Trace Paley - Wiener Theorem

w Statement of the theorem 1.1. Let G be a reductive p-adic group. A smooth representation (~', E) of the group G on a complex vector space E is called a G-module. Usually we shorten the notation and write w or E. Let d~(G) be the category of G-modules, Irr G the set of equivalence classes of irreducible G-modules, and R (G) the Grothendieck group of G-modules of fnite length; R(G) is a free abelian group with basis Irr G. We fix a minimal parabolic subgroup PoC G and its Levi decomposition P0 = Mo" Uo. By a standard Levi subgroup we mean a subgroup M _D Mo which is a Levi component of the parabolic subgroup P = M-Po (notation M < G). For any standard Levi subgroup M < G the functors iGu : d/t(M)-, d~(G) and rM~ : d~ (G)-~ d~ (M) define morphisms i~ : R (M)-~ R (G), rM~ : R (G)-~ R (M) (see [2, w or [1, 2.5]). Let xlt(G)C {t~: G-~C*} be the group of unramified characters of G. It acts naturally on Irr G and R(G) by if: 1r ~ ~Tr. This group has a natural structure of complex algebraic group (isomorphic to (C*)d). 1.2. Let ~(G) be the Hecke algebra of G (algebra of locally constant complex valued measures on G with compact support). Each measure h ~ ~(G) defines a linear form fh : R(G)-~C by/,(~-) = tr 7r(h). It is easy to see that the form f = fh satisfies the following conditions: (i) For any standard Levi subgroup M < G and or E IrrM, the function ~b ~ f(icM (~bo')) is a regular function on the complex algebraic variety xlt(M). (ii) There exists an open compact subgroup K C G which dominates f, i.e., f is nonzero only on the G-modules E, which have a nontriviai space E K of K-invariant vectors. We want to prove that the conditions (i)-(ii) characterize the trace forms {fh}.