Whittaker vectors and conical vectors

Abstract

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.