Nilpotent Orbits in Representation Theory

Kostant section, universal centralizer, and a modular derived Satake equivalence

“…We remark that π rs isétale by [J3,Lemma 13.4]. On the other hand, the quotient morphism t rs → t rs /W isétale by Lemma 2.3.3(1) and [J3,Remark 12.8], hence the morphism g rs × trs/W t rs → g rs is alsoétale. We deduce that ϕ rs isétale.…”

“…This follows from [J3,Proposition 7.13]. Alternatively, one can prove this result directly using the facts that χ separates semi-simple G-orbits (since they are closed, see [Bo,Proposition 11.8]), that C G (x) • is reductive if x ∈ g is semi-simple (see [Bo,Proposition 13.19]) and that the nilpotent cone of a connected reductive group is irreducible (see [J3,Lemma 6.2]).…”

“…We deduce that ϕ rs isétale. Moreover, comparing the fibers of π rs (see [J3,Lemma 13.3]) and of the projection g rs × trs/W t rs → g rs (see Lemma 2.3.3(1)) over closed points, we obtain that ϕ rs induces a bijection at the level of closed points. Using [SP, Tag 04DH], we deduce that ϕ rs is an isomorphism.…”

“…First we observe that regular nilpotent elements exist: this follows from the fact that the nilpotent cone in g has dimension dim(g)−r (see [J3,Theorem 6.4]) and consists of finitely many G-orbits (see [J3, §2.8] and references therein). Then the claim follows from [Sp,Lemma 5.8].…”

Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

“…Since π is a principal W -bundle, we have End(π !Ql Y ) =Q l [W ] (see for example [3], Lemma 12.9). It follows that End(ϕ !Q lX ) =Q l [W ].…”