zbMATH — the first resource for mathematics

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.

14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
Full Text: DOI arXiv
[1] R. Bezrukavnikov, The dimension of the fixed point set on affine flag manifolds , Math. Res. Lett. 3 (1996), 185–189. · Zbl 0874.20033 · doi:10.4310/MRL.1996.v3.n2.a5
[2] T. Chang and T. Skjelbred, The topological Schur lemma and related results , Ann. of Math. (2) 100 (1974), 307–321. JSTOR: · Zbl 0288.57021 · doi:10.2307/1971074 · links.jstor.org
[3] P. Deligne, Théorie de Hodge, II , Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57. · Zbl 0219.14007 · doi:10.1007/BF02684692 · numdam:PMIHES_1971__40__5_0 · eudml:103914
[4] –. –. –. –., ”Poids dans la cohomologie des variétés algébriques” in Proceedings of the International Congress of Mathematicians (Vancouver, B.C., Canada, 1974) , Canad. Math. Congress, Montreal, 1975, 79–85.
[5] M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem , Invent. Math. 131 (1998), 25–83. · Zbl 0897.22009 · doi:10.1007/s002220050197
[6] ——–, Purity of equivalued affine Springer fibers , · Zbl 1133.22013 · doi:10.1090/S1088-4165-06-00200-7 · arxiv.org
[7] M. Hall, Combinatorial Theory , 2nd ed., Wiley-Intersci. Ser. Discrete Math., Wiley, New York, 1986.
[8] J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups , Math. Surveys Monogr. 43 , Amer. Math. Soc., Providence, 1995. · Zbl 0834.20048
[9] D. Kazhdan and G. Lusztig, A topological approach to Springer’s representations , Adv. in Math. 38 (1980), 222–228. · Zbl 0458.20035 · doi:10.1016/0001-8708(80)90005-5
[10] –. –. –. –., Fixed point varieties on affine flag manifolds , Israel J. Math. 62 (1988), 129–168. · Zbl 0658.22005 · doi:10.1007/BF02787119
[11] R. Kottwitz, Isocrystals with additional structure, II , Compositio Math. 109 (1997), 255–339. · Zbl 0966.20022 · doi:10.1023/A:1000102604688
[12] –. –. –. –., Transfer factors for Lie algebras , Represent. Theory 3 (1999), 127–138. · Zbl 1044.22011 · doi:10.1090/S1088-4165-99-00068-0
[13] S. Kumar, ”An introduction to ind -varieties,” Appendix B of ”Infinite Grassmannians and moduli spaces of \(G\)-bundles” in Vector Bundles On Curves –.New Directions (Cetraro, Italy, 1995) , Lecture Notes in Math. 1649 , Springer, Berlin, 1997, 33–38. · Zbl 0884.14021 · doi:10.1007/BFb0094424
[14] R. P. Langlands, Les débuts d’une formule des traces stables , Publ. Math. Univ. Paris VII 13 , Université de Paris VII, U.E.R. de Mathématiques, Paris, 1983.
[15] R. P. Langlands and D. Shelstad, On the definition of transfer factors , Math. Ann. 278 (1987), 219–271. · Zbl 0644.22005 · doi:10.1007/BF01458070 · eudml:164291
[16] G. Laumon, Fibres de Springer et Jacobiennes compactifiees , · Zbl 1097.14027 · doi:10.1007/978-0-8176-4532-8_9 · arxiv.org
[17] ——–, Sur le lemme fondamental pour les groupes unitaires , · arxiv.org
[18] G. Lusztig, Affine Weyl groups and conjugacy classes in Weyl groups , Transform. Groups 1 (1996), 83–97. · Zbl 0870.20028 · doi:10.1007/BF02587737
[19] G. Lusztig and J. M. Smelt, Fixed point varieties on the space of lattices , Bull. London Math. Soc. 23 (1991), 213–218. · Zbl 0779.14004 · doi:10.1112/blms/23.3.213
[20] D. S. Sage, A construction of representations of affine Weyl groups , Compositio Math. 108 (1997), 241–245. · Zbl 0896.20030 · doi:10.1023/A:1000167027904
[21] –. –. –. –., The geometry of fixed point varieties on affine flag manifolds , Trans. Amer. Math. Soc. 352 (2000), 2087–2119. JSTOR: · Zbl 1057.14060 · doi:10.1090/S0002-9947-99-02295-3 · links.jstor.org
[22] J.-P. Serre, Corps Locaux , 2nd ed., Publ. Inst. Math. Univ. Nancago 8 , Hermann, Paris, 1968.
[23] I. R. Shafarevich, On some infinite-dimensional groups, II , Math. USSR-Izv. 18 (1982), 185–194. · Zbl 0491.14025 · doi:10.1070/IM1982v018n01ABEH001379
[24] J.-L. Waldspurger, Le lemme fondamental implique le transfert , Compositio Math. 105 (1997), 153–236. · Zbl 0871.22005 · doi:10.1023/A:1000103112268
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.