A Schr\"odinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;\alpha)$

TL;DR

This paper develops Schr"odinger and Fock models for the exceptional Lie superalgebra D(2,1;α), introducing an intertwining isomorphism analogous to the Segal-Bargmann transform, expanding the representation theory of Lie superalgebras.

Contribution

It constructs new infinite-dimensional irreducible representations for D(2,1;α) and introduces an intertwining isomorphism similar to the Segal-Bargmann transform.

Findings

Constructed Schr"odinger and Fock models for D(2,1;α)

Introduced an intertwining isomorphism analogous to Segal-Bargmann transform

Extended representation theory for exceptional Lie superalgebras

Abstract

We construct two infinite-dimensional irreducible representations for $D(2,1;\alpha)$: a Schr\"odinger model and a Fock model. Further, we also introduce an intertwining isomorphism. These representations are similar to the minimal representations constructed for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. The intertwining isomorphism is the analogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type.