Stable distributions and nilpotent orbital integrals

TL;DR

This paper investigates the structure of stable, nilpotent-supported invariant distributions on the Lie algebra of certain reductive groups over non-archimedean fields, proving some conjectures under specific characteristic conditions.

Contribution

It proves several conjectures about stable nilpotent orbital integrals for quasi-split, adjoint, simple groups over fields with large residual characteristic.

Findings

Proved some conjectures on stable nilpotent orbital integrals.

Established conditions under which the conjectures hold.

Enhanced understanding of invariant distributions in p-adic harmonic analysis.

Abstract

Let G be a connected reductive group defined over a non-archimedean local field of characteristic 0. We assume G is quasi-split, adjoint and absolutly simple. Let g be the Lie algebra of G. We consider the space of the invariant distributions on g(F), which are stable and supported by the set of nilpotent elements of g(F). Magdy Assem has stated several conjectures which describe this space. We prove some of these conjectures, assuming that the residual characteristic of F is ''very large'' relatively to G.