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Infinite-dimensional classical groups of finite $$r$$-rank: description of representations and asymptotic theory. (English. Russian original) Zbl 0545.22020
Funct. Anal. Appl. 18, 22-34 (1984); translation from Funkts. Anal. Prilozh. 18, No. 1, 28-42 (1984).
The admissible unitary representations of the following three series of groups are considered; $$G=\mathrm{SO}_ 0(p,\infty)$$, $$\mathrm{U}(p,\infty)$$ and $$\mathrm{Sp}(p,\infty)$$, $$p=0,1,2,...$$. The number $$p$$ is called the rank of the group. In paragraph 1, results are given in the case of rank 0. In paragraph 4, these results are generalized to the case of rank $$p\geq 1.$$
Let $$G(n)$$ be groups $$\mathrm{SO}_ 0(p,n-p)$$, $$\mathrm{U}(p,n-p)$$, and $$\mathrm{Sp}(p,n-p)$$, respectively. Considering $$G$$ as the inductive limit of $$G(n)$$, some approximation results are given in paragraph 5 and these are applied to the classification of representations of rank 0 groups.
Reviewer: Y. Asoo

##### MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods
##### Keywords:
unitary representations; rank
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##### References:
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