Howe duality and the trace formula
type: Item
instance of: scholarly article
title: Howe duality and the trace formula
zbMATH Open document ID: 1008.22008
author: Zhengyu Mao
author: Steve Rallis
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: One way to study a conjectured correspondence between automorphic representations of two groups \(G, G'\) is by the comparison of two ``trace formulas''. Each of the two trace formulas has a geometric side and a representation-theoretic side. The comparison of the geometric sides amounts to the matching of local orbital integrals, with, in particular, the ``fundamental lemma'', saying that corresponding Hecke functions are matching. The authors of the present paper observed that, when \(G, G'\) form a dual reductive pair and the correspondence is the Howe duality, then Howe duality for spherical functions may be used to give an easy proof for the fundamental lemma. This is done in the paper for the dual pair \(\text{SO}(2n+1), \widetilde {\text{SL}}(2)\). The authors also consider the dual pair \(\text{SO}(2n+1), \widetilde {\text{Sp}}(2n)\), treated by \textit{M. Furusawa} in [J. Reine Angew. Math. 466, 87-110 (1995; Zbl 0827.11032)]. They introduce trace formulas involving a theta series on \(\text{SO}(2n+1) \times \widetilde{\text{Sp}}(2n)\) and prove the matching of local integrals and the fundamental lemma, using again Howe duality for spherical functions.
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 1463734
zbMATH Keywords: automorphic representations
zbMATH Keywords: trace formulas
zbMATH Keywords: dual reductive pair
zbMATH Keywords: Howe duality
zbMATH Keywords: theta series
zbMATH Keywords: spherical functions
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: pjm.1999.191.329
OpenAlex ID: W2054721073
DOI: 10.2140/PJM.1999.191.329
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description: scientific article; zbMATH DE number 1463734