The orbit structure of Dynkin curves

Abstract

Abstract. Let G be a simple algebraic group over an algebraically closed field k; assume that char k is zero or good for G. Let B be the variety of Borel subgroups of G and let e ∈ Lie G be nilpotent. There is a natural action of the centralizer C G (e) of e in G on the Springer fibre B e = {B ′ ∈ B | e ∈ Lie B ′ } associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case B e is a Dynkin curve. We give a complete description of the C G (e)-orbits in B e . In particular, we classify the irreducible components of B e on which C G (e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.

One-parameter contractions of Lie-Poisson brackets

“…Calculating centralisers g e of subregular nilpotent elements in type E, on GAP or by hand, one can show that g e does not contain an Abelian subalgebra of codimension 1. Together with Theorem 5.5 and equation (5•1), this fact provides an additional explanation for [GHR,Theorem 2.4(a)(i)]. That result states that for g simply laced, D i contains an open B-orbit if and only if r i = 1.…”

“…In the simply laced case g e does not contain Abelian subalgebras of codimension 1. For the remaining Lie algebras, [GHR,Theorem 2.4. (a)(ii)] provides the following answer.…”

Compatible Poisson brackets associated with 2‐splittings and Poisson commutative subalgebras of S(g)

“…In the remaining cases, , is abelian and still . This is possible if and only if is of type with and , see [6] and [27, Proposition 5.13]. As a Richardson element in , we take ; next and .…”

Adjoint Action of Automorphism Groups on Radical Endomorphisms, Generic Equivalence and Dynkin Quivers

“…The groups H V and H W consist of matrices of the form Using an isomorphism f with b = 0, We identify W 1 with W 2 and V 1 with V 2 . We claim that with some restriction on c 12 , there exists maps x : W 1 → W 2 and y : V 1 → V 2 such that c 11 − c 12 x + yc 12 is surjective. Now by the structure of X(d ), with respect to the fixed basis, the map x : V 1 → V 2 can be any quadratic matrix.…”

“…The theorem of Richardson holds for parabolic Lie algebras of any reductive Lie-algebra, and in type A, elements with dense orbits have been explicitly constructed by Brüstle, Hille, Ringel and Rörhle using representations of quivers [5], and in the classical types by Baur using a different approach [1]. These results and methods have been extended in various directions, see for example [3,11,12,14,19].…”