The investigator will work on a number of topics which arise in the study of harmonic analysis for reductive groups over nonarchimedian local fields. The unifying theme for these topics is their relation to homogeneity statements. At their most basic level, homogeneity questions ask: given two distributions, what is the largest set of functions on which the two distributions will agree? The first topic concerns the homogeneity question for invariant distributions supported on the compact elements in the group. This is the last unproven (in general) homogeneity statement, and it has implications for the "fundamental lemma" (which is a conjecture). The second topic concerns questions of stability for distributions supported on the unipotent set. The final topic concerns the meaning of the constants occurring in the Harish-Chandra-Howe local character expansion. The topics discussed above fall under the rubric of harmonic analysis on Lie groups. This area of mathematics, which is solidly rooted in physics, was pioneered by Harish-Chandra beginning some fifty years ago. Since the 1970s, much of the work in this area has been directed toward questions which arise naturally in the Langlands program (which seeks to relate, in a strong sense, number theory, representation theory, and harmonic analysis). For example, Harish-Chandra studied the distributions which arise when one integrates over a conjugacy class. From the perspective of the Langlands program, it is also important to study the distributions which arise when one integrates over certain stable conjugacy classes, that is, classes which become conjugate over an algebraic closure of the ground field. The investigator hopes his program of research will contribute to future progress in this area.
