Algebra Seminar: Thiel -- Hyperplane arrangements associated to symplectic quotient singularities

Ulrich Thiel (University of Sydney) 

Friday 22 June, 12-1pm, Place: Carslaw 375 

Title: Hyperplane arrangements associated to symplectic quotient singularities 

Abstract: Namikawa associated to any conic symplectic singularity a hyperplane
arrangement which is deeply intertwined with its geometry.  For example, Bellamy proved
that for a symplectic quotient singularity the cohomology of the complement of this
arrangement encodes the number of minimal models of the singularity.  For the symplectic
singularity associated to a complex reflection group we were able to prove that the
Namikawa arrangement coincides with the degenericity locus of the number of torus fixed
points of the corresponding Calogero-Moser deformation.  This has a series of remarkable
consequences, especially it proves a conjecture by Bonnafe and Rouquier.  Using
representation theory and sophisticated computer algebraic methods, we could compute
this arrangement explicitly for several exceptional complex reflection groups.  The
arrangements seem to be of a new kind, and many more are out there.  This is joint work
with Gwyn Bellamy (Glasgow) and Travis Schedler (London), and with Cedric Bonnafe
(Montpellier).