Spherical functions of the principal series representations of \(Sp(2,\mathbb{R})\) as hypergeometric functions of \(C_2\)-type (Q679149)

In this article we determine explicitly the systems of differential equations satisfied by spherical functions with nontrivial \(K\)-types of the principal series and the generalized principal series representations of \(Sp(2,\mathbb{R})\). 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 \(G\), \(P_0=M_0A_0N_0\) be a parabolic subgroup of \(G\). Let \(H_\pi\) be an admissible representation of \(G\) and \((\tau,V_\tau)\), \((\eta,V_\eta)\) be irreducible representations of \(K\) which is contained in \(H_\pi\). We call elements of \(\Hom_K(V_\tau, C^\infty_\eta (K \setminus G)) \simeq C^\infty_\eta (K\setminus G)\otimes_K V^*_\tau =C^\infty_{ \eta, \tau}(K\setminus G/K)\) spherical functions of type-\((\eta,\tau)\), where \(C^\infty_\eta(K\setminus G)\) is the space of smooth sections of the homogeneous vector bundle over \(K\setminus G\) associated to \(V_\eta\) and \(V^*_\tau\) is the contragredient representation of \(V_\tau\). Let \(\varphi\in \Hom_{({\mathfrak g},K) }(H_\pi,C^\infty_\eta(K\setminus G))\) and \(i\in \Hom_K(V_\tau, H_\pi)\), then \(\varphi\circ i\) is a spherical function attached to \(H_\pi\). There are many studies on the system of differential equations satisfied by spherical functions for 1-dimensional \(K\)-types. Moreover they are generalized as the Weyl group invariant commuting differential operators with continuous parameters, which are introduced by generalizing root multiplicities. In this article we treat \(Sp( 2,\mathbb{R})\) as \(G\) and the principal series and the generalized principal series representation as \(H_\pi\). We call \(H_{\pi_0}=\text{Ind}^G_P\) (a character of \(P)\) the principal series representation and \(H_{\pi_1}= \text{Ind}^G_{P_J} (\sigma)\) the generalized principal series representation. Here \(P=MAN\) is a minimal parabolic subgroup of \(G\), \(P_J=M_jA_jN_j\) is the Jacobi parabolic subgroup of \(G\) and \(\sigma\) is a tensor product of a discrete series representation of \(M_J\) and a character of \(A_JN_J\). We give explicit formulas of the systems of differential equations satisfied by spherical functions of \(H_{\pi_0}\) and \(H_{\pi_1}\) if \(H_\pi\) has the infinitesimal character, its spherical function is the eigenfunction of elements of \(Z({\mathfrak g})\), the center of the universal enveloping algebra \(U({\mathfrak g})\). \(Z({\mathfrak g})\) for \(G=Sp(2,\mathbb{R})\) is generated by two elements. One is the Casimir element of order 2, the other is of order 4. It is difficult to calculate the radial part of the latter operator with respect to \(KAK\)-decomposition. We avoid the difficulty by using shift operators, which are defined by means of the Schmid operator. Its name comes from the property of shifting the parameter \(K\)-types. Moreover, this method is useful for studying the reducibility of the differential equations for \(H_{\pi_1}\). We choose \(\tau,\eta\) from \(K\)-types of minimal dimension in \(H_\pi\), which is 1- or 2-dimensional. We can obtain spherical functions for higher dimensional \(K\)-types from those for minimal dimensional \(K\)-types and shift operators in principle. We shall give series expansions and integral representations for the solutions of the system of the differential equations of \(H_{\pi_1}\).