Frobenius Splitting of Cotangent Bundles of Flag Varieties and Geometry of Nilpotent Cones

Let G be a semisimple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Let U be the unipotent part of a Borel subgroup B ⊂ G and u the Lie algebra of U . Springer [15] has shown for good primes, that there is a B-equivariant isomorphism U → u, where B acts through conjugation on U and through the adjoint action on u (for G = SLn one has the well known equivariant isomorphism X 7→ X − I between unipotent and nilpotent upper triangular matrices). Fix a good prime p. Then there is an isomorphism of homogeneous bundles X = G × U → G × u, where the latter can be identified with the cotangent bundle T (G/B) of G/B. Motivated in part by [11] we establish a link between the G-invariant form χ on the Steinberg module St = H(G/B, (p − 1)ρ) and Frobenius splittings [14] of the cotangent bundle T (G/B): The representation H(G/B, 2(p−1)ρ) is a quotient of the functions H(X,OX) on X (here H(G/B,M) denotes the G-module induced from the B-module M and ρ half the sum of the roots R opposite to the roots of B). There is a natural map