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Unitary representations and complex analysis. (English) Zbl 1143.22002
Casadio Tarabusi, Enrico (ed.) et al., Representation theory and complex analysis. Lectures given at the C.I.M.E. summer school, Venice, Italy, June 10–17, 2004. Berlin: Springer (ISBN 978-3-540-76891-3/pbk). Lecture Notes in Mathematics 1931, 259-344 (2008).
One goal of representation theory is to understand abstractly all the possible ways that a group \(G\) can act by linear transformations on a vector space \(V\). What this exactly means depends on the context. For topological groups (like Lie groups), one is typically interested in continuous actions on topological vector spaces. Using ideas from the spectral theory of linear operators, it is sometimes possible (at least in nice cases) to build such representations from irreducible representations, which play the role of scalar operators on one-dimensional spaces in linear algebra. The goal of these notes is to describe a geometric framework for some basic questions in representation theory for noncompact reductive Lie groups.
In Section 2, Compact Groups and the Borel-Weil theorem, the author recalls the simplest example – the Borel-Weil theorem describing irreducible representations of a compact group. In Section 3, Examples for \(\text{SL}(2,\mathbb R)\), some examples of representations of \(\text{SL}(2,\mathbb R)\) are presented in order to develop some feeling about what infinitesimal equivalence, minimal globalizations, and so on look like in examples. In Section 4, Harish-Chandra Modules and Globalization, some general facts about representations of real reductive groups are briefly recalled. In order to get some feeling for the various globalization functors, in Section 5, Real Parabolic Induction and the Globalization Functors, the author computes them in the setting of parabolically induced representations.
In Section 6, Examples of Complex Homogeneous Spaces, the author examines the complex homogeneous spaces for reductive groups that are used to construct representations. The author makes extensive use of the structure theory for complex reductive Lie algebras, and for that purpose it is convenient to have at our disposal a complex reductive group. This means a complex Lie group that is also a reductive group. The central idea in these notes is to construct representations of a real reductive group \(G\) by starting with a measurable complex flag variety \(X= G/L\) and using \(G\)-equivariant holomorphic vector bundles on \(X\).
One goal of Section 7, Dolbeault Cohomology and Maximal Globalizations, is to find a reasonable extension of the Borel-Weil theorem to noncompact reductive groups. The author considers representations of \(G\) on Dolbeault cohomology spaces \(H^{p,q}(X,\mathcal V)\). It is entirely natural to look for something like a pre-Hilbert space structure on the group representations, that might be completed to a unitary group representation. Theorem 7.27 of this section provides a large family of group representations with unitary Harish-Chandra modules. The Wong theorem guarantees that each representation provided by Theorem 7.27 is the maximal globalization of its Harish-Chandra module. Maximal globalizations never admit \(G\)-invariant pre-Hilbert space structures (unless they are finite-dimensional). We need something analogous to the Wong theorem that produces minimal globalizations instead. This means that we need to identify the dual of the topological vector space \(H^{p,q}(X,\mathcal V)\).
In Section 8, Compact Supports and Minimal Globalizations, the author is interested in the Dolbeault cohomology, which is a quotient of subspaces of a simple space of forms. The problem of computation of dual spaces of subspaces and quotients of topological vector spaces is considered in this section. In Sections 7 and 8, many representations are identified with spaces related to smooth functions and distributions on manifolds.
In Section 9, Invariant Bilinear Forms and Maps between Representations, the author uses these realizations to describe Hermitian forms on the representations. The original goal is to understand invariant bilinear forms on minimal globalization representations. Once the minimal globalizations are identified geometrically, one can at least offer a language for discussing this problem using standard functional analysis.
In the last Section 10, Open Questions are considered.
For the entire collection see [Zbl 1132.22001].

22E46 Semisimple Lie groups and their representations
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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