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 characters of Demazure modules, the submodules of a finite dimensional representation generated by an extremal weight under the action of the nilradical of a Borel subalgebra.

Victor Kac pointed out that Demazure's original proof of the character formula has a serious gap, as Proposition 11 of Section 2 of Demazure (1974a) is false. Anderson (1985) gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by Ramanan & Ramanathan (1985) and Mehta & Ramanathan (1985). Joseph (1985) gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. Kashiwara (1993) proved a refined version of the Demazure character formula that Littelmann (1995) conjectured (and proved in many cases).

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
• Kashiwara, Masaki (1993), "The crystal base and Littelmann's refined Demazure character formula", Duke Mathematical Journal, 71 (3): 839–858, doi:10.1215/S0012-7094-93-07131-1, ISSN 0012-7094, MR1240605
• Littelmann, Peter (1995), "Crystal graphs and Young tableaux", Journal of Algebra, 175 (1): 65–87, doi:10.1006/jabr.1995.1175, ISSN 0021-8693, MR1338967
• Mehta, V. B.; Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for Schubert varieties", Annals of Mathematics. Second Series, 122 (1): 27–40, doi:10.2307/1971368, ISSN 0003-486X, MR799251
• 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