  We use the G-invariant non-degenerate form on the Steinberg module to
Frobenius split the cotangent bundle of a flag variety in good prime
characteristics. This was previously only known for the general linear group.
Applications are a vanishing theorem for pull back of line bundles to the
cotangent bundle (proved for the classical groups and G_2 by Andersen and
Jantzen and in characteristic zero by B. Broer (for all groups)), normality and
rational singularities for the subregular nilpotent variety and good
filtrations of the global sections of pull backs of line bundles to the
cotangent bundle, which in turn implies good filtrations of cohomology of
induced representations.
