Contragredients and a multiplicity one theorem for general Spin groups

Abstract

Each orthogonal group Opnq has a nontrivial GLp1q-extension, which we call GPinpnq. The identity component of GPinpnq is the more familiar GSpinpnq, the general Spin group. We prove that the restriction to GPinpn ´1q of an irreducible admissible representation of GPinpnq over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for GSpinpnq. The case for GPinpnq is the analogue of the theorem for Opnq proven by Aizenbud, Gourevitch, Rallis and Schiffmann, and the case for GSpinpnq the one for SOpnq by Waldspurger.