  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.
