Dual Pairs inPin(p, q) and Howe Correspondences for the Spin Representation

In this paper we obtain examples of dual pairs in the groupPin(V W) by considering the inverse images of the subgroupsO(V) IdWandIdV O(W) under the double covering map π:Pin(V W) → O(V W). The main technical results are the definition of the group of determinant graded double covers ofO(V) and the fact that the mapW π − 1(O(V) IdW) defines a homomorphism from the Witt group to this group when the ground fieldksatisfiesk*/(k*)2 2and −1 is not a square ink. We also show that the spin representation ofPin(V W) sets up Howe correspondences for each of these dual pairs whenk = .