Let $(X,ω)$ be an integral symplectic manifold and let $(L,\nabla)$ be a quantum line bundle, with connection, over $X$ having $ω$ as curvature. With this data one can define an induced symplectic manifold $(\widetilde {X},ω_{\widetilde {X}})$ where $dim \widetilde {X} = 2 + dim X$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\widetilde {X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\widetilde {X}$ is the coadjoint orbit of some larger Lie group $G$. We show that this is the case if $G$ is a non-compact simple Lie group with a finite center and $K$ is the maximal compact subgroup of $G$. The coadjoint orbit $X$ arises (Borel-Weil) from the action of $K$ on $\p$ where $\g= \k +\p$ is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\widetilde {X},ω_{\widetilde {X}})\cong (Z,ω_Z)$ where $Z$ is a non-zero minimal "nilpotent" coadjoint orbit of $G$. This is applied to show that the split forms of the 5 exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.
We give a representative of every coadjoint orbit of the odd symplectic group. Our argument follows that used for the Poincarè group but the details differ.
We compute the symplectic volume of the symplectic reduced space of the product of N coadjoint orbits of a compact connected Lie group G. We compare our result with the result of Suzuki and Takakura , who study this in the case G = SU(3) starting from geometric quantization.
We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.
Let G be a noncompact real semisimple Lie group. The regular coadjoint orbits of G can be partitioned into a finite set of types. We show that on each regular orbit, the Iwasawa decomposition induces a left-invariant foliation which is isotropic with respect to the Kirillov symplectic form. Moreover, the leaves are affine subspaces of the dual of the Lie algebra, and the dimension of the leaves depends only on the type of the orbit. When G is a split real form, the foliations induced from the Iwasawa decomposition are actually Lagrangian fibrations with a global transverse Lagrangian section.
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 …
In this paper we explain how the cocycle of the momentum map of the action of the affine symplectic group on $\mathbb{R}^{2n}$ gives rise to a coadjoint orbit of the odd real symplectic group with a modulus.
Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$
具有固定Jordan结构的所有线性变换的集合$J$是同构于$GL(N,C)$的余伴随轨道$\mathcal O (J)$的辛流形。任何线性变换都可以沿着其特征空间投影到补维的（至少一个）坐标子空间。像的Jordan结构$\tilde J$由原像的Jordan结构$J$定义，因此投影$\mathcal O (J)\to \mathcal O (\tilde J)$是辛流形的映射.证明了该投影的纤维$\mathcal E$是线性辛空间，映射$\mathcal O(J) \to \mathcal E \times \mathcal O (\tilde J)$是双有理辛类。该过程的迭代给出了$\mathcal O (J)$和线性辛空间之间的同构，线性辛空间是投影的所有纤维的直积。$\mathcal O(J)$上的达布坐标是所讨论的线性空间上的正则坐标的拉回。
The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to orbit method and construct nontrivial irreducible Harish-Chandra modules for certain coadjoint orbits. Our examples include new geometric construction of representations associated to certain orbits of real exceptio…
We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliations. In particular, we show that for a (strict) Lie 2-group the coadjoint orbits of the units in the dual of its Lie 2-algebra are symplectic groupoids, meaning that the classical Kostant-Kirillov-Souriau symplectic forms on these special coadjoint orbits are multiplicative.
For a complex Lie group $G$ with a real form $G_0\subset G$, we prove that any Hamiltionian automorphism $ϕ$ of a coadjoint orbit $\mathcal O_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal O_0$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal O$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary G-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian G-spaces, which (as we show) unfortunately fails to mirror the situation where more than one G-module "quantizes" a given Hamiltonian G-space. This paper offers evidence that the situation is remedied by working in the category of *prequantum* G-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces, and establish Frobenius reciprocity …
The asymptotics of characters $χ_{kλ}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$. This phenomenon generalises to coadjoint orbits of central extensions of loop groups $\widehat{LG}$ and of diffeomorphisms of the circle $\widehat{\rm Diff}(S^1)$. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contrib…
We give a proof of the fact that a simply-connected symplectic homogeneous space $(M,ω)$ of a connected Lie group $G$ is the universal cover of a coadjoint orbit of a one-dimensional central extension of $G$. We emphasise the rôle of symplectic group cocycles and the relationship between such cocycles, left-invariant presymplectic structures on $G$ and central extensions of $G$; in particular, we show that integrability of a central extension of $\mathfrak{g}$ to a central extension of $G$ is equivalent to integrability of a representative Chevalley-Eilenberg 2-cocycle of $\mathfrak{g}$ to a s…
For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $ω_ξ^s$, where $ξ\in \mathfrak{t}^*_+$ is in the positive Weyl chamber and $s \in \mathbb{R}$. The symplectic form $ω_ξ^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $ξ$. The cohomology class of $ω_ξ^s$ is independent of $s$ for a fixed value of $ξ$. In this paper, we show that as $s\to -\infty$, the symplectic volume of $ω_ξ^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T \cong G/B$. This s…
We show that the symplectic reduction of the dynamics of $N$ point vortices on the plane by the special Euclidean group $\mathsf{SE}(2)$ yields a Lie--Poisson equation for relative configurations of the vortices. Specifically, we combine symplectic reduction by stages with a dual pair associated with the reduction by rotations to show that the $\mathsf{SE}(2)$-reduced space with non-zero angular impulse is a coadjoint orbit. This result complements some existing works by establishing a relationship between the symplectic/Hamiltonian structures of the original and reduced dynamics. We also find…
We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.
