Analytic extension techniques for unitary representations of Banach-Lie groups - Merigon, Stephane; Neeb, Karl Hermann (International Mathematics Research Notices, 2012)
Let (G,θ) be a Banach–Lie group with involutive automorphism θ, be the θ-eigenspaces in the Lie algebra of G, and H=(Gθ)0 be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup S⊆G of the form , where W is an open Ad(H)-invariant convex cone in and the polar map is a diffeomorphism. Any such semigroup carries an involution * satisfying . Our central result, generalizing the Lüscher–Mack theorem for finite-dimensional groups, asserts that any locally bounded *-representation with a dense set of smooth vectors defines by “analytic continuation” a unitary representation of the simply connected Lie group Gc with Lie algebra . We also characterize those unitary representations of Gc obtained by this construction. With similar methods, we further show that semibounded unitary representations extend to holomorphic representations of complex Olshanski semigroups.

Categories of unitary representations of Banach-Lie supergroups and restriction functors - Merigon, Stephane; Neeb, Karl Hermann; Salmasian, Hadi (Pacific Journal of Mathematics, 2012)
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.
Keywords: Banach–Lie supergroups; unitary representations; smooth vectors; analytic vectors

Integrability of unitary representations on reproducing kernel spaces - Merigon, Stephane; Neeb, Karl Hermann; Olafsson, Gestur (Representation Theory, 2015)
Keywords: Unitary representations; finite and infinite dimensional Lie groups; manifolds; reproducing kernel