Several variable spectral theory and complex structures
by
Daniel Beltita
Institute of Mathematics of the Romanian Academy, Bucharest, Romania

An important class of representation spaces in the representation theory is the one of Hilbert spaces of holomorphic functions on complex homogeneous spaces. In particular, in order to investigate such representations of infinite-dimensional Lie groups, a method to construct invariant complex structures on infinite-dimensional homogeneous spaces is needed.

Our approach to the latter problem makes use of functional calculi for several (not necessarily commuting) variable, the most important example of such an object being the celebrated Weyl functional calculus for tuples of self-adjoint operators. It turns out that, in order to construct invariant complex structures on homogeneous spaces of a Banach-Lie group with the Lie algebra g, it suffices to have a sort of ``triangular decomposition''

A=A-+A0+A+

of the complexification A of g. We get such a decomposition using a suitable functional calculus whiose support is the disjoint union of the sets

(-S), {0}, S

where S is a closed subsemigroup of a finite-dimensional real vector space.

A typical application of this construction is provided by Grassmann and flag manifolds in C^*-algebras.

Date received: May 15, 2002