{R. Howe} and \textit{A. Moy} [Harish-Chandra homomorphism for \(p\)-adic groups, Reg. Conf. Ser. Math. 59 (1985; Zbl 0593.22014)].
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 743972
zbMATH Keywords: Chevalley group
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: Harish-Chandra homomorphism
zbMATH Keywords: irreducible unitary representation
zbMATH Keywords: \(\sigma\)-spherical Hecke algebra
zbMATH Keywords: \(q\)-congruence maximal compact subgroup
zbMATH Keywords: Levi component
zbMATH Keywords: parabolic subgroup
author: Zhonghu Chen
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 743972