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Integral models of representations of the current groups of simple Lie groups. (English. Russian original) Zbl 1202.22019

Russ. Math. Surv. 64, No. 2, 205-271 (2009); translation from Usp. Mat. Nauk 64, No. 2, 5-72 (2009).
The authors construct and study integral models of representations of current groups. Although these models are not as general as the Fock model they allow to give much simpler proofs of irreducibility of representations and lead to a new explicit interpretation of continuous tensor products. The main part of their paper is devoted to a detailed construction of a new model of representation for the class of groups \(P\) which are the semidirect product of a locally compact group \(P_0\) and the group \(\mathbb R\) acting on \(P_0\) as a one-parameter group of automorphisms. This class includes in particular the maximal parabolic subgroups of simple groups of rank one, that is of the groups \(SO(n, 1), SU(n, 1)\) and \(Sp(n, 1)\). Since the authors can extend the integral model of the representation of the current group \(P^X\) to an arbitrary locally compact group \(G\) containing \(P\), they are able in the case of the groups \(G = SO(n, 1)\) and \(G = SU(n, 1)\) to extend the representations of current groups of their maximal parabolic subgroups to representations of current groups \(G^X\). In the case \(G = Sp(n, 1)\) the corresponding current group has no representation. A separate section is devoted to the case of the group \(SL_{2}(\mathbb R) \cong SU(1, 1)\) in which \(P\) is the group of triangular matrices. A key role in the constructions which are carried out by the authors is played by a \(\sigma\)-finite measure which is an infinite dimensional analogue of the Lebesgue measure in the space of Schwartz distributions. The authors claim that their paper is a preparatory step toward a monograph devoted to representations of current groups.

MSC:

22E46 Semisimple Lie groups and their representations
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22D12 Other representations of locally compact groups
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
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