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

الأصل · EN

Let π be a Speh module of GL(2n,R) based on a discrete series of GL(2,R). The aim of this paper is to build a chain complex of π by direct sum of auto-duals standard modules, aligneq:abstract 0→ π→ X₀→ → Xᵢϕᵢ Xᵢ₊₁→→ 0. 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 Iₙ of the symmetric group Sₙ. Under this parametrization one can show that the inversion of the Bruhat order in Iₙ coincide with the Vogan order defined over the set of irreducible representations of GL(R). This allows us to reduce the construction of the chain complex to the study of combinatorial properties of the Bruhat order on Iₙ. In the last chapter we show that the chain complex of π, for n≤ 4, is θ-exact i.e. the twisted trace of ϕᵢ₊₁/imϕᵢ 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, Paquets d'Arthur des Groupes classiques et unitaires.