Frobenius splitting of thick flag manifolds of Kac-Moody algebras

TL;DR

This paper demonstrates that thick flag manifolds of Kac-Moody algebras are Frobenius split using Plücker relations, leading to new insights into their global sections, bases, and normality.

Contribution

It extends Frobenius splitting results from thin to thick flag manifolds and describes their global sections and basis structures.

Findings

Thick flag manifolds are Frobenius split.

Global sections of line bundles are characterized as conjectured.

Thick Schubert varieties are projectively normal.

Abstract

We explain that the Pl\"ucker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag manifolds to the thick case. As a consequence, we provide a description of the global sections of line bundles of a thick Schubert variety as conjectured in Kashiwara-Shimozono [Duke Math. J. 148 (2009)]. This also yields the existence of a compatible basis of thick Demazure modules, and the projective normality of the thick Schubert varieties.