Demazure module: Difference between revisions

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In mathematics, the '''Demazure character formula''' is a generalization of the [[Weyl character formula]] for the characters of finite dimensional representations of [[semisimple Lie algebra]]s, introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year=1974a|year2=1974b|loc2loc=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.

==History of the proof==

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

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

==Statement==

==References==