We define the group analogue of birational sheets, a construction performed by Losev for reductive Lie algebras. For G semisimple simply connected, we describe birational sheets in terms of Lusztig-Spaltenstein induction and we prove that they form a partition of G, and that they are unibranch varieties with smooth normalization by means of a local study.

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 universal equivariant quantization with respect to any group acting on it by C ˆ-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 the 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 the 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.

We prove that the closure of every Jordan class J in a semisimple simply connected complex algebraic group G at a point x with Jordan decomposition x=rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent, we also show that the closure of J around x is smoothly equivalent to the closure of a Jordan class in Lie(G) around exp−1x. For G simple we apply these results in order to determine a (non‐exhaustive) list of smooth sheets in G, the complete list of regular Jordan classes whose closure is normal and Cohen–Macaulay, and to prove that all sheets and Lusztig strata in normalSLnfalse(double-struckCfalse) are smooth.

Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M, O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

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