Simon M. Goodwin

Highest Weight Theory for Finite W-Algebras

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step toward the classification of finite-dimensional irreducible modules. We also introduce an analogue of the BGG category . Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.

PRINCIPAL W-ALGEBRAS FOR GL(m|n)

We consider the (finite)

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

W

Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type

-algebra

On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

W_{m|n}

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

attached to the principal nilpotent orbit in the general linear Lie superalgebra

Translation for finite -algebras

\mathfrak{gl}_{m|n}(\mathbb C)

On conjugacy of unipotent elements in finite groups of Lie type

. Our main result gives an explicit description of

FINITE ORBIT MODULES FOR PARABOLIC SUBGROUPS OF EXCEPTIONAL GROUPS

W_{m|n}

The generic character table of a Sylow

as a certain truncation of a shifted version of the Yangian