conjectures for unitary representations
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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