Représentations des espaces tordus sur un groupe réductif connexe p-adique

Description

Let F be a locally compact non-Archimedean field, of any characteristic. Let G be a connected reductive group defined over F, and G^ be a twisted G-space also defined over F. The set G^(F) is assumed to be non-empty, and it is endowed with the topology defined by F. We fix a character (i.e. a continuous homomorphism in C^) of G(F). In this memoir, we study the theory of (complex, smooth) -representations of G^(F), from that of representations of G(F). An -representation of G^(F) is given by a representation (,V) of G(F) and a map from G^(F) into the group of C-automorphisms of V, such that (xy) = (x) ()()(y) for all G^(F) and all x, yG(F). If the underlying representation of G(F) is admissible, we can define the character _ of , which is a distribution on G^(F). The main results proved in this memoir are: itemize if is of finite length, then the distribution _ is given by a locally constant function on the open set of (quasi-)regular elements in G^(F); the scalar Paley-Wiener theorem, which describes the image of the Fourier transform – the map which associate to a compactly supported locally constant function on G^(F) the linear form _() on a suitable Grothendieck group; the spectral density theorem, which describes the kernel of the Fourier transform. itemize