Speaker: David Vogan (MIT)

Title: Arthur representations and the unitary dual

Abstract: The view from 10,000 meters is that the unitary dual of a reductive algebraic group over
a local field should consist of the representations whose existence Arthur conjectured in
the 1980s, together with others arising by deformation. Here is a precise conjecture in
that direction:

     CONJECTURE. Suppose G is a real reductive algebraic group, and π is a unitary
     representation of G having integral infinitesimal character. Then π is an Arthur
     representation.

The conjecture is true for many classical groups up to rank 6. My guess is that it is true
for ALL classical groups (and that the technology of the experts is sufficient to prove that).
I have checked that it is true for all representations of split G₂; for all but two representa-
tions of split F₄; for all but six representations of split E₇; and all but 27 representations
of split E₈. It is true for the complex forms of G₂, F₄, and E₆ .

I will say a little about why the conjecture is plausible, and where the counterexamples
come from.

Related elementary problem: suppose H ⊂ G is complex reductive, and γ in X ^* (H) is
a weight. Give a simple algorithm to calculate all the maximal proper Levi subgroups
L ⊃ H so that γ is in the Q-span of the roots of L.