Coordinate rings of \(G/P\) are Koszul (Q1903020)

The aim of this paper is to prove the following theorem. Let \(G\) be a simply connected, semisimple algebraic group over \(k\) (algebraically closed, of any characteristic) and let \(P\) be a parabolic subgroup of \(G\). If \(L\) is an ample line bundle on \(G/P\), then \(R_L= \bigoplus_{n\geq 0}H^0(G/P, L^{\otimes n})\) is a Koszul ring. This theorem generalizes a result due to Ramanathan. The proof uses Kempf's methods to show the required cohomology vanishing.