Let ${\cal O}$ be the orbit of $η\in{\frak g}^*$ under the coadjoint action of the compact Lie group $G$. We give two formulae for calculating the action integral along a closed Hamiltonian isotopy on ${\cal O}$. The first one expresses this action in terms of a particular character of the isotropy subgroup of $η$. In the second one is involved the character of an irreducible representation of $G$.
The rational cohomology of a coadjoint orbit ${\cal O}$ is expressed as tensor product of the cohomology of other coadjoint orbits ${\cal O}_k$, with $ \hbox{dim} {\cal O}_k< \hbox{dim} {\cal O}$.
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 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.
Our goal is to find a representative of each orbit of the coadjoint action of the generalized Galile group on the dual of its Lie algebra. Our line of argument follows that of Cushman and van der Kallen, but differs in the details.
We find an upper bound for the Gromov width of coadjoint orbits of compact Lie groups with respect to the Kirillov Kostant Souriau form by computing certain Gromov Witten invariants, the approach presented here is closely related to the one used by Gromov in his celebrated nonsqueezing theorem.
We find an upper bound for the Gromov width of coadjoint orbits of U(n) with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated non-squeezing theorem.
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 describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,ω)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegener…
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.
Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_lambda through lambda in Lie(T)^* is canonically a symplectic manifold. Therefore we can ask the question about its Gromov width. In many known cases the Gromov width is exactly the minimum over the set {< alpha_j^{\vee},lambda > ; alpha_j^{\vee} a coroot and < alpha_j^{\vee},lambda > positive}. We show that the Gromov width of coadjoint orbits of the unitary group and of most of the coadjoint orbits of the special orthogonal group is at least the above minimum. The proof uses the torus action coming from t…
Let G be an n-dimensional semisimple compact and connected Lie group acting on both the Lie algebra g of G and its dual g*. We show that a nondegenerate Killing form of G induces an Ad*-equivariant isomorphism of g onto g* which, in turn, induces by passage to quotients a symplectomorphism between adjoint and coadjoint orbit spaces of G.
We prove that the Gromov width of coadjoint orbits of the symplectic group is at least equal to the upper bound known from the works of Zoghi and Caviedes. This establishes the actual Gromov width. Our work relies on a toric degeneration of a coadjoint orbit to a toric variety. The polytope associated to this toric variety is a string polytope arising from a string parametrization of elements of a crystal basis for a certain representation of the symplectic group.
We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group. We show that this result follows from positivity results of Gromov-Witten invariants and the fact that the quantum product of Schubert classes can never be zero.
Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_λ through λin the dual of the Lie algebra of T, is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width. In many known cases the width is exactly the minimum over the positive results of pairing λwith coroots: min{< α_j^{\vee},λ> ; α_j^{\vee} is a coroot and < α_j^{\vee},λ> is positive}. We will show that the Gromov width for regular coadjoint orbits of the special orthogonal group is at least this minimum. The proof uses the torus action coming from the Gelfand-Tsetlin s…
We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of $\mathbb{R}^3$ associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. If the vorticity density has discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative …
We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.
We use the Gelfand-Tsetlin pattern to construct an effective Hamiltonian, completely integrable action of a torus T on an open dense subset of a coadjoint orbit of the unitary group. We then identify a proper Hamiltonian T-manifold centered around a point in the dual of the Lie algebra of T. A theorem of Karshon and Tolman says that such a manifold is equivariantly symplectomorphic to a particular subset of R^2D. This fact enables us to construct symplectic embeddings of balls into certain coadjoint orbits of the unitary group, and therefore obtain a lower bound for their Gromov width. Using t…
We show how to construct a resolution of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularise generic symplectic quotients. Given a manifold with a Hamiltonian action of a compact Lie group, symplectic reduction at a coadjoint orbit which is transverse to the moment map produces a symplectic orbifold. If moreover the points of this coadjoint orbit are regular elements of the Lie coalgebra, that is, their stabiliser is a maximal torus, the result for torus quotients may be applied to obtain a desingularisation…
We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of a compact Lie group. We show that all the faces of such an orbitope are exposed. The face structure is studied by means of the momentum map and it is shown that every face is again a coadjoint orbitope. Up to conjugation the faces are completely determined by the momentum polytope and can be described in a simple way in terms of root data. Finally we consider the complex geometry of the coadjoint orbit and we prove that the submanifolds of the orbit that are extreme sets of a face are exactly the closed orbits of parabolic …
In the paper we study the coadjoint orbits of the group $\mathrm{UT}(n,K)$ associated with involutions. We obtain a formula for dimension of the orbit. We construct a polarization for the canonical element of orbit. We find a system of generators in the defining ideal of orbit.
We correspond to any factor algebra of the unitriangular Lie algebra with respect to a regular ideal some permutation. In terms of this permutation one can construct a diagram, that allows to calculate index and maximal dimension of coadjoint representation.
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…
I give an answer to the question ``Which groups have compact coadjoint orbits?''. Whilst I thought that the answer, which is straightforward, must be in the literature, I was unable to find it. This note aims to rectify this. It is also a plea: If the result is already published then I would like to be told the reference.
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.
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…
Let G be a simple, noncompact, connected, real Lie group with finite center, and K a maximal compact subgroup of G. We assume that G/K is Hermitian. Using GIT methods derived from the generalized eigenvalue problem, we compute a set of affine equations describing the moment polyhedron of the projection on k* of holomorphic coadjoint orbits of G.
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.
We give a uniform proof for the conjectured Gromov width of coadjoint orbits of all compact connected simple Lie groups, by analyzing simplices in Newton-Okounkov bodies.
We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is simple as a Poisson algebra iff O is semisimple.
Actions of $U(n)$ on $U(n+1)$ coadjoint orbits via embeddings of $U(n)$ into $U(n+1)$ are an important family of examples of multiplicity free spaces. They are related to Gelfand-Zeitlin completely integrable systems and multiplicity free branching rules in representation theory. This paper computes the Hamiltonian local normal forms of all such actions, at arbitrary points, in arbitrary $U(n+1)$ coadjoint orbits. The results are described using combinatorics of interlacing patterns; gadgets that describe the associated Kirwan polytopes.
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.
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 simpl…
Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups.
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.
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…
For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^1$-bundle one obtains central extensions for certain groups of diffeomorphisms of $M$. We can realize $Gr_{m-2}(M)$ as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians $SGr_{2k}(M)$ as coadjoint orbits in the group…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at http://arxiv.org/abs/math.MG/0210189, and "Tangent bundles to sub-Riemannian groups", math.MG/0307342, available at http://arxiv.org/abs/math.MG/0307342 .
We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, we identify the moment map for a Hamiltonian circle action on the reduced product. Finally, we make use of results of Adams, Harnad, and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.
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 …
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…
This paper is devoted to the construction of a hyperkaehler structure on the complexification of any Hermitian-symmetric affine coadjoint orbit O of a semi-simple L*-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kaehler structure of O. By the identification of the complexification of O with the cotangent space of O induced by Mostow's Decomposition Theorem, this leads to the existence of a hyperkaehler structure on the cotangent space of O compatible with Liouville's symplectic form and whose restriction to the zero…
Let mathcal{O}_lambda be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of mathcal{O}_lambda by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of \mathcal{O}_lambda and it makes explicit the dependence on λand a parameter in Lie(T)^*.
The topology of the embedding of the coadjoint orbits of the unitary group U(H) of an in-finite dimensional complex Hilbert space H, as canonically determined subsets of the B-space T_s of symmetric trace class operators, is investigated. The space T_s is identified with the B-space predual of the Lie-algebra L(H)_s of the Lie group U(H). It is proved, that orbits con-sisting of symmetric operators with finite rank are (regularly embedded) closed submanifolds of T_s. An alternative method of proving this fact is given for the `one-dimensional' orbit, i.e. for the projective Hilbert space P(H).…
For any coadjoint orbit $G/L$, we determine all useful terms of the associated Savelyev-Seidel morphism defined on $H_{-*}(ΩG)$. Immediate consequences are: (1) the dimension of the kernel of the natural map $π_*(G)\otimes \mathbb{Q}\rightarrow π_*(Ham(G/L))\otimes \mathbb{Q}$ is at most the semi-simple rank of $L$, and (2) the Bott-Samelson cycles in $ΩG$ which correspond to Peterson elements are solutions to the min-max problem for Hofer's max-length functional on $ΩHam(G/L)$. The proof is based on Bae-Chow-Leung's recent computation of Ma'u-Wehrheim-Woodward morphism for the moment correspo…
We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p^+)+k+(p^-) where p^+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. Af…
