[Paper Review] Symplectic and Hamiltonian properties of holomorphic coadjoint orbits

Guillaume Deltour|arXiv (Cornell University)|Jan 20, 2011

Advanced Algebra and Geometry20 references3 citations

TL;DR

This thesis establishes a complete system of linear inequalities characterizing the moment map image (projection of holomorphic coadjoint orbits) for real reductive Lie groups G, particularly focusing on noncompact, simple, Hermitian symmetric spaces. Using invariant complex structures on the noncompact part of the Lie algebra and combinatorics of well-covering pairs in the Weyl group, the author derives explicit moment polytope equations, generalizing Ressayre's criterion and providing a computational framework for SU(p,q), Sp(2n,R), and SO*(2n).

ABSTRACT

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G with finite center, such that G/K is a Hermitian symmetric space, where K is a maximal compact subgroup of G. Holomorphic coadjoint orbits are a generalization of the Hermitian symmetric space G/K. In this thesis, we prove that the McDuff's symplectomorphism on Hermitian symmetric spaces, has an analogous for holomorphic coadjoint orbits. Then, using this symplectomorphism and recent GIT arguments from Ressayre, we compute the equations of the projection of the O, relatively to the maximal compact subgroup K.

Motivation & Objective

• To characterize the moment map image (projection of holomorphic coadjoint orbits) for noncompact real reductive Lie groups.
• To generalize Ressayre's criterion for moment polytopes to holomorphic coadjoint orbits in noncompact settings.
• To provide a systematic computational method for determining the moment polytope equations using well-covering pairs and Weyl group combinatorics.
• To apply the results to classical groups such as Sp(2n,R), SU(p,q), SO*(2n), and SO(2p,2), particularly in the Hermitian symmetric case.
• To establish a connection between the complex structure on the noncompact part of the Lie algebra and the moment polytope equations.

Proposed method

• Utilizes the decomposition 𝔤 = 𝔨 ⊕ 𝔭 for a real reductive Lie group G with K a maximal compact subgroup.
• Introduces K-invariant complex structures on 𝔭 via the adjoint action of elements in the center of 𝔨, defining 𝔭⁺ and 𝔭⁻ as complex vector spaces.
• Applies the Corollary E from the author’s prior work to derive moment polytope inequalities using well-covering pairs (C(w,w′,0), λ) in the set 𝒫₀(𝔭⁻).
• Employs the Chevalley formula for computing cup products in the cohomology of the complete flag variety of GLₙ(ℂ), essential for verifying well-covering conditions.
• Performs systematic computation of dominant, indivisible, 𝔭-admissible one-parameter subgroups λ of the complexified maximal torus Tℂ.
• Derives the moment polytope Δ_K(𝒪_Λ) as the solution set of inequalities ⟨wλ, ξ⟩ ≤ ⟨w₀w′λ, Λ⟩ for all such well-covering pairs.

Experimental results

Research questions

• RQ1 What is the complete system of linear inequalities that characterizes the moment map image of a holomorphic coadjoint orbit under the action of a maximal compact subgroup?
• RQ2 How can Ressayre's criterion for moment polytopes be extended to holomorphic coadjoint orbits in noncompact real reductive Lie groups?
• RQ3 What is the role of K-invariant complex structures on the noncompact part 𝔭 of the Lie algebra in determining the moment polytope?
• RQ4 Which combinatorial data—specifically well-covering pairs in the Weyl group—determine the moment polytope for classical groups?
• RQ5 Can explicit moment polytope equations be computed for specific groups such as SU(p,q), Sp(2n,R), and SO*(2n)?

Key findings

• The moment polytope Δ_K(𝒪_Λ) of a holomorphic coadjoint orbit is characterized by the system of inequalities ⟨wλ, ξ⟩ ≤ ⟨w₀w′λ, Λ⟩ for all well-covering pairs (C(w,w′,0), λ) in 𝒫₀(𝔭⁻).
• For G simple, noncompact, Hermitian symmetric, and with finite center, the moment polytope is fully described by the inequalities in Theorem G.
• The author computes all dominant, indivisible, 𝔭-admissible one-parameter subgroups λ for classical groups including Sp(2n,R), SU(p,q), SO*(2n), and SO(2p,2).
• The method successfully computes the moment polytope for SU(n,1), SO*(6), SO*(8), and SU(2,2), demonstrating the framework's effectiveness.
• The work proves that certain Horn-type triplets (I,J,L) appear as index triplets in the moment polytope equations for SU(n,1), linking the problem to Horn's conjecture.
• The complex structure on 𝔭⁻ corresponds to the negative noncompact roots of 𝔤, and this structure is essential for the derivation of the moment polytope equations.