Tamely Ramified Langlands Correspondence

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The investigator attempts to classify square-integrable tamely ramified representations of split reductive p-adic groups G and their inner forms, by means of homomorphisms from the tame Weil-Deligne group into the dual group of G. This project is part of the local Langlands conjecture for such groups. The representations are partitioned into finite sets, called ``L-packets". The method is a p-adic analogue of Lusztig's Jordan decomposition for finite reductive groups. From a functorial point of view, the tame L-packets are bijective lifts of unipotent L-packets for quasi-split groups and their inner forms. Properties of the L-packets, such as formal degrees and Whittaker models, will also be investigated.Symmetry is an effective way to study mathematical and physical objects. A Group is a collection of symmetries, and Representation Theory is the study of how these symmetries manifest in different ways. As a simple example, there is a group of sixty symmetries consisting of permutations of five objects, and this group also manifests as the sixty symmetries of an icosohedron, and of the fullerene molecule. The investigator studies certain infinite groups of infinite dimensional symmetries. In the past forty years, mathematicians and physicists have guessed that there may be deep relations between such symmetries and fundamental, as yet undiscovered, properties of numbers. The investigator attempts to confirm part of these conjectures, and make them more explicit.