Explicit Local Langlands Correspondences

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The Local Langlands correspondence is a conjectural interaction between two kinds of symmetry: the structure of complex Lie groups and local Galois theory.The work in this proposal will establish new cases of the explicit local Langlands conjecture, with special emphasis on groups such as E8, which arise from icosahedral symmetry. The proposer will build on his previous work in this direction by first completing an ongoing project aimed at explicitly verifying the tame Langlands correspondence and second by extending the scope to verify the wild correspondence in some cases. Symmetry is a fundamental concept in Mathematics.Representation Theory is the study of how symmetry manifests in different ways. This proposal concerns the interactions of two quite different kinds of symmetry, one of them geometric, as in the icosahedron, and the other algebraic, arising from the symmetries among roots of polynomials. This interaction of symmetries is a branch of mathematical development whose roots go back to Euclid's Elements. Part of the Broader Impact of this proposal is to recognize and share this historical component of the local Langlands correspondence, by using the common language of ancient mathematics to unify disparate mathematical communities. For example, not everyone can understand E8, but it has roots in Euclid, which can be, and at one time was studied by a wide population, for diverse extra-mathematical reasons that are important for all educated citizens.