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In this article, we prove that for any indecomposable dominant character of amaximal torus $T$ of a simple adjoint group $G$ such that there is a Coxeterelement $w \in W$ for which $X(w)^{ss}_T(\mathcal L_\chi) \neq \emptyset$. Iffurther, for any dominant character $\chi_1$ of $T$ such that $\chi_1\lneqq\chi$ with respect to the dominant ordering, $dim(H^0(G/B, \mathcalL_{\chi_1})^T) < dim (H^0(G/B, \mathcal L_\chi)^T)$, then the graded algebra$\oplus_{d \in \mathbb Z_{\geq 0}}H^0(G/B, \mathcal L_\chi^{\otimes d})^T$ is apolynomial ring in $r$ variables where $r\geq 2$.

When is the ring of $T$ invariants of the homogeneous coordinate ring of $G/B$ a polynomial algebra- connection with the Coxeter elements