A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GLn

Let k be an algebraically closed field of characteristic $p>0$, let $G={\mathrm{GL}}_n$ be the general linear group over k, let P be a parabolic subgroup of G, and let $u_P$ be the Lie algebra of its unipotent radical. We show that the Kumar-Lauritzen-Thomsen splitting of the cotangent bundle $G{\times }^P{u}_P$ of $G/P$ has top degree $(p-1)\dim (G/P)$. The component of that degree is therefore given by the $(p-1)$-th power of a function f. We give a formula for f and deduce that it vanishes on the exceptional locus of the resolution $G{\times }^P{u}_P\to \bar{O}$ where $\bar{O}$ is the closure of the Richardson orbit of P. As a consequence we obtain that the higher cohomology groups of a line bundle on $G{\times }^P{u}_P$ associated to a dominant weight are zero. The splitting of $G{\times }^P{u}_P$ given by $f^{p-1}$ can be seen as a generalisation of the Mehta-Van der Kallen splitting of $G{\times }^{B}u$.