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.
The former PI of this grant Avraham Aizenbud aimed the following: 1. Find a new approach for proving the Gelfand property (joint with Dmitry Gourevitch) 2. Study representation growth of profinite groups (joint with Nir Avni). He made a good progress toward his goal and the results appeared in the following papers and preprints: --Papers to appear 1. The wave front set of the Fourier transform of algebraic measures Avraham Aizenbud, Vladimir Drinfeld, to appear in Israel J. of Math 2. Twisted homology for the mirabolic nilradical Avraham Aizenbud, Dmitry Gourevitch, Siddhartha Sahi, to appear in Israel J. of Math. 3. Derivatives for smooth representations of GL(n,R) and GL(n,C) Avraham Aizenbud, Dmitry Gourevitch, Siddhartha Sahi, to appear in Israel J. of Math. --Preprints in the arxiv 1. z-Finite distributions on p-adic groups Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag, Alexander Kemarsky 2. Vanishing of certain equivariant distributions on p-adic spherical spaces, and non-vanishing of spherical Bessel functions Avraham Aizenbud, Dmitry Gourevitch, Alexander Kemarsky 3. Vanishing of certain equivariant distributions on spherical spaces Avraham Aizenbud, Dmitry Gourevitch 4. arXiv:1307.0371 [pdf, other] Representation Growth and Rational Singularities of the Moduli Space of Local Systems Avraham Aizenbud, Nir Avni He has been also actively disseminating his results through conferences and seminars. After taking over the grant 1100943, the current PI Julee Kim continues to study representation theory of $p$-adic groups, especially, the PI has been studying the asymptotic behavior of orbital integrals and also characters. Characters and orbital integrals appear in trace formula on spectral side and geometric side respectively. In applying trace formula for problems arising automorphic forms, it is important to understand these objects. The following are products for this reporting period 2013-14 by the PI: In a joint work with Sug Woo Shin and Nicolas Templier, we aim to study limit multiplicities in automorphic setting using these results. The following preprints (joint with Shin and Templier) are in preparation: Local Constancy of Characters and Applications Asymptotic behavior of Orbital Integrals and Equidistributions The PI has written the following paper on Whittaker functions of certain induced representations, which is solicited for a volume dedicated to James Cogdell. Generalized Casselman-Shalika formula The PI also participated in various conferences and seminars to give talks disseminating the results in the above papers. In particular, she participated in the following conference and gave a general audience talk: Connections for Women: Model Theory, Arithmetic Geometry and Number Theory February 10-11, 2014 at MSRI, Berkeley, California During the IAP (Independent Activity Period) in January at MIT, the PI ran the Directed Reading Program for undergradutes. This is a program where a pair of graduate mentor and undergraduate mantee are reading more specialized math books in depth. The PI also participated in MSRP (MIT Summer Research Program) as a faculty mentor. MSRP is a summer research program for under represented minorities students from outside MIT.