Character formulas for general Kac-Moody algebras. (Formules de caractères pour les algèbres de Kac-Moody générales.) (French) Zbl 0683.17010

Mathieu, Olivier

Centre National de la Recherche Scientifique (France). Astérisque, 159-160. Paris: Société Mathématique de France. 267 p. FF 185.00; $ 31.00 (1988).

The author extends Demazure’s character formula to any Kac-Moody algebra (not necessarily symmetrizable). From this, using an argument of G. Heckman, he gets the Weyl character formula. Underlying these results, there is the identification of the characters with some Euler-Poincaré characteristic dimensions and the proof of vanishing theorems for the cohomology of semi-ample line bundles over the Schubert varieties. This machinery also allows him to prove a generalization of the Bott-Borel- Weyl theorem and Kempf’s theorem, as well as some properties of the Schubert varieties.

Reviewer: F.Levstein

MSC:

17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17-02	Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15	Grassmannians, Schubert varieties, flag manifolds
17B65	Infinite-dimensional Lie (super)algebras
14F99	(Co)homology theory in algebraic geometry

Keywords:

Kac-Moody algebra; Weyl character formula; Euler-Poincaré characteristic dimensions; vanishing theorems; cohomology of semi-ample line bundles; Schubert varieties; generalization of the Bott-Borel-Weyl theorem; Kempf’s theorem