Narrow Results

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.

For this, we study the Corwin–Greenleaf multiplicity function for Hermitian symmetric spaces G/K. First, we prove that for any G-coadjoint orbit and any K-coadjoint orbit if . Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.

Second, we find a necessary and sufficient condition for by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions.

(1) The restriction π|_H is discretely decomposable in the sense of Kobayashi.

(2) The momentum map is proper.

In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

Let G_ℂ be a complex simple Lie group, G_U a compact real form, and the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit of G_U, the intersection of with a coadjoint orbit of G_ℂ is either an empty set or a single orbit of G_U if is isomorphic to a complex symmetric space.

Zuckerman’s derived functor module of a semisimple Lie group $G$ yields a unitary representation $\pi$ which may be regarded as a geometric quantization of an elliptic orbit ${𝒪}$ in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations $\pi$ of the indefinite unitary group $G=U(p,q)$ and a family of subgroups $H$ of $G$ such that the restriction $\pi {|}_H$ is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of $A_q(\lambda )$ with respect to reductive subgroups, II, Ann. of Math.147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci.41 (2005) 497–549), where $\pi$ is not necessarily tempered and $H$ is not necessarily compact. We prove that the corresponding moment map $\mu :{𝒪}\to {𝔥}^{\ast }$ is proper, determine the image $\mu ({𝒪})$, and compute the Corwin–Greenleaf multiplicity function explicitly.

Let $K$ be a closed connected subgroup of the unitary group $U(d),d\in {\mathbb{N}}^{*}$ and let ${ℍ}_{{𝒱}}$ be the $(2d+1)$-dimensional Heisenberg group (${𝒱}\simeq {ℂ}^d$). We consider the semidirect product $G=K⋉{ℍ}_{{𝒱}},$ such that $(K,{ℍ}_{{𝒱}})$ is a Gelfand pair. Let $𝔤\supset 𝔨$ be the respective Lie algebras of $G$ and $K$ and $q:{𝔤}^{*}\to {𝔨}^{*}$ be the natural projection. It was pointed out by Lipsman, that the unitary dual $Ĝ$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits ${𝔤}^{‡/G}$ (see [10]). Let $\pi \in Ĝ$ be a generic representation of $G$ and let $\tau \in \widehat{K}$. To these representations we associate, respectively, the admissible coadjoint orbit ${{𝒪}}^G\subset {𝔤}^{*}$ and ${{𝒪}}^K\subset {𝔨}^{*}$ (via the Lipsman’s correspondence). We denote by $\chi ({{𝒪}}^G,{{𝒪}}^K)$ the number of $K$-orbits in ${{𝒪}}^G\cap q^{-1}({{𝒪}}^K),$ which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity $m_{\pi }(\tau )$ of an irreducible $K$-module $\tau$ occurring in the restriction $\pi {|}_K$ could be read from the coadjoint action of $K$ on ${{𝒪}}^G\cap q^{-1}({{𝒪}}^K).$ In this paper, we show that

Let $N$ be a simply connected nilpotent Lie group, and let $K$ be a connected compact subgroup of the automorphism group, $Aut(N),$ of $N.$ Let $G:=K⋉N$ be the semidirect product (of $K$ and $N$). Let $𝔤\supset 𝔨$ be the respective Lie algebras of $G$ and $K$ and $q:{𝔤}^{\ast }\to {𝔨}^{\ast }$ be the natural projection. It was pointed out by Lipsman, that the unitary dual $\widehat{G}$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits ${𝔤}^{‡}/G$ (see [R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl.59 (1980) 337–374]). Let $\pi \in \widehat{G}$ be a generic representation of $G$ and let $\tau \in \widehat{K}.$ To these representations we associate, respectively, the admissible coadjoint orbit ${{𝒪}}^G\subset {𝔤}^{\ast }$ and ${{𝒪}}^K\subset {𝔨}^{\ast }$ (via the Lipsman’s correspondence). We denote by $\chi ({{𝒪}}^G,{{𝒪}}^K)$ the number of $K$-orbits in ${{𝒪}}^G\cap q^{-1}({{𝒪}}^K),$ which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity $m_{\pi }(\tau )$ of an irreducible $K$-module $\tau$ occurring in the restriction $\pi {|}_K$ could be read from the coadjoint action of $K$ on ${{𝒪}}^G\cap q^{-1}({{𝒪}}^K).$ Under some assumptions on the pair $(K,N),$ we prove that for a class of generic representations $\pi \in \widehat{G},$ one has

The general model of an arbitrary spin massive particle in any dimensional space–time is derived on the basis of Kirillov–Kostant–Souriau approach. It is shown that the model allows consistent coupling to an arbitrary background of electromagnetic and gravitational fields.

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

The dynamical realizations of the Schrödinger group are analyzed. Under the assumption that the "mass" parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes the standard Newton equations.

We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson–Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for the field equations instead of separation of variables. The results obtained are illustrated by the example of the three-dimensional rotation group.