  Let $X$ denote a flag variety of type $A$ or type $C$. We construct a
canonical Frobenius splitting of $X \times X$ which vanishes with maximal
multiplicty along the diagonal. This way we verify a conjecture by Lakshmibai,
Mehta and Parameswaran 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.
