Demazure module

The Demazure character formula was introduced by (Demazure 1974b, theorem 2). 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. Andersen (1985) gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by Ramanan & Ramanathan (1985) and Mehta & Ramanathan (1985). Joseph (1985) gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. Kashiwara (1993) proved a refined version of the Demazure character formula that Littelmann (1995) conjectured (and proved in many cases).

The Demazure character formula is

${\displaystyle {\text{Ch}}(F(w\lambda ))=\Delta _{1}\Delta _{2}\cdots \Delta _{n}e^{\lambda }}$

Here:

• w is an element of the Weyl group, with reduced decomposition w = s₁...s_n as a product of reflections of simple roots.
• λ is a lowest weight, and e^λ the corresponding element of the group ring of the weight lattice.
• Ch(F(wλ)) is the character of the Demazure module F(wλ).
• P is the weight lattice, and Z[P] is its group ring.
• ${\displaystyle \rho }$ is the sum of fundamental weights and the dot action is defined by ${\displaystyle w\cdot u=w(u+\rho )-\rho }$.
• Δ_α for α a root is the endomorphism of the Z-module Z[P] defined by

${\displaystyle \Delta _{\alpha }(u)={\frac {u-s_{\alpha }\cdot u}{1-e^{-\alpha }}}}$

and Δ_j is Δ_α for α the root of s_j