Arthur representations and the unitary dual
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 pi is a unitary representation of G having integral infinitesimal character. Then pi is an Arthur representation. Since unitary duals of many real groups are now known, it is interesting to investigate the status of this conjecture. As I write this abstract in June, I have checked that the conjecture is true for all representations of split G_2; for all but at most two representations of split F_4; and for all but at most six representations of split E7. I hope to have a cleaner statement by August.