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Unitary structure in representations of infinite-dimensional groups and a convexity theorem. (English) Zbl 0534.17008

An essential feature when classifying complex semisimple Lie algebras has been the Cartan matrix. So one could try to go the way back: what sort of matrices do allow to define topological groups (with given Cartan matrix) such that the situation becomes as similar to the case of semi-simple Lie groups as to produce analogous results? This program has been carried out by the authors and others for several years.
In this paper, they show that the Kac-Moody algebra \(\mathfrak g(A)\) associated to a symmetrizable generalized Cartan matrix \(A\) carries a contravariant Hermitian form which is positive-definite on all root spaces. They deduce that every integrable highest weight \(\mathfrak g(A)\)-module \(L(A)\) carries a contravariant positive-definite Hermitian form. This allows to define the moment map and prove a generalization of the Schur-Horn-Kostant-Heckman-Atiyah-Pressley convexity theorem. The proofs are based on an identity which also gives estimates for the action of \(\mathfrak g(A)\) on \(\mathfrak g(A)\) and \(L(A)\).

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
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References:

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