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

In this paper, we give a complete picture of Howe correspondence for the setting ($O(E, b), Pin(E, b), \Pi$), 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 $\Pi$ is the spinorial representation of $Pin(E, b)$. More precisely, for a dual pair ($G, G'$) in $O(E, b)$, we determine explicitly the nature of its preimages $(\tilde{G}, \tilde{G'})$ in $Pin(E, b)$, and prove that apart from some exceptions, $(\tilde{G}, \tilde{G'})$ is always a dual pair in $Pin(E, b)$; then we establish the Howe correspondence for $\Pi$ with respect to $(\tilde{G}, \tilde{G'})$.