Characters and related distributions on reductive p-adic groups, distinguised representations

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This proposal involves the study of representations of reductive p-adic groups. These groups are matrix groups whose matrix entries belong to certain fields that are important in number theory. There are three different components to the proposed research. The first component involves finding ways to classify some of the representations of p-adic groups. The current classification of some families of representations appears to be more complicated than it needs to be, and we plan to investigate ways of simplifying the classification. The second component involves studying properties of the characters of representations of reductive p-adic groups. The characters are functions that play an important role in harmonic analysis on p-adic groups. Our goal is to derive new formulas for the characters of particular kinds of representations. The third component involves the study of spherical characters of distinguished representations. Such representations have nice symmetry properties relative to involutions of the group. We hope to find formulas for spherical characters of some families of distinguished representations.