Vogan duality for nonlinear type B

Abstract

Let G = S p i n [ 4 n + 1 ] \mathbb {G}=\mathrm {Spin}[4n+1] be the connected, simply connected complex Lie group of type B 2 n B_{2n} and let G = S p i n ( p , q ) G=\mathrm {Spin}(p,q) ( p + q = 4 n + 1 ) (p+q=4n+1) denote a (connected) real form. If q ∉ { 0 , 1 } q \notin \left \{0,1\right \} , G G has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by G ~ = S p i n ~ ( p , q ) \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 G ~ \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 G ~ \tilde {G} .