A characterization of the stable invariant integral (Q1085285)

The study of the invariant integral and its Fourier transform is a central problem in harmonic analysis on a reductive Lie group G, e.g., in Harish-Chandra's work on the Plancherel formula and in the theory of the Selberg trace formula. The stabilized invariant integral of a function f relative to a \(\theta\)-stable Cartan A is the sum over w in \(W\setminus W_ C\) of the sign \(\epsilon\) (w) times the conjugate of the invariant integral \(F^ A_ f\) by w. Discrete series representations of G are partitioned into ''stable'' subsets as follows. Let T be a \(\theta\)-stable compact Cartan subgroup of G. For each regular integral \(\chi\), let \(\hat G_ d(\chi)\) be the set of cardinality \(W\setminus W_ C\) consisting of those representations \(\omega\) with infinitesimal character equal to \(\chi\). The sum \(\Theta_{\Omega}\) of the characters of the representations in such a set defines a central eigendistribution on G, which is stable in the sense of \textit{D. Shelstad} [Compos. Math. 39, 11-45 (1979; Zbl 0431.22011)]. Stabilized sums of the functions \(\Delta_ T \Theta_{\omega}\) are \(W_ C\)-skew and allow Fourier expansion of \(C^{\infty}\) functions on T which are \(W_ C\)-skew. Shelstad determines a ''pointwise'' characterization of the invariant integrals for functions f in the Schwartz space of G. The main result of the author is a characterization in terms of Fourier transforms of stabilized invariant integrals for f in \(C_ c^{\infty}(G)\). His proof uses explicit inversion formulae of Herb for the invariant integrals and a Paley-Wiener type theorem due to Clozel and Delorme. Their results are reviewed in the present paper.