Stephen DeBacker

Admissible Invariant Distributions on Reductive -adic Groups

Introduction Fourier transforms on the Lie algebra An extension and proof of Howes Theorem Theory on the group Bibliography List of symbols Index.

Homogeneity results for invariant distributions of a reductive p-adic group

Let k denote a complete nonarchimedean local field with finite residue field. Let G be the group of k-rational points of a connected reductive linear algebraic group defined over k. Subject to some conditions, we establish a range of validity for the Harish-Chandra–Howe local expansion for characters of admissible irreducible representations of G. Subject to some restrictions, we also verify two analogues of this result.

Parametrizing nilpotent orbits via Bruhat-Tits theory

Let k denote a field with nontrivial discrete valuation. We assume that k is complete with perfect residue field. Let G be the group of k-rational points of a reductive, linear algebraic group defined over k. Let g denote the Lie algebra of G. Fix r ∈ R. Subject to some restrictions, we show that the set of distinguished degenerate Moy-Prasad cosets of depth r (up to an equivalence relation) parametrizes the nilpotent orbits in g.

On some generic very cuspidal representations

Let G be a reductive p -adic group. Given a compact-mod-center maximal torus S ⊂ G and sufficiently regular character χ of S , one can define, following Adler, Yu and others, a supercuspidal representation π ( S , χ ) of G . For S unramified, we determine when π ( S , χ ) is generic, and which generic characters it contains.

Lectures on Harmonic Analysis for Reductive p-adic Groups

In his paper The characters of reductive -adic groups [14] Harish-Chandra outlines his philosophy about harmonic analysis on reductive -adic groups. According to this philosophy, there are two distinguished classes of distributions on the group: orbital integrals and characters. Similarly, there are two classes of distributions on the Lie algebra which are interesting: orbital integrals and their Fourier transforms. The real meat of his philosophy states that we ought to treat orbital integrals on both the group and its Lie algebra in the “same way” and similarly, we should think of characters and the Fourier transform of orbital integrals in the same way. This philosophy has many manifestations (see, for example, Robert Kottwitz’s excellent article [12]). In this series of lectures, we will examine the various distributions discussed above and discuss one of the deepest connections between them: the Harish-Chandra–Howe local character expansion. Because of requests on the part of participants, I will spend much time reviewing the basics of -adic fields and discussing some of the uses of Moy-Prasad filtrations in the representation theory of reductive -adic groups. As such, at a small cost in terms of generality, I will concentrate on those techniques which use this theory. These lectures and the notes are meant as an informal and elementary introduction to the material. For complete, rigorous proofs, please see the references. Very little of the material in this set of lectures is original. I have borrowed heavily from the work and lectures of Harish-Chandra, Roger Howe, Robert Kottwitz, Allen Moy, Gopal Prasad, Paul Sally, Jr., and J.-L. Waldspurger, among others. I thank the organizers of this conference, in particular, Eng-Chye Tan and Chen-Bo Zhu, for inviting me and allowing me to present this series of tutorials.

Stability of character sums for positive-depth, supercuspidal representations

We re-write the character formul{\ae} of Adler and the second-named author in a form amenable to explicit computations in

Stable Distributions Supported on the Nilpotent Cone for the Group G 2

p

A generalization of a result of Kazhdan and Lusztig

-adic harmonic analysis, and use them to prove the stability of character sums for a modification of Reeders conjectural positive-depth, unramified, toral supercuspidal L-packets.

Homogeneity of Certain Invariant Distributions on the Lie Algebra of p-adic GLn

Assuming that p is sufficiently large, we describe the stable distributions supported on the set of nilpotent elements for p-adic G2.

What is Old is New Again: A Systemic Approach to the Challenges of Calculus Instruction

Kazhdan and Lusztig showed that every topologically nilpotent, regular semisimple orbit in the Lie algebra of a simple, split group over the field C((t)) is, in some sense, close to a regular nilpotent orbit. We generalize this result to a setting that includes most quasisplit p-adic groups.