Special functions associated with K -types of degenerate principal series of Sp ( n , C )

Abstract

We study $K$-types of degenerate principal series of $Sp(n,\mathbb{C})$ by using two realizations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the noncompact 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.