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.

My first question : Is there such a theory ?

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).