Positivity and Kleiman transversality in equivariant K-theory of   homogeneous spaces
  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. 
Dave Anderson, Stephen Griffeth, and Ezra Miller