Slodowy slice associated to a nilpotent element

Let $f$ be nilpotent in a finite-dimensional complex simple Lie algebra $\mathfrak{g}$, and let $(e,h,f)$ be a chosen ${\mathfrak{s}\mathfrak{l}}_2$-triple. Write ${\mathfrak{g}}^e=\{x\in \mathfrak{g}:[e,x]=0\}$. The Slodowy slice for this chosen triple is the affine subspace $S_f=f+{\mathfrak{g}}^e\subseteq \mathfrak{g}\mathrm{.}$ For the fixed nondegenerate invariant form $B$ from def-cft-root-data-for-principal-w-levels, it is identified with $B(f,\cdot )+B({\mathfrak{g}}^e,\cdot )\subseteq {\mathfrak{g}}^{\ast }$. This definition does not assert transversality to adjoint orbits or independence from the chosen triple.

Root normalization and dual data for principal W levels

The affine-slice definition is complete for supplied $(e,h,f)$, and the local Jacobson-Morozov theorem supplies such a triple for every nilpotent $f$. No transversality or independence from the chosen triple is asserted here.