Equivariant ( K -)homology of affine Grassmannian and Toda lattice
Abstract
For an almost simple complex algebraic group
G
with affine Grassmannian
G
r
G
=G(C((t)))/G(C[[t]])
we consider the equivariant homology
H
G(C[[t]])
(G
r
G
)
, and
K
-theory
K
G(C[[t]])
(G
r
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
G
ˇ
, and we relate the spectrum of
K
-homology ring to the universal group-group centralizer of
G
and of
G
ˇ
. 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
G
r
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.