Poisson structures on complex flag manifolds associated with real forms

Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of ${\mathcal S}(\mathfrak g)$ on coadjoint orbits. This concerns, in particular, Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras.

Configuration Poisson groupoids of flags

Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${\mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of P…

旗簇的泊松退化轨迹

The Poisson degeneracy locus of a flag variety

本研究针对配备标准泊松结构的复半单李群完全旗簇的泊松退化轨迹展开系统分析，给出组合描述与结构刻画，证明其不可约分支同构于复射影直线的幂次，并将结果推广至部分旗簇，明确了最高维数的组合与代数表示论意义。

Equivariant mirrors and the Virasoro conjecture for flag manifolds

We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.

作为余伴随轨道的加权非线性旗流形

Weighted nonlinear flag manifolds as coadjoint orbits

本文研究加权非线性旗的Frechet流形几何，推广了加权非线性Grassmannians，并在辛流形环境下利用加权迷向非线性旗描述了Hamilton微分同胚群的一类余伴随轨道。

Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

The purpose of this note is to define tri-moment maps for certain manifolds that carry closed non-degenerate 4-forms and an $Sp(1)^n$-action. Examples include quaternionic vector spaces and flag manifolds. We show how this map can be used ro reduce such manifolds to the ones with fewer symmetries. The images of such maps for quaternionic flag manifolds, which are defined using the Dieudonné determinant, resemble the polytopes from the complex case.