Minimal Nilpotent Orbits of type D and E

TL;DR

The paper shows that the closures of minimal nilpotent adjoint orbits in types $D_n$ and $E_6$ can be realized as affinizations of cotangent bundles to specific homogeneous spaces: $\overline{\mathcal{O}}_{min}^{D_n}\cong T^*(SL_{n-1}/[P,P])^{aff}$ with $P$ corresponding to $(1,1,n-3)$, and $\overline{\mathcal{O}}_{min}^{E_6}\cong T^*(SL_4/P^u)^{aff}$ with $P^u$ the unipotent radical of $P_{(2,2)}$. The proofs combine embeddings of Lie algebras, decompositions of abelian nilradicals, and Joseph-ideal/hereditary-filter arguments to match the orbit closures with Hamiltonian reductions of cotangent bundles, supported by isomorphisms involving $\mathcal{D}$-modules and associated graded algebras. The work leverages generalizations inspired by a $D_4$ case and results from LSS88, and hints at a similar paradigm for type $E_7$. These identifications illuminate the symplectic geometry of minimal nilpotent orbits and relate their closures to well-structured affine varieties with symplectic singularities.

Abstract

We first show the closure of the minimal nilpotent adjoint orbit Omin^{D_n} in so_{2n} is isomorphic to the affinization of T^*(SL_{n-1}/[P,P]) where P is the parabolic subgroup P_{(1,1,n-3)} of SL_{n-1}(C). Then we prove that the closure of the minimal nilpotent adjoint orbit Omin^{E_6} of the complex simple Lie algebra E_6 is isomorphic to the affinization of T^*(SL_4/P^u) where P^u is the unipotent radical of the parabolic subgroup P_{(2,2)} of SL_4(\C). In the end we will formulate a similar result for type E_7.

Theorems & Definitions (11)

• Theorem 2.1
• Lemma 2.2
• proof
• proof : Another explanation of Theorem \ref{['Dmain']}
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• Conjecture 3.4