Analytic extension techniques for unitary representations of Banach-Lie groups

TL;DR

This paper extends analytic continuation techniques for unitary representations from finite-dimensional groups to Banach-Lie groups, establishing a correspondence between *-representations of Olshanski semigroups and unitary representations of complexified groups.

Contribution

It generalizes the L"uscher--Mack Theorem to Banach-Lie groups and characterizes the unitary representations obtained via analytic continuation from *-representations of Olshanski semigroups.

Findings

Any locally bounded *-representation with smooth vectors extends to a unitary representation of the complexified group.

The construction characterizes which unitary representations of the complexified group arise from Olshanski semigroup representations.

Semibounded unitary representations extend holomorphically to complex Olshanski semigroups.

Abstract

Let $(G,\theta)$ be a Banach--Lie group with involutive automorphism $\theta$, $\g = \fh \oplus \fq$ be the $\theta$-eigenspaces in the Lie algebra $\g$ of $G$, and $H = (G^\theta)_0$ 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 $H \times W \to S, (h,x) \mapsto h \exp x$ is a diffeomorphism. Any such semigroup carries an involution * satisfying $(h\exp x)^* = (\exp x) h^{-1}$. Our central result, generalizing the L\"uscher--Mack Theorem for finite dimensional groups, asserts that any locally bounded *-representation $\pi \: S \to B(\cH)$ 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