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Equivariant ($K$-)homology of affine Grassmannian and Toda lattice

Roman Bezrukavnikov, Michael Finkelberg, Ivan Mirković

Abstract

For an almost simple complex algebraic group $G$ with affine Grassmannian $Gr_G= G(C((t)))/G(C[[t]])$ we consider the equivariant homology $H^{G(C[[t]])}(Gr_G)$, and $K$-theory $K^{G(C[[t]])}(Gr_G)$. They both have a commutative ring structure, with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of $K$-homology ring to the universal group-group centralizer of $G$ and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the ($K$)-homology ring, and thus a Poisson structure on its spectrum. We identify this structure with the standard one on the universal centralizer. The commutative subring of $G(C[[t]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant $K$-ring of the affine Grassmannian Steinberg variety. The equivariant $K$-homology of $Gr_G$ is equipped with a canonical base formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin-Loktev fusion product of $G(C[[t]])$-modules.

Equivariant ($K$-)homology of affine Grassmannian and Toda lattice

Abstract

For an almost simple complex algebraic group with affine Grassmannian we consider the equivariant homology , and -theory . They both have a commutative ring structure, with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group , and we relate the spectrum of -homology ring to the universal group-group centralizer of and of . If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the ()-homology ring, and thus a Poisson structure on its spectrum. We identify this structure with the standard one on the universal centralizer. The commutative subring of -equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant -ring of the affine Grassmannian Steinberg variety. The equivariant -homology of is equipped with a canonical base formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin-Loktev fusion product of -modules.

Paper Structure

This paper contains 42 sections, 15 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notations and statements of the results
  4. Kostant slices and Steinberg sections
  5. The universal centralizers
  6. Isogenies
  7. Symplectic structures
  8. Affine blow-ups
  9. Poisson structures
  10. Flat group sheaves
  11. Equivariant Borel-Moore Homology
  12. Equivariant $K$-theory
  13. Calculations in rank 1
  14. ${\mathfrak{Z}}_G^G$ and ${\mathfrak{B}}_G^G$
  15. ${\mathfrak{Z}}_{\mathfrak{g}}^G$ and ${\mathfrak{B}}_{\mathfrak{g}}^G$
  16. ...and 27 more sections

Key Result

Proposition 2.7

The Poisson structure on ${\mathfrak{B}}_{\mathfrak{g}}^{\mathfrak{g}}-{\mathbf{D}}$ (resp. ${\mathfrak{B}}_G^{\mathfrak{g}}-{\mathbf{D}}, {\mathfrak{B}}_{\mathfrak{g}}^G-{\mathbf{D}},\ {\mathfrak{B}}_G^G-{\mathbf{D}},\ {\mathfrak{B}}_G^{\check G}{}-{\mathbf{D}}$) extends to the global Poisson struc

Theorems & Definitions (23)

  • Proposition 2.7
  • Proposition 2.8
  • Remark 1
  • Proposition 2.10
  • Theorem 2.12
  • Remark 2.13
  • Theorem 2.15
  • Proposition 4.2
  • Lemma 4.4
  • Lemma 5.1
  • ...and 13 more