Explicit Demazure character formula for negative dominant characters

TL;DR

This paper establishes an explicit Demazure character formula for negative dominant characters in semisimple algebraic groups, linking cohomology of line bundles on Schubert varieties to Demazure operators and providing a new basis for their kernels.

Contribution

It introduces a new explicit formula for Demazure characters involving cohomology of Schubert varieties and constructs a basis for the kernels of Demazure operators.

Findings

Proves a character formula relating cohomology on Schubert varieties to Demazure characters.

Provides a basis for the intersection of kernels of Demazure operators.

Connects top cohomology modules with dual characters in the context of algebraic groups.

Abstract

In this paper, we prove that for any semisimple simply connected algebraic group $G$, for any regular dominant character $\lambda$ of a maximal torus $T$ of $G$ and for any element $\tau$ in the Weyl group $W$, the character $e^{\rho}\cdot char(H^{0}(X(\tau), \mathcal{L}_{\lambda-\rho}))$ is equal to the sum $\sum_{w\leq \tau}char(H^{l(w)}(X(w),\mathcal{L}_{-\lambda}))^{*})$ of the characters of dual of the top cohomology modules on the Schubert varieties $X(w)$, $w$ running over all elements satisfying $w\leq \tau$. Using this result, we give a basis of the intersection of the Kernels of the Demazure operators $D_{\alpha}$ using the sums of the characters of $H^{l(w)}(X(w),\mathcal{L}_{-\lambda})$, where the sum is taken over all elements $w$ in the Weyl group $W$ of $G$.