  Let G be a semisimple algebraic group over an algebraically closed field of
positive characteristic p. Generalizing the construction of the PBW filtration
on Weyl modules for G we construct a G-stable filtration on tensor products of
Weyl modules which we call the induced PBW filtration. We use this filtration
to give some purely representation-theoretic conditions which are equivalent to
the existence of a Frobenius splitting of the double flag variety $G/B \times
G/B$ that maximally compatibly splits the diagonal. In particular, this gives a
sufficient condition for Wahl's conjecture to hold for G and we use this
criterion to prove that Wahl's conjecture holds in type G2 for p at least 11.
