BPS SHEAF FOR COMMUTING VARIETIES

In this short note, we give a description of the BPS sheaf of the triple commuting variety of a reductive Lie algebra. It is shown to be isomorphic to a direct sum of constant sheaves on the cube of the center of the Lie algebra. The multiplicity of the constant sheaf is given by the number of distinguished nilpotent orbits in the Lie algebra. This answers a conjecture of the author, motivated by cohomological Donaldson-Thomas theory, in this particular case. This study relies on the virtual smallness of the approximation by schemes of the good moduli space morphism, which we establish in this paper for general (weakly) symmetric representations of reductive groups. The virtual smallness gives a lower bound on the perverse filtration, which also allows us to give a partial identification of the BPS sheaf for symmetric representations of reductive groups.