Equivariant homology and K-theory of affine Grassmannians and Toda lattices
Equivariant homology and K-theory of affine Grassmannians and Toda lattices
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DOI：
10.1112/s0010437x04001228
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发表时间：
2005-05-01
影响因子：
1.8
通讯作者：
Mirkovic, I
中科院分区：
文献类型：
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作者：
Bezrukavnikov, R;Finkelberg, M;Mirkovic, I
对于仿射Grassmanian Gr(G)=G((C((T)/G(C[[t]]))的几乎单复代数群G，我们考虑等变同调H-G(C[[t]])(GRG)和K-理论KG(C[[t]])(GRG)。它们都具有关于卷积的交换环结构。我们用朗兰兹对偶群G的泛群-代数中心子来确定同调环的谱，并将K-同调环的谱与G和G的泛群-群中心子联系起来。如果加上圈旋转等差，我们就得到了(K-)同调环的一个非对易形变，从而得到了它的谱上的泊松结构。我们将这种结构与万能中心子上的标准结构联系起来。点的G(C[[t]])-等变同调的交换子环产生一个极化，它与Kostant的Toda格可积系有关。我们还计算了仿射Grassman-Steinberg簇的等变K-环。GRG的等变K-同调具有由简单等变倒向凝聚层类构成的标准基。它们的卷积也是逆的，并且与G(C[[t]])-模的Feigin-Loktev融合积有关。
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]])(GrG) and K-theory KG(C[[t]])(GrG). 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, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of 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 relate this structure to 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 GrG is equipped with a canonical basis 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.