Let $G$ be your favorite simple complex Lie group, and $P\subset G$ your favorite parabolic subgroup. We can identify $T^*G/P$ with the space of pairs $$\{(gP,x)\in G/P\times \mathfrak g | x\perp \operatorname{Ad}_g(\mathfrak p)\}$$ where $\perp$ denotes perpendicularity in the Killing form. Thus, we have the second projection $p_2:T^*G/P\to \mathfrak g$; when $P=B$, this is the famous Springer map.

Now, let $e$ be your favorite nilpotent in $\mathfrak g$, and let $e,h,f$ be a completion of this to a $\mathfrak{sl}_2$ triple (which exists by Jacobson-Morozov). Then $S=e+\ker(\operatorname{ad}_f)\subset \mathfrak g$ is an affine subspace of $\mathfrak g$ transverse to the orbit $G\cdot e$ called the **Slodowy slice** to $e$.

It's a theorem that $p_2^{-1}(S)$ is a smooth symplectic variety (it's actually a symplectic reduction of $T^*G/P$ by the action of a nilpotent subgroup $M\subset G$ at a regular value of the moment map), and it's one that I like very much.

Does this variety have an agreed-upon name?

`$p$`

in work by Premet and others. Also, a small nitpick if someone's favorite nilpotent is the zero element; this is the (usually trivial) case where Jacobson-Morosov doesn't apply. $\endgroup$ – Jim Humphreys Jan 10 '11 at 14:19