Minimal representations of the metaplectic Lie supergroup and the super Segal-Bargmann transform

TL;DR

This paper develops minimal models of the metaplectic Lie supergroup's representation, connecting Schr"odinger and Fock models, and introduces a super Segal-Bargmann transform extending the classical version.

Contribution

It constructs new minimal Schr"odinger and Fock models for the metaplectic Lie supergroup and introduces a super Segal-Bargmann transform extending classical theory.

Findings

Fock model of the minimal representation enables a new Fock model of the metaplectic representation.

Constructed an intertwining super Segal-Bargmann transform extending classical results.

Connected Schr"odinger models of minimal and metaplectic representations.

Abstract

We construct a Schr\"odinger model and a Fock model of a minimal representation of the metaplectic Lie supergroup $\mathrm{Mp}(2m|2n,2n)$. Then, we show that the Schr\"odinger model of the minimal representation leads to an already known Schr\"odinger model of the metaplectic representation of $\mathrm{Mp}(2m|2n,2n)$. Therefore, the Fock model of the minimal representation allows us to construct a Fock model of this metaplectic representation. We then construct an intertwining super Segal-Bargmann transform which extends the classical Segal-Bargmann transform.