Vogan duality for nonlinear type B

Crofts, Scott

doi:10.1090/S1088-4165-2011-00398-8

Vogan duality for nonlinear type B
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by Scott Crofts

Represent. Theory 15 (2011), 258-306

DOI: https://doi.org/10.1090/S1088-4165-2011-00398-8

Published electronically: March 24, 2011

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Abstract:

Let $\mathbb {G}=\mathrm {Spin}[4n+1]$ be the connected, simply connected complex Lie group of type $B_{2n}$ and let $G=\mathrm {Spin}(p,q)$ $(p+q=4n+1)$ denote a (connected) real form. If $q \notin \left \{0,1\right \}$, $G$ has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by $\tilde {G}=\widetilde {\mathrm {Spin}}(p,q)$. The main purpose of this paper is to describe a symmetry in the set of genuine parameters for the various $\tilde {G}$ at certain half-integral 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 $\tilde {G}$.

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