title: Exotic group C*-algebras of higher rank Lie groups
Creators: name: Dabeler, Antje
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2023
Subjects: subject: 510 Mathematics

description: An exotic group C*-algebra is a C*-algebra that lies naturally between the reduced and the universal group C*-algebra. In this thesis we study the existence of such C*-algebras for the special linear groups over the complex numbers and for the symplectic group, as examples of connected simple Lie groups with real rank greater or equal than 2. In both cases we are able to show the existence of a continuum of exotic group C*-algebras. In joint work with Emilie Elkiær, Maria Gerasimova and Tim de Laat we studied induction of unitary representations from an open subgroup H in a locally compact group G. We observed that in this case exotic group C*-algebras of H give rise to exotic group C*-algebras of G. If H further has the Kunze-Stein property, it is moreover possible to deduce the existence of representations of G with specific integrability properties from the existence of such representations of H.
description: 1 Introduction 1 2 Completions and unitary duals 7 2.1 Ideal completions and Lp-completions . . . . . . . . . . . . . . . . . . . . 7 2.2 Exotic group C∗-algebras of discrete groups . . . . . . . . . . . . . . . . . 14 2.3 Kunze–Stein property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Strategy to find exotic group C∗-algebras . . . . . . . . . . . . . . . . . . 23 3 Structure theory of Lie groups 26 3.1 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Gelfand pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Representation theory of Lie groups 35 4.1 Parabolic induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Infinitesimal character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Asymptotic expansion of spherical representations . . . . . . . . . . . . . 40 4.4 Representation theory of SL(n,C) . . . . . . . . . . . . . . . . . . . . . . 45 4.4.1 Stein complementary series . . . . . . . . . . . . . . . . . . . . . . 45 4.4.2 Vogan’s description of the unitary dual of GL(n,C) . . . . . . . . . 47 4.4.3 Duflo’s description of parabolically induced representations . . . . 48 5 Exotic group C∗-algebras 53 5.1 Exotic group C∗-algebras of SL(2,C) . . . . . . . . . . . . . . . . . . . . . 53 5.2 Representations with minimal decay . . . . . . . . . . . . . . . . . . . . . 54 5.3 Exotic group C∗-algebras of SL(n,C) . . . . . . . . . . . . . . . . . . . . . 55 5.3.1 Some low-dimensional cases . . . . . . . . . . . . . . . . . . . . . . 55 5.3.2 Integrability properties of the Stein complementary series . . . . . 59 5.4 Exotic group C∗-algebras of Sp(2,C) . . . . . . . . . . . . . . . . . . . . . 63 5.5 Lp+-representations of colored Neretin groups . . . . . . . . . . . . . . . . 66 6 Outlook 71 6.1 Other higher rank Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2 Induction from closed subgroups . . . . . . . . . . . . . . . . . . . . . . . 72 Bibliography 74