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Let G be a complex simple Lie group, and let U ⊆ G be a maximal compact subgroup. Assume that G admits a homogenous space X = G/Q = U/K which is a compact Hermitian symmetric space. Let L ? X be the ample line bundle which generates the Picard group of X. In this paper we study the restrictions to K of the family (H0(X,Lk))k∈N of irreducible Grepresentations. We explicitly describe the moment polytopes for the moment maps X ? ... associated to positive integer multiples of the Kostant-Kirillov symplectic form on X, and we use these, together with an explicit characterization of the closed KC-orbits on X, to find the decompositions of the spaces H0(X,Lk). We also construct a natural Okounkov body for L and the Kaction, and we identify it with the smallest of the moment polytopes above. In particular, the Okounkov body is a convex polytope. In fact, we even prove the stronger property that the semigroup defining the Okounkov body is finitely generated [ABSTRACT FROM AUTHOR]

We consider the integrable system with three degrees of freedom for which V. V. Sokolov and A. V. Tsiganov specified the Lax pair. The Lax representation generalizes the L-A pair found by A. G. Reyman and M. A. Semenov-Tian-Shansky for the Kovalevskaya gyrostat in a double field. We give explicit formulas for the additional first integrals K and G (independent almost everywhere), which are functionally related to the coefficients of the spectral curve for the Sokolov-Tsiganov L-A pair. Using this form of the additional integrals K and G and the Kharlamov parametric reduction, we analytically present two invariant four-dimensional submanifolds where the induced dynamical system is Hamiltonian (almost everywhere) with two degrees of freedom. The system of equations specifying one of the invariant submanifolds is a generalization of the invariant relations for the integrable Bogoyavlensky case (rotation of a magnetized rigid body in homogeneous gravitational and magnetic fields). We use the method of critical subsystems to describe the phase topology of the whole system. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of the Liouville tori both inside the subsystems and in the whole system. [ABSTRACT FROM AUTHOR]

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples. [ABSTRACT FROM AUTHOR]

Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π1(M) ≅ π1(M/G); if additionally ϕ is proper, then π1(M) ≅ π1 (ϕ-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π1(M) ≅ π1(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π1(M) ≅ π1(ϕ-1(G⋅a)) ≅ π1(ϕ-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then for all a ∈ ϕ(M), where is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π1(M/G) ≅ π1(ϕ-1(G⋅a)/G) for all a ∈ ϕ(M). [ABSTRACT FROM AUTHOR]

Abstract: In this article we verify Dufloʼs conjecture for with respect to a Borel subgroup. It turns out that the non-standard notion weakly proper (faiblement propre) in the formulation of Dufloʼs conjecture is more reasonable than the standard notions proper and proper over image. [Copyright &y& Elsevier]

We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple ( M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold ( M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric. [ABSTRACT FROM AUTHOR]

We give a new formula which writes down the moment map for holomorphic isometric action on a Kähler manifold using the motion of a charged particle when the action of is Hamiltonian. [ABSTRACT FROM AUTHOR]

Consider an effective Hamiltonian circle action on a compact symplectic 2n-dimensional manifold (M, ω). Assume that the fixed set MS1 is minimal, in two senses: It has exactly two components, X and Y, and dim(X) + dim(Y) = dim(M) - 2. We prove that the integral cohomology ring and Chern classes of M are isomorphic to either those of ℂℙn or (if n ≠ 1 is odd) to those of ${\Tilde{G}}_2({\mathbb R}^{n+2})$, the Grassmannian of oriented two-planes in ℝn+2. In particular, Hi(M;ℤ) = Hi(ℂℙn; ℤ) for all i, and the Chern classes of M are determined by the integral cohomology ring. We also prove that the fixed set data of M agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer ℤk for any k > 2. The same conclusions hold when MS1 has exactly two components and the even Betti numbers of M are minimal, that is, b2i(M) = 1 for all i ∈ {0, ..., ½dim(M)}. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions. [ABSTRACT FROM AUTHOR]

Going back to Kirwan and others, there is an established theory that uses moment map techniques to study actions by complex reductive groups on Kähler manifolds. Work of P. Heinzner, G. Schwarz, and H. Stötzel has extended this theory to actions by real reductive groups. In this paper, we apply these techniques to actions of the real group SU(1,1) ⊂ SL(2, ℂ) on a certain complex manifold of dimension two. More precisely, because of the SU(2)-invariant hyperkähler structure on this manifold, we are able to study a family of actions which includes and 'interpolates' two well-known actions of SL(2, ℂ): the adjoint action on the orbit of a semisimple element of 픰픩(2, ℂ), and the action of SL(2, ℂ) on the cotangent bundle of the flag variety of SL(2, ℂ). [ABSTRACT FROM AUTHOR]

Abstract: We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in preserved by a two-torus symmetry in terms of tri-symplectic geometry of four-manifolds. [Copyright &y& Elsevier]

A relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini–Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a real compact connected simple Lie group, and its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in , or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in and products of conjugacy classes in contain an open set, in terms of the highest multiplicities of eigenvalues. For and we show -singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in . [Copyright &y& Elsevier]

Abstract: Any unitary irreducible representation π of a Lie group G defines a moment set , subset of the dual of the Lie algebra of G. Unfortunately, does not characterize π. If G is exponential, there exists an overgroup of G, built using real-analytic functions on , and extensions of any generic representation π to such that characterizes π. In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions. [Copyright &y& Elsevier]

Abstract: A way to separate irreducible unitary representations for a Lie group by moment sets is to use an infinite-dimensional overgroup and extensions of each representation to a representation of , in such a manner that the moment set of characterizes . In this paper we propose a universal overgroup , which is an infinite-dimensional Fréchet–Lie group. We extend each to a Hamiltonian action of . The moment set of characterizes . [ABSTRACT FROM AUTHOR]

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal. [ABSTRACT FROM AUTHOR]

A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux’s example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed. [ABSTRACT FROM AUTHOR]

We consider a connected symplectic manifold M acted on properly and in a Hamiltonian fashion by a connected Lie group G. If G is compact, then we characterize the symplectic manifolds whose squared moment map is constant. We also give a sufficient condition for G to admit a symplectic orbit. Then we study the case when G is a non-compact Lie group proving splitting results for symplectic manifolds. [ABSTRACT FROM AUTHOR]

Abstract: Let be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group . Let be the symplectic quotient at value 0 of the moment map . The space may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over and the -invariant subspace of the quantum Hilbert space over . In this paper, without any regularity assumption on the quotient , we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck’s constant of a modified map of the above isomorphism under a “metaplectic correction” of the two quantum Hilbert spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we construct complete Hamiltonian minimal submanifolds in complex hyperbolic space with -symmetry which are not minimal in the usual sense. [Copyright &y& Elsevier]

The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equivariant version of this problem, in the presence of a linear action of a reductive group on M. [ABSTRACT FROM AUTHOR]

Abstract: We study invariants on symplectic representations of a connected reductive group. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration over an affine space with rational generic fibers. Of particular interest are representations where these fibers are points. We show that they are cofree. Our main tool is a symplectic version of the local structure theorem. [Copyright &y& Elsevier]

We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equation-free multiscale algorithms for these systems. The moment map itself is deterministic and attempts to capture the implied probability distribution of the dynamics. By choosing situations where the lowdimensional dynamics can be understood a priori, we evaluate the moment map. Despite requiring the evolution of an ensemble to define the map, this can be an efficient numerical tool, as the map opens up the possibility of bifurcation analyses and other high level tasks being performed on the system. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath. [ABSTRACT FROM AUTHOR]

We collect some properties of the moment map relative to the isometric and holomorphic action of a compact Lie group G on a (compact) Kähler (Einstein) manifold; in particular, we study some invariants which only depend on the cohomology class of the invariant Kähler form and then specialize to the complex projective space when the group G is simple and acts linearly. [ABSTRACT FROM AUTHOR]

Abstract: We study the flatness for the moment map associated to the cotangent bundle of the space of representations of a quiver Q. If the associated moment map of a root is flat, then we call the root a flat root. We first study the flat roots in the fundamental set of a quiver Q. Then we give an explicit description of all the flat roots of Q. This description is obtained by using the natural class of -reflections, which are introduced in this paper. We also show that there are only a finite number of flat roots in each orbit under the Weyl-group of the quiver Q. [Copyright &y& Elsevier]

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin showed that the moduli space M of solutions of the self-duality equations on a compact Riemann surface of genus g > 1 has a hyper-Kähler structure. In particular M is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which are missing in Hitchin''s paper. Next we apply Quillen''s determinant line bundle construction to show that M admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above. [Copyright &y& Elsevier]

Abstract: Suppose a compact Lie group acts on a polarized complex projective manifold . Under favorable circumstances, the Hilbert–Mumford quotient for the action of the complexified group may be described as a symplectic quotient (or reduction). This paper addresses some metric aspects of this identification, by analyzing the relationship between the Szegö kernel of the pair and that of the quotient. [Copyright &y& Elsevier]