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Parabolic induction, categories of representations and operator spaces. (English) Zbl 1355.22003

Doran, Robert S. (ed.) et al., Operator algebras and their applications. A tribute to Richard V. Kadison. AMS special session on operator algebras and their applications: a tribute to Richard V. Kadison, San Antonio, TX, USA, January 10–11, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1948-6/pbk; 978-1-4704-3500-4/ebook). Contemporary Mathematics 671, 85-107 (2016).
The paper under review fits into the authors’ project to study representation theory of real reductive groups by operator algebraic methods. More precisely, they investigate adjunction relations between functors that realize parabolic induction and restriction in various categories of representations, in the spirit of Frobenius reciprocity and Bernstein’s Second Adjoint theorem.
First, the authors compare categories of operator space modules and Hilbert spaces representations, of more common use in the representation theory of groups. They establish that both categories have the same irreducible objects in the case of type I C*-algebras. Functors between these categories are discussed and a sufficient condition is given for certain functors to have a left adjoint.
Next, the discussion focuses on the case of parabolic induction, previously studied by the authors and the reviewer in the framework of Hilbert C*-modules [Compos. Math. 152, No. 6, 1286–1318 (2016; Zbl 1346.22005)], now in the context of operator modules. A completely bounded version of Frobenius Reciprocity is proved, which contrasts with the Hilbert module situation, where only local adjunctions between parabolic induction and restriction hold. The case of \(\mathrm{SL}(2,\mathbb{R})\) is presented in some detail as an illustration.
Frobenius Reciprocity in concerned with right adjunction for the functor of parabolic induction. A result of Bernstein, known as the Second Adjoint Theorem and valid in the \(p\)-adic realm, says that a right adjoint is obtained by considering opposite parabolic restriction. In the final section of the article, the authors identify, again in the case of \(\mathrm{SL}(2,\mathbb{R})\), right adjoints for the parabolic induction functor between categories of smooth Fréchet modules over Harish-Chandra Schwartz algebras and of operator modules.
For the entire collection see [Zbl 1347.46001].

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
46L07 Operator spaces and completely bounded maps
46H15 Representations of topological algebras

Citations:

Zbl 1346.22005
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