Entire vectors and holomorphic extension of representations

Abstract:

Let $G$ be a connected, simply connected real Lie group and let $U$ be a representation of $G$ in a complete, locally convex, topological vector space $\mathcal {J}$. If $G$ is solvable, it can be canonically embedded in its complexification ${G_c}$. A vector $v \in \mathcal {J}$ is said to be entire for $U$ if the map $g \to {U_g}v$ of $G$ into $\mathcal {J}$ is holomorphically extendible to ${G_c}$. The space of entire vectors is an invariant subspace of the space of analytic vectors. $U$ is said to be holomorphically extendible iff the space of entire vectors is dense. In this paper we consider the question of existence of holomorphic extensions We prove Theorem. A unitary representation $U$ is holomorphically extendible to ${G_C}$ iff $G$ modulo the kernel of $U$ is type $R$ in the sense of Auslander-Moore [1]. In the process of proving the above results, we develop several interesting characterizations of entire vectors which generalize work of Goodman for solvable Lie groups and we prove a conjecture of Nelson concerning the relationship between infinitesimal representations of Lie algebras and representations of the corresponding Lie groups.