Categories of unitary representations of Banach-Lie supergroups and restriction functors

TL;DR

This paper establishes foundational properties of smooth and analytic unitary representations of Banach-Lie supergroups, including restriction functors and category isomorphisms, extending finite-dimensional results to the infinite-dimensional setting.

Contribution

It proves that restriction functors are well-defined for Banach-Lie supergroups and shows an isomorphism between analytic and smooth representation categories, extending finite-dimensional theories.

Findings

Restriction of representations to subsupergroups is well-defined.

Analytic representations form a subcategory of smooth representations.

An analytic realization of the oscillator representation is provided.

Abstract

We prove that the categories of smooth and analytic unitary representations of Banach--Lie supergroups are well-behaved under restriction functors, in the sense that the restriction of a representation to an integral subsupergroup is well-defined. We also prove that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach--Lie supergroups. They extend the known results for finite dimensional Lie supergroups. In the infinite dimensional case the proofs require several new ideas. As an application, we give an analytic realization of the oscillator representation of the restricted orthosymplectic Banach--Lie supergroup.