Spherical representations of Lie supergroups

TL;DR

This paper extends the Cartan-Helgason theorem to reductive symmetric supergroups, computes the Harish-Chandra c-function for these spaces, and demonstrates self-duality of spherical representations in a specific type.

Contribution

It generalizes classical representation theory results to the supergroup setting and computes key harmonic analysis functions for symmetric superspaces.

Findings

Generalization of the Cartan-Helgason theorem to supergroups

Computation of the Harish-Chandra c-function for symmetric superspaces

Proof that all spherical representations are self-dual in type AIII|AIII

Abstract

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an application, we show that all spherical representations are self-dual in type AIII|AIII.