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In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety X , with values in the space of Picard Lie algebroid over X . As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition.

Considering the Poincaré group ISO(d−1,1) in any dimension d>3, we characterise the coadjoint orbits that are associated with massive and massless particles of discrete spin. We also comment on how our analysis extends to the case of continuous spin.

Let GG be a finitely generated nilpotent group. The representation zeta function ζG(s)\\zeta _G(s) of GG enumerates twist isoclasses of finite-dimensional irreducible complex representations of GG. We prove that ζG(s)\\zeta _G(s) has rational abscissa of convergence α(G)\\alpha (G) and may be meromorphically continued to the left of α(G)\\alpha (G) and that, on the line {s∈C∣Re(s)=α(G)}\\{s\\in \\mathbb {C} \\mid \\mathrm {Re}(s) = \\alpha (G)\\}, the continued function is holomorphic except for a pole at s=α(G)s=\\alpha (G). A Tauberian theorem yields a precise asymptotic result on the representation growth of GG in terms of the position and order of this pole. We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form G(O)\\mathbf {G}(\\mathcal {O}), where G\\mathbf {G} is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring O\\mathcal {O} of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of G\\mathbf {G}, independent of O\\mathcal {O}.

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra${\\mathcal J}$ , we exploit the generalized distribution determined by the Jordan product on the dual${\\mathcal J}^{\\star}$to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of${\\mathcal J}$ , in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in${\\mathcal J}^{*}$lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on${\\mathcal J}$ , the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of${\\mathcal J}$ , it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

The Planet Nine hypothesis encompasses a body of about 5–8 Earth’s masses whose orbital plane would be inclined to the ecliptic by one or two tens of degrees and whose perihelion distance would be as large as about 240–385 astronomical units. Recently, a couple of his epigones have appeared: Planet X and Planet Y. The former is similar to a minor version of Planet Nine in that all its physical and orbital parameters would be smaller. Instead, the latter would have a mass ranging from that of Mercury to Earth’s and a semimajor axis within 100–200 astronomical units. By using realistic upper bounds for the orbital precessions of Saturn, one can obtain insights on their position which, for Planet Nine, appears approximately confined around its aphelion. Planet Y can only be a Mercury-sized object at no less than about 125 astronomical units, while Planet X appears to be ruled out. Dedicated data reductions by modeling such perturber(s) are required to check the present conclusions, to be intended as hints of what might be detectable should planetary ephemerides include them. A probe on the same route of Voyager 1 would be perturbed by Planet Nine by about 20–40 km after some decades.

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in the space of solutions of linear partial differential equations with a set of noncommuting symmetry operators. The approach is implemented for the generalized nonlinear Schrödinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of the noncommutative reduction of the nonstationary nonlinear Schrödinger equation on the motion group E(2) of the two-dimensional plane R2. In this particular case, we come to the usual (1+1)-dimensional nonlinear Schrödinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrödinger equation on the four-dimensional exponential solvable group.

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman’s theorem is intimately related to the representation theory of the quantum group Uqsl2, in the so-called crystal regime q→0. On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit r→∞ of a Brownian motion on the hyperbolic space H3=SL2(C)/SU2. This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation Uqħsl2 of the Jimbo–Drinfeld quantum group which isolates the role of curvature r and that of the Planck constant ħ. The simple relationship between parameters is q=e-r. The semi-classical limits ħ→0 are the Poisson–Lie groups dual to SL2(C) with varying curvatures r∈R+. We also construct classical and quantum random walks, drawing a full picture which includes Biane’s quantum walks and the construction of Bougerol–Jeulin. Taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory (ħ>0) and to the Bougerol–Jeulin construction in the classical world (ħ=0). All these results are neatly in accordance with the philosophy of Kirillov’s orbit method.