## Equivariant $$K$$-theory of affine flag manifolds and affine Grothendieck polynomials.(English)Zbl 1173.19004

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.

### MSC:

 19L47 Equivariant $$K$$-theory 14M17 Homogeneous spaces and generalizations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties

### Keywords:

$$K$$-theory; affine flag manifold; Grothendieck polynomial
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### References:

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