 
Summary: Canonical normalization of weighted characters
and a transfer conjecture
James Arthur*
§1. Introduction
Suppose that G is a connected reductive algebraic group over a local field F of characteristic 0. If unit(G)
is any irreducible unitary representation of G(F), the character
f  fG() = tr (f) , f H(G),
is an invariant linear form on the Hecke algebra H(G) of G(F). Among the irreducible characters, there is a
special place for the induced characters
fM () = fG(G
) = tr IP (, f) , unit(M).
Here M is a Levi subgroup of G, that will remain fixed throughout the paper, P P(M) is a parabolic subgroup
of G over F with Levi component M, and G
= IP () is the corresponding induced representation of G(F).
As the notation suggests, fM () is independent of P. Another elementary property is that fM () is an analytic
function of a real variable in the vector space
ia
M = i Hom X(M)F , R
that parametrizes the unramified unitary twists of . (We are following standard notation here and below; the
reader can consult [5, §12] for more explanation.)
