Special functions associated with K-types of degenerate principal series of ${\rm Sp}(n,\mathbb{C})$

TL;DR

This paper investigates special functions linked to K-types of degenerate principal series representations of the complex symplectic group, using two models to derive hypergeometric and Bessel function solutions.

Contribution

It introduces explicit descriptions of K-finite vectors in these representations via hypergeometric and Bessel functions using two different realizations.

Findings

K-finite vectors expressed as hypergeometric solutions in the compact model.

K-finite vectors involving Bessel functions in the non-compact model.

Establishment of explicit formulas connecting representation theory and special functions.

Abstract

We study $K$-types of degenerate principal series of ${\rm Sp}(n,\mathbb{C})$ by using two realisations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the non-compact picture via an appropriate partial Fourier transform. In the first case we find a family of $K$-finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of $K$-finite vectors whose expressions involve Bessel functions.