3:00–4:20 pm Eckhart Hall, Room 202
Quantum–Classical Analogies in Harmonic Analysis and the Langlands Program
Many of the objects that one studies in representation theory are understood to be "quantum" analogs of some "classical" symplectic space. I will survey such analogies in the context of harmonic analysis on Lie groups and their homogeneous spaces. I will then shift to certain quotient spaces, like the space of conjugacy classes for a group (which, in the context of the Langlands program, are related to trace formulas), and discuss examples where the "classical" geometry of cotangent bundles reflects very interesting "quantum" operators, giving rise to a very promising approach to the functoriality conjecture "beyond endoscopy."
This will be a general audience talk, with large (but not complete) overlap with a specialist talk I gave a couple of years ago; the highlight will be a theorem on Fourier transform on the space of (n x n)-matrices, so it should be pretty understandable.