Residues formulas for the push-forward in K-theory, the case of $$\mathbf{G}_2/P$$
Tóm tắt
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
$$\mathbf{G}_2$$
. One of them,
$$\mathbf{G}_2/P_2$$
corresponding to the shorter root, embeds in the Grassmannian Gr(2, 7). We find its fundamental class in the equivariant K-theory
$$K^\mathbb {T}(Gr(2,7))$$
. This allows to derive a residue formula for the push-forward. It has significantly different character compared to the classical groups case. The factor involving the fundamental class of
$$\mathbf{G}_2/P_2$$
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
$$\mathbf{G}_2/B$$
as well.
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