Inversion formula for twisted orbital integrals on real reductive Lie groups
In this sequel to his article [ibid. 145, 374-454 (1997; Zbl 0877.22002)], the author proves an inversion formula for twisted orbital integrals on a reductive Lie group \(G\). These integrals are obtained by integrating a smooth function \(f\) with compact support on the twisted conjugacy class of \(x\), i.e. the set of all \(gx\theta (g^{-1})\) for \(g\in G\), where \(\theta\) is a finite order automorphism of \(G\). The main result of the paper is a Fourier inversion formula, expressing the orbital integral as a sum of integrals of \(\Theta_\chi (f) \Psi_\chi (x)\) (simplified notation), where the variable \(\chi\) runs over some character group, and the summation runs over the set of conjugacy classes of Cartan subspaces. The \(\theta_\chi\)'s are invariant eigendistributions under the action of biinvariant differential operators, and the \(\Psi_\chi\)'s are specific orbital functions. The construction and asymptotic study of the \(\Psi_\chi\)'s make up the most important part of the present paper.
scientific article; zbMATH DE number 1045911