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Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

Part of: Lie groups

Published online by Cambridge University Press:  01 June 2011

Heiko Gimperlein ,
Bernhard Krötz  and
Henrik Schlichtkrull
Heiko Gimperlein
Affiliation:
Leibniz Universität Hannover, Institut für Analysis, Welfengarten 1, D-30167, Hannover, Germany (email: gimperlein@math.ku.dk)
Bernhard Krötz
Affiliation:
Leibniz Universität Hannover, Institut für Analysis, Welfengarten 1, D-30167, Hannover, Germany (email: kroetz@math.uni-hannover.de)
Henrik Schlichtkrull
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark (email: schlicht@math.ku.dk)
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Abstract

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In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

Keywords

real Lie groupanalytic representationHarish-Chandra moduleglobalization

MSC classification

Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields 22E46: Semisimple Lie groups and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1581 - 1607
Copyright
Copyright © Foundation Compositio Mathematica 2011

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