In mathematics, a Demazure module, introduced by Demazure (1974a, 1974b), is a submodule of a finite-dimensional representation generated by an extremal weight space under the action of a Borel subalgebra.
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.
A Demazure module is the b-submodule of V generated by the weight space of an extremal vector wλ, 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 algebras, except that one now has 2 cases as one can consider the submodules generated by either the Borel subalgebra b or its opposite subalgebra.
Victor Kac pointed out that Demazure's proof has a serious gap, as it depends on (Demazure 1974a, Proposition 11, section 2), which is false; see (Joseph 1985, section 4) for Kac's counterexample.
Joseph (1985) gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques.