Demazure module

In mathematics, the Demazure character formula is a generalization of the Weyl character formula for the characters of finite dimensional representations of semisimple Lie algebras, introduced by Demazure (1974a, 1974b, theorem 2). Demazure's formula gives the character of the submodule of a finite dimensional representation generated by an extremal weight under the action of the unipotent radical of a Borel subalgebra.

Victor Kac pointed out that Demazure's original proof of the character formula was incorrect: Proposition 11 of Section 2 of Demazure (1974a) is false. Anderson (1985) later gave a proof of Demazure's character formula using the geometry of Schubert varieties, and Joseph (1985) gave a proof for generic highest weight modules using Lie algebra techniques.

References

• Andersen, H. H. (1985), "Schubert varieties and Demazure's character formula", Inventiones Mathematicae, 79 (3): 611–618, doi:10.1007/BF01388527, ISSN 0020-9910, MR782239
• Demazure, Michel (1974a), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I, 7: 53–88, ISSN 0012-9593, MR0354697
• Demazure, Michel (1974b), "Une nouvelle formule des caractères", Bulletin des Sciences Mathématiques. 2e Série, 98 (3): 163–172, ISSN 0007-4497, MR0430001
• Joseph, Anthony (1985), "On the Demazure character formula", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 18 (3): 389–419, ISSN 0012-9593, MR826100
• Ramanan, S.; Ramanathan, A. (1985), "Projective normality of flag varieties and Schubert varieties", Inventiones Mathematicae, 79 (2): 217–224, doi:10.1007/BF01388970, ISSN 0020-9910, MR778124