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Unitary representations of infinite dimensional pairs (G,K) and the formalism of R. Howe. (English) Zbl 0724.22020

Representation of Lie groups and related topics, Adv. Stud. Contemp. Math. 7, 269-463 (1990).
[For the entire collection see Zbl 0714.00007.]
In this paper an infinite dimensional Lie group is by definition an inductive limit of finite dimensional Lie groups with the inductive limit topology. The author exclusively considers a certain set of classical groups such as \(G_ n=GL(n,{\mathbb{R}})\) where the map \(G_ n\to G_{n+1}\) is the obvious one. The new idea is to study infinite dimensional pairs (G,K) where \(K=\lim_{\to} K_ n\) plays the rôle of a maximal compact subgroup in the finite dimensional case. For various pairs the author constructs irreducible unitary representations, proves their inequivalence, gives explicit decompositions of tensor powers, calculates the spherical functions and studies approximations of representations by representations of the finite dimensional components. The most prominent example is the Weil representation of the infinite dimensional symplectic group. This paper is a rich source of examples for everyone working in this field.
Reviewer: A.Deitmar (Bonn)

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

Citations:

Zbl 0714.00007