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Method of holomorphic extensions in the theory of unitary representations of infinite-dimensional classical groups. (English. Russian original) Zbl 0668.22008
Funct. Anal. Appl. 22, No. 4, 273-285 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 23-37 (1988).
Let G be one of the groups GL($$\infty,C)$$, U($$\infty)\times U(\infty)$$ and Mot($$\infty)$$, where $$GL(\infty,C)=\cup GL(n,C)$$, $$U(\infty)=\cup U(n)$$, and Mot($$\infty)$$ is the semidirect product of U($$\infty)$$ and P($$\infty)$$ $$(P(\infty)=\cup P(n)$$ and P(n) is the space of Hermitian $$n\times n$$ matrices). The author constructs a wide class X of unitary representations of the groups G. It is proved that representations from X are irreducible and pairwise inequivalent. The representations are constructed in the following way. The Lie algebra $${\mathfrak g}$$ of G is imbedded into some Lie algebra $${\mathfrak g}^*$$ of polynomial currents. The “unitary” representations of $${\mathfrak g}^*$$ with highest weights are constructed. They are restricted onto $${\mathfrak g}$$. It is proved that the representations of G, corresponding to the constructed representations of $${\mathfrak g}$$, are irreducible. The proof is based on the fact, that a unitary representation of the “compact” subgroup K can be extended to a unitary representation of some larger group $$K^*$$.
Reviewer: A.Klimyk

##### MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties
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