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Restrictions of unitary representations of real reductive groups. (English) Zbl 1072.22008
Anker, Jean-Philippe (ed.) et al., Lie theory. Unitary representations and compactifications of symmetric spaces. Basel: Birkhäuser (ISBN 0-8176-3526-2/hbk ). Progress in Mathematics 229, 139-207 (2005).
This expository manuscript gives an overview on unitary representation of real reductive Lie groups based on courses the author gave at the European School on Group Theory (Odense, 2000). Representations are considered as admissible continuous group homomorphisms \(\pi : G \to \text{ U} (\mathcal H)\), where \(\mathcal H\) is some Hilbert space and admissible means that \(\pi\) is unitarily equivalent to a discrete sum of irreducible unitary representations \(\sigma_{i}\) of \(G\) and dim Hom\(_{G}(\sigma_{i}, \pi|_{G}) < \infty\) for any index \(i\). After the first two introductory sections branching laws are discussed, mainly through various examples involving the group SL\( (2,{\mathbb R})\), outlining the classification of its irreducible unitary representations due to Bargmann [V. Bargmann, Ann. Math. 48, 568–640 (1947; Zbl 0045.38801)]. Important notions for the topic under investigation are that of \(({\mathbb g}, K)\)-modules, infinitesimally unitary representations and infinitesimal discrete decompositions. To every infinitesimally discretely decomposable restriction an associated variety is assigned which serves as an approximation of modules of Lie algebras. Section 6 explains how concepts coming from microlocal analysis lead to a criterion for the admissibility of unitary representations. The main idea is to consider the singularity spectrum of the hyperfunction character of an irreducible representation. The strategy for proving the (non-)existence of a continuous spectrum in the irreducible decomposition of the restriction \(\pi|_{G'}\), where \(G'\) is a reductive subgroup of \(G \) and \(\pi\) is an irreducible unitary representation of \(G\), is to pass from \(\pi|_{G'}\) to a restriction of its character tr\((\pi)\) to some subgroup and from there to a restriction of a holomorphic function to a complex submanifold. Section 7 investigates a family of irreducible unitary representations of a reductive Lie group \(G\) derived from elliptic coadjoint orbits. The members of this family are realized in dense subspaces of Dolbeault cohomology groups of some class of equivariant holomorphic line bundles. Alternatively, they can be expressed as Zuckerman’s derived functor modules. In the final section of the manuscript the author highlights the importance of restrictions for representation theory. This goes along the common theme of trying to understand a mathematical object by decomposition into atoms, i.e. irreducible representations. In the context of representation theory this approach needs to be complemented by a change of viewpoint, which here means the selection of a subgroup \(G'\) of \(G\). So, the method the author suggests is to study irreducible representations of \(G\) by investigating the restrictions to \(G'\) and decomposing these restrictions. More than a 100 references accompany this rich exposition of unitary representation of real reductive Lie groups.
For the entire collection see [Zbl 1067.22001].

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