E-polynomials of character varieties

Zamora, Alfonso

doi:10.15517/tvetzr54

Autores/as

Alfonso Zamora Universidad Politécnica de Madrid, Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones (Escuela Técnica Superior de Ingenieros Informáticos), Madrid, Spain Autor/a https://orcid.org/0000-0002-8340-5366

DOI:

https://doi.org/10.15517/tvetzr54

Palabras clave:

Variedades de caracteres, e-polinomios, Estructuras de Hodge mixtas, Simetría especular topológica, Dualidad de Langlands

Resumen

Este trabajo contiene las ideas básicas y construcciones sobre e-polinomios en variedades de caracteres y el estado del arte de cierta investigación en el campo, junto con algunas nuevas direcciones. En él introducimos las estructuras de Hodge mixtas y los E-polinomios, junto con una serie de técnicas aritméticas (contar puntos sobre cuerpos finitos) y geométricas (estratificaciones en tipos parabólicos) para calcularlos. Incluimos un ejemplo completo del cálculo del e-polinomio para la GL₃-variedad de caracteres del grupo libre. Finalmente, extendemos la estratificación geométrica en tipos parabólicos a un grupo reductivo G general para obtener expresiones motívicas explícitas para la G-variedad de caracteres, y reducir ciertas conjeturas en simetría especular topológica para estos espacios de móduli.

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