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

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.