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Homology of affine Springer fibers in the unramified case. (English) Zbl 1162.14311
Let $$G$$ be a complex connected reductive algebraic group with Lie algebra $$\mathfrak g$$ and let $$T$$ be a maximal torus of $$G$$ with Lie algebra $$\mathfrak t$$. Let $$F={\mathbb C}((\epsilon))$$ be the field of formal Laurent series and $${\mathfrak O}={\mathbb C}[[t]]$$ be its ring of integers, the formal power series. For any $$\mathbb C$$-algebra $$R$$, let $$G(R)$$ denote the $$R$$-rational points of the algebraic group $$G$$, etc. Then the affine Grassmannian is, by definition, $$X=G(F)/G({\mathfrak O})$$. For $$\gamma\in{\mathfrak g}(F)$$, let $X_\gamma:= \{xG({\mathfrak O})\in X:\text{Ad}(x^{-1})(\gamma)\in{\mathfrak g}({\mathfrak O})\}$ be the affine Springer fiber. Now take any regular $$\gamma\in{\mathfrak t}({\mathfrak O})$$. Then $$T$$ acts on $$X_\gamma$$ and, moreover, $$X_\gamma$$ is a finite-dimensional ind-subvariety of $$X$$. One of the main results of the paper under review determines the $$T$$-equivariant homology $$H^T_*(X_\gamma)$$ with complex coefficients under the assumption that the singular homology $$H_*(X_\gamma)$$ with complex coefficients is pure in the sense of mixed Hodge theory. In fact, the authors conjecture that $$H_*(X_\gamma)$$ is always pure.
Assume for simplicity that $$G$$ is adjoint and let $$(H,s)$$ be an endoscopic data for $$G$$, i.e., $$H$$ is a complex connected reductive algebraic group and $$s\in\hat T\subset\hat G$$ such that the dual group $$\hat H$$ is the centralizer of $$s$$ in $$\hat G$$, where $$\hat T$$ is the dual torus in the dual group $$\hat G$$. In particular, $$H$$ and $$G$$ share the same maximal torus $$T$$. Thus we can view $$\gamma$$ also as an element $$\gamma_H\in{\mathfrak t}({\mathfrak O})\subset{\mathfrak h}({\mathfrak O}), \mathfrak h$$ being the Lie algebra of $$H$$. Let $$X^H_{\gamma H}\subset H(F)/H({\mathfrak O})$$ denote the associated affine Springer fiber. Then, the second main result of the paper asserts that, under the assumption that $$H_*(X_\gamma)$$ and $$H_*(X^H_{\gamma H})$$ are pure, there is a homomorphism $H^T_i(X_\gamma)\to H^T_{i-2r}(X^H_{\gamma H}),$ which becomes an isomorphism after a certain localization, where $$r$$ is an explicitly defined nonnegative integer depending upon the root systems of $$G$$ and $$H$$ and the element $$\gamma$$. The above homomorphism has various equivariance properties. There does not seem to be any direct connection between the ind-varieties $$X_\gamma$$ and $$X^H_{\gamma H}$$ and hence the above homomorphism is rather surprising.
Corresponding results for the affine Springer fiber $$Y_\gamma$$ in the full affine flag variety $$Y:= G(F)/\mathcal B$$ are also obtained in the paper, where $$B\subset G$$ is a fixed Borel subgroup containing $$T$$, $$e\colon G({\mathfrak O})\to G$$ is the homomorphism obtained from the evaluation at 0 and $${\mathcal B}:= e^{-1}(B)$$. In addition, they explicitly describe the action of the affine Weyl group on $$H_*(Y_\gamma)$$ constructed by Lusztig.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14L30 Group actions on varieties or schemes (quotients)
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