Principal series Whittaker functions on \(\text{Sp}(2;{\mathbb{R}})\). Explicit formulae of differential equations (Q2705879)

Let \(G\) be the real symplectic group of rank 2, \(\text{Sp}(2;{\mathbb{R}})\). Let \((\pi,H_\pi)\) be an irreducible admissible principal series of \(G\) with respect to the minimal parabolic subgroup \(P=MAN\). For a continuous character \(\eta\colon N\to {C}^*\), let \(C^\infty_\eta(N\backslash G)\) be the space of complex-valued \(C^\infty\)-functions of \(G\) satisfying \(f(n,g)=\eta(n)\cdot f(g)\) for any \(n\in {N}\), \(g\in G\). \(C^\infty_\eta(N\backslash G)\) is a \((g,K)\)-module via the right regular action of \(G\). The intertwining space \(\text{Hom}_{g,K}(H_\pi, C^\infty_\eta(N\backslash G))\) is the space of algebraic Whittaker vectors. Let \((\delta,V_\delta)\) \((\delta\in\hat K)\) be a \(K\)-type that occurs with multiplicity one in \(H_\pi\). Elements in the image of the map NEWLINE\[NEWLINE\text{Hom}_{g,K}(H_\pi,C^\infty_\eta(N\backslash G))\to C^\infty_\eta(N\backslash G)\otimes V^*_\deltaNEWLINE\]NEWLINE are called Whittaker functions with \(K\)-type \(\delta^*\) belonging to \(\pi\). The restrictions of Whittaker functions to the split component \(A\) satisfy a system of partial differential equations.NEWLINENEWLINENEWLINEIn this paper the authors compute these equations explicitly. One of the differential equations in question is given by the Casimir operator. The other operators are gradient type operators called Schmid operators. The latter are differential operators commuting with the Casimir operator in \(C^\infty_\eta(N\backslash G)\). The results of this paper have applications in number theory and in the theory of the quantum mechanical Toda lattice associated to the root system \(C_2\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00022].