Arthur, James Towards a local trace formula. (English) Zbl 0765.22010 Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 1-23 (1989). [For the entire collection see Zbl 0747.00038.]The subject matter of this paper is the representation theory of reductive algebraic groups over local fields. As D. Kazhdan has pointed out the diagonal embedding of \(G(F)\) in \(G(F)\times G(F)\) is very analogous to that of \(G(k)\) in \(G(k_ \mathbb{A})\) where \(k\) is a global field and \(k_ \mathbb{A}\) its ring of ideles. The author analyses the spectral decomposition of \(L^ 2(G(F))\) under the action of \(G(F)\times G(F)\) using Harish-Chandra’s theory of Eisenstein integrals and Plancherel theorem. He exploits a truncation argument similar to that used in his work on the Selberg trace formula and a formula of Waldspurger for the inner product of two Eisenstein integrals. He does not give a full proof of this analogue of the trace formula and explains the technical difficulties which have still to be overcome. The consequence would be an identity associated with a sum of ‘orbital integrals’ associated with ‘all Levi subgroups’ and a sum of distributions associated with certain representations of the Levi subgroups. He then shows how the formula can be brought into invariant form. The conjectural formula would be a far-reaching generalization of a theorem of J.-L. Waldspurper. Reviewer: S.J.Patterson (Göttingen) Cited in 1 Document MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg) Keywords:reductive algebraic groups; local fields; diagonal embedding; spectral decomposition; Eisenstein integrals; Plancherel theorem; Selberg trace formula; representations Citations:Zbl 0747.00038 PDFBibTeX XMLCite \textit{J. Arthur}, in: Algebraic analysis, geometry, and number theory: proceedings of the JAMI inaugural conference, held at Baltimore, MD, USA, May 16-19, 1988. Baltimore: Johns Hopkins University Press. 1--23 (1989; Zbl 0765.22010)