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On the \(E\)-polynomials of a family of \({\text {Sl}}_n\)-character varieties

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Abstract

We find the \(E\)-polynomials of a family of twisted character varieties \({\mathcal {M}}({\text {Sl}}_n)\) of Riemann surfaces by proving they have polynomial count, and applying a result of Katz regarding the counting functions. To count the number of \(\mathbb {F}_q\)-points of these varieties as a function of \(q,\) we invoke a formula from Frobenius. Our calculations make use of the character tables of \({\text {Sl}}_n(q),\) partially computed by Lehrer, and a result of Hanlon on the Möbius function of a fixed subposet of set-partitions. We compute the Euler characteristic of the \({\mathcal {M}}({\text {Sl}}_n)\) with these polynomials, and show they are connected.

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Notes

  1. Most of the results in this paper can be found in the author’s Ph.D. thesis [26].

  2. Also: the biggest \((\frac{|H|}{\chi (1)})\) is attained by the character with smallest degree \(\chi (1).\)

  3. The restriction of the trivial character \(1_G\) is \(1_H,\) hence it cannot have more than one component.

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Acknowledgments

The author wishes to thank the University of Texas at Austin, for having kept their doors open at all times, and the Harrington Foundation, for financial support, and many stimulating meetings with a diverse community of scholars from every corner of knowledge. This is part of his Ph.D. thesis, written under the supervision of Rodríguez-Villegas, to whom the author is greatly indebted, as to Vaaler, Voloch, Lehrer and specially Ben-Zvi for the assistance provided on the topic. Helpful suggestions from Miatello, García Fernández, Hausel, Jeronimo, Quallbrunn, Cueto and Mautner were greatly appreciated. The author is particularly grateful for the orientation given by Adduci. Special thanks are extended to members of FKC for their feedback, to Dickenstein, Cortiñas, Lalin, D’Andrea, Cattani, Fauring and Cuckierman, for their trust and encouragment, and to social worker Ramírez for her endless patience.

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  1. Department of Mathematics, FCEyN-UBA, Ciudad Universitaria Pab 1, 1428, Buenos Aires, Argentina

    Martin Mereb

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Mereb, M. On the \(E\)-polynomials of a family of \({\text {Sl}}_n\)-character varieties. Math. Ann. 363, 857–892 (2015). https://doi.org/10.1007/s00208-015-1183-2

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