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

We study $K$-types of degenerate principal series of $\mathrm{S}\mathrm{p}(n,\mathbb{C})$ 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.