A holomorphic version of Borel-Weil-Bott theorem
In this paper the authors give a geometric realization of representations of compact Lie groups by using holomorphic objects. Let \(M=G/Q\) be a generalized flag manifold where \(G\) is a connected complex reductive Lie group, and \(Q\) a parabolic subgroup with Levi part \(L\). Gindikin and Wong consider the Stein manifold \(X=G/L\), the fibration \(\pi:X\to M\) and a complex of holomorphic relative differential forms on \(X\), along the fibers of \(\pi\). The resulting holomorphic cohomology is canonically isomorphic to the Dolbeault cohomology on \(M\). Thus, they obtain a purely holomorphic realization of finite-dimensional representations of \(G\). On the other hand, they also prove a Hodge-type theorem in the holomorphic setting. They define the notion of holomorphic harmonic forms. At each point these differential forms are multiples of some differential monomials which involve a linear subspace of the exterior algebra of the relative cotangent space related to a certain Weyl group element \(\omega\). They call differential forms of this sort ``Schubert \(\omega\)-forms''. When \(M\) is complete or is compact Hermitian symmetric, they give a geometrical characterization of these Schubert forms.
scientific article; zbMATH DE number 1066079
scientific article