K-Types and Harmonic Analysis on p-Adic Reductive Groups

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AbstractKim The PI will study the theory of types, and their applications in Harmonicanalysis on p-adic groups. First, jointly with Jiu-Kang Yu, the PI willconstruct K-types in certain tame case and show that they are types in thesense of Bushnell-Kutzko. These types have Hecke algebras associated tothem. In the second project, the PI will study the structure ofthese Hecke algebras. Lastly, the PI will consider various applications ofthese K-types in Harmonic analysis. Recent progress in the theory oftypes suggests that the Harmonic analysis on p-adic groups can becompletely understood via harmonic analysis on the associated Hecke algebras.These projects are necessary steps towards advancing the theory of types.In the tame case, one can expect them to lead to a significantly improvedunderstanding of representation theory of p-adic groups, andthey should also prove useful in the more general situation.Representation theory is a major mathematical technique for exploiting thepresence of symmetry. For example, the structure of the hydrogen atom, oneof the fundamental computations in quantum mechanics, is controlled byrepresentation theory. Another way of thinking about representation theoryis as a generalization of the theory of eigenvalues and matrices that manypeople see in a college course on basic linear algebra. The investigatorstudies certain infinite groups of infinite dimensional symmetries.This project addresses important questions in representation theory,which potentially has applications to number theory.