Universal filtered quantizations of nilpotent Slodowy slices

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $\mathbb{C}^\times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn--Namikawa--Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply-laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally we apply this result to give a presentation of the subregular finite $W$-algebra in type B as a quotient of a shifted Yangian.