Representation Theory of Reductive Groups over Local Fields

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The proposed research is on representation theory of reductive groups over local fields, i.e. groups like the group of invertible matrices over fields like the field of real or p-adic numbers. The proposed problems have direct applications in the theory of automorphic forms and in the Langlands program. They also have analytic and geometric aspects. The problems are divided into three topics: Harmonic analysis, representation theory of groupoids and the integrability theorem. Here are brief descriptions of those topics: The PI views harmonic analysis on spherical spaces as a generalization of representation theory. His aim is to transfer certain fundamental results from representation theory to the realm of harmonic analysis on spherical spaces.The notion of a groupoid is an interesting generalization of the notion of group. The PI proposes to re-build the representation theory of p-adic groups for the case of groupoids. The integrability theorem is a theorem from the theory of D-modules which has powerful applications in the theory of invariant distributions on real manifolds, which in turn is an important ingredient of representation theory. The theory of D-modules, however, is not applicable to the p-adic case. Based on a previous partial result, the PI proposes to provide an analog of the integrability theorem for the p-adic case.The propose project is about representation theory and harmonic analysis. A model example of the problems in this project can be the Fourier series. The Fourier series is a decomposition of a function on the circle as a sum of imaginary exponent (which are closely related to the trigonometric functions sine and cosine). These exponents change in a very simple way when you rotate the circle. The problems that the PI studies are, in a sense, a generalization of this construction for higher dimensional cases. In general, group theory can be viewed as the study of symmetries of mathematical objects, representation theory - as the study of symmetries of vector spaces, and harmonic analysis - as the study of spaces of functions over geometric objects that possess symmetries. The geometric objects that are studied are real and p-adic manifolds. Real manifolds are geometric objects that locally look like a line, a plain, a three dimensional space (like the one we live in) or a higher dimensional space. p-adic manifolds are certain analogues of real manifolds. Representation theory of real groups and harmonic analysis on real spaces have various applications in geometry, analysis and subsequently in physics, signal processing, image processing and biology. Both the p-adic and the real case have many important applications in number theory. More specifically, in the theory of automorphic forms and in the Langlands program.