Representations of twisted spaces on a connected reductive p-adic group

Let F be a locally compact non-Archimedean field, of any characteristic. Let G be a connected reductive group defined over F, and G(sic) be a twisted G-space also defined over F. The set G(sic)(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-x) of G(F). In this memoir, we study the theory of (complex, smooth) w-representations of (F), from that of representations of G(sic)(F). An w-representation of G(sic)(F) is given by a representation (pi, V) of G(F) and a map II from G(sic) (F) into the group of C-automorphisms of V, such that II(x . delta . y) = pi(x) o II(delta) o (w pi)(y) for all 5 is an element of G(sic)(F) and all x, y is an element of G (F). If the underlying representation 71 of G(F) is admissible, we can define the character Theta(Pi) of Pi, which is a distribution on G(sic)(F). The main results proved in this memoir are: - if pi is of finite length, then the distribution Theta(Pi) is given by a locally constant function on the open set of (quasi-)regular elements in G(sic)(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 phi on G(sic)(F) the linear form Pi bar right arrow Theta(Pi) (phi) on a suitable Grothendieck group; - the spectral density theorem, which describes the kernel of the Fourier transform.