Local Langlands correspondence for reductive p-adic groups

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This proposal is about interactions between Representation Theory and Number Theory, as predicted by the conjectural Local Langlands Correspondence (LLC) for general reductive groups. The LLC is a body of predicted relations between the representation theory of local Galois groups and the representation theory of reductive groups over a p-adic field. The goals of the work in this proposal are aimed toward an explicit understanding of the LLC. There are four topics: i) Extending the proposer's earlier work introducing Geometric Invariant Theory to study representations of p-adic groups, in particular the recently-constructed 'epipelagic' representations. ii) Study of discrete Langlands parameters, including proof of a conjectured inequality for adjoint Swan conductors, atypical behavior at small primes, parameters for Yu's representations, classification of depths and mass formulas. iii) Uniqueness results for the LLC. iv) Jordan decomposition of depth zero representations and LLC.The mathematics in this proposal is rooted in two ancient topics of mathematics: Representation Theory (the study of symmetry) and Number Theory (numerical solutions of equations). Though these appear to be two quite different areas of mathematics, the Local Langlands Correspondence predicts surprising relations between them. Roughly speaking, it says that certain kinds of infinite dimensional symmetries should correspond to certain equations whose solutions have a related collection of finite dimensional symmetries. The aims of the proposal are first, to discover and explicitly verify new and interesting examples of the LLC, and second, to use the predictions of the LLC to make new discoveries in Number Theory and Representation Theory.