Birational sheets in reductive groups

Abstract

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.

Spherical birational sheets in reductive groups

“…a nilpotent orbit in g) is rigid if and only if it is birationally rigid if and only if it is {1} (resp. {0}), see [1,Example 3.4]. Moreover, sheets coincide with sheets in g and in G, see [1,Corollary 5.4].…”

“…The image of γ is the closure of a single orbit O ∈ N /G, and Ind g l O L := O is the orbit induced from O L . It only depends on the pair (l, O L ), not on the parabolic subgroup P chosen to define (1). If O ∈ N /G cannot be induced from a nilpotent orbit O L in a proper Levi subalgebra l g, then O is said to be rigid.…”

“…birationally rigid). All these notions are independent of the chosen parabolic subgroup P , see [1,Lemma 3.5].…”

“…Remark 3.4. Thanks to the bijective correspondences between parabolic subgroups, Levi subgroups, unipotent classes in G and parabolic subalgebras, Levi subalgebras, nilpotent orbits in g, we have that γ in (1) is birational if and only if γ in (2)is so, see [1,Remark 3.4]. Definition 3.5.…”

“…In this paper we deal with this problem with respect to spherical orbits both in the setting of conjugacy classes in G with simply-connected derived subgroup and in the setting of adjoint orbits in g. We recall the definition of birational sheet in g from [16] and in G from [1]. A birational sheet is a certain union of G-orbits and is contained in a sheet, hence the property of being spherical is preserved along birational sheets.…”