A note on \(L\)-packets
As the author is a master at communicating technical ideas effectively to the general mathematician, it would be more enlightening to read the introduction of his paper rather than this review of it. Nevertheless, we indicate very briefly what the paper contains, and we cannot do better than essentially fish out parts of the description given in the paper. The paper is partly expository and also discusses a conjectural characterization of \(L\)-packets attached to a general \(p\)-adic group. Recall that for a connected, reductive algebraic group \(G\) over a local field \(F\) of characteristic \(0\), Langlands conjectured that the set \(\Pi(G)\) of equivalence classes of irreducible representations of \(G(F)\) breaks up into finite subsets (parametrized by so-called Langlands parameters) called \(L\)-packets. Once this local Langlands correspondence is established, the problem is to classify the representations in any packet in terms of data attached to the corresponding Langlands parameter. This problem is still open for a general group when \(F\) is non-Archimedean -- it was done for \(\text{GL}(n)\) by \textit{M. Harris} and \textit{R. Taylor} [The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)] and by \textit{G. Henniart} [Invent. Math. 139, No. 2, 439--455 (2000; Zbl 1048.11092)]. In the paper under review, Arthur gives, among other things, a conjectural formula for the number of elements in an \(L\)-packet. He also formulates a precise conjecture characterizing \(L\)-packets. His stress is on Langlands-Shelstad's transfer factors. A certain equivariance property for automorphisms of endoscopic data plays a key (although implicit) part in his analysis. We do not state this technical property precisely here but point out that this replaces the nonexistence of a section for the principal \(\text{U}(1)\)-bundle of transfer families over the space of endoscopic data in an isomorphism class. At the end, Arthur provides some evidence for the general conjecture by applying it to special cases, such as the norm 1 group of a quaternion algebra.
scientific article; zbMATH DE number 5510991