CALCULUS OF PRINCIPAL SERIES WHITTAKER FUNCTIONS ON SL(n,R)

We study Whittaker functions for the principal series representation of SL(n,R). We derive a system of partial differential equations characterizing our Whittaker functions. We give explicitly power series solutions at the regular singularity of the system, and integral representations of unique moderate growth Whittaker function. Introduction In this paper we explicitly determine the radial parts of Whittaker functions for the principal series representations of SL(n,R), by extending our previous works for n = 3, 4 ([MIO], [HnIO]). According to the celebrated result of Shalika [Sha], Fourier expansions (along a maximal parabolic subgroup) of cusp forms on GL(n) can be described in terms of Whittaker functions. Therefore precise information of local Whittaker functions plays important roles in various aspects of automorphic forms on GL(n), for example, analysis of local zeta integrals. The well-known formula of Shintani [Shi] asserts that the class one Whittaker functions over non-archimedean fields can be written by using the character of finite dimensional representations of GL(n,C). Combined with some facts on representation theory of GL(n,C), one can perform unramified computation for non-archimedean zeta integrals. This is a first step for a study of automorphic L-functions via Rankin-Selberg method. Our interest here is archimedean Whittaker functions. Let G = SL(n,R) and N the standard maximal unipotent subgroup of G consisting of upper triangular matrices. We fix a nondegenerate unitary character η of N and consider the induced representation C-IndN(η), whose representation space is C∞ η (N\G) = {φ ∈ C∞(G,C) | φ(xg) = η(x)φ(g) for all (x, g) ∈ N ×G}. For an irreducible admissible representation π of G, a realization of π in C∞ η (N\G) is called a Whittaker model of π. When π is the principal series representation πI,ν (see Definition 2.1), Kostant [Ko, §5] proved that the dimension of the intertwing space W(πI,ν , η) = Hom( C,K)(πI,ν , C∞ η (N\G)) is n!, the order of the Weyl group Sn. Here K = SO(n) is a maximal compact subgroup of G and g = gl(n,R). For a nonzero vector v in πI,ν and a nonzero intertwiner Φ in W(πI,ν , η), we call the function Φ(v) the Whittaker function. We will give explicit formula of Φ(v) for the vectors v belonging to minimal K-types of πI,ν . Date: April 27, 2009.