Hostname: page-component-6dbcb7884d-4vmf6 Total loading time: 0 Render date: 2025-02-14T06:14:59.228Z Has data issue: false hasContentIssue false
  • English
  • Français

A new characterization of the Bruhat decomposition

Published online by Cambridge University Press:  22 January 2016

Yoshifumi Kato
Yoshifumi Kato*
Affiliation:
Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

By an algebraic homogeneous space, we mean the factor space X = G/P, where G is a simply-connected, complex, semi-simple Lie group and P is a parabolic subgroup of G. Many typical manifolds such as the projective spaces and the Grassmann varieties belong to this class of manifolds. For instance, the Grassmann variety G(k, n) can be expressed as SL(n + 1, C)/P, where P is a maximal parabolic subgroup of SL(n + 1, C) leaving a suitable k + 1 dimensional subspace invariant. In this paper, we devote ourselves to study the Bruhat decomposition of an algebraic homogeneous space X = G/P.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 86 , June 1982 , pp. 131 - 153
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Bernstein, I. N. I., Gel’fand, M. and Gel’fand, S. I., Schubert cells and cohomology of the spaces G/P , Russ. Math. Surveys, 28 (1975), 1–26.CrossRefGoogle Scholar
[ 2 ] Bernstein, I. N. I., Gel’fand, M. and Gel’fand, S. I., Differential operators on the base affine space and a study of g-modules, Lie groups and their representations, Halsted Press (1975), 21–64.Google Scholar
[ 3 ] Dixmier, J., Enveloping algebras, North-Holland Mathematical Library, Vol. 14.Google Scholar
[ 4 ] Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxter group and determination of the relative Möbius function, Inventiones Math., 39 (1977), 187–198.CrossRefGoogle Scholar
[ 5 ] Kato, Y., On the vector fields on an algebraic homogeneous space, preprint.CrossRefGoogle Scholar
[ 6 ] Kostant, B., Lie algebra cohomology and generalized Borel-Weil theorem, Ann. Math., 74(2) (1961), 329–387.CrossRefGoogle Scholar
[ 7 ] Kostant, B., Lie algebra cohomology and generalized Schubert cells, Ann. Math., 77(1) (1963), 72–144.CrossRefGoogle Scholar
[ 8 ] Sommese, A. J., Holomorphic vector fields on compact kaehler manifolds, Math. Ann., 210 (1974), 75–82.CrossRefGoogle Scholar
[ 9 ] Varadarajan, V. S., Lie groups, Lie algebras and their representations, Prentice-Hall, Inc.Google Scholar
[10] Wallach, N., Harmonic analysis on homogeneous spaces, Marcel Dekker, Inc.Google Scholar
[11] Warner, G., Harmonic analysis on semisimple Lie groups I, Springer Verlag.Google Scholar