For a local field \(F\), and reductive algebraic group \(G\) defined over \(F\), the “classical” Local Langlands Correspondence attempts to parametrize the irreducible admissible representations of \(G(F)\) in terms of the (purely arithmetic) Weil-Deligne group \(W_ F\) and the (purely algebraic) complex dual group \(G^ \vee\); cf. A. Borel’s article [in Proc. Symp. Pure Math. 33, No. 2, 27-61 (1979; Zbl 0412.10017)]. For \(p\)-adic fields (as opposed to archimedean \(F\)), this conjecture has not yet been proved, but it has been reformulated and refined to a detailed collection of (conjectural) relationships between \(p\)-adic representation theory and the geometry of the space of “Langlands parameters” [cf. the work of D. Kazhdan and G. Lusztig, Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)]. In the case of real groups, the (original) predicted parametrization of representations was proved by Langlands himself. However, most of the deeper relations suggested by the \(p\)-adic theory (between real representation theory and geometry on the space of real Langlands parameters) no longer holds, and it is the purpose of recent work of Adams, Barbasch and the author to remedy this situation, i.e. to redefine the space of real Langlands parameters so as to recover these relationships between the representation theory and geometry [cf. J. Adams, D. Barbasch and the author, The Langlands classification and irreducible characters for real reductive groups. (Birkhäuser, 1992; Zbl 0756.22004)]. In this new reformulation of the Langlands correspondence, Weil groups are not really used, and it is no longer clear how to divide the conjecture into arithmetic and algebraic aspects. Moreover, the mere statement of these reformulations requires more technical background than this reviewer feels comfortable with, and even an expert might find this “survey paper” rough going. Nevertheless, since this approach seems to tie in nicely with Arthur’s conjectured multiplicity formulas for \(L\) (and \(A\)) packets of automorphic representations, and also provides the “geometry” missing in Langlands’ original parameter space, this paper clearly deserves careful attention and study.
For the entire collection see [Zbl 0773.00011].