Residues formulas for the push-forward in K-theory, the case of G2/P

Abstract

We study residue formulas for push-forward in the equivariant K-theory of homogeneous spaces. For the classical Grassmannian the residue formula can be obtained from the cohomological formula by a substitution. We also give another proof using symplectic reduction and the method involving the localization theorem of Jeffrey-Kirwan. We review formulas for other classical groups, and we derive them from the formulas for the classical Grassmannian. Next we consider the homogeneous spaces for the exceptional group G2. One of them, G2/P2 corresponding to the shorter root, embeds in the Grassmannian Gr(2, 7). We find its fundamental class in the equivariant K-theory K T (Gr(2, 7)). This allows to derive a residue formula for the push-forward. It has significantly different character comparing to the classical groups case. The factor involving the fundamental class of G2/P2 depends on the equivariant variables. To perform computations more efficiently we apply the basis of K-theory consisting of Grothendieck polynomials. The residue formula for push-forward remains valid for the homogeneous space G2/B as well.

Darondeau-Pragacz formulas in complex cobordism

“…Zielenkiewicz [41] gave formulas of the integrals over the Grassmannians as iterated residues at infinity. Using the iterated residues at zero and infinity, Rimanyi and Szenes [33,Lemma 6.2], Weber and Zielenkiewicz [39] gave similar formulas for Grassmannians in the equivariant K-theory. Thus many authors treat essentially the same subject.…”