Workshop on Representation Theory and Geometry

A symplectic singularity has a natural stratification by Poisson leaves, which are symplectic, hence even-dimensional.  Perverse sheaves constructible with respect to this stratification are semisimple if the coefficients are in characteristic zero, but with characteristic p coefficients the categories can be complicated.  I will discuss two cases where this category has been understood.  For the nilpotent cone of $GL(n)$, Mautner showed that perverse sheaves are equivalent to representations of a Schur algebra $S(n,n)$, which is a highest weight category and is its own Ringel dual.  In joint work with Mautner, we showed that perverse sheaves on an affine hypertoric variety form a highest weight category which is Ringel dual to the same category for the symplectic dual hypertoric variety.  Time permitting, I will discuss what might be true for other symplectic singularities.