  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 $G$.

Algebraic Geometry