Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Abstract

Abstract. Let G be a simply connected semisimple algebraic group with Lie algebra g, let G 0 ⊂ G be the symmetric subgroup defined by an algebraic involution σ and let g 1 ⊂ g be the isotropy representation of G 0 . Given an abelian subalgebra a of g contained in g 1 and stable under the action of some Borel subgroup B 0 ⊂ G 0 , we classify the B 0 -orbits in a and we characterize the sphericity of G 0 a. Our main tool is the combinatorics of σ-minuscule elements in the affine Weyl group of g and that of strongly orthogonal roots in Hermitian symmetric spaces.

Fully commutative elements and spherical nilpotent orbits

“…Proof. Subsets of orthogonal roots Γ ⊂ Φ + giving rise to non-spherical nilpotent orbits were studied in [6], [7], and classified in [3,Proposition 3.7]. In particular, if Gx Γ is not spherical, since Φ is not of tye G 2 we only have the following possibilities:…”

Nilpotent orbits of height 2 and involutions in the affine Weyl group

“…Given S ⊂ Φ a strongly orthogonal subset, set [12,Lemma 6.1]). Let S ⊂ Φ be a strongly orthogonal subset, then e S ∈ N , and ht(e S ) 4.…”

“…then by[12, Lemma 5.2] the matrix ( α, β ∨ ) α,β∈ S0 is a generalized Cartan matrix of finite or affine type. Notice that every node of the corresponding Dynkin diagram is connected with the node associated to γ, and that by construction the degree of γ as a vertex of this Dynkin diagram is α∈S0 γ, α ∨ .…”