Spherical representations of Lie supergroups

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.

37 pages; title changed; substantially revised version; accepted for publication, J. Func. Anal. (2014)

References in corpus (5)

• Classification of Topological Insulators and Superconductors
• Invariant Berezin integration on homogeneous supermanifolds
• Integration on Non-Compact Supermanifolds
• The Harish-Chandra isomorphism for reductive symmetric superpairs
• Cartan--Helgason theorem, Poisson transform, and Furstenberg--Satake compactifications

Cited by in corpus (2)

• The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators
• Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak{q}(n)$