A holomorphic version of Borel-Weil-Bott theorem
type: Item
instance of: scholarly article
title: A holomorphic version of Borel-Weil-Bott theorem
zbMATH Open document ID: 0945.22007
author: S. G. Gindikin
author: Hon-Wai Wong
published in: Advances in Mathematics
publication date: 2000-10-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 1066079
zbMATH Keywords: compact Lie groups
zbMATH Keywords: generalized flag manifold
zbMATH Keywords: complex reductive Lie group
zbMATH Keywords: Stein manifold
zbMATH Keywords: holomorphic relative differential forms
zbMATH Keywords: holomorphic cohomology
zbMATH Keywords: Dolbeault cohomology
zbMATH Keywords: holomorphic harmonic forms
zbMATH Keywords: Schubert forms
reviewed by: L. Barchini
MaRDI profile type: Publication
full work available at URL: aima.1997.1656
OpenAlex ID: W2058632369
cites work: Q3993373
cites work: Structure of representations generated by vectors of highest weight
cites work: Q4107261
cites work: Q3964792
cites work: Q3956111
cites work: Complex manifolds without potential theory. (With an appendix on the geometry of characteristic classes). 2nd ed
cites work: Holomorphic realization of \(\overline{\partial}\)-cohomology and constructions of representations
cites work: Integral geometry on k-dimensional planes
cites work: Q3998601
cites work: Q3769563
cites work: Lie algebra cohomology and the generalized Borel-Weil theorem
cites work: Lie algebra cohomology and generalized Schubert cells
cites work: Espaces Homogènes De Stein Des Groupes De Lie Complexes
cites work: Representations of real reductive Lie groups
DOI: 10.1006/AIMA.1997.1656
Recommended article: Q4715741
Recommended article: Q4807915
Recommended article: A holomorphic version of Landau's theorem
Recommended article: Bolzano's theorems for holomorphic mappings
Recommended article: On the Bott-Borel-Weil theorem
Recommended article: A complex Borel-Bernstein theorem
Recommended article: The weinstein conjecture in the presence of holomorphic spheres
Recommended article: Q4286643
Recommended article: Q5655598
Recommended article: Q5380014
description: scientific article; zbMATH DE number 1066079