The generalized Whittaker functions for several admissible representations of \(Sp(2,\mathbb{R})\)
This note studies the generalized Whittaker functions attached to various types of admissible representations \(\pi\) of the rank two symplectic group \(G=Sp(2,{\mathbb R})\), including the large discrete series. The Whittaker functions in question transform under a character \(\eta\) of the unipotent radical of the Siegel parabolic \(P\) and also by a character of the identity component of the stabilizer subgroup \(SO(\eta)\) in the Levi of \(P\). For given \(K\)-type (\(K\) a fixed maximal parabolic subgroup), these functions are determined by restriction to the split maximal torus \(A\) of \(G\). Under certain assumptions on both the character \(\eta\) and the representation \(\pi\), the author obtains an explicit system of differential equations for the restriction of such a generalized Whittaker function to \(A\) by using the \(K\)-equivariant shift operators defined by Schmid. He moreover observes that this system of differential equations is completely integrable, and in certain cases he expresses the solution in terms of integrals of products of \(J\)-Bessel functions of the first kind and hypergeometric functions. In all cases treated, he computes the Mellin transform of this integral, which arises as the archimedean part of the Rankin-Selberg integral for the spinor \(L\)-function obtained by Andrianov, and he shows that this Mellin transform is a product of Gamma functions.
admissible representations of the rank two symplectic group
scientific article; zbMATH DE number 1160952