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K-theory Schubert calculus of the affine Grassmannian

Part of: Projective and enumerative geometry Algebraic combinatorics

Published online by Cambridge University Press:  26 January 2010

Thomas Lam ,
Anne Schilling  and
Mark Shimozono
Thomas Lam
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: tfylam@math.harvard.edu)
Anne Schilling
Affiliation:
Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA (email: anne@math.ucdavis.edu)
Mark Shimozono
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (email: mshimo@vt.edu)
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Abstract

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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 where G=SLn, 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, calledK-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 by means of Kashiwara’s thick affine flag manifold.

Keywords

affine GrassmannianK-theorySchubert calculussymmetric functionsGKM condition

MSC classification

Secondary: 05E05: Symmetric functions and generalizations 14N15: Classical problems, Schubert calculus
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 811 - 852
Copyright
Copyright © Foundation Compositio Mathematica 2010

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