David Vogan (MIT)

Near the heart of Langlands' notion of functoriality is a classification of irreducible representations of a reductive group G over a local field. This classification was proven by Langlands for archimedean fields, but remains conjectural for non-archimedean fields. If the classification is well understood, then one can say exactly what the "functorial lifting" of a group representation ought to be. To a first approximation, tempered representations should be those arising as functorial lifts of unitary characters of tori. (This is precisely correct in the archimedean case). This can be regarded as a "reason" for the fundamental role of tempered representations in the theory of automorphic forms.

Arthur asked what the next larger class of naturally unitary representations should be (arising in the theory of automorphic forms). He identified them as the functorial lifts of one-dimensional unitary characters of groups locally isomorphic to SL(2) times a torus. From the point of view of functoriality, it is entirely reasonable that such representations should be unitary, and should appear in automorphic forms; what is much more amazing is Arthur's conjecture that one needs nothing more for the L^2 theory of automorphic forms.

I will explain enough about the local Langlands conjecture to formulate Arthur's conjectures more precisely. The idea is this: the local Langlands conjecture concerns a smooth algebraic variety X on which the dual group acts with finitely many orbits. Each orbit corresponds to an L-packet; individual representations in the L-packet correspond to equivariant local systems on the orbit. The precise statement of the local Langlands conjecture is in terms of equivariant D-modules on X. The additional notion needed to formulate Arthur's conjectures is that of the "characteristic cycle" of a D-module.

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