Transfer factors for Lie Algebras

Let $G$ be a quasi-split connected reductive group over a local field of characteristic $0$, and fix a regular nilpotent element in the Lie algebra $\mathfrak g$ of $G$. A theorem of Kostant then provides a canonical conjugacy class within each stable conjugacy class of regular semisimple elements in $\mathfrak g$. Normalized transfer factors take the value $1$ on these canonical conjugacy classes.