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Inspired by Kirillov's orbital method, we propose a conjecture (Conjecture 1.7), which suggests an explicit and exhaustive construction of characters for finite pattern groups via coadjoint orbits. First, we establish the construction for irreducible characters of degree q for finite pattern groups. Second, we verify the conjecture for the case of finite pattern groups contained in U 4 (F q) , where U n stands for the full unitriangular group of rank n. Finally, we classify irreducible characters for G n (F q) , a generalization of Heisenberg group, where G n is the pattern subgroup of U n , associated to the closed set { (1 , m) , (k , n) , (i , n + 1 − i) | 1 < m , k < n , 1 ⩽ i ⩽ [ n 2 ] } , 3 ⩽ n ∈ N. Hence Conjecture 1.7 and the analogous conjectures of Higman, Lehrer and Isaacs holds for G n (F q). [ABSTRACT FROM AUTHOR]

We identify pointed vortex loops in the plane with low dimensional nonlinear flags decorated with volume forms. We show how submanifolds of such decorated nonlinear flags can be identified with coadjoint orbits of the area pre- serving diffeomorphism group and relate them to coadjoint orbits of pointed vortex loops. The subgroup of the dihedral group preserving the vorticity data plays a role in the description of these coadjoint orbits.

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described. [ABSTRACT FROM AUTHOR]

Let G be a unipotent algebraic group defined over a p -adic field of characteristic zero. We denote by G the set of rational points of G. It is a p -adic Lie group with Lie algebra denoted by . Let π be an irreducible unitary representation of G in a Hilbert space ℋ π , f be a linear form on and be a polarization at f. We denote by χ f a character of H = exp () related to f. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors (ℋ π − ∞ ) H , χ f of  π. [ABSTRACT FROM AUTHOR]

Let G = 1 + A be a finite pattern group, where A is the associated pattern algebra over a finite field F q. For any monomial matrix T in A t , we explicitly construct an irreducible character Φ T of G , corresponding to T. Furthermore, if A is symmetric (in the sense of Definition 3.2), we can determine the largest character degree of 1 + A using this construction. We also propose a conjecture that an explicit one-to-one correspondence between irreducible representations and coadjoint orbits exists for any finite pattern groups. [ABSTRACT FROM AUTHOR]

We use the underlying structure of then coadjoint orbits of a semidirect product of a connected Lie group and a vector space to obtain families of Lagrangian submanifolds in the adjoint orbits of complex semisimple Lie groups with respect to the symplectic hermitian form. This construction is a generalization of a type of semi-direct orbit previously studied by the authors.

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We give a classification of irreducible representations of generalized Heisenberg groups K n ( F q) , n ≥ 5 , which is the pattern group associated to the closed set {(1, i), (2, j), (s, n − 1), (t, n) | 2 ≤ i ≤ n, 3 ≤ j ≤ n, 3 ≤ s < n − 1, 3 ≤ t < n}. In light of the conjectures of Higman, Lehrer and Isaacs for unitriangular groups, this result shows that the number of irreducible characters of K n ( F q) with a fixed degree is a polynomial in q−1 with non-negative integer coefficients. [ABSTRACT FROM AUTHOR]

In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds. [ABSTRACT FROM AUTHOR]

Let F be a nonarchimedean local field of characteristic zero and G be a unipotent algebraic group defined over F. The set of rational points of G , denoted by G , is a p -adic Lie group. Let g be the Lie algebra of G. Now let H be a normal closed subgroup of G , χ be a unitary character of H and π be an irreducible unitary representation of G in a Hilbert space H π. The aim of this paper is the determination of the space formed by the χ -semi-invariant vectors of π. [ABSTRACT FROM AUTHOR]

ABSTRAK Dalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari melalui orbit coadjoint buka di titik tertentu pada ruang dual dari aljabar Lie . Aksi dari grup Lie pada ruang dual menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine tepat mempunyai dua buah orbit coadjoint buka. Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine berdimensi dan untuk kasus grup Lie lainnya. ABSTRACT In this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

Do co-adjoint orbits of Lie groups support a Kähler structure? We study this question from a point of view derived from coherent states. We examine three examples of Lie groups: the Weyl–Heisenberg group, SU(2) and SU(1, 1). In cases, where the orbits admit a Kähler structure, we show that coherent states give us a Kähler embedding of the orbit into projective Hilbert space. In contrast, squeezed states (which like coherent states, also saturate the uncertainty bound) only give us a symplectic embedding. We also study geometric quantisation of the co-adjoint orbits of the group SUT(2, ℝ) of real, special, upper triangular matrices in two dimensions. We glean some general insights from these examples. Our presentation is semi-expository and accessible to physicists. [ABSTRACT FROM AUTHOR]

Let G n : = U (n) ⋉ H n be the semidirect product of the unitary group acting by automorphisms on the Heisenberg group H n. It was pointed out in [5], that the unitary dual G n ^ of G n is homeomorphic to the space of admissible coadjoint orbits g n ‡ / G n of G n. One of the important subset of G n ^ is what is called the cortex of G n , cor (G n) , which is the set of all π ∈ G n ^ that cannot be Hausdorff separated from the identity representation 1 G n of G n. In the present paper, we use the orbit method of Lipsman to determine the cortex of G n. [ABSTRACT FROM AUTHOR]

This paper gives explicit constructions of all irreducible representations of unipotent radicals N n 1 , n 2 , n 3 (F q) of the standard parabolic subgroups P n 1 , n 2 , n 3 (F q) of GL n (F q) , corresponding to the ordered partition (n 1 , n 2 , n 3) of n. The construction gives a bijection between coadjoint orbits of N n 1 , n 2 , n 3 (F q) and irreducible representations inducing from degree 1 characters in the sense of Boyarchenko's construction. The result shows that the number of irreducible characters of N n 1 , n 2 , n 3 (F q) with a fixed degree is a polynomial in q - 1 with nonnegative integer coefficients and verifies analogue conjectures of Higman, Lehrer, and Isaacs. [ABSTRACT FROM AUTHOR]

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. [ABSTRACT FROM AUTHOR]

For every semisimple coadjoint orbit O of a complex connected semisimple Lie group yG, we obtain a family of g -invariant products *h on the space of holomorphic functions on O. For every semisimple coadjoint orbit O of a real connected semisimple Lie group G, we obtain a family of G-invariant products *h on a space A(O) of certain analytic functions on O by restriction. A(O), endowed with one of the products „, is a G-Fréchet algebra, and the formal expansion of the products around h = 0 determines a formal deformation quantization of O, which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions. [ABSTRACT FROM AUTHOR]

Let ℍ n be the (2 n + 1) -dimensional Heisenberg group and let n be the n -dimensional torus acting on ℍ n by automorphisms. We consider the semidirect product group G n : = n ⋉ ℍ n. The cortex, cor (G n) , of G n is the set of all unitary irreducible representations π in the unitary dual G n ̂ of G n that cannot be Hausdorff separated from the identity representation 1 G n of G n. In this paper, we describe explicitly the cortex (cor (G n)) of G n using the coadjoint orbits of the group. [ABSTRACT FROM AUTHOR]

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described. [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]

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]

We describe coadjoint orbits associated to the motion of codimension one singular vorticities in ideal fluids, e.g. vortex sheets in 3D. We show that these coadjoint orbits can be identified with a certain class of decorated nonlinear Grassmannians, that consist of codimension one submanifolds belonging to an isodrast (i.e., enclosing a given volume, provided their homology class vanishes), endowed with a closed 1-form of a given type. Such isodrasts are defined via an integrable distribution associated to a manifold endowed with a volume form, similar to the isodrastic distribution [35] in symplectic setting. With the choice of a Riemannian metric, the expression of the orbit symplectic form is shown to take a particularly simple expression reminiscent of Darboux coordinates. In general we get coadjoint orbits of the universal central extension of the group of exact volume preserving diffeomorphisms and we give a necessary and sufficient condition under which the central extension is not needed. We also focus on the case in which we get coadjoint orbits of the group of volume preserving diffeomorphisms, which is relevant for the ideal fluid, and discuss how our results extend the class of coadjoint orbits of exact vortex sheets considered in [25]. [ABSTRACT FROM AUTHOR]

We prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair due to Holm and Marsden, and the ideal fluid dual pair due to Marsden and Weinstein. [ABSTRACT FROM AUTHOR]

We consider the semidirect product G = K × V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product h, i. We denote by Ĝ the unitary dual of G (note that we identify each representation π ∈ Ĝ to its classes [π]) and by g‡/G the space of admissible coadjoint orbits, where g is the Lie algebra of G. It was pointed out by Lipsman that the correspondence between g‡/G and Ĝ is bijective. Under some assumption on G, we prove that the Lipsman mapping Θ : g‡/G -→ Ĝ O -→ πO is a homeomorphism. [ABSTRACT FROM AUTHOR]

Motivated by the work of Leznov-Mostovoy [17], we classify the linear deformations of standard 2n-dimensional phase space that preserve the obvious symplectic o(n)-symmetry. As a consequence, we describe standard phase space, as well as T∗Sn and T∗Hn with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in Rn+2. [ABSTRACT FROM AUTHOR]

Abstract We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincaré group, and discuss the limitations and extent of this result to other groups. [ABSTRACT FROM AUTHOR]

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits. • We study the Toda flow on Hessenberg elements of real, split simple Lie algebras. • Nontrivial invariants of the maximal compact subgroup K in involution are constructed. • The formula for the number ν of such K-invariants involves only Lie-algebraic data. • Dimension of generic Hessenberg coadjoint orbits = 2 ν. • Toda is integrable on generic Hessenberg orbits for the classical simple Lie algebras. [ABSTRACT FROM AUTHOR]

The relativistic (Poincaré and conformal) symmetries of classical elementary systems are briefly discussed and reviewed. The main framework is provided by the Hamiltonian formalism for dynamical systems exhibiting symmetry described by a given Lie group. The construction of phase space and canonical variables is given using the tools from the coadjoint orbits method. It is indicated how the “exotic” Lorentz transformation properties for particle coordinates can be derived; they are shown to be the natural consequence of the formalism.

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant. [ABSTRACT FROM AUTHOR]

By using a projective connection over the space of two-dimensional affine connections, we are able to show that the metric interaction of Polyakov two-dimensional gravity with a coadjoint element arises naturally through the projective Ricci tensor. Through the curvature invariants of Thomas and Whitehead, we are able to define an action that could describe dynamics to the projective connection. We discuss implications of the projective connection in higher dimensions as related to gravitation. [ABSTRACT FROM AUTHOR]

For a Lie groupoid G with Lie algebroid A , we realize the symplectic leaves of the Lie–Poisson structure on A ∗ as orbits of the affine coadjoint action of the Lie groupoid J G ⋉ T ∗ M on A ∗ , which coincide with the groupoid orbits of the symplectic groupoid T ∗ G over A ∗ . It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang–Mills field. [ABSTRACT FROM AUTHOR]

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie-Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD. [ABSTRACT FROM AUTHOR]