Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0811.1351

The orbit structure of the Gelfand-Zeitlin group on n x n matrices

Mark Colarusso
View original source ↗Read source document ↗Export metadata ↓

Abstract

In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq \mathbb{C}^{n\choose 2}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the authors show that $A$-orbits of dimension ${n\choose 2}$ form Lagrangian submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit structure of $A$ on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all $A$-orbits of dimension ${n\choose 2}$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory.

Explore related subjects

math.SGmath.AG

Keep this discovery

BibTeXRIS

Mark Colarusso. 2009-03-31. The orbit structure of the Gelfand-Zeitlin group on n x n matrices. https://arxiv.org/abs/0811.1351

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗