Un revêtement de l'arbre de GL2 d'un corps local

Abstract

Let F be a non-archimedean local field with residue class field k. Put G=GL2(F), Γ=PGL2(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex \(\tilde X\), equipped with an action of G × Γ and with a G × Γ-equivariant simplicial projection \(\pi :\tilde X \to X\) (for the trivial action of Γ on X). We prove that the cohomology with compact support \(H_c^1 (\tilde X,\mathbb{C})\) contains nontrivial representations of G (in particular positive level supercuspidal representations).