$\chi$-admissible subalgebras of $\sl_{pn}(\C)$ and finite $W$-algebras

TL;DR

This paper explores new admissible subalgebras of () associated with a nilpotent element e, generalizing known results about finite W-algebras and Slodowy slices in a specific Lie algebra setting.

Contribution

It constructs non-isomorphic admissible subalgebras in () and proves their associated graded algebras are isomorphic to () analogues of Slodowy slices, extending prior results.

Findings

Constructed new admissible subalgebras in () not derived from the Dynkin grading.

Proved the associated graded algebra is isomorphic to an analogue of the Slodowy slice.

Generalized Premet and Gan-Ginzburg's results to this specific case.

Abstract

Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading for e, we recover the finite W-algebra associated to e and it is well-known that gr(H) is isomorphic to \C[S] as a graded Poisson algebra where S is the Slodowy slice of e and gr(H) is the graded algebra associated to the Kazhdan filtration. In this paper, we consider the case where g =\sl_{pn}(\C) and e consists of p Jordan blocks all of the same size n. Here, the only good grading for e is the Dynkin grading and we construct admissible subalgebras non isomorphic to the one derived from this good grading. For these algebras m, we prove that gr(H) is isomorphic to \C[S], generalizing Premet and Gan-Ginzburg's result in this particular case, where S denotes an analogue to the Slodowy slice.