×

Canonical normalization of weighted characters and a transfer conjecture. (English) Zbl 0906.11021

Let \(G\) be a connected reductive group over a local field \(F\) of characteristic zero. Among the distributions occurring in the trace formula, there are weighted characters \[ J_M^P (\pi,f):f\mapsto \text{tr} (J_M(\pi,P) I_P(\pi,f)). \] Here \(M\) is a Levi component, \(P\) a parabolic having \(M\) as Levi factor and \(\pi\) is an irreducible unitary representation of \(M\). The induced representation of \(\pi\) is \(I_P (\pi)\), which explains the notation \(I_P (\pi,f)\) for \(f\) a smooth function on \(G\) of compact support. The weight factor \(J_M (\pi,P)\) is a nonscalar operator on the space of \(I_P(\pi)\) constructed from the basic intertwining operators of parabolically induced representations.
Let \(\lambda\) be a character of \(M\) and \(\pi_\lambda= \pi\otimes \lambda\). Then the function \(\lambda\mapsto J_M^P (\pi_\lambda,f)\) may have poles. Moreover, it appears inconvenient that \(J_M^P\) depends on the choice of a parabolic \(P\). In the papers on the trace formula, J. Arthur solved these problems by multiplying \(J_M^P\) with a scalar normalizing factor \(r\). There is, however, no canonical such factor, so the terms in the trace formula incorporate choices of normalizing factors, a fact which becomes a serious problem when comparing trace formulas for different groups.
The goal of the present paper is to normalize the weighted characters in a different way. Instead of choosing normalizing factors, Arthur makes up new operators by means of Harish-Chandra’s \(\mu\)-functions, i.e. he uses Plancherel densities to arrive at a more canonical distribution \(I_M\). He shows that these do not depend upon the choice of a parabolic. He finally states a conjecture on how these distributions should be describable by data from endoscopic groups. This last important transfer conjecture should, if verified, make way for inductive arguments in comparing trace formulas.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11R39 Langlands-Weil conjectures, nonabelian class field theory
22E35 Analysis on \(p\)-adic Lie groups
PDFBibTeX XMLCite