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

Source abstract

We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] 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.