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 
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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/

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