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Transactions of the American Mathematical Society

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Entire vectors and holomorphic extension of representations. II

Author: Richard Penney
Journal: Trans. Amer. Math. Soc. 191 (1974), 195-207
MSC: Primary 22E45
Part I: Trans. Amer. Math. Soc. 198 (1974), 107-121
MathSciNet review: 0364556
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Abstract: Let G be a connected, simply connected Lie group and let $ {G_c}$ be its complexification. Let U be a unitary representation of G. The space of vectors v at which U is holomorphically extendible to $ {G_c}$ is denoted $ \mathcal{H}_\infty ^\omega (U)$. In [9] we characterized those U for which $ \mathcal{H}_\infty ^\omega $ is dense. In the present work we study $ \mathcal{H}_\infty ^\omega $ as a topological vector space, proving e.g., that $ \mathcal{H}_\infty ^\omega $ is a Montel space if U is irreducible and G is nilpotent. We prove a representation theorem for $ (\mathcal{H}_\infty ^\omega )'$ which yields a Bergman kernel type theorem for G. As an application we give a necessary and sufficient condition for the set of holomorphic functions on certain solvmanifolds to separate points.

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Keywords: Analytic vector, Lie group, solvmanifold
Article copyright: © Copyright 1974 American Mathematical Society

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