On Kostant Sections and Topological Nilpotence

TL;DR

This paper constructs a subset of the Lie algebra for certain reductive groups over local fields that uniquely identifies conjugacy classes and exhibits favorable properties under endoscopic transfer, extending previous results.

Contribution

It generalizes earlier work by defining a Kostant section-like subset for broader classes of groups and fields, with good transfer properties in the p-adic case.

Findings

Provides a new subset of the Lie algebra that intersects each stable, regular, topologically nilpotent class exactly once.

Shows the characteristic function of this set behaves well under endoscopic transfer for p-adic fields.

Extends prior results under weaker hypotheses.

Abstract

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.