E-series of character varieties of non-orientable surfaces
Annales de l'Institut Fourier, Online first, 36 p.

In this paper we are interested in two kinds of stacks associated to a compact non-orientable surface Σ. (A) We consider simply the quotient stack of the space of representations of the fundamental group of Σ to GL n . (B) We choose a set S of k-punctures of Σ and a generic k-tuple of semisimple conjugacy classes of GL n , and we consider the stack of anti-invariant local systems on the orientation cover of Σ with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincaré series). In case (B), unexpectedly (see Remark 1.9), when the Euler characteristic of Σ is even, our formulas turn out to be closely related to those arising from the character varieties of punctured compact orientable surfaces studied in [14] and [15].

Dans cet article nous nous intéressons à deux types de champs associés à une surface compacte non-orientable Σ. (A) On considère simplement le champ quotient de l’espace des représentations du groupe fondamental de Σ dans GL n . (B) On choisit un ensemble S de k points de Σ et un k-uplet générique de classes de conjugaison semsimples de GL n et on considère le champ des systèmes locaux anti-invariants sur le revêtement d’orientation de ΣS avec monodromies locales dans les classes de conjugaison choisies. On calcule le nombre de points de ces champs sur un corps fini et on en déduit une formule pour leur E-série (une certaine spécialisation de la série de Poincaré mixte). Dans le cas (B), étonnamment (voir Remarque 1.9), lorsque la caractéristique d’Euler de Σ est paire, nos formules sont très proches de celles provenant des variétés de caractères de surfaces compactes orientables épointées étudiées dans [14] et [15].

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3540
Classification: 00X99
Keywords: Character varieties, non-orientable surfaces.
Letellier, Emmanuel 1; Rodriguez-Villegas, Fernando 2

1 IMJ-PRG, Université Paris Cité (France)
2 ICTP Trieste (Italy)
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Letellier, Emmanuel; Rodriguez-Villegas, Fernando. E-series of character varieties of non-orientable surfaces. Annales de l'Institut Fourier, Online first, 36 p.

[1] Baird, T. J.; Wong, M. L. E-polynomials of character varieties for real curves (https://arxiv.org/abs/2006.01288)

[2] Behrend, Kai A. The Lefschetz trace formula for algebraic stacks, Invent. Math., Volume 112 (1993) no. 1, pp. 127-149 | DOI | MR | Zbl

[3] Benyash-Krivets, V.; Chernousov, V. I. Representation varieties of the fundamental groups of compact non-orientable surfaces, Math. Sb., Volume 188 (1997), pp. 42-92

[4] Cheng, S. Character varieties with non-connected structure groups (https://arxiv.org/abs/1912.04360)

[5] Deligne, Pierre La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974) no. 43, pp. 273-307 | DOI | MR | Zbl

[6] Deligne, Pierre Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974) no. 44, pp. 5-77 | DOI | MR | Zbl

[7] Ellerman, D. The number of direct-sum decompositions of a finite vector space (https://arxiv.org/abs/1603.07619)

[8] Faddeev, L. D.; Kashaev, R. M. Quantum dilogarithm, Modern Phys. Lett. A, Volume 9 (1994) no. 5, pp. 427-434 | DOI | MR | Zbl

[9] Frobenius, G.; Schur, I. Über die reellen Darstellungen der endlichen Gruppen., Berl. Ber., Volume 1906 (1906), pp. 186-208

[10] Fulman, Jason; Guralnick, Robert Conjugacy class properties of the extension of GL(n,q) generated by the inverse transpose involution, J. Algebra, Volume 275 (2004) no. 1, pp. 356-396 | DOI | MR | Zbl

[11] Gordon, Cameron; Rodriguez-Villegas, Fernando On the divisibility of #Hom(Γ,G) by |G|, J. Algebra, Volume 350 (2012), pp. 300-307 | DOI | MR | Zbl

[12] Gow, R. Properties of the characters of the finite general linear group related to the transpose-inverse involution, Proc. London Math. Soc. (3), Volume 47 (1983) no. 3, pp. 493-506 | DOI | MR | Zbl

[13] Hausel, M.; Mereb, T.; Rodriguez-Villegas, F. Mirror symmetry in the character table of SL n (𝔽 q ) (in preparation)

[14] Hausel, Tamás; Letellier, Emmanuel; Rodriguez-Villegas, Fernando Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J., Volume 160 (2011) no. 2, pp. 323-400 | DOI | MR

[15] Hausel, Tamás; Letellier, Emmanuel; Rodriguez-Villegas, Fernando Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math., Volume 234 (2013), pp. 85-128 | DOI | MR | Zbl

[16] Hausel, Tamás; Rodriguez-Villegas, Fernando Mixed Hodge polynomials of character varieties, Invent. Math., Volume 174 (2008) no. 3, pp. 555-624 (With an appendix by Nicholas M. Katz) | DOI | MR | Zbl

[17] Jacobsthal, Ernst Sur le nombre d’éléments du groupe symétrique S n dont l’ordre est un nombre premier, Norske Vid. Selsk. Forh., Trondheim, Volume 21 (1949) no. 12, pp. 49-51 | MR

[18] Laszlo, Yves; Olsson, Martin The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. Inst. Hautes Études Sci. (2008) no. 107, pp. 109-168 | DOI | MR | Zbl

[19] Laszlo, Yves; Olsson, Martin The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. Inst. Hautes Études Sci. (2008) no. 107, pp. 169-210 | DOI | MR | Zbl

[20] Macdonald, I. G. Affine root systems and Dedekind’s η-function, Invent. Math., Volume 15 (1972), pp. 91-143 | DOI | MR | Zbl

[21] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR

[22] Mellit, Anton Integrality of Hausel–Letellier–Villegas kernels, Duke Math. J., Volume 167 (2018) no. 17, pp. 3171-3205 | DOI | MR | Zbl

[23] Mozgovoy, Sergey A computational criterion for the Kac conjecture, J. Algebra, Volume 318 (2007) no. 2, pp. 669-679 | DOI | MR | Zbl

[24] Rodríguez-Villegas, Fernando Counting colorings on varieties, Publ. Mat., Volume 51 (2007), pp. 209-220 (Proceedings of the Primeras Jornadas de Teoría de Números) | DOI | MR | Zbl

[25] Schaffhauser, Florent Lectures on Klein surfaces and their fundamental group, Geometry and quantization of moduli spaces (Adv. Courses Math. CRM Barcelona), Birkhäuser/Springer, Cham, 2016, pp. 67-108 | DOI | MR | Zbl

[26] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl

[27] Waterhouse, William C. The number of congruence classes in M n (F q ), Finite Fields Appl., Volume 1 (1995) no. 1, pp. 57-63 | DOI | MR | Zbl

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