Invariant Distributions on p-adic Lie Algebras

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Technical Description: Let G be a connected reductive p-adic group, and let L(G) be its Lie algebra. The proposal is to study the G-invariantdistributions on L(G) which are Fourier transforms of invariant distributions with compactly generated support. This class of distributions includes the Fourier transforms of orbital integrals, which are important for the theory of characters on the group G. Harish-Chandra proved that these distributions are given by functions which are locally constant on the set of regular elements of L(G), and can be normalized to be locally bounded on L(G). When G is a real Lie group, the restrictions of these functions to Cartan subalgebras have simple formulas because they satisfy differential equations. In the p-adic case, there are analogous formulas for the Fourier transforms of orbital integrals restricted to Cartan subalgebras of L(G), but they are only valid for large enough sufficiently regular elements in the Cartan subalgebra. These formulas can be used to develop a theory of the constant term analogous to that for the real case. One goal of this proposal is to use these formulas at infinity to prove global bounds for the restrictions of Fourier transforms of orbital integrals to Cartan subalgebras. In order to do this it is necessary to control behavior at infinity uniformly as regular elements approach singular hyperplanes. A further goal of this proposal is to suitably generalize this work to the class of distributions obtained as Fourier transforms of invariant distributions with compactly generated support. Non-technical Description: The theory of Fourier series and Fourier transforms was developed, starting in the 18th century, to study functions of a real variable. The idea is to write an arbitrary function as a sum or integral of the well-understood trigonometric functions. This theory today has many applications in the sciences, in engineering, and in mathematics. Many of the basic ideas involved in Fourier analysis can be extended to analyze functions on any space with sufficient symmetry. One class of spaces of special interest in physics and many areas of mathematics is linear algebraic groups and their Lie algebras, the analysis on the Lie algebra being a linearization of the analysis on the group. These groups and algebras can be realized as matrices. Classically, the entries of the matrices are real or complex numbers. However there is also an interesting theory when the entries come from other fields, in particular the fields of p-adic numbers. These fields are important in number theory, and the study of p-adic groups has many applications to number theory. In the classical situation, much of the analysis involves the use of differential equations. This tool is not available in the p-adic case. The goal of this proposal is to analyze the behavior at infinity of certain important distributions on p-adic Lie algebras. The classical results are available as motivation, but the techniques of proof are necessarily completely different.