The theta correspondence for similitudes
type: Item
instance of: scholarly article
title: The theta correspondence for similitudes
zbMATH Open document ID: 0870.22011
DOI: 10.1007/BF02762709
author: Brooks Roberts
published in: Israel Journal of Mathematics
publication date: 1997-09-03T00:00:00Z
review text: Let \(X\) be a nondegenerate finite dimensional quadratic space and \(Y\) a nondegenerate finite dimensional symplectic space over a nonarchimedean local field. The restriction of the oscillator (or Weil-) representation of the metaplectic cover of \(Sp(X\otimes Y)\) to the inverse image of \(O(X)Sp(Y)\) under the covering map gives rise to the local theta correspondence of admissible representations of \(H=O(X)\) and of \(G=Sp(Y)\). This correspondence has been extensively studied because of its importance in the (global) theory of automorphic forms as well as in the local representation theory. In particular, the fundamental conjectures of Roger Howe about (strong) duality and multiplicity preservation have been proven by Howe and Waldspurger with the exception of the case of residual characteristic \(2\). In applications to the theory of automorphic forms and to Shimura varieties, however, one often needs corresponding statements for groups of orthogonal and symplectic similitudes instead of isometries. The attempt to generalize the results to this situation faces the difficulty that the inverse images of the groups of similitudes of \(X\) and \(Y\) in the metaplectic cover of the group of similitudes of \(X\otimes Y\) do not commute. Barthel studied this problem and found a correspondence between certain sets of representations of either group. The present article reconsiders the problem with a different method, studying representations \(\Omega\) of \(G'\times H\) and \(\omega\) of the subgroup of pairs of equal similitude norm, where \(G'\) is either the symplectic similitude group of \(Y\) or its metaplectic cover, depending on the parity of the dimension of \(X\). Both representations have been studied previously. The detailed investigation of the resulting duality properties that is undertaken here has been missing so far and is an extremely useful tool for further investigations in this area. The author first gives a proof of the fact that Howe duality and…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 902626
zbMATH Keywords: Weil representation
zbMATH Keywords: oscillator representation
zbMATH Keywords: Howe duality
zbMATH Keywords: theta correspondence
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 902626