In his '91 paper, Lusztig defines a collection of simple perverse sheaves that correspond to the canonical basis; these are defined using a pushforward construction, and from the definition, it's easy to see that their singular support is contained in the set of nilpotent representations of the doubled quiver (thinking of reps of the doubled quiver as the cotangent bundle of reps of the quiver with an orientation).
In finite type, this is just all equivariant perverse sheaves on the space of quiver representations. But in other types, it gets trickier. For example, for the Kronecker quiver and dimension vector (1,1), I only have two canonical basis vectors, but I have infinitely many equivariant perverse sheaves. Lusztig only wants me to take the constant sheaf on the 0 rep, and the constant sheaf on all reps, forgetting all in between.
On the other hand, in this case, and in fact in all affine type A examples, I see that nilpotent characteristic cycle is the only thing I need to impose (one way to see this is that if I take the cyclic orientation, there are only finitely many nilpotent orbits, each of which is equivariantly simply connected, and Lusztig's sheaves cover all the IC sheaves of those with nilpotent conormals).
Does this carry over to the general case? Are there any equivariant perverse sheaves on the space of quiver representations which aren't in Lusztig's class, but do have nilpotent characteristic variety?