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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Entire vectors and holomorphic extension of representations


Author: Richard Penney
Journal: Trans. Amer. Math. Soc. 198 (1974), 107-121
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1974-99938-X
Part II: Trans. Amer. Math. Soc. 191 (1974), 195-207
MathSciNet review: 0364556
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1974-99938-X
Keywords: Analytic vector, type-$ R$, Lie group, holomorphic extension
Article copyright: © Copyright 1974 American Mathematical Society