On the dual of complex Olshanskii semigroups

2000/06/26 by Bernhard Kroetz
Mathematics · #math.RT #math.OA

paper · pdf

published as Math. Z. 237(3) (2001), 505-529 · 19 pages, to appear in Mathematische Zeitschrift

arxiv created 2000/06/26 · arxiv updated 2009/11/30

Abstract

Let G be a connected Lie group and G its unitary dual. We are interested in the part Λ⊂ G which corresponds to the unitary highest weight representations of G. Then there are several topologies on Λ: The euclidean topology T_E which comes from the identification of Λ with the set of highest weights, the induced topology T_I induced from the Fell topology on G and finally a natural topology T_S which comes from the hull kernel topology of certain CCR C^*-algebras which are related to the holomorphic extemsion of unitary highest weight representations to complex Olshanskii semigroups S. One of the main results in this paper is the inclusion chain T_S⊂ T_I⊂ T_E. Further we exhibit very large interesting subspaces of Λ where these topologies coincide. Finally we show that the Borel structures on Λ induced from the three different topologies coincide.

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