An explicit integral representation of Whittaker functions on \(\text{Sp}(2;\mathbb{R})\) for the large discrete series representations
The author remarks that it is striking that the Whittaker functions for the discrete series representations of \(\text{SL} (2,\mathbb{R})\) are much simpler functions in comparison to the general case. He sets out to determine whether a similar phenomenon takes place for \(\text{Sp} (2,\mathbb{R})\) (of rank 2). He develops a theory of such functions and determines which differential equations they satisfy. He shows also that for generic discrete series representations the space of solutions is of dimension 4; otherwise, and in general, it is of dimension 0 or 8. The functions of moderate growth in this space form a subspace of dimension 0 or 1. He is then able to determine this function, when it exists, as a simple integral involving a confluent hypergeometric function. This is done by verifying that it satisfies the differential equations.
\(\text{Sp} (2,\mathbb{R})\) (of rank 2)
scientific article; zbMATH DE number 650900
scientific article