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 $\lambda$ 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 $G$.