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==Demazure modules==
Suppose that ''g'' is a complex semisimple Lie algebra, with a [[Borel subalgebra]] ''b'' containing a [[Cartan subalgebra]] ''h''. An irreducible finite-dimensional representation ''V'' of ''g'' splits as a sum of eigenspaces of ''h'', and the highest weight space is 1-dimensional and is an eigenspace of ''b''. The [[Weyl group]] ''W'' acts on the weights of ''V'', and the conjugates ''w''λ of the highest weight vector λ under this action are the extremal weights, whose weight spaces are all 1-dimensional.
A Demazure module is the ''b''-submodule of ''V'' generated by the weight space of an extremal vector ''w''λ, so the Demazure submodules of ''V'' are parametrized by the Weyl group ''W''.
There are two extreme cases: if ''w'' is trivial the Demazure module is just 1-dimensional, and if ''w'' is the element of maximal length of ''W'' then the Demazure module is the whole of the irreducible representation ''V''.
Demazure modules can be defined in a similar way for highest weight representations of [[Kac–Moody algebra]]s, except that one now has 2 cases as one can consider the submodules generated by either the Borel subalgebra ''b'' or its opposite subalgebra. In the finite-dimensional these are exchanged by the longest element of the Weyl group, but this is no longer the case in infinite dimensions as there is no longest element.
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*{{Citation | last1=Andersen | first1=H. H. | title=Schubert varieties and Demazure's character formula | doi=10.1007/BF01388527 | mr=782239 | year=1985 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=79 | issue=3 | pages=611–618}}
*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Désingularisation des variétés de Schubert généralisées | url=http://www.numdam.org/item?id=ASENS_1974_4_7_1_53_0 | series=Collection of articles dedicated to [[Henri Cartan]] on the occasion of his 70th birthday, I | mr=0354697 | year=1974a | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=7 | pages=53–88}}
*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Une nouvelle formule des caractères | mr=0430001 | year=1974b| journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=98 | issue=3 | pages=163–172}}
*{{Citation | last1=Joseph | first1=Anthony | title=On the Demazure character formula | url=http://www.numdam.org/item?id=ASENS_1985_4_18_3_389_0 | mr=826100 | year=1985 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=18 | issue=3 | pages=389–419}}
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