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

The induced PBW filtration, Frobenius splitting of double flag varieties, and Wahl's conjecture