Special pieces in exceptional Lie algebras
Paul Levy (Lancaster University)
Wednesday 9th October 16:00-17:00 Maths 311B
Abstract
In connection with the Springer correspondence, Lusztig defined an important subset of the nilpotent orbits in a simple Lie algebra, called the special orbits. To each special orbit is associated an open subset of its closure, called a special piece; the special pieces partition the nilpotent cone. A long-standing conjecture of Lusztig, open in exceptional types, is that each special piece is the quotient of a smooth variety by a certain finite group H.
In this talk I will outline a proof of the conjecture. The first step is the establishment of a "local version", which holds in a suitable transverse slice. In each case, the transverse slice is isomorphic to the quotient of a vector space by H. The local version allows us to establish smoothness of a certain H-cover of the special piece, therefore establishing the conjecture. Along the way, we observe various interesting symplectic quotient singularities appearing as transverse slices between nilpotent orbits in exceptional Lie algebras.
This is joint work with Fu, Juteau, Sommers and Yu.
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