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

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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 ∈ W for which X(w)ssₜ(Lχ) ≠. If further, for any dominant character χ₁ of T such that χ₁ χ with respect to the dominant ordering, dim(H⁰(G/B, Lᵪ₁)ᵗ) < dim (H⁰(G/B, Lχ)ᵗ), then the graded algebra ⊕d ∈ Z≥ ₀H⁰(G/B, Lχ⊗ ᵈ)ᵗ is a polynomial ring in r variables where r≥ 2.