ON p-ADIC GENERALIZED WHITTAKER FUNCTIONS ON GSp(4)

Let G be a connected reductive algebraic group defined over a non-Archimedean local field F . The study of Whittaker functions on G has a long history. For example, if G = GL(2), an explicit formula for the Whittaker function is well-known (See [5]). Shintani proved an explicit formula for the case G = GL(n) and expressed it by Schur polynomials in [16]. Casselman and Shalika generalized Shintani’s result to the case where G is split in [3]. Li further generalized it to the case where G is quasi-split in [8]. Reeder [14] etc. also studied p-adic Whittaker functions. However, the Whittaker functions we will consider may be a little different from theirs because our interest is the functions as the Fourier coefficients of the Eisenstein series described as follows. In this paper, we mainly consider the local theory of Whittaker functions. However, we review the global theory of the Fourier expansion of the Eisenstein series for the Borel group of G = GSp(4) to describe our motivation of this study. Though the notion of the Fourier expansion has yet to be established for general reductive groups, it is known that there is such a notion for G = GSp(4). Let k be a number field and A its adele ring. We use the notation Gk, GA for the group of k-rational points and the adelization of G respectively. Let B be the Borel subgroup of G whose unipotent radical N is generated by matrices in the form