Vogan duality for nonlinear type \(B\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1246.22017
DOI: 10.1090/S1088-4165-2011-00398-8
author: B. Scott Crofts
publication date: 2012-06-07T00:00:00Z
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 6043179
zbMATH Keywords: character duality
zbMATH Keywords: nonlinear groups
zbMATH Keywords: Kazhdan-Lusztig theory
zbMATH Keywords: extended Hecke algebra
zbMATH Keywords: spin groups
zbMATH Keywords: genuine representations
MaRDI profile type: Publication
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title: Vogan duality for nonlinear type \(B\)
published in: Representation Theory
review text: Let \(\mathbb G=\mathrm{Spin}(4n+1)\) be the connected, simply connected complex (reductive) Lie group of type \(B_{2n}\), and let \(G=\mathrm{Spin}(p,q)\) with \(p+q=4n+1\) denote a real form of \(\mathbb G\). If \(q\notin\{0,1\}\) then \(G\) has a nonlinear two-fold covering group \(\widetilde G=\widetilde{\mathrm{Spin}}(p,q)\). The main result of this paper is a certain special symmetry in the set of genuine representation parameters for the various \(\widetilde G\) at half-integral (non-singular) infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized Hecke modules and ultimately results in a character multiplicity duality for the genuine characters of \(\widetilde G\).NEWLINENEWLINEGiven a (genuine) irreducible admissible representation \(\pi\) of \(\widetilde G\), let \(X(\pi)\) be the standard module that contains \(\pi\) as its Langlands subquotient. Because the character of a standard module is more readily computable than that of an irreducible, it is of interest to study the composition factors of \(X(\pi)\) via expressions of the form NEWLINE\[NEWLINE X(\pi)=\sum_{\sigma}m(\sigma,\pi)\sigma.NEWLINE\]NEWLINE Here the \(\sigma\) run over the \textit{block} of representations containing \(\pi\): a certain finite set of irreducible representations of \(\widetilde G\) with the same infinitesimal character as that of \(\pi\). The \(m(\sigma,\pi)\) are nonnegative integers. This sum may be interpreted as an identity in an appropriate Grothendieck group, or as a character identity. Similarly, we have the problem of finding an expression for \(\pi\) in terms of the standard representations (now with possibly negative coefficients) NEWLINE\[NEWLINE\pi=\sum_{\sigma}M(\sigma,\pi)X(\sigma).NEWLINE\]NEWLINE Given a block \(\mathcal B=\{\pi_1,\pi_2,\dots,\pi_r\}\) of representations of \(\widetilde G\) (or their parameters), the author defines a group \(\widetilde G'\) and a block \(\mathcal B^{\vee}=\{\sigma_1,\dots,\sig…
reviewed by: Annegret Paul
description: scientific article; zbMATH DE number 6043179