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We consider the moment map m: ℙVn → iu(n) for the action of GL(n) on Vn = ⊗2(ℂn)* ⊗ ℂn, and study the critical points of the functional Fn = ∥m∥2: ℙVn → ℝ. Firstly, we prove that [μ] ∈ ℙVn is a critical point if and only if Mμ = cμI + Dμ for some cμ ∈ ℝ and Dμ ∈ Der(μ), where m ([ μ ]) = M μ ∥ μ ∥ 2 . Then we show that any algebra μ admits a Nikolayevsky derivation ϕμ which is unique up to automorphism, and if moreover, [μ] is a critical point of Fn, then ϕ μ = − 1 c μ D μ . Secondly, we characterize the maxima and minima of the functional F n : A n → R , where A n denotes the projectivization of the algebraic varieties of all the n-dimensional associative algebras. Furthermore, for an arbitrary critical point [μ] of F n : A n → R , we obtain a description of the algebraic structure of μ. Finally, we classify the critical points of F n : A n → R for n = 2 and n = 3, respectively. [ABSTRACT FROM AUTHOR]

We consider the moment map m : ℙ V n → i (n) for the action of GL (n) on V n = ⊗ 2 (ℂ n) ∗ ⊗ ℂ n , and study the functional F n = ∥ m ∥ 2 restricted to the projectivizations of the algebraic varieties of all n -dimensional Leibniz algebras L n and all n -dimensional symmetric Leibniz algebras S n , respectively. First, we give a description of the maxima and minima of the functional F n : L n → ℝ , proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point [ μ ] of F n : S n → ℝ , we characterize the structure of [ μ ] by virtue of the nonnegative rationality. Finally, we classify the critical points of F n : S n → ℝ for n = 2 , 3 , respectively. [ABSTRACT FROM AUTHOR]

Let $\mathbf {D}$ be a bounded homogeneous domain in ${\mathbb {C}}^n$. In this note, we give a characterization of the Stein domains in $\mathbf {D}$ which are invariant under a maximal unipotent subgroup N of $Aut(\mathbf {D})$. We also exhibit an N -invariant potential of the Bergman metric of $\mathbf {D}$ , expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case. [ABSTRACT FROM AUTHOR]

Let T m be the maximal torus of a set of m × m unitary diagonal matrices. Let T be a collection of all maps that rigidly rotate every circle of latitude of the sphere with a fixed angle. T is also a maximal torus, and we shall prove in this paper that T is the topological limit inf of T m . [ABSTRACT FROM AUTHOR]

We study the variety of complex n -dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras. [ABSTRACT FROM AUTHOR]

Let G be a compact connected Lie group, (X , ω , μ) a Hamiltonian G-manifold with moment map μ , and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices (A , u) near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface Σ ˚ , and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on Σ ˚ ≅ [ 0 , + ∞) × S 1 is finite, then the A-twisted differential d A u (r , θ) is uniformly bounded for all (r , θ) ∈ [ 0 , + ∞) × S 1 . [ABSTRACT FROM AUTHOR]

For B n the n-dimensional unit ball and D n its Siegel unbounded realization, we consider Toeplitz operators acting on weighted Bergman spaces with symbols invariant under the actions of the maximal Abelian subgroups of biholomorphisms T n (quasi-elliptic) and T n × R + (quasi-hyperbolic). Using geometric symplectic tools (Hamiltonian actions and moment maps) we obtain simple diagonalizing spectral integral formulas for such kinds of operators. Some consequences show how powerful the use of our differential geometric methods are. [ABSTRACT FROM AUTHOR]

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags. [ABSTRACT FROM AUTHOR]

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface M in a complex Calabi–Yau manifold (that is, a complex n -manifold with a nowhere-vanishing holomorphic n -form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in R n . We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces M bounding compact domains Ω ⊂ Z. That is, we study critical points of the volume functional A (M) where the ordinary volume V (Ω) is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere S 2 n − 1 ⊂ C n does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in "most" directions but non-negative in directions corresponding to deformations of S 2 n − 1 by holomorphic diffeomorphisms. We are led to conjecture a "minimax" characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case n = 3 and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in Z using only a closed "pseudoconvex" real 3 -form on M. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact 3 -forms on M and the moment map for the action of the diffeomorphisms of M. [ABSTRACT FROM AUTHOR]

We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu's bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian H will give rise to the non-vanishing Gromov–Witten invariant. [ABSTRACT FROM AUTHOR]

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on T ∗ G u / K 0 . We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of T ∗ G u / K 0 . As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2). [ABSTRACT FROM AUTHOR]

For any natural numbers k and l < k, it is defined the action of the algebraic torus (C∗)k on CPN-1, where N = ..., which is given as the l-symmetric power representation of (C∗)k in (C∗)N and the standard action of (C∗)N on CPN-1. It is well known question about description of toric manifolds arising from this action. In this note we solve this problem for k = 2n and l = n, that is we describe the orbits of this action and their closures, that is the corresponding toric manifolds. In this context we also discuss the (C*)2n-action on the complex Grassmann manifolds G2n,n by the Pl¨ucker embedding G2n,n → CPN-1, where N = .... The explicit expressions of torus orbit closures we obtain can be further used in description of singularities of toric varieties. [ABSTRACT FROM AUTHOR]

For any natural numbers k and l < k, it is defined the action of the algebraic torus (C∗)k on CPN-1, where N = ..., which is given as the l-symmetric power representation of (C∗)k in (C∗)N and the standard action of (C∗)N on CPN-1. It is well known question about description of toric manifolds arising from this action. In this note we solve this problem for k = 2n and l = n, that is we describe the orbits of this action and their closures, that is the corresponding toric manifolds. In this context we also discuss the (C*)2n-action on the complex Grassmann manifolds G2n,n by the Pl¨ucker embedding G2n,n → CPN-1, where N = .... The explicit expressions of torus orbit closures we obtain can be further used in description of singularities of toric varieties. [ABSTRACT FROM AUTHOR]

Let X be a compact connected orientable Cauchy–Riemann (CR) manifold with the action of a compact Lie group G. Under natural pseudoconvexity assumptions we show that the CR Guillemin–Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of 0 by the momentum map. We also show that "quantization commutes with reduction" for Sasakian manifolds. [ABSTRACT FROM AUTHOR]

Both statistical phase space (SPS), which is Γ = T * R 3 N of N -body particle system, and kinetic theory phase space (KTPS), which is the cotangent bundle T * P (Γ) of the probability space P (Γ) thereon, carry canonical symplectic structures. Starting from this first principle, we provide a canonical derivation of thermodynamic phase space (TPS) of nonequilibrium thermodynamics as a contact manifold in two steps. First, regarding the collective observation of observables in SPS as a moment map defined on KTPS, we apply the Marsden–Weinstein reduction and obtain a mesoscopic phase space in between KTPS and TPS as a (infinite-dimensional) symplectic fibration. Then we show that the reduced relative information entropy defines a generating function that provides a covariant construction of a thermodynamic equilibrium as a Legen-drian submanifold. This Legendrian submanifold is not necessarily graph-like. We interpret the Maxwell construction of equal-area law as the procedure of finding a continuous, not necessarily differentiable, thermodynamic potential and explain the associated phase transition by identifying the procedure with that of finding a graph selector in symplecto-contact geometry and in the Aubry-Mather theory of dynamical system. [ABSTRACT FROM AUTHOR]

In this paper, we obtain explicit branching laws for all irreducible unitary representations of G=\operatorname {Spin}(N,1) when restricted to a parabolic subgroup P. The restriction turns out to be a finite direct sum of irreducible unitary representations of P. We also verify Duflo's conjecture for the branching laws of discrete series representations of G with respect to P. That is to show: in the framework of the orbit method, the branching law of a discrete series representation is determined by some geometric behavior of the moment map for the corresponding coadjoint orbit. [ABSTRACT FROM AUTHOR]

In this article, we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold. [ABSTRACT FROM AUTHOR]

We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of R 4 × R 4 phase space. [ABSTRACT FROM AUTHOR]

Suppose given a Hamiltonian and holomorphic action of G = U (2) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit O for G, under transversality assumptions we shall consider two naturally associated 'conic' reductions. One, which will be denoted M ¯ O G , is taken with respect to the action of G and the cone over O ; another, which will be denoted M ¯ ν T , is taken with respect to the action of the standard maximal torus T ⩽ G and the ray R + ı ν along which the cone over O intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit. [ABSTRACT FROM AUTHOR]

Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

The Stark problem is a completely integrable system which describes the motion of an electron in a constant electric field and subject to the attraction of a proton. In this paper we show that in the planar case after Levi-Civita regularization the bounded component of the energy hypersurfaces of the Stark problem for energies below the critical value can be interpreted as boundaries of concave toric domains. [ABSTRACT FROM AUTHOR]

A new way of describing weighted poly-Bergman spaces and true weighted poly-Bergman spaces on the upper half-plane Π emerges using a system of coordinates induced by the R + -action on Π , the moment map associated with this action, and the standard symplectic structure. We give an isomorphic description of such spaces using Romanovski polynomials. We also provide a unitary description of Toeplitz operators on these spaces whose symbols are associated with the moment map. [ABSTRACT FROM AUTHOR]

This article surveys the results in three recent papers by the present authors, which study the metric, differential and projective geometry of the Poincaré half space and the Poincaré disk of a C*-algebra. [ABSTRACT FROM AUTHOR]

The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration A of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of A . For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope. [ABSTRACT FROM AUTHOR]

Let (X , T 1 , 0 X) be a compact connected orientable CR manifold of dimension 2 n + 1 with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group G action. Under certain natural assumptions about the group G action, we define G-equivariant Szegő kernels and establish the associated Boutet de Monvel–Sjöstrand type theorems. When X admits also a transversal CR S 1 action, we study the asymptotics of Fourier components of G-equivariant Szegő kernels with respect to the S 1 action. [ABSTRACT FROM AUTHOR]

Suppose that a compact and connected Lie group G acts on a complex Hodge manifold M in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle A on M. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map. [ABSTRACT FROM AUTHOR]

In this note, we study in a finite dimensional Lie algebra g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone C x . Assuming that g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying [ h , x ] = x for which C x pointed. Given x, we show that such elements h can be constructed in such a way that ad h defines a 5-grading, and characterize the cases where we even get a 3-grading. [ABSTRACT FROM AUTHOR]

We compare existence and equivariance phenomena for weak moment maps and homotopy moment maps in multisymplectic geometry. [ABSTRACT FROM AUTHOR]

Let (X , T 1 , 0 X) be a compact connected orientable strongly pseudoconvex CR manifold of dimension 2 n + 1 , n ≥ 1 . Assume that X admits a connected compact Lie group G action and a transversal CR S 1 action, we compute the coefficients of the first two lower-order terms of the equivariant Szegő kernel asymptotic expansions with respect to the S 1 action. [ABSTRACT FROM AUTHOR]

The aim of this paper is to explain how to parametrize the equations of the facets of Kirwan polytopes using the notion of Ressayre's pairs. [ABSTRACT FROM AUTHOR]

We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia [5] which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation. [ABSTRACT FROM AUTHOR]

Associated with a Clifford system on ℝ2l, a Spin(m + 1) action is induced on ℝ2l. An isoparametric hypersurface N in S2l−1 of OT-FKM (Ozeki, Takeuchi, Ferus, Karcher and Münzner) type is invariant under this action, and so is the Cartan-Münzner polynomial F(x). This action is extended to a Hamiltonian action on ℂ2l. We give a new description of F(x) by the moment map μ : ℂ 2 l → k * , where k ≅ o (m + 1) is the Lie algebra of Spin(m + 1). It also induces a Hamiltonian action on ℂP2l−1. We consider the Gauss map G of N into the complex hyperquadric ℚ2l−2(ℂ) ⊂ ℂP2l−1, and show that G(N) lies in the zero level set of the moment map restricted to ℚ2l−2(ℂ). [ABSTRACT FROM AUTHOR]

The moment map is a mathematical expression of the concept of the conservation associated with the symmetries of a Hamiltonian system. By considering topological properties of the moment map we are lead to the definition of the abstract moment map. This map is defined from G-manifold M to dual Lie algebra of G. We will interested study maps from G-manifold M to spaces that are more general than dual Lie algebra of G. These maps help us to reduce the dimension of a manifold much more. [ABSTRACT FROM AUTHOR]

If ω is a closed G-invariant 2-form and μ is a moment map, we obtain necessary and sufficient conditions for equivariant prequantizability that can be computed in terms of the moment map μ. Our main result is that G-equivariant prequantizability is related to the fact that the moment map μ should be quantized for certain vectors on the Lie algebra of G. We also compute the obstructions to lift the action of G to a prequantization bundle of ω. Our results are valid for any compact and connected Lie group G. [ABSTRACT FROM AUTHOR]

This is the second of the series of papers on the classification of six-dimensional closed monotone symplectic manifold admitting a semifree Hamiltonian S 1 -action. In [Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Int. J. Math.6 1950032], we dealt with the case where at least one of the extremal fixed point is isolated and proved that every such manifold is Kähler Fano. In this paper, we show that if the maximal and the minimal fixed components are both two-dimensional, then the manifold is S 1 -equivariantly symplectomorphic to some Kähler Fano manifold with a certain holomorphic Hamiltonian S 1 -action. We also give a complete list of such Fano manifolds together with an explicit description of the S 1 -actions. [ABSTRACT FROM AUTHOR]

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags. [ABSTRACT FROM AUTHOR]

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations—the generalised Monge–Ampère equation, the almost Hitchin system, and the Calabi–Yang–Mills equations. These are all perturbations of already existing equations. Our construction for the generalised Monge–Ampère equation is conditioned on a conjecture from algebraic geometry. In addition, we prove that for small values of the perturbation parameters, some of these equations have solutions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra g , we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of S (g) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in g ≃ g ∗ . [ABSTRACT FROM AUTHOR]