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Explicit Demazure character formula for negative dominant characters

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