Positivity in $T$-equivariant $K$-theory of flag varieties associated to Kac-Moody groups II

TL;DR

This paper proves sign-alternation of structure constants in the torus-equivariant K-theory of flag varieties for symmetrizable Kac-Moody groups, extending previous finite case results and confirming a conjecture for affine Grassmannians.

Contribution

It generalizes sign-alternation results from finite to Kac-Moody flag varieties and confirms a conjecture for affine Grassmannians.

Findings

Sign-alternation of structure constants proven for Kac-Moody flag varieties.

Extension of finite case results to infinite-dimensional Kac-Moody groups.

Confirmation of a conjecture regarding signs in the affine Grassmannian.

Abstract

We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated to an arbitrary symmetrizable Kac-Moody group $G$, where $P$ is any parabolic subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian.