Derivatives and Asymptotics of Whittaker functions

Let F be a p-adic field, and Gn one of the groups GL(n, F ), GSO(2n−1, F ), GSp(2n, F ), or GSO(2(n − 1), F ). Using the mirabolic subgroup or analogues of it, and related “derivative” functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of Gn, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by Zn the center of Gn, and by Nn the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in L2(ZnNn\Gn) in terms of these characters. This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in L2(ZnNn\Gn) are the generic discrete series; we prove it for the group Gn. Introduction Let Gn be the points of one of the groups GL(n), GSO(2n− 1), GSp(2n), or GSO(2(n− 1)) over a p-adic field K. The main result (Theorem 2.1) of this work describes the asymptotic behaviour of the restriction of Whittaker functions to the standard maximal torus, in terms of a family of characters which is minimal in some sense. From results of [L-M], this restriction can be described for split reductive groups in terms of cuspidal exponents. Here, after having defined analogues of the mirabolic subgroup for the groups Gn, and the corresponding derivative functors, following [C-P] (where the case of completely reducible derivatives is treated for GL(n)), we choose to describe the restriction of Whittaker functions to the torus in terms of central exponents of the derivatives. This description, inspired by [B], is better adapted to understanding when the Whittaker model of a unitary generic representation is a subspace of L(ZnNn\Gn) (these representations are conjectured to be generic discrete series by Lapid and Mao). In the first section, we review the groups in question and define their mirabolic subgroups. We also give a decomposition of the unipotent radical of the standard Borel subgroup, and a description of how nondegenerate characters of this radical behave with respect to this decomposition. In Section 2, we give properties of the derivative functors, and use them to prove our asymptotic expansion of Whittaker functions, which is Theorem 2.1. In Section 3, we characterize generic representations with Whittaker model included in L(ZnNn\Gn) in terms of central exponents of the derivatives, in Corollary 3.1. We then prove in Theorem 3.2 the conjecture 3.5 of [L-M]. Nadir Matringe, University of East Anglia, School of Mathematics, Norwich, UK, NR4 7TJ. Email: [email protected]