# Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. I will refer to a nice paper of De-Graaf-Elashvili for more concrete definitions and summary of results.

Now, the classification of sheets is usually given by data that involves a pair $(\mathfrak{l},e_0)$ where $\mathfrak{l}$ is a Levi subalgebra and $e_0$ is a rigid nilpotent orbit in the Levi subalgebra $\mathfrak{l}$. I am hoping that there is a theory surrounding an affine analog of this situation. For ex, I would like to replace the Levi subalgebra $\mathfrak{l}$ by pseudo-Levi subalgebras of $\mathfrak{g}$. By pseduo-Levi subalgebras, I mean the sub-algebras that arise as centralizers of semi-simple elements in the lie algebra. Outside of lie algebras of Type A, this is a richer set of possibilities than the set of Levi subalgebras.

1. My first question : Is there such a theory ?

2. My second question : If such a theory exists, I would like to know if there is a good analog of the notion of a "Dixmier sheet". In the usual setting, Dixmier sheets are the sheets that contain a semi-simple element.

Clarification (added after Paul Levy's response below) :

My comment about pseudo-Levi subalgebras entering the picture is really more of an aesthetic guess. Feel free to ignore it. My question is really about whether there is a theory of sheets for the affine Lie algebra $\hat{g}$ or the Loop Group $\hat{G}$. Optimistically, I'd also like a definition of an analog of a Dixmier sheet in this setting. If such a definition exists, then is every sheet a Dixmier sheet ? (This latter statement is known to be false for the lie algebra setting outside of Type A)

Clarification 2 :

I really should have posed two separate questions.

1) Is there a theory of sheets with pseudo-levis ?

2) Is there an affine analog of the theory of sheets ?

Paul Levy's response (pointing to the works of Carnovale-Esposito) does answer 1). So, I will provisionally accept this. It also seems like the fair thing to do since I posed a hybrid/confusing question in the first place. I am still hoping someone can answer 2).

Levi subalgebras appear in the classification of sheets because they are the centralizers of semisimple elements of the Lie algebra. So given any element of ${\mathfrak g}$, we can associate a pair $({\mathfrak l},{\mathcal O}_{\mathfrak l})$ where ${\mathfrak l}$ is the centralizer of the semisimple part, and ${\mathcal O}_{\mathfrak l}$ is the $L$-orbit of the nilpotent part.
Pseudo-Levi subalgebras arise as centralizers of semisimple elements (of finite order) of the group. So a natural candidate for a theory of sheets along the lines you suggest is to consider the action of $G$ on itself by conjugation. Under mild assumptions on the characteristic, the set of unipotent orbits of $L$ is in bijection with the set of nilpotent orbits of ${\mathfrak l}$, so this does correspond to your question. This has been studied, see Carnovale and Esposito On sheets of conjugacy classes in reductive groups (published in IMRN in 2012 but also available on the arXiv). In particular their main theorem identifies the analogues of decomposition classes in this setting (somewhat more precisely phrased than just pairs $(L,{\mathcal O}_{L})$ with $L$ a pseudo-Levi and ${\mathcal O}_{L}$ a rigid unipotent orbit in $L$) and they remark at the end that one can therefore classify the sheets in $G$ and determine their dimensions. (This was deferred to a later work, but I'm not sure whether it has appeared yet.) Their main theorem also states that a sheet contains a semisimple element if and only if ${\mathcal O}_{L}$ is trivial, which answers your question about Dixmier sheets. (Every semisimple element of $G$ lies in a unique sheet.)
• I am using Eric Sommers' terminology when I call things pseudo-Levi subalgebras (from this paper : imrn.oxfordjournals.org/content/1998/11/539.extract ). He uses it both in the lie algebra and the lie group settings ( pseudo-Levi subalgebras are $Z_g(s)$ and pseudo-Levi subgroups are $Z_G(s)$). – Aswin Jan 30 '16 at 23:18
• Also, I really did want an affine analog of the theory of sheets ie, sheets in the affine Lie algebra $\hat{g}$ or the Loop Group. My quip about the pseudo-Levis was more of a guess about them being involved in such a theory. Perhaps I should have split the two and asked it as two questions to avoid confusion .. – Aswin Jan 30 '16 at 23:20