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Unitary representations of infinite-dimensional pairs $$(G,K)$$ and the formalism of R. Howe. (English. Russian original) Zbl 0532.22019
Sov. Math., Dokl. 27, 290-294 (1983); translation from Dokl. Akad. Nauk SSSR 269, 33-36 (1983).
Let $$G$$ and $$K$$ be the inductive limits $$\cup_n \mathrm{U}(n)$$ and $$\cup_n \mathrm{SO}(n)$$ of the classical groups $$\mathrm{U}(n)$$ and $$\mathrm{SO}(n)$$. A unitary representation of $$G$$ is called “admissible for the pair $$(G,K)$$” if its restriction to $$K$$ fulfills certain conditions. The author studies and constructs admissible representations for the pair $$(G,K)$$ and for several other pairs of inductive limits of classical or symmetric groups.

##### MSC:
 22D10 Unitary representations of locally compact groups 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties