gamma-functions of representations and lifting

Let F be a local non-Archimedean field and let psi : F --> C-x be a non-tri vial additive character of F. To this data one associates a meromorphic fun ction gamma (psi) : pi \ --> gamma (psi)(pi) on the set of irreducible repr esentations of the group GL(n, F) in the following way. Consider the invari ant distribution Phi (psi) := psi (tr(g))\det(g)\(n)\dg\ on GL(n, F), where \dg\ denotes a Haar distribution on GL(n, F). Although the support of Phi (psi) is not compact, it is well known that for a generic irreducible repre sentation pi of GL(n, F) the action of Phi (psi) in pi is well defined and thus it defines a number gamma (psi)(pi). These gamma-functions and the ass ociated L-functions were studied by J. Tate for n = 1 and by R. Godement an d H. Jacquet for arbitrary n. Now let G be the group of points of an arbitrary quasi-split reductive alge braic group over F and let G(V) be the Langlands dual group. Let also rho : G(V) --> GL(n,C) be a finite-dimensional representation of GV. The local L anglands conjectures predict the existence of a natural map l(rho) : Irr(G) --> Irr(GL(n, F)). Assuming that a certain technical condition on p (guara nteeing that the image of I, does not lie in the singular set of gamma (psi )) is satisfied, one can consider the meromorphic function gamma (psi),(rho ) On Irr(G), setting gamma (psi),(rho)(pi) = gamma (psi)(l(rho)(pi)). The main purpose of this paper is to propose a general framework for an exp licit construction of the functions gamma (psi),(rho). Namely, we propose a conjectural scheme for constructing an invariant distribution Phi (psi),(r ho) on G, whose action in every rr E Irr(G) is given by multiplication by g amma (psi),(rho). Surprisingly, this turns out to be connected with certain geometric analogs of M. Kashiwara's crystals (cf. [BeK]). We work out several examples in detail. As a byproduct we obtain a conjectu ral formula for the lifting l(rho)(theta) where theta is a character of an arbitrary maximal torus in GL(n, F). In some of these examples we also give a definition of the corresponding local L-function and state a conjectural rho -analogue of the Poisson summation formula for Fourier transform. This conjecture implies that the corresponding global L-function has meromorphi c continuation and satisfies a functional equation.