The Langlands’ programme, which has given direction to the theory of automorphic forms for the last 40 years consists in essence of two parts. One is the comparison of automorphic forms on different groups; this is a question which is primarily analytic in its nature. The second consists of a comparison between families of automorphic forms and Galois representations, a question fundamentally arithmetic in nature. In both directions there have been important and far-reaching advances of late. The paper under review is a very important contribution to the first of the two classes of problems. Langlands himself has stressed on several occasions the importance of a comparison between arbitrary groups and the general linear groups (“her All-embracing Majesty” in the phrase of Hermann Weyl [The classical groups, their invariants and representations (Oxford 1939; Zbl 0020.20601), p. 136]. Kirillov theory provides a very rich representation theory for this family that singles it out amongst all others.
In this paper this goal is achieved for the families of classical groups over a number field \(k\). It represents the culmination of some 30 years’ work in which the techniques of Converse Theorems and of the application of Eisenstein series in the establishment of the analytic properties of the appropriate L-series have been developed. Each of these families has an L-group also of classical type which has a natural embedding into a general linear group. The authors prove that the corresponding Langlands functoriality exists for globally generic cuspidal representations and has all the expected properties. In particular they can characterize the image of the lifting map for these representations completely. They also explain, in an appendix, how the condition globally generic can be lifted.
The proof of the main theorem – the statement of which is a mere one and a half lines – is long and eclectic. It is eclectic in the sense that it makes extensive use of previous results. It is also itself technically involved. The method of Converse Theorems requires that one has a matching of L-functions between a candidate representation and that of a given representation. To achieve this a delicate analysis of the local representations is required and it is this that takes up the bulk of this paper. It is an interplay between local and global methods. For example the authors use global methods to prove the elegant dimension relation conjectured by C. Mœglin, a statement about local representations. The main theorem also implies that one can introduce a theory of conductors for cuspidal representations of classical groups. It also gives new insights into the formulation of the Ramanujan conjecture for these families of groups.