Multiplicity one theorem for general Spin groups: The Archimedean case

Abstract Let $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$ ) be a general Spin group (resp., a general Pin group) associated with a nondegenerate quadratic space V of dimension n over an Archimedean local field F . For a nondegenerate quadratic space W of dimension $n-1$ over F , we also consider $\operatorname {GSpin}(W)$ and $\operatorname {GPin}(W)$ . We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ( $\operatorname {GSpin}(V), \operatorname {GSpin}(W)$ ) and also for a pair of groups ( $\operatorname {GPin}(V), \operatorname {GPin}(W)$ ); namely, we prove that the restriction to $\operatorname {GSpin}(W)$ (resp., $\operatorname {GPin}(W)$ ) of an irreducible Casselman–Wallach representation of $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$ ) is multiplicity free.