Spherical Functions of the Principal Series Representations of $Sp(2, \mathbf R)$ as Hypergeometric Functions of $\mathrm C_2$-Type

Abstract

In this article we determine explicitly the systems of differential equations satisfied by spherical functions with non-trivial K-typQs of the principal series and the generalized principal series representations of Sp(2, R). Then we obtain series expansions and integral formulas of spherical functions of the generalized principal series representation. We shall define spherical functions. Let G be a real reductive Lie group and K be maximal compact subgroup of G, Po=MoA0No be a parabolic subgroup of G. Let Hn be an admissible representation of G and (r, Vr), (#, Vri) be irreducible representations of K which is contained in HK. We call elements of Horru( Vr, C7(K\G))^ C?(K\G)®KV? = C~,r(K\G/K) spherical functions of type-(#, r), where CTM(K\G) is the space of smooth sections of the homogeneous vector bundle over K\G associated to V? and V? is the contragredient representation of VT. Let 0E:Horri(ai K}(Hn, CTM(K\G}} and i^ Horrid Vr, HTC then °i is a spherical function attached to H*. There are many studies on the system of differential equations satisfied by spherical functions for 1-dimensional K-typzs. Moreover they are generalized as the Weyl group invariant commuting differential operators with continuous parameters, which are introduced by generalizing root multiplicities (cf. [DGl], [DG2], [HI], [HO], [Ko], [OO], [Opl], [Op2], [Os], [OS], [Sh]). On the other hand, there are few studies for vector-valued spherical functions. Besides, spherical functions are rarely calculated in explicit forms except for rank one cases. Therefore it is interesting to study vector-valued spherical functions of higher rank Lie group explicitly.