Γ-functions of Representations and Lifting

Let F be a local non-archimedian field and let ψ : F → C be a nontrivial additive character of F . To this data one associates a meromorphic function γψ : π 7→ γψ(π) on the set of irreducible representations of the group GL(n, F ) in the following way. Consider the invariant distribution Φψ := ψ(tr(g))| det(g)|n|dg| on GL(n, F ), where |dg| denotes a Haar distribution on GL(n, F ). Although the support of Φψ is not compact, it is well-known that for generic irreducible representation π of GL(n, F ) the action of Φψ in π is well-defined and thus it defines a number γψ(π). These gamma-functions and the associated L-functions were studied by J. Tate for n = 1 and by R. Godement and H. Jacquet for arbitrary n. Let now G be the group of points of an arbitrary quasi-split reductive algebraic group over F and let G be the Langlands dual group. Let also ρ : G → GL(n,C) be a finite-dimensional representation of G. The local Langlands conjectures predict the existence of a natural map lρ : Irr(G) → Irr(GL(n, F )). Assuming that a certain technical condition on ρ (guaranteeing that the image of lρ does not lie in the singular set of γψ) is satisfied, one can consider the meromorphic function γψ,ρ on Irr(G), setting γψ,ρ(π) = γψ(lρ(π)). The main purpose of this paper is to propose a general framework for an explicit construction of the functions γψ,ρ. Namely, we propose a conjectural scheme for constructing an invariant distribution Φψ,ρ on G, whose action in every π ∈ Irr(G) is given by multiplication by γψ,ρ. Surprisingly, this turns out to be connected with certain geometric analogs of M.Kashiwara’s crystals (cf. [3]). We work out in detail several examples. As a byproduct we obtain a conjectural formula for the lifting lρ(θ) where θ 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 ρ-analogue of the Poisson summation formula for Fourier transform. This conjecture implies that the corresponding global L-function has meromorphic continuation and satisfies a functional equation.