Demazure and Weyl-Kac Character Formulas

Abstract

We give a proof of the Demazure character formula (in an arbitrary Kac-Moody setting) and, as a consequence, obtain the Weyl-Kac character formula for an arbitrary Kac-Moody Lie algebra. For a dominant integral weight λ ∈ D_ℤ, let L^max (λ) be the maximal integrable highest weight $\mathfrak{g}$-module, as defined in 2.1.5, and, for any w ∈ W, let v_wλ. ∈ L^max(λ) be an extremal weight vector of weight wA., which is unique up to scalar multiples. Then, the Demazure module L ^max _w is, by definition, the U($\mathfrak{b}$)-span of υ _wλ . The Demazure character formula determines the character of L ^max _w (λ) as a T-module. The proof uses algebro-geometric techniques.