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\(K\)-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
Software:
SageMath
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