Vogan duality for nonlinear type B

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 $\tilde{G}$.