Narrow Results

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 $𝔤$. Let $\pi$ be an irreducible unitary representation of $G$ in a Hilbert space ${{\mathscr{H}}}_{\pi }$, $f$ be a linear form on $𝔤$ and $𝔥$ be a polarization at $f$. We denote by ${\chi }_f$ a character of $H=exp(𝔥)$ related to $f$. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors $({{\mathscr{H}}}_{\pi }^{-\infty }{)}^{H,{\chi }_f}$ of $\pi$.

We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M, with volume form μ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M. Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang–Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of μ. The magnetic field evolves by a quadratically nonlinear "Euler" equation, which may also be regarded as describing geodesic flow on SDiff(M, μ). Static solutions are obtained. For uniform μ, an infinite sequence of local conserved charges beginning with the Hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang–Mills theory and matrix field theories, whose gauge-invariant phase-space is conjectured to be a coadjoint orbit of the diffeomorphism group of a noncommutative space.

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.

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p - 1 (in which case we only obtain a conditional statement) in the dual space W* under the induced action of G. As a corollary we prove that the algebra of invariants k[W*]^G is trivial.

Let G be a Lie group, H a closed subgroup and M the homogeneous space G/H. Each representation Ψ of H determines a G-equivariant principal bundle on M endowed with a G-invariant connection. We consider subgroups of the diffeomorphism group Diff(M), such that, each vector field admits a lift to a preserving connection vector field on . We prove that . This relation is applicable to subgroups of the Hamiltonian groups of the flag varieties of a semisimple group G. Let M_Δ be the toric manifold determined by the Delzant polytope Δ. We put φ_b for the loop in the Hamiltonian group of M_Δ defined by the lattice vector b. We give a sufficient condition, in terms of the mass center of Δ, for the loops φ_b and to be homotopically inequivalent.

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.

Let ${ℍ}_n$ be the $(2n+1)$-dimensional Heisenberg group and let ${𝕋}^n$ be the $n$-dimensional torus acting on ${ℍ}_n$ by automorphisms. We consider the semidirect product group $G_n:={𝕋}^n⋉{ℍ}_n.$ The cortex, $\mathrm{\text{cor}}(G_n),$ of $G_n$ is the set of all unitary irreducible representations $\pi$ in the unitary dual $\widehat{G_n}$ 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 ($\mathrm{\text{cor}}(G_n)$) of $G_n$ using the coadjoint orbits of the group.

We study the transverse Poisson structures to adjoint orbits in complex simple Lie algebras with special emphasis on the subregular and minimal orbits.