Functional Equations of Whittaker Functions on p-Adic Groups

Abstract

Introduction. The Fourier coefficients of Eisenstein series on a rational tube domain have Euler product expansions, and one can view the Euler factors as functions of the weight of the Eisenstein series. Our main result is the existence of a functional equation for these functions, the local singular series. To prove this we observe that the local singular series is the value of a Whittaker function W attached to a class 1 principal series representation PS (X) induced from a character X on P, a maximal parabolic subgroup of G. That is, W transforms under left translation by elements of the unipotent radical, N, of P according to a character 4. The crucial point is to show that for suitable 4 there is only one operator, up to scalar multiple, that intertwines PS(x) with the space of Whittaker functions for 4. Functional equations then follow from the existence of intertwining operators corresponding to certain elements of the Weyl group. In contrast to [9] and [12], in dealing with Whittaker models we work with a horocycle N that is not maximal. This is necessary because we are dealing with degenerate principal series representations. In a separate paper, we apply the functional equations to the computation of local singular series for Baily's exceptional modular group. We now summarize the contents of the present paper. In Section 1 we introduce the principal series representations PS(X) on G and generic characters. By means of the Frobenius Reciprocity Law and a filtration on the space PS(x) deriving from the cellular decomposition of G with respect to a pair of parabolic subgroups, we reduce the uniqueness theorem to a study of certain integrals. In Section 2 we prove a technical lemma about p-adic fiber spaces and use it to show that the integrals mentioned above must all vanish. In Section 3 we prove the uniqueness theorem (3.2). Then we show how to obtain Whittaker maps as Cauchy principal value integrals. This gives rise to analytic families of distributions on G. As a consequence of