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 ), P i n ( E , b ),π), where O ( E , b ) is a real orthogonal group, P i n ( E , b ) is the two-fold Pin-covering of O ( E , b ), and π is the spinorial representation of P i n ( E , b ). More precisely, for a dual pair (G, G^') in O ( E , b ), we determine explicitly the nature of its preimages (G̃, G̃^̃'̃) in P i n ( E , b ), and prove that apart from some exceptions, (G̃, G̃^̃'̃) is always a dual pair in P i n ( E , b ); then we establish the Howe correspondence for π with respect to (G̃, G̃^̃'̃) .