Generalized Whittaker functions on GSp(2,R) associated with indefinite quadratic forms

We study the generalized Whittaker models for G = GSp(2,R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series representation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions. 0. Introduction. Let G = GSp(2) be the symplectic group with similitude defined over the field Q of rational numbers. When we wish to write down the Fourier expansion of automorphic forms on GA along its Siegel parabolic subgroup, we necessitate not only the Whittaker models but also the generalized Whittaker models (see Section 1 for details). Our concern in this paper is the local theory of generalized Whittaker models at the real place, which still lies in an intermediate state. For example, in the paper [PS], which had circulated since the late 1970s, I. I. Piatetski-Shapiro stated the multiplicity free theorem of such models for G := GR = GSp(2,R) without a proof ([PS, Theorem 3.1]). However nobody seems to establish it up to today. For the generalized Whittaker model for G associated with a definite binary quadratic form, there are some results supporting Piatetski-Shapiro’s assertion. Besides H. Yamashita’s result [Y] in a general setting, there are several detailed studies for specific kinds of representations of G ([Ni], [Mi-1], [Mi-2], [Is]), where the multiplicity free results as well as explicit formulae of generalized Whittaker functions are obtained (see Subsection 8.3). On the other hand, little is known about the generalized Whittaker models associated 2000 Mathematics Subject Classification. Primary 11F46; Secondary 22E30.