Dual pairs in the Pin-group and duality for the corresponding spinorial representation

Abstract

In this paper, we give a complete picture of Howe correspondence for the setting ($O(E,b),Pin(E,b),\hat{I}$), where $O(E,b)$ is an orthogonal group (real or complex), $Pin(E,b)$ is the two-fold Pin-covering of $O(E,b)$, and $\hat{I}$ is the spinorial representation of $Pin(E,b)$. More precisely, for a dual pair ($G,G^{\prime }$) in $O(E,b)$, we determine explicitly the nature of its preimages $(\tilde{G},\widetilde{G^{\prime }})$ in $Pin(E,b)$, and prove that apart from some exceptions, $(\tilde{G},\widetilde{G^{\prime }})$ is always a dual pair in $Pin(E,b)$; then we establish the Howe correspondence for $\hat{I}$ with respect to $(\tilde{G},\widetilde{G^{\prime }})$.

22 pages