A proof of Wahl's conjecture in the symplectic case

Let X denote a flag variety of type A or type C. We construct a canonical Frobenius splitting of X x X which vanishes with maximal multiplicity along the diagonal. This way we verify a conjecture by Lakshmibai, Mehta and Parameswaran [4] in type C, and obtain a new proof in type A. In particular, we obtain a proof of Wahl's conjecture in type C, and a new proof in type A. We also present certain cohomological consequences.