Minimal Nilpotent Orbits of type D and E

TL;DR

This paper characterizes the closures of minimal nilpotent orbits in Lie algebras of types D and E, showing they are isomorphic to the affinizations of certain cotangent bundles of homogeneous spaces.

Contribution

It establishes explicit isomorphisms between minimal nilpotent orbit closures and affinizations of cotangent bundles for types D and E Lie algebras, extending known results.

Findings

Closure of Omin^{D_n} is isomorphic to the affinization of T^*(SL_{n-1}/[P,P])

Closure of Omin^{E_6} is isomorphic to the affinization of T^*(SL_4/P^u)

Proposes a similar formulation for type E_7

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.