Abstract
We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties \(\mathcal{M}_{n}\) of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type \(\text{GL}(n,\mathbb{F}_{q})\) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of \(\mathcal{M}_{n}\): an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.
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Alvis, D.: The duality operation in the character ring of a finite Chevalley group. Bull. Am. Math. Soc., New Ser. 1(6), 907–911 (1979)
Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond., Ser. A 308, 523–615 (1982)
Beilinson, A., Drinfeld V.: Quantization of Hitchins Integrable System and Hecke Eigensheaves. http://www.math.uchicago.edu/∼arinkin/langlands/ (ca. 1995) (preprint)
Bott, R., Tolman, S., Weitsman, J.: Surjectivity for Hamiltonian loop group spaces. Invent. Math. 155(2), 225–251 (2004)
Crawley-Boevey, W., Van den Bergh, M.: Absolutely indecomposable representations and Kac–Moody Lie algebras. With an appendix by Hiraku Nakajima. Invent. Math. 155(3), 537–559 (2004)
Curtis, C.: Truncation and duality in the character ring of a finite group of Lie type. J. Algebra 62(2), 320–332 (1980)
Danilov, V.I., Khovanskiĭ, A.G.: Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 50(5), 925–945 (1986)
Deligne, P.: Théorie de Hodge II. Publ. Math., Inst. Hautes Étud. Sci. 40, 5–47 (1971)
Deligne, P.: Théorie de Hodge III. Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974)
Deligne, P.: La conjecture de Weil. I. Publ. Math., Inst. Hautes Étud. Sci. 43, 273–307 (1974)
Deligne, P.: La conjecture de Weil II. Publ. Math., Inst. Hautes Étud. Sci. 52, 313–428 (1981)
Earl, R., Kirwan, F.: The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface. J. Lond. Math. Soc., II. Ser. 60(3), 835–846 (1999)
Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. In: Moduli of Curves and abelian Varieties. Aspects Math., vol. E33, pp. 109–129. Vieweg, Braunschweig (1999)
Faltings, G.: p-adic Hodge theory. J. Am. Math. Soc. 1(1), 255–299 (1988)
Fontaine, J.-M., Messing, W.: p-adic periods and p-adic étale cohomology. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985). Contemp. Math., vol. 67, pp. 179–207. Am. Math. Soc., Providence, RI (1987)
Franz, M., Weber, A.: Weights in cohomology and the Eilenberg–Moore spectral sequence. Ann. Inst. Fourier (Grenoble) 55(2), 673–691 (2005)
Freed, D., Quinn, F.: Chern–Simons theory with finite gauge group. Commun. Math. Phys. 156(3), 435–472 (1993)
Frobenius, F.G.: Über Gruppencharaktere (1896). In: Gesammelte Abhandlungen III. Springer, Berlin–Heidelberg (1968)
Fulton, W.: Introduction to toric varieties. Ann. Math. Stud. The William H. Roever Lectures in Geometry, vol. 131. Princeton University Press, Princeton, NJ (1993)
Garsia, A.M., Haiman, M.: A remarkable q,t-Catalan sequence and q-Lagrange inversion. J. Algebr. Comb. 5(3), 191–244 (1996)
Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58(3), 300–321 (1985)
Green, J.A.: The characters of the finite general linear groups. Trans. Am. Math. Soc. 80, 402–447 (1955)
Getzler, E.: Mixed Hodge Structures of Configuration Spaces. arXiv:alg-geom/9510018 (preprint)
Gothen, P.B.: The Betti numbers of the moduli space of rank 3 Higgs bundles. Int. J. Math. 5, 861–875 (1994)
Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9, 119–221 (1957)
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Publ. Math., Inst. Hautes Étud. Sci. 32, 5–361 (1967)
Hanlon, P.: The fixed point partition lattices. Pac. J. Math. 96, 319–341 (1981)
Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles over curves. Math. Ann. 212, 215–248 (1975)
Hausel, T.: Vanishing of intersection numbers on the moduli space of Higgs bundles. Adv. Theor. Math. Phys. 2, 1011–1040 (1998) arXiv:math.AG/9805071
Hausel, T. : Quaternionic geometry of matroids. Cent. Eur. J. Math. 3(1), 26–38 (2005) arXiv:math.AG/0308146
Hausel, T.: Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve. In: Bogomolov, F., Tschinkel, Y. (eds.) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol. 235. Birkhäuser, Boston (2005). arXiv:math.AG/0406380
Hausel, T.: Betti numbers of holomorphic symplectic quotients, via arithmetic Fourier transform. Proc. Natl. Acad. Sci. USA 103(16), 6120–6124 (2006) arXiv: math.AG/0511163
Hausel, T.: S-duality in hyperkähler Hodge theory. arXiv:0709.0504v1 (to appear in: “The many facets of geometry: a tribute to Nigel Hitchin”, OUP)
Hausel, T.: Arithmetic harmonic analysis, Macdonald polynomials and the topology of the Riemann–Hilbert monodromy map (in preparation)
Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties (in preparation)
Hausel, T., Sturmfels, B.: Toric hyperkähler varieties. Doc. Math. 7, 495–534 (2002) arXiv:math.AG/0203096
Hausel, T., Thaddeus, M.: Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles. J. Am. Math. Soc. 16, 303–329 (2003) arXiv:math.AG/0003094
Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality and Hitchin systems. Invent. Math. 153(1), 197–229 (2003) arXiv:math.AG/0205236
Hausel, T., Thaddeus, M.: Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles. Proc. Lond. Math. Soc. 88, 632–658 (2004) arXiv:math.AG/0003093
Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987)
Hua, J.: Counting representations of quivers over finite fields. J. Algebra 226(2), 1011–1033 (2000)
Ito, T.: Stringy Hodge numbers and p-adic Hodge theory. Compos. Math. 140(6), 1499–1517 (2004)
Jeffrey, L.: Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds. Duke Math. J. 77, 407–429 (1995)
Kac, V.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)
Kac, V.: Root systems, representations of quivers and invariant theory. In: Invariant Theory (Montecatini, 1982). Lect. Notes Math., vol. 996, pp. 74–108. Springer, Berlin (1983)
Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007) arXiv:hep-th/0604151
Katz, N.M.: Rigid Local Systems. Ann. Math. Studies, vol. 139. Princeton University Press, Princeton, NJ (1996)
Laumon, G.: Comparaison de caractéristiques d’Euler–Poincaré en cohomologie l-adique. C. R. Acad. Sci., Paris, Sér I, Math. 292(3), 209–212 (1981)
Liebeck, M.W., Shalev, A.: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159(2), 317–367 (2005)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn., Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1995)
Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. J. Reine Angew. Math. 544, 61–82 (2002)
McCleary, J.: A User’s Guide to Spectral Sequences, 2nd edn. Cambridge University Press, Cambridge (2001)
Mednykh, A.D.: Determination of the number of nonequivalent coverings over a compact Riemann surface. Soviet Mathematics Doklady 19, 318–320 (1978)
Meinrenken, A.: Witten’s formulas for intersection pairings on moduli spaces of flat G-bundles. Adv. Math. 197(1), 140–197 (2005)
Mumford, D.: The Red Book of Varieties and Schemes. Springer, Berlin–Heidelberg (1999)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edition. Springer, Berlin (1994)
Newstead, P.E.: Introduction to Moduli Problems and Orbit Spaces. Tata Inst. Bombay. Narosa Publishing House, New Delhi (1978)
Peters, C., Steenbrink, J.: Mixed Hodge Structures. Ergeb. Mathematik, vol. 52. Springer, Berlin (2008)
Racaniére, S.: Kirwan map and moduli space of flat connections. Math. Res. Lett. 11, 419–433 (2004)
Reineke, M.: The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152, 349–368 (2003)
Rodriguez-Villegas, F.: Counting colorings on varieties. In: Proceedings of Primeras Jornadas de Teoría de Números (de Espanya), Villanova i la Geltrú, Spain, 2005. Publ. Mat., Barc., vol. extra 2007. Universitat Autònoma de Barcelona, Barcelona (2007)
Serre, J.-P.: Topics in Galois Theory. Jones and Bartlett Publishers, Boston (1992)
Seshadri, C.S.: Geometric reductivity over arbitrary base. Adv. Math. 26(3), 225–274 (1977)
Simpson, C.T.: Nonabelian Hodge theory. In: Proceedings of the International Congress of Mathematicians, vols. I, II (Kyoto, 1990), pp. 747–756. Math. Soc. Jap., Tokyo (1991)
Van den Bogaart, T., Edixhoven, B.: Algebraic stacks whose number of points over finite fields is a polynomial. In: Number Fields and Function Fields – Two Parallel Worlds. Progr. Math., vol. 239, pp. 39–49. Birkhäuser, Boston, MA (2005)
Wang, C.-L.: Cohomology theory in birational geometry. J. Differ. Geom. 60(2), 345–354 (2002)
Zagier, D.: Elementary aspects of the Verlinde formula and of the Harder–Narasimhan–Atiyah–Bott formula. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993). Israel Math. Conf. Proc., vol. 9, pp. 445–462. Bar-Ilan Univ., Ramat Gan (1996)
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Hausel, T., Rodriguez-Villegas, F. Mixed Hodge polynomials of character varieties. Invent. math. 174, 555–624 (2008). https://doi.org/10.1007/s00222-008-0142-x
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DOI: https://doi.org/10.1007/s00222-008-0142-x