arXiv subjects

Nicolás Arancibia

Publications and source records attributed to Nicolás Arancibia.

3 recordsLinked to original sources

arXiv2021

Characteristic cycles, micro local packets and packets with cohomology

Relying on work of Kashiwara-Schapira and Schmid-Vilonen, we describe the behaviour of characteristic cycles with respect to the operation of geometric induction, the geometric counterpart of taking parabolic or cohomological induction in representation theory. By doing this, we are able to describe to some extent the characteristic cycle associated to an induced representation, in terms of the characteristic cycle of the representation being induced. More precisely, under the hypothesis that the infinitesimal character is regular (and dominant), we show that the characteristic cycle of an induced representation splits in two terms. We describe the first term precisely, but we are not able to do the same for the second one. What we are able to say, is that this second term is supported on the boundary of the space generated by the inclusion in the flag variety of $G$, of the flag variety of the Levi subgroup. As a consequence, we prove that the cohomology packets defined by Adams and Johnson are micro-packets, that is to say that the cohomological constructions of Adams-Johnson are particular cases of the sheaf-theoretic ones of Adams-Barbasch-Vogan.

math.RT↗

arXiv2017

Paquets d'Arthur des groupes classiques et unitaires

Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted $\mathbf{GL}_N$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.

math.RT↗

arXiv2015

Construction d'un complexe différentiel pour des modules de Speh $θ$-invariants

Let $π$ be a Speh module of $\mathbf{GL}(2n,\mathbb{R})$ based on a discrete series of $\textbf{GL}(2,\mathbb{R})$. The aim of this paper is to build a chain complex of $π$ by direct sum of auto-duals standard modules, \begin{align}\label{eq:abstract} 0\rightarrow π\rightarrow X_{0}\rightarrow \cdots\rightarrow X_{i}\xrightarrow{ϕ_{i}} X_{i+1}\rightarrow\cdots\rightarrow 0. \end{align} The standard modules in the previous chain are the auto-duals standard modules which occurs in the Johnson's resolution of $π$, they are parameterized by the set of involutions $\mathfrak{I}_{n}$ of the symmetric group $\mathfrak{S}_{n}$. Under this parametrization one can show that the inversion of the Bruhat order in $\mathfrak{I}_{n}$ coincide with the Vogan order defined over the set of irreducible representations of $\mathbf{GL}(\mathbb{R})$. This allows us to reduce the construction of the chain complex to the study of combinatorial properties of the Bruhat order on $\mathfrak{I}_{n}$. In the last chapter we show that the chain complex of $π$, for $n\leq 4$, is $θ$-exact i.e. the twisted trace of $\kerϕ_{i+1}/\text{im}ϕ_{i}$ is trivial. This allows us to write the twisted trace of $π$ as a linear combination of twisted traces of standard modules, wich implies for $π$ one of the the main results of the paper, $\textit{Paquets d'Arthur des Groupes classiques et unitaires}$.

math.RT↗