Correspondence between Representations of the Dual Reductive Pair

Ilya Piatetski-shapiro, M. Teicher

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Abstract

We state a connection between the Weil representation of the dual reductive pair O(n) and SL 2 , and prove that they have eigenvectors in common. 1. Introduction In this exposition we continue to study the Weil representation of Sp(2n; k) that we studied in "Commutativity of the Schur algebra". There we were interested in the decomposition of its induced representation whereas here we are interested in the irreducible components of its restriction to the product of the dual reductive pair SL 2 and O(n): Dual reductive pairs are playing an important role in the study of representation theory. We shall prove here a result on the correspondence between representations of a dual reductive pair, based on Schur theory. We recall the definition of a dual reductive pair. Definition. A dual reductive pair for Sp(2n; k) is a pair G; H of reductive subgroups of Sp(2n; k): Each is the full centralizer of the other. (See Howe [1].) 2. Construction of the dual reductive pair O(n); SL 2 We are ... Correspondence between Representations of the Dual Reductive Pair. Available from: https://www.researchgate.net/publication/2814240_Correspondence_between_Representations_of_the_Dual_Reductive_Pair [accessed Jan 3, 2016].
Original languageAmerican English
Pages (from-to)145-148
JournalBeitrage zur Algebra und Geometrie
Volume38
StatePublished - 1997

Bibliographical note

Printed also (with permission)
in: Sitzungsberichte der Berliner Mathematischen Gesellschaft 1993-
1996 (1998), p. 103-106

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