  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.
