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

TL;DR

This paper introduces the induced PBW filtration on tensor products of Weyl modules for semisimple algebraic groups in positive characteristic, linking it to Frobenius splitting of double flag varieties and proving Wahl's conjecture for type G2 when p ≥ 11.

Contribution

It constructs a new G-stable filtration called the induced PBW filtration and relates it to Frobenius splitting and Wahl's conjecture, providing new criteria and proofs.

Findings

Established a criterion for Frobenius splitting of double flag varieties.

Proved Wahl's conjecture for type G2 when p ≥ 11.

Connected representation-theoretic conditions to geometric properties.

Abstract

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.