FUNCTORS GIVEN BY KERNELS, ADJUNCTIONS AND DUALITY

Let X-1 and X-2 be schemes of finite type over a field of characteristic 0. Let Q be an object in the category D-mod(X-1 x X-2) and consider the functor F : D-mod(X-1) -> D-mod(X-2) defined by Q. Assume that F admits a right adjoint also defined by an object P in D-mod(X-1 x X-2). The question that we pose and answer in this paper is how P is related to the Verdier dual of Q. We subsequently generalize this question to the case when X-1 and X-2 are no longer schemes but Artin stacks, where the situation becomes much more interesting.