Analytic extension techniques for unitary representations of Banach-Lie groups

2011/02/01 by Stéphane Merigon, Merigon, Stéphane, Karl-Hermann Neeb +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT) #math.FA #math.RT

paper · pdf · doi:10.48550/arxiv.1102.0213

26 pages

arxiv created 2011/02/01 · arxiv updated 2011/02/02

Abstract

Let (G,θ) be a Banach--Lie group with involutive automorphism θ, \g = \fh ⊕ \fq be the θ-eigenspaces in the Lie algebra \g of G, and H = (G^θ)₀ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup S \subeq G of the form S = H exp(W), where W is an open \Ad(H)-invariant convex cone in \fq and the polar map H × W → S, (h,x) ↦ h exp x is a diffeomorphism. Any such semigroup carries an involution * satisfying (hexp x)^* = (exp x) h^-1. Our central result, generalizing the Lüscher--Mack Theorem for finite dimensional groups, asserts that any locally bounded *-representation π S → B(\cH) with a dense set of smooth vectors defines by "analytic continuation" a unitary representation of the simply connected Lie group G_c with Lie algebra \g_c = \fh + i \fq. We also characterize those unitary representations of G_c obtained by this construction. With similar methods, we further show that semibounded unitary representations extend to holomorphic representations of complex Olshanski semigroups

Related

• Integrability of unitary representations on reproducing kernel spaces
• On the dual of complex Olshanskii semigroups
• On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
• Categories of unitary representations of Banach-Lie supergroups and restriction functors
• On the existence of regular vectors