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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]

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]

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]

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]

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]

Abstract: We determine conditions under which two Hamiltonian torus actions on a symplectic manifold are homotopic by a family of Hamiltonian torus actions, when is a toric manifold and when is a coadjoint orbit. [Copyright &y& Elsevier]

Abstract: Let G be the semidirect product where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let be a coadjoint orbit of G associated by the Kirillov–Kostant method of orbits with a unitary irreducible representation π of G. We consider the case when the corresponding little group is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on where . By dequantizing π we then construct a symplectomorphism between the orbit and the product where is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on which is adapted to the representation π in the sense of [B. Cahen, Quantification d''une orbite massive d''un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I 325 (1997) 803–806]. In particular we recover well-known results for the Poincaré group. [Copyright &y& Elsevier]

Abstract: Geometrical optics is extended so as to provide a model for spinning light rays via the coadjoint orbits of the Euclidean group characterized by color and spin. This leads to a theory of “geometrical spinoptics” in refractive media. Symplectic scattering yields generalized Snell–Descartes laws that include the recently discovered optical Hall effect. [Copyright &y& Elsevier]

Abstract: A d-dimensional rational polytope is a polytope whose vertices are located at the nodes of lattice. Consider the number of points inside the inflated with coefficient of inflation . The Ehrhart polynomial of counts the number of such lattice points inside the inflated and (may be) at its faces (including vertices). In Part I [A.L. Kholodenko, New string for old Veneziano amplitudes. I. Analytical treatment, J. Geom. Phys. 55 (2005) 50–74] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sense of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) physical models reproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g. those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on scattering, etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [M. Vergne, Convex polytopes and quanization of symplectic manifolds, Proc. Natl. Acad. Sci. 93 (1996) 14238–14242]. [Copyright &y& Elsevier]

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H, as canonically determined subsets of the space Ts of symmetric trace-class operators, is investigated. The space Ts is identified with the B-space predual of the Lie-algebra L(H)s of the Lie group U. It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of Ts. Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics.An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P(H). Closeness of all the orbits lying in Ts is also proved. [Copyright &y& Elsevier]

The asymptotics of characters χ k λ (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 L G ˆ and of diffeomorphisms of the circle Diff ˆ (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 contribution which can be interpreted as a formal DH orbital integral. For some Virasoro modules, our results match formal DH integrals recently computed by Stanford and Witten. In this case, the k -scaling has the same origin as the one which gives rise to classical conformal blocks. Furthermore, we consider reduced spaces of Virasoro coadjoint orbits and we suggest a new invariant which replaces symplectic volume in the infinite dimensional situation. We also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and we obtain DH-type expressions which do not correspond to any Virasoro coadjoint orbits. We introduce volume functions V (x) corresponding to formal DH integrals over coadjoint orbits of the Virasoro algebra and show that they are related by the Hankel transform to spectral densities ρ (E) recently studied by Saad, Shenker and Stanford. [ABSTRACT FROM AUTHOR]