On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ? G or Q ? K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.     

Noise and Dissipation on Coadjoint Orbits

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.     

Adjoint and Coadjoint Orbits of the Poincaré Group

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincaré group.The author (R.C) was partially supported by European Community funding for the Research and Training Network MASIE (HPRN-CT-2000-00113).     

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Separation of Coadjoint Orbits of Generalized Diamond Lie Groups

Mathematical Notes - Let $$G$$ be a type I connected and simply connected generalized diamond Lie group defined as the semidirect product of a $$d$$ -dimensional Abelian Lie group $$N$$ with...     

Unitary Representations and Coadjoint Orbits for a Group of Germs of Real Analytic Diffeomorphisms

The method of coadjoint orbits is developed for the group of real analytic germs of diffeomorphisms φ with φ(0) = 0 and φ′(0) = 1. The form of all infinite dimensional coadjoint orbits is described. Classes U of unitary representations are constructed. In the case n = 2 these representations are related to coadjoint orbits.     

Coadjoint Orbits and Time-Optimal Problems for Step- $$2$$ Free Nilpotent Lie Groups

Mathematical Notes -     

Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem

Doklady Mathematics - 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...     

Symmetric Pairs for Quantized Enveloping Algebras

Let θ be an involution of a semisimple Lie algebra g, let g^θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a quantum analog of U(g^θ) which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to U(g^θ).     

Information Geometry for Symmetric Diffusions

The idea of Amari's information geometry is extended to a certain family of symmetric diffusions. It provides us a method of introducing a Hilbert–Riemannian structure to a given family of probability measures. The Riemannian metric associated with `an infinite-dimensional version of' the Fisher information, the geodesics with respect to a pair of certain `dual' affine connections, and the energies along them are computed.     

Law of Large Numbers for Branching Symmetric Hunt Processes with Measure-Valued Branching Rates

We establish weak and strong laws of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state dependent. Our work is motivated by recent work on the strong law of large numbers for branching symmetric Markov processes by Chen and Shiozawa (J Funct Anal 250:374–399, 2007) and for branching diffusions by Engländer et al. (Ann Inst Henri Poincaré Probab Stat 46:279–298, 2010). Our results can be applied to some interesting examples that are covered by neither of these papers.     

Orbits in Degenerate Compactifications of Symmetric Varieties

Let G be a simply connected semisimple algebraic group and let H ⁰ be the subgroup of points fixed under an involution of G. If V is an irreducible representation with a line L of vectors fixed by H ⁰ we consider the closure of the G-orbit of L in . We describe the G-orbits of this closure and we prove that the normalization of this variety is homeomorphic to the variety itself.     

Symmetric Periodic Orbits and Uniruled Real Liouville Domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.     

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras. Presented by Susan Montgomery.     

On the Geometry of Dual Pairs

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non-zero vector has a positive multiple which is the sum of two v-dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc-connectedness of the numerical range determined by v.     

Symmetric Stable Laws and Stable-Like Jump-Diffusions

Asymptotic expansions are obtained for finite-dimensional symmetricstable distributions. They are used to prove the existence ofcontinuous transition probability densities of stable and stable-likejump-diffusions, and to construct local multiplicative asymptoticsandglobal two-sided estimates for these densities. As a consequence,the distribution of the first passage times for stable jump-diffusionsis estimated and the integral test for the limsup behaviourof their sample paths as t 0 is provided. 1991 MathematicsSubject Classification: 60E07, 60G17, 60J35, 47D07.     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.