Asymptotics of spherical superfunctions on rank one Riemannian symmetric superspaces

TL;DR

This paper computes the Harish-Chandra c-function for rank-one symmetric superspaces, revealing its meromorphic nature and asymptotic behavior, with special cases expressed via Jacobi polynomials, advancing understanding of spherical superfunctions.

Contribution

It provides explicit formulas for the c-function and asymptotic expansions of spherical superfunctions on rank-one symmetric superspaces, extending classical results to the supersymmetric setting.

Findings

c-function expressed in terms of Euler Gamma functions

c-function poles are shifted into the right half-plane compared to the even case

full asymptotic expansion of spherical superfunctions derived

Abstract

We compute the Harish-Chandra $c$-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces $X=G/K$ in terms of Euler $\Gamma$ functions, proving that it is meromorphic. Compared to the even case, the poles of the $c$-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on $X$. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.