Asymptotic and descent formulas for weighted orbital integrals

TL;DR

This paper refines Arthur's asymptotic formulas for weighted orbital integrals using residue calculus, extends them to Schwartz space, and derives descent formulas to simplify the analysis of invariant distributions.

Contribution

It introduces a residue calculus approach to Arthur's formulas, extends the formulas to Schwartz space, and establishes descent formulas for differential equations and coefficients.

Findings

Explicit determination of some coefficients in Fourier transforms of invariant distributions

Extension of asymptotic formulas to Schwartz space

Descent formulas reducing cases to elliptic elements

Abstract

We rewrite Arthur's asymptotic formula for weighted orbital integrals on real groups with the aid of a residue calculus and extend the resulting formula to the Schwartz space. Then we extract the available information about the coefficients in the decomposition of the Fourier transforms of Arthur's invariant distributions I_M(\gamma) in terms of standard solutions of the pertinent holonomic system of differential equations. This allows us to determine some of those coefficients explicitly. Finally, we prove descent formulas for those differential equations, for their standard solutions and for the aforementioned coefficients, which reduce each of them to the case that \gamma is elliptic in M.