Where Stands Functoriality Today?

The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, Takagi, and Artin and others; and the representation theory of semisimple Lie groups in the form given to it by Harish-Chandra. In both theories the statements are deep and general and the proofs difficult, highly structured, and incisive. Historical antecedents and contemporary influences aside, both were in large part created by the power of one or two mathematicians. Whether for intrinsic reasons or because of the impotence of the mathematicians who have attempted to solve its problems, the fate of functoriality has been different, and the theory of automorphic forms remains in 1997 as it was in 1967: a diffuse, disordered subject driven as much by the availability of techniques as by any high esthetic purpose. Preoccupied with other matters I have drifted away from the field, so that I certainly have no remedy to offer. None the less, having had two occasions to address the question posed in the title, I have tried to understand something of the techniques that have led to progress on the questions central to functoriality, their successes and their limitations, as well as the new circumstances in the theory of automorphic forms: problems and notions that once seemed peripheral to me and whose importance I failed to appreciate are now central, either because of their intrinsic importance or because of their accessibility. Partly as an encouragement to younger, fresher mathematicians to take up the problem of functoriality, for that is one of the purposes of this school, but also partly as an idle reflection as to what I myself might undertake if I returned to it, I would like to respond to the title in broad terms, personal and certainly diffident and uncertain. My own mathematical experience and observation strongly suggest that progress is almost always the result of sustained awareness of the principal issues supplemented by some specific, concrete insight: begged, borrowed, or stolen or, happiest of all, distilled in one’s own alembic. I offer no insights. Initially there were two principal issues: functoriality itself, the relation between automorphic forms on different groups; and the identification of motivic L-functions, thus those associated to algebraic varieties over number fields, of which the zeta function, Artin Lfunctions, and the Hasse-Weil L-functions are the primitive examples, with automorphic L-functions, of which the zeta function and Hecke L-functions—of all types—are the first examples. The first issue arose and could be broached in the context of nonabelian harmonic analysis: representation theory and the trace formula. The second arose elsewhere but could