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

Abstract

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