K-theory Schubert calculus of the affine Grassmannian

Details K-theory Schubert calculus of the affine Grassmannian K-theory Schubert calculus of the affine Grassmannian

2009-01-12

arxiv.org/abs/0901.1506v3

Compositio Mathematica 146 Issue 4 (2010) 811-852

Mathematics

Combinatorics

38 pages

Scientific paper

10.1112/S0010437X09004539

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 NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case G = 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 Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations using Kashiwara's thick affine flag manifold.

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