Narrow Results

Let G_ℝ be a real form of a complex, semisimple Lie group G. Assume is an even nilpotent coadjoint G_ℝ-orbit. We prove a limit formula, expressing the canonical measure on as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with .

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 = K ⋉ ℝⁿ, where K is a compact connected subgroup of O(n) acting on ℝⁿ by rotations. Let 𝔤 ⊃ 𝔨 be the respective Lie algebras of G and K, and pr : 𝔤* → 𝔨* the natural projection. For admissible coadjoint orbits and , we denote by the number of K-orbits in , which is called the Corwin–Greenleaf multiplicity function. Let π ∈ Ĝ and be the unitary representations corresponding, respectively, to and by the orbit method. In this paper, we investigate the relationship between and the multiplicity m(π, τ) of τ in the restriction of π to K. If π is infinite-dimensional and the associated little group is connected, we show that if and only if m(π, τ) ≠ 0. Furthermore, for K = SO(n), n ≥ 3, we give a sufficient condition on the representations π and τ in order that .

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.

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary $G$-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian $G$-spaces, which (as we show) unfortunately fails to mirror the situation where more than one $G$-module “quantizes” a given Hamiltonian $G$-space. This paper offers evidence that the situation is remedied by working in the category of prequantum$G$-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces and establish Frobenius reciprocity as well as the “induction in stages” property.

This paper deals with the construction of noncommutative phase spaces as coadjoint orbits of noncentral extensions of Galilei and Para-Galilei groups in two-dimensional space. The noncommutativity is due to the presence of a dual magnetic field B* in the Galilei case and of a magnetic field B in the Para-Galilei case.

Toward a unified algebraic theory for mean-field Hamiltonian describing paired- and unpaired-mode effects, in this paper, we propose a generalized Hartree–Bogoliubov mean-field Hamiltonian in terms of fermion pair and creation-annihilation operators of the $\mathrm{\text{SO}}(2N+1)$ Lie algebra. We diagonalize the generalized Hartree–Bogoliubov mean-field Hamiltonian and throughout its diagonalization we can first obtain the unpaired mode amplitudes which are given by the self-consistent field parameters appeared in the Hartree–Bogoliubov theory together with the additional self-consistent field parameter in the generalized Hartree–Bogoliubov mean-field Hamiltonian and by the parameter specifying the property of the $\mathrm{\text{SO}}(2N+1)$ group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such results. We construct the Killing potential in the coset space $\frac{\mathrm{\text{SO}}(2N)}{U(N)}$ on the Kähler symmetric space which is equivalent to the generalized density matrix. We show another approach to the fermion mean-field Hamiltonian based on such a generalized density matrix. We derive an $\mathrm{\text{SO}}(2N+1)$ generalized Hartree–Bogoliubov mean-field Hamiltonian operator and a modified Hartree–Bogoliubov eigenvalue equation. We discuss on the mean-field theory related to the algebraic mean-field theory based on the generalized density matrix and the coadjoint orbit leading to the nondegenerate symplectic form.

Representations of solvable Lie groups are realized and classified by geometric quantization of coadjoint orbits through positive polarizations.

Let G be a nilpotent connected and simply connected Lie group, K an analytic subgroup of G and π a unitary and irreducible representation of G. We study in this paper a variant of the Frobenius reciprocity for the restricted representation π_|K of π on K. It consists in proving that generically, the multiplicity of any isotopic component involved in the central canonical disintegration of π_|K coincides with the dimension of a certain space of generalized tempered distributions which are semi-invariant under the action of a subgroup of K. This problem was considered and partially solved in our previous work ¹.