Dual pairs in pin(p,q) and Howe correspondences for the spin representation

In this paper we obtain examples of dual pairs in the group Pin(V x W) by considering the inverse images of the subgroups O(V) x Id(W) and Id(V) x O(W) under the double covering map pi: Pin(V x W) --> O(V x W). The main technical results are the definition of the group of determinant graded double covers of O(V) and the fact that the map W bar right arrow pi(-1)(O(V) x Id(W)) defines a homomorphism from the Witt group to this group when the ground field k satisfies k*/(k*)(2) congruent to Z(2) and -1 is not a square in k. We also show that the spin representation of Pin(V x W) sets up Howe correspondences for each of these dual pairs when k = R. (C) 1998 Academic Press.