Dual pair \(G_{\mathcal J}\times PGL_ 2 G_{\mathcal J}\) is the automorphism group of the Jordan algebra \({\mathcal J}\)
type: Item
instance of: scholarly article
title: Dual pair \(G_{\mathcal J}\times PGL_ 2 G_{\mathcal J}\) is the automorphism group of the Jordan algebra \({\mathcal J}\)
zbMATH Open document ID: 0858.22015
DOI: 10.1007/BF01231530
author: Gordan Savin
published in: Inventiones Mathematicae
publication date: 1995-01-26T00:00:00Z
full work available at URL: https://eudml.org/doc/144233
review text: Let \(G\) be a simple, split, simply laced group over a nonarchimedean local field \(F\) of characteristic different from 2. Let \(J(\chi_{\text{sr}})\) be the spherical representation of \(G\) with Satake parameter \(\chi_{\text{sr}}\). The author computes the wavefront set of \(J(\chi_{\text{sr}})\) by computing the wavefront set of the representation \(\pi_0\) constructed by \textit{D. Kazhdan} and \textit{G. Savin} [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his 60th birthday, Pt. I (Tel Aviv 1989), Israel Math. Conf. Proc. 2, 209-233 (1990; Zbl 0737.22008)] and shows that if \(G\) is of types \(A_{2n-1}\), \(D_n\) or \(E_n\) \((n=6,7,8)\), then \(J(\chi_{\text{sr}})\) is a minimal representation of \(G\). Let \(\mathcal G\) be a (split) Jordan algebra and \(G_{\mathcal G}\) be the automorphism group of the Jordan algebra \(\mathcal G\). Using methods of \textit{S. Sahi} [Invent. Math. 110, 409-418 (1992; Zbl 0779.22006)], the author obtains results for the dual pair \(G_{\mathcal G} \times \text{PGL}_2\). This generalizes in a natural way results of \textit{R. Howe} [J. Funct. Anal. 32, 297-303 (1979; Zbl 0408.22018)] and others in the classical situation, i.e., when \(\mathcal G\) is the algebra of symmetric \(n\times n\) matrices over \(F\). By this generalization it is shown that the dual pair is \(O(n)\times \text{SL}_2\) and the minimal representation is the Weil representation.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17C30
zbMATH DE Number: 703880
zbMATH Keywords: spherical representation
zbMATH Keywords: wavefront set
zbMATH Keywords: automorphism group
zbMATH Keywords: Jordan algebra
zbMATH Keywords: dual pair
zbMATH Keywords: minimal representation
zbMATH Keywords: Weil representation
reviewed by: Zhonghu Chen
MaRDI profile type: Publication
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OpenAlex ID: W2055373141
description: scientific article; zbMATH DE number 703880