zbMATH — the first resource for mathematics

Orbital integrals and distributions. (English) Zbl 1248.11037
Arthur, James (ed.) et al., On certain \(L\)-functions. Conference on certain \(L\)-functions in honor of Freydoon Shahidi on the occasion of his 60th birthday, West Lafayette, IN, USA July 23–27, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-5204-0/pbk). Clay Mathematics Proceedings 13, 107-115 (2011).
The objective of this note under review is to correct and generalize the results in [J.-P. Labesse, Cohomology, stabilization and base change. Astérisque. 257. Paris: Société Mathématique de France (1999; Zbl 1024.11034)], Appendix A, which concern the base change for unitary groups. Here the authors consider only the archimedean setting.
After recalling Labesse’s formalism of twisted spaces and twisted orbital integrals. They introduced the Lefschetz number and the Lefschetz function; the latter is defined by regularization via Arthur’s multiplier method, the proof is unconditional thanks to the recent work of P. Delorme and P. Mezo [“A twisted invariant Paley-Wiener theorem for real reductive groups”, Duke Math. J. 144, No. 2, 341–380 (2008; Zbl 1189.22005)]. The Lefschetz function is nonzero only when the technical Assumption 2.1 is satisfied.
Thereafter, the authors give a correct and generalized proof of Théorème A.1.1 in [Labesse (loc. cit.)]. These results are then specialized to unitary groups as in op. cit.
For the entire collection see [Zbl 1214.11007].

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22F30 Homogeneous spaces