Local normal forms for multiplicity free $U(n)$ actions on coadjoint orbits

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups $K=U(n), SO(n)$. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniques of Poisson-Lie theory and the geometric crystals of …

Geometry of coadjoint orbits and multiplicity-one branching laws for symmetric pairs

Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results…

A completely integrable system on $G_2$ coadjoint orbits

We construct a Gelfand-Zeitlin system on a one-parameter family of $G_2$ coadjoint orbits that are multiplicity-free Hamiltonian $SU(3)$-spaces. Using this system we prove a lower bound for the Gromov width of these orbits. This lower bound agrees with the known upper bound.

The Duistermaat-Heckman Measure for the Coadjoint Orbits of Compact Semisimple Lie Groups

We apply the Guillemin-Lerman-Sternberg theorem to reprove a formula of Heckman for the Duistermaat-Heckman measure associated to the coadjoint action of $T$, a maximal torus of a compact semisimple Lie group $G$, on a regular coadjoint $G$-orbit in the dual space of the Lie algebra of $G$. This formula is, in an appropriate sense, a limiting case of the Kostant multiplicity formula.

余切提升作用的辛法空间

The Symplectic Normal Space of a Cotangent-Lifted Action

本研究针对李群等距光滑真作用提升下的光滑黎曼流形余切丛，刻画了任意点的辛法空间，证明其可分解为线性空间余切丛与余伴随轨道约化辛线性空间的直和，且该刻画可仅用底流形群作用与余伴随表示表述，还探讨了相关特例。

Woodward乘积自由U(2)流形上的Lagrangian Floer理论

Lagrangian Floer theory on Woodward's multiplicity-free $U(2)$-manifolds

本文研究Woodward提出的流形家族，利用辛割应用toric Lagrangian Floer理论，识别非可移除Lagrangian子流形并证明Fukaya子category的开闭映射是同构。