E-polynomials of generic $\operatorname{GL}_n \rtimes \langle \sigma \rangle $-character varieties: unbranched case
[E-polynômes des variétés de $\operatorname{GL}_n \rtimes \langle \sigma \rangle $-caractère générique : le cas non ramifié]
Shu, Cheng1
1Institute for Theoretical Sciences, Westlake University, Hangzhou 310030, China
Annales de l'Institut Fourier, Online first, 42 p.

For any unbranched double covering of a compact Riemann surface, we study the associated character varieties that are unitary in the global sense, which we call $\operatorname{GL}_n \rtimes \langle \sigma \rangle $-character varieties. We introduce $k>0$ punctures on the surface, restrict the monodromies around the punctures to generic semi-simple conjugacy classes in $\operatorname{GL}_n$, and compute the E-polynomials of these character varieties using the character table of $\operatorname{GL}_n(q)$. The result is expressed as the inner product of certain symmetric functions. We are then led to a conjectural formula for the mixed Hodge polynomial, which is built out of (modified) Macdonald polynomials, their self-pairings, and self-pairings of wreath Macdonald polynomials.

Pour tout revêtement non ramifié de degré deux d’une surface de Riemann compacte, nous étudions les variétés de caractère associées qui sont unitaires dans le sens global, et nous les appellerons variétés de $\operatorname{GL}_n \rtimes \langle \sigma \rangle $-caractère. Nous enlevons $k$ points de la surface donnée, exigeons que les monodromies autour des points enlevés appartiennent aux classes de conjugaison semi-simples génériques dans $\operatorname{GL}_n$, et calculons les E-polynômes de ces variétés de caractère en utilisant la table des caractères de $\operatorname{GL}_n(q)$. Le résultat est exprimé comme le produit scalaire de certaines fonctions symétriques. Nous proposons une formule conjecturale pour le polynôme de Hodge mixte, qui est construit à partir des polynômes de Macdonald, de leurs produits scalaires, et des produits scalaires des polynômes de Macdonald en couronne.

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DOI : 10.5802/aif.3793
Classification : 14M35, 20C33, 05E05
Keywords: character varieties, mixed Hodge polynomial, symmetric functions
Mots-clés : variétés de caractère, polynôme de Hodge mixte, fonctions symétriques

Shu, Cheng  1

1 Institute for Theoretical Sciences, Westlake University, Hangzhou 310030, China 
Shu, Cheng. E-polynomials of generic $\operatorname{GL}_n \rtimes \langle \sigma \rangle $-character varieties: unbranched case. Annales de l'Institut Fourier, Online first, 42 p.
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