# Journal of the European Mathematical Society

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**Volume 13, Issue 1, 2011, pp. 57–84**

**DOI: 10.4171/JEMS/244**

Published online: 2010-10-21

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

^{[1]}, Stephen Griffeth

^{[2]}and Ezra Miller

^{[3]}(1) University of Michigan, Ann Arbor, USA

(2) University of Minnesota, Minneapolis, USA

(3) Duke University, Durham, USA

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.

*Keywords: *Flag variety, equivariant K-theory, Kleiman transversality, homological transversality, Schubert variety, Borel mixing space, rational singularities, Bott–Samelson resolution

Anderson Dave, Griffeth Stephen, Miller Ezra: Positivity and Kleiman transversality in equivariant *K*-theory of homogeneous spaces. *J. Eur. Math. Soc.* 13 (2011), 57-84. doi: 10.4171/JEMS/244