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

Stéphane Merigon, Karl-Hermann Neeb and Hadi Salmasian

Vol. 257 (2012), No. 2, 431–469

DOI: 10.2140/pjm.2012.257.431

Abstract

We prove two results which show that the categories of smooth and analytic unitary representations of a Banach–Lie supergroup are well-behaved. The first result states that the restriction functor corresponding to any homomorphism of Banach–Lie supergroups is well-defined. The second result shows 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.

Keywords

Banach–Lie supergroups, unitary representations, smooth vectors, analytic vectors

Mathematical Subject Classification 2010

Primary: 17B65, 22E66

Milestones

Received: 29 June 2011

Revised: 25 August 2011

Accepted: 20 December 2011

Published: 4 July 2012