# zbMATH — the first resource for mathematics

$$K$$-theory Schubert calculus of the affine Grassmannian. (English) Zbl 1256.14056
Summary: We construct the Schubert basis of the torus-equivariant $$K$$-homology of the affine Grassmannian of a simple algebraic group $$G$$, using the $$K$$-theoretic nil Hecke ring of B. Kostant and S. Kumar [J. Differ. Geom. 32, No. 2, 549–603 (1990; Zbl 0731.55005)]. This is the $$K$$-theoretic analogue of a construction of D. Peterson in equivariant homology. For the case where $$G=\text{SL}_n$$, the $$K$$-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called $$K$$-$$k$$-Schur functions, whose highest-degree term is a $$k$$-Schur function. The dual basis in $$K$$-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of T. Lam [J. Am. Math. Soc. 21, No. 1, 259–281 (2008; Zbl 1149.05045)]. In addition, we give a Pieri rule in $$K$$-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

##### MSC:
 14N15 Classical problems, Schubert calculus 05E05 Symmetric functions and generalizations 19E08 $$K$$-theory of schemes
SageMath
Full Text:
##### References:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.