Method of holomorphic extensions in the theory of unitary representations of infinite-dimensional classical groups (Q1118052)

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 \(K^*\).