The PI will study the theory of types, and their applications in Harmonic analysis on p-adic groups. First, jointly with Jiu-Kang Yu, the PI will construct K-types in certain tame case and show that they are types in the sense of Bushnell-Kutzko. These types have Hecke algebras associated to them. In the second project, the PI will study the structure of these Hecke algebras. Lastly, the PI will consider various applications of these K-types in Harmonic analysis. Recent progress in the theory of types suggests that the Harmonic analysis on p-adic groups can be completely 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 improved understanding of representation theory of p-adic groups, and they should also prove useful in the more general situation.
Representation theory is a major mathematical technique for exploiting the presence of symmetry. For example, the structure of the hydrogen atom, one of the fundamental computations in quantum mechanics, is controlled by representation theory. Another way of thinking about representation theory is as a generalization of the theory of eigenvalues and matrices that many people see in a college course on basic linear algebra. The investigator studies certain infinite groups of infinite dimensional symmetries. This project addresses important questions in representation theory, which potentially has applications to number theory.