Does the preimage of the Slodowy slice in $T^*G/P$ have a name? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:26:12Z http://mathoverflow.net/feeds/question/51613 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51613/does-the-preimage-of-the-slodowy-slice-in-tg-p-have-a-name Does the preimage of the Slodowy slice in $T^*G/P$ have a name? Ben Webster 2011-01-10T02:06:08Z 2011-06-14T22:32:26Z <p>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.</p> <p>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 <strong>Slodowy slice</strong> to $e$. </p> <p>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. </p> <blockquote> <p>Does this variety have an agreed-upon name?</p> </blockquote> http://mathoverflow.net/questions/51613/does-the-preimage-of-the-slodowy-slice-in-tg-p-have-a-name/67810#67810 Answer by Ben Webster for Does the preimage of the Slodowy slice in $T^*G/P$ have a name? Ben Webster 2011-06-14T22:32:26Z 2011-06-14T22:32:26Z <p>In my paper "<a href="http://front.math.ucdavis.edu/0909.1860" rel="nofollow">Singular blocks of parabolic category O and finite W-algebras</a>", these are called "S3-varieties." S3 is for "Slodowy-Springer-Spaltenstein."</p>