Holomorphic representations of nilpotent Lie groups |
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Authors: | Roe Goodman |
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Affiliation: | Department of Mathematics, Rutgers, The State University, New Brunswick, New Jersey 08903 USA |
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Abstract: | Let G be a connected, simply-connected complex nilpotent Lie group, and Gr ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of GR to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of GR, we study the convex cone of entire functions on G whose restrictions to GR are positive-definite, and determine the minimal order of growth at infinity of such functions. |
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