Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

TL;DR

This paper proves conjectures about sign patterns of structure constants in equivariant K-theory of flag varieties, using a homological transversality principle and vanishing theorems to simplify computations and establish sign behavior.

Contribution

It introduces an equivariant homological Kleiman transversality principle and a vanishing theorem for subvarieties with rational singularities, advancing understanding of structure constants in equivariant K-theory.

Findings

Confirmed sign alternation conjectures for structure constants.

Reduced coefficient computations to Euler characteristics via transversality.

Established vanishing results for subvarieties with rational singularities.

Abstract

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.