Let (G, θ) be a Banach-Lie group with involutive automorphism θ, g = h ⊕ q 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 ⊆ G of the form S = H exp(W ), where W is an open Ad(H)invariant convex cone in q and the polar map H × W → S, (h, x) → 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 Lüscher-Mack Theorem for finite dimensional groups, asserts that any locally bounded *representation π : S → B(H) with a dense set of smooth vectors defines by "analytic continuation" a unitary representation of the simply connected Lie group Gc with Lie algebra gc = h + iq. 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.

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.

Abstract. Let g be a Banach-Lie algebra and τ : g → g an involution. Write g = h⊕q for the eigenspace decomposition of g with respect to τ and g c := h⊕iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g c . The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M . The kernel is assumed to satisfy a natural invariance condition with respect to an infinitesimal action β : g → V(M ) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group H with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K ∈ C −∞ (M × M ) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism β : g → V(M ). As a consequence, we get a generalization of the Lüscher-Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.

We give two integrability criteria for representations of Banach-Lie algebras as skew-symmetric unbounded operators on a dense domain of a Hilbert space. One of them is based on analytic vectors.

We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain $V\oplus i\Omega$ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of $GL(\Omega)$. Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density for the Plancherel measure involves quotients of $\Gamma$-functions and the $c$-function for a symmetric cone of smaller rank.Comment: 22 page