This thesis focuses on two problems on l-modular representation theory of p-adic groups. Let F be a non-archimedean local field of residue characteristic p different from l. In the first part, we study block decomposition of the category of smooth modular representations of GL(n; F) and its inner forms.We want to reduce the description of a positive-level block to the description of a 0-level block (of a similar group) seeking equivalences of categories. Using the type theory of Bushnell-Kutzko in the modular case and a theorem of category theory, we reduce the problem to find an isomorphism between two intertwining algebras. The proof of the existence of such an isomorphism is not complete because it relies on a conjecture that we state and we prove for several cases. In the second part we generalize the construction of metaplectic group and Weil representation in the case of representations over un integral domain. We define a central extension of the symplectic group over F by the multiplicative group of an integral domain. We prove that it satisfies the same properties as in the complex case.

(2015). Représentations l-modulaires des groupes p-adiques. Décomposition en blocs de la catégorie des représentations lisses de GL(m,D), groupe métaplectique et représentation de Weil. (Tesi di dottorato, Università degli Studi di Milano-Bicocca, 2015).

Représentations l-modulaires des groupes p-adiques. Décomposition en blocs de la catégorie des représentations lisses de GL(m,D), groupe métaplectique et représentation de Weil

CHINELLO, GIANMARCO
2015

Abstract

This thesis focuses on two problems on l-modular representation theory of p-adic groups. Let F be a non-archimedean local field of residue characteristic p different from l. In the first part, we study block decomposition of the category of smooth modular representations of GL(n; F) and its inner forms.We want to reduce the description of a positive-level block to the description of a 0-level block (of a similar group) seeking equivalences of categories. Using the type theory of Bushnell-Kutzko in the modular case and a theorem of category theory, we reduce the problem to find an isomorphism between two intertwining algebras. The proof of the existence of such an isomorphism is not complete because it relies on a conjecture that we state and we prove for several cases. In the second part we generalize the construction of metaplectic group and Weil representation in the case of representations over un integral domain. We define a central extension of the symplectic group over F by the multiplicative group of an integral domain. We prove that it satisfies the same properties as in the complex case.
Sécherre, Vincent
Modular representations; p-adic groups; Weil representation; metaplectic group; block decomposition
MAT/02 - ALGEBRA
English
French
7-set-2015
27
2014/2015
Matematica
Università degli Studi di Milano-Bicocca
In Francia non esistono i cicli di dottorato, ho messo il corrispettivo italiano che iniziava nel 2011.
(2015). Représentations l-modulaires des groupes p-adiques. Décomposition en blocs de la catégorie des représentations lisses de GL(m,D), groupe métaplectique et représentation de Weil. (Tesi di dottorato, Università degli Studi di Milano-Bicocca, 2015).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/123569
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