Equivariant \(K\)-theory of affine flag manifolds and affine Grothendieck polynomials
Let \(G\) be an affine Lie group, \(B\) a Borel subgroup and \(X= G/B\) the corresponding affine flag manifold which is decomposed into the union of affine Schubert varieties \(X_w = \overline{BwB/B}\). \(X\) is an infinite-dimensional (not quasi-compact) scheme over \(\mathbb C\) and its \(B\)-orbits are parametrized by the elements of the Weyl group \(W\). Each \(B\) orbit is a locally closed subscheme with finite codimension and is isomorphic to the scheme \(\mathbb A^\infty = \text{Spec} (\mathbb C(x_1,x_2,\dots,))\). The \(K\)-group is decomposed into the product \(K_B(X) \cong \prod_{w\in W} K_B(pt)[\mathcal{O}_{X^w}]\). The group \(K_B(pt)\) is isomorphic to the group ring \(\mathbb Z[P]\) of the weight lattice \(P\) of a maximal torus of \(B\). Similar to the finite-dimensional case there is a homomorphism \(\mathbb Z[P] \otimes \mathbb Z[P] \cong K_B(pt) \otimes K_B(pt) \to K_B(X)\) which factors through the equivariant Atiyah-Hirzebruch homomorphism \(\mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P] \to K_B(X)\), where \(\mathbb Z[P]^W\) is the ring of invariants with respect to the action of the Weyl group \(W\). In the affine case, this morphism in injective but not surjective, not all \([{\mathcal O}_{X^w}]\) are in the image of this morphism. Nevertheless, after localization by a generator \(\delta\) of null roots, taking tensor product with the subring \(\mathbb Q[\delta]\) we get the main result of the paper: Theorem 4.4: all the elements \([{\mathcal O}_{X^w}]\) are in \(\mathbb Q[\delta] \bigotimes_{\mathbb Z[e^{\pm\delta}]} K_B(X)\). The authors call the elements of \(R \bigotimes_{\mathbb Z[e^{\pm\delta}]} \mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P]\) the affine Grothendieck polynomials.
scientific article