A Demazure character is the formal character of a Demazure module, a Borel subalgebra submodule generated by an extremal weight space in a finite-dimensional
irreducible representation of a complex
semisimple Lie algebra. Let 
be the irreducible highest-weight Lie
algebra representation of dominant integral weight 
, let 
belong to the Weyl group, and
let 
be the Borel subalgebra corresponding to the
positive roots used to define the highest weight.
The Demazure module is
 |
where 
is the universal enveloping algebra
and 
is the weight space of weight 
. Its character records the dimensions
of its weight spaces:
 |
where 
is a formal symbol for weight 
, rather than a numerical exponential
function (Demazure 1974). For the longest element of the Weyl
group, the Demazure module is the whole irreducible
representation, whose character is given by the Weyl
character formula.
In type A, the characters of polynomial Demazure modules for the general linear group 
are the key polynomials. Replacing the formal symbols
by monomials 
expresses the character as a generally
nonsymmetric polynomial (Mason 2009).
