Unitary representations of nilpotent super Lie groups

TL;DR

This paper extends classical representation theory to nilpotent super Lie groups, providing a geometric parametrization of irreducible unitary representations and generalizing the Stone-von Neumann theorem.

Contribution

It introduces a method to construct irreducible unitary representations of nilpotent super Lie groups via induction from special subgroups, generalizing Kirillov theory.

Findings

Representation parametrization by coadjoint orbits

Construction of representations from polarizing subgroups

Generalization of Stone-von Neumann theorem

Abstract

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.