×

zbMATH — the first resource for mathematics

Demazure-Weyl formulas, and generalization of the Borel-Weil-Bott theorem. (Formules de Demazure-Weyl, et généralisation du théorème de Borel-Weil-Bott.) (French) Zbl 0602.17008
Let \(A\) be a generalized Cartan matrix and \(\mathfrak g\) the Kac-Moody Lie algebra associated to \(A\). One of the results of this paper is to compute the character of the maximal integrable highest weight modules over \(\mathfrak g\) in terms of the Weyl group of \(\mathfrak g\) by proving first the Demazure formula. This computation was made by Weyl when \(A\) is a Cartan matrix and by Kac when \(A\) is a symmetrizable generalized Cartan matrix.
Another important result is the computation of the cohomology of \(G/B\) with values in \(\mathfrak L(\lambda)\) generalizing therefore the Borel-Weil-Bott theorem. Here \(G/B\) stands for a ringed space coinciding with the usual flag variety when \(A\) is a Cartan matrix and \({\mathfrak L}(\lambda)\) stands for the invertible sheaf of \(G/B\).
Since the use of the Casimir (whose existence is equivalent to the symmetrisability of \(A\)) is no longer possible, the author develops some difficult technics in algebraic geometry. The first argument is the fact that the Demazure variety is Frobenius splittable in non-zero characteristic (this notion, and its utilisation for semisimple Lie algebras is due to Mehta, Ramanan and Ramanathan). The second argument is a variation of a Serre’s theorem for a pair of varieties whose normalisations are homeomorphisms (this allows to go through the absence of a flag variety).

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
14L15 Group schemes
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
PDF BibTeX XML Cite