On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

We consider the finite $W$-algebra U(\g,e) associated to a nilpotent element e \in \g in a simple complex Lie algebra \g of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that U(\g,e) always has a 1-dimensional representation, when \g is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U(\g) whose associated variety is the coadjoint orbit corresponding to $e$.Comment: 14 pages, minor changes

Weyl submodules in restrictions of simple modules

Let F be an algebraically closed field of characteristic p>0. Suppose that SL_{n-1}(F) is naturally embedded into SL_n(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SL_{n-1}(F) can be embedded into the restriction L(\omega)\downarrow_{SL_{n-1}(F)}, where L(\omega) is a simple SL_n(F)-module. This allows us to construct new primitive vectors in L(\omega)\downarrow_{\SL_{n-1}(F)} from any primitive vectors in the corresponding Weyl modules. Some examples are given to show that this result actually works

Elementary invariants for centralizers of nilpotent matrices

We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the Lie algebra gl_N(C). In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova (math.RT/0610049).Comment: 12 page

Graded decomposition numbers for cyclotomic Hecke algebras

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.Comment: 57 pages; final versio