# Sheets in Lie algebras are classified by the pair $(\mathfrak{l},O)$. How about sheets in Lie groups?

Borho in this paper showed that a sheet (i.e. a suitable union of adjoint orbits of the same dimension) in a complex semisimple Lie algebra $\mathfrak{g}$ is classified by a pair $(\mathfrak{l},O)$ where $\mathfrak{l}$ is a Levi subalgebra of $\mathfrak{g}$ and $O$ is a rigid nilpotent orbit of $\mathfrak{l}$.

How does the story go in the case of a sheet in the corresponding group $G$? I guess I need to replace the Levi algebra with a pseudo-Levi group. I'd appreciate if you could suggest a reference to look at.

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I'm not aware of any closely related group-theoretic formulation, but in any case Borho's paper has open access at the German archive: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002097583 – Jim Humphreys Dec 2 '11 at 15:06

## 1 Answer

There is a paper by Carnovale and Esposito, see 1011.5791. As the comment section of the link says, it seems to go back to Lusztig.

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Sorry for the oversight. I think I noticed the first version of this preprint but didn't go back to it. (As they point out in their addendum, some of the ideas involved were introduced earlier by Lusztig). I'm not sure how far this analogue of the Lie algebra case goes. Probably the mild restriction on characteristic is there mainly to get a natural bijection between the nilpotent orbits and unipotent classes as in characteristic 0 – Jim Humphreys Dec 6 '11 at 0:46