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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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