Event Details

Lusztig's special pieces conjecture

Let \g be a simple Lie algebra over \C. The set of special nilpotent orbits plays a key role in relation to the representation theory of \g and its Weyl group. To each special orbit \0 is attached a union of orbits called a \emph{special piece}; the set of all nilpotent elements is the disjoint union of special pieces.Lusztig's special pieces conjecture, posed in 1998, states that each special piece in an exceptional Lie algebra is isomorphic to the quotient of a smooth variety by a finite group depending on the special orbit. Although this is a direct extension of a result for classical Lie algebras proved earlier by Kraft and Procesi, the finite group appearing for exceptional \g can be more "interesting" than in classical types, and this has given the conjecture an air of mystery.

In this talk I will outline (mostly by means of examples) two proofs of the conjecture, both depending on a recently established local version: that a suitable transverse slice in the special piece is the quotient of a vector space by the same finite group. This is joint work involving collaborators Fu, Juteau, Sommers and Yu.