An Introduction to the GGP conjectures - I

Abstract

The speaker will try to give an introduction to the GGP conjectures, keeping in mind that he will be speaking to a very mixed audience some of whom may be seeing representation theory of groups over local fields for the first time. I will try not to presume much beyond a basic introduction to representation theory of finite groups over complex numbers, and familiarity with p-adic fields, and p-adic groups. There will be four lectures whose outline I give below.
Lecture 1: Branching laws illustrated with some finite dimensional examples, emphasizing the need of a parametrization, Gelfand pairs, strong Gelfand pairs. Automorphic representations and period integrals, Local-global principle, L-functions.
Lecture 2: Review of Classical groups in general, and their classification over local and global fields; their parabolics and Levi subgroups, Whittaker models, degenerate Whittaker models, Bessel and Fourier-Jacobi models, the last will need a bit of the Weil representations.
Lecture 3: A bit of representation theory of groups over local fields, parabolic induction, cuspidal representations. Review of the Local Langlands correspondence, L-functions and epsilon factors. L-packets, the Jacquet-Langlands correspondence, The GGP conjectures: both local and global conjectures.
Lecture 4: Spill-over from the last lecture, and finish with some low dimensional examples, including the fundamental work of Waldspurger; illustrative examples from finite fields.
References:
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