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?

  • $\begingroup$ Does your question make sense with suitable modifications over an algebraically closed field of arbitrary characteristic? Slodowy slices often come up in characteristic $p$ 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
  • $\begingroup$ This should all work over any field where $P$ and $e$ are defined, though maybe I'm missing something silly. As for the case of the 0 nilpotent, if you take f to be 0, then everything works, and e=h=f=0 satisfy the Chevalley relations of $\mathfrak{sl}_2$. $\endgroup$ – Ben Webster Jan 10 '11 at 17:47

In my paper "Singular blocks of parabolic category O and finite W-algebras", these are called "S3-varieties." S3 is for "Slodowy-Springer-Spaltenstein."

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    $\begingroup$ So, I guess you can accept this answer as soon as someone else starts calling them S3-varieties as well. It's amazing you wrote an entire paper just for the possibility of extra MO reputation. $\endgroup$ – Mike Skirvin Jun 15 '11 at 0:00
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    $\begingroup$ And since intent can be difficult to judge on the internet, let me say that my comment above should not be taken seriously. $\endgroup$ – Mike Skirvin Jun 15 '11 at 0:03
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    $\begingroup$ This could be confused with Serre's S3 condition in commutative algebra. $\endgroup$ – Joel Kamnitzer Jun 15 '11 at 1:42
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    $\begingroup$ That's true. Serre, Springer, Spaltenstein and Steinberg really should have thought of that. $\endgroup$ – Ben Webster Jun 16 '11 at 7:14

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