Asymptotic and descent formulas for weighted orbital integrals

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 IM (γ) 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 γ is elliptic in M . Introduction Weighted orbital integrals are the local contributions to the geometric side of the Arthur-Selberg trace formula. They are defined for connected reductive linear algebraic groups G over a local field F . A weighted orbital integral is a certain distribution supported on a conjugacy class {xγx | x ∈ G(F )} in the group G(F ) of F -rational points and depends on a Levi R-subgroup M of G containing γ. Its value on a test function f is denoted JM(γ, f), and it is tempered in the sense that it extends to continuous linear functionals on Harish-Chandra’s Schwartz space C(G) of rapidly decreasing functions on G(R). The precise definition will be recalled in the section 1. This paper is a contribution to the calculation of the Fourier transform of weighted orbital integrals. For a survey of this subject we refer to [15]. It is important