no. 4
E-series of character varieties of non-orientable surfaces
[E-série des variétés de caractères des surfaces non-orientées]
Letellier, Emmanuel1 ; Rodriguez-Villegas, Fernando2
1 IMJ-PRG, Université Paris Cité (France)
2 ICTP Trieste (Italy)
Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1385-1420.

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 Σ∖S 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 [13] et [14].

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 [13] and [14].

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3540
Classification : 00X99
Keywords: Character varieties, non-orientable surfaces.
Mot clés : Variétés de caractères, surfaces non-orientées.

Letellier, Emmanuel 1 ; Rodriguez-Villegas, Fernando 2

1 IMJ-PRG, Université Paris Cité (France)
2 ICTP Trieste (Italy)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2023__73_4_1385_0,
     author = {Letellier, Emmanuel and Rodriguez-Villegas, Fernando},
     title = {E-series of character varieties of non-orientable surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1385--1420},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {4},
     year = {2023},
     doi = {10.5802/aif.3540},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3540/}
}
TY  - JOUR
AU  - Letellier, Emmanuel
AU  - Rodriguez-Villegas, Fernando
TI  - E-series of character varieties of non-orientable surfaces
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 1385
EP  - 1420
VL  - 73
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3540/
DO  - 10.5802/aif.3540
LA  - en
ID  - AIF_2023__73_4_1385_0
ER  - 
%0 Journal Article
%A Letellier, Emmanuel
%A Rodriguez-Villegas, Fernando
%T E-series of character varieties of non-orientable surfaces
%J Annales de l'Institut Fourier
%D 2023
%P 1385-1420
%V 73
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3540/
%R 10.5802/aif.3540
%G en
%F AIF_2023__73_4_1385_0
Letellier, Emmanuel; Rodriguez-Villegas, Fernando. E-series of character varieties of non-orientable surfaces. Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1385-1420. doi : 10.5802/aif.3540. https://aif.centre-mersenne.org/articles/10.5802/aif.3540/

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

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

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

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

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

[9] 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

[10] 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

[11] 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

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

[13] 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

[14] 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

[15] 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

[16] 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

[17] 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

[18] 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

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

[20] 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

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

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

[23] 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

[24] 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

[25] Shu, C. Character varieties with non-connected structure groups (https://arxiv.org/abs/1912.04360)

[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

Cité par Sources :