Construction d'un complexe diff\'erentiel pour des modules de Speh $\theta$-invariants

Let $\pi$ 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 $\pi$ by direct sum of auto-duals standard modules, \begin{align}\label{eq:abstract} 0\rightarrow \pi\rightarrow X_{0}\rightarrow \cdots\rightarrow X_{i}\xrightarrow{\phi_{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 $\pi$, 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 $\pi$, for $n\leq 4$, is $\theta$-exact i.e. the twisted trace of $\ker\phi_{i+1}/\text{im}\phi_{i}$ is trivial. This allows us to write the twisted trace of $\pi$ as a linear combination of twisted traces of standard modules, wich implies for $\pi$ one of the the main results of the paper, $\textit{Paquets d'Arthur des Groupes classiques et unitaires}$.