Abstract of "Unitary representations of reductive groups." The philosophy of coadjoint orbits of Kirillov and Kostant predicts that most irreducible unitary representations of reductive Lie groups should arise by (parabolic or cohomological) induction from the (still undefined) unipotent representations. Each reductive group should have just finitely many unipotent representations, and these should be related to the (finitely many) nilpotent orbits of the coadjoint representation. This proposal has two parts. The first (partly joint with Susana Salamanca-Riba) concerns a precise version of the Kirillov-Kostant prediction. The goal is to characterize nicely a small set of unitary representations with the property that every unitary representation can be obtained from it by induction. (The set will include all unipotent representations, and also various complementary series.) The second (partly joint with William Graham) concerns an idea for constructing unipotent representations. More than fifty years ago, I. M. Gelfand set forth a program of abstract harmonic analysis, a very general way to study mathematical problems with symmetry. Such problems appear in physics, and in almost every part of mathematics. One of the most fundamental examples arises in connection with musical sounds. In that case the symmetry is passage through time: any two times are indistinguishable in terms of what kinds of sounds can appear. Gelfand's harmonic analysis amounts to decomposing a sound into the "pure tones" produced by a tuning fork (which change in an extremely simple way with the passage of time). Gelfand showed that a similar analysis was possible in the presence of more complicated symmetry. The role of the pure tones is played by "irreducible unitary representations." This project continues work done by many people on irreducible unitary representations. The central idea is this. The formal definition of an irreducible unitary representation involves linear a lgebra and Euclidean geometry, in the form of what are called Hilbert spaces and unitary operators. These are exactly the objects needed to formulate quantum mechanics. In physics, quantum mechanical systems arise from Newtonian ones by an (imperfectly understood) process known as quantization. An idea going back to Kirillov and Kostant in the 1960s is that irreducible unitary representations should also arise by "quantization" of some simple "Newtonian" analogues. These Newtonian analogues are fairly well understood, but the problem of quantizing them is difficult. On the other hand, similar quantization problems appear in many parts of mathematics and physics, so there is no shortage of examples to consider and ideas to try.
