Topics in Harmonic Analysis on Reductive p-adic Groups

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract DeBacker The investigator will continue his research into a number of topics in harmonic analysis for Lie groups over nonarchimedian fields. The first goal is to establish Murnaghan-Kirillov theory for depth zero supercuspidal representations. At its most basic level, Murnaghan-Kirillov theory asks for a connection between the supercuspidal representations of our group and the Fourier transforms of certain coadjoint orbital integrals. Because of their intimate connection to finite groups of Lie type, the problem of establishing Murnaghan-Kirillov theory for these representations reduces to the problem of associating regular semisimple orbital integrals to generalized Green functions. The second objective is to investigate questions about stability. For example, it would be useful to explicitly understand, in a uniform way via Bruhat-Tits theory, the space of stable distributions supported on the nilpotent set. Thanks to various homogeneity results, this problem can be addressed by associating (as above) regular semisimple orbital integrals to generalized Green functions. Harmonic analysis on Lie groups traces its roots to the following problem from physics: Describe the motion of a plucked guitar string. Eventually, people realized that this problem --- and rather more pure problems like calculating Gauss sums or studying the density of primes in arithmetic progressions --- could be understood by studying certain well-behaved functions on the circle (or other groups). These well-behaved functions are called characters, and the resulting theory is called harmonic analysis. By the 1930s mathematicians had a firm understanding of harmonic analysis on many types of groups (for example, compact or abelian groups). During the 1940s problems from relativistic physics led people to think about harmonic analysis on a more general class of groups, called Lie groups. Initiated by the work of Bargmann, Gelfand--Naimark, and Harish-Chandra, the goal was, as for the guitar problem, to understand functions on the group by studying characters. Thanks mostly to Harish-Chandra this goal was largely realized. Based at least partially on his understanding of this work, in the late 1960s Langlands was led to formulate his program; this program is a vast, remarkable web of conjectures and ideas --- a kind of mathematical theory of everything. For example, the celebrated works of Harris--Taylor, Kim--Shahidi, and Lafforgue provide a small sampling of the deep results it anticipates. As harmonic analysis on Lie groups plays a central role in our understanding of many of the problems in this area, the investigator hopes his research will contribute to future progress.