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

Let k be an algebraically closed field of characteristic p > 0 , let G = 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 × 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 × P u P → O ‾ where 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 × P u P associated to a dominant weight are zero. The splitting of G × P u P given by f p − 1 can be seen as a generalisation of the Mehta-Van der Kallen splitting of G × B u .