Multiplicity One Theorem for General Spin Groups: The Archimedean Case

Let $\text{GSpin}(V)$ (resp. $\text{GPin}(V)$) 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 $\text{GSpin}(W)$ and $\text{GPin}(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($\text{GSpin}(V),\text{GSpin}(W)$) and also for a pair of groups ($\text{GPin}(V),\text{GPin}(W)$); namely, we prove that the restriction to $\text{GSpin}(W)$ (resp. $\text{GPin}(W)$) of an irreducible Casselman-Wallach representation of $\text{GSpin}(V)$ (resp. $\text{GPin}(V)$) is multiplicity free.