Demazure module: Difference between revisions

Revision as of 15:49, 3 March 2014

Revision as of 20:45, 8 March 2014

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In mathematics, a '''Demazure module''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year1=1974a|year2=1974b}}, is a submodule of a finite -dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The '''Demazure character formula''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year=1974b|loc=theorem 2}}, gives the characters of Demazure modules, and is a generalization of the [[Weyl character formula]].

The dimension of a Demazure module is a polynomial in the highest weight, called a '''Demazure polynomial'''.

==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''.

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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.

==Demazure character formula==