Narrow Results

Let $(G,H)$ be a symmetric pair for a real semisimple Lie group $G$ and $(G,H_0)$ its associated pair. For each irreducible square integrable representation $\pi$ of $G$ so that its restriction to $H$ is admissible, we find an irreducible square integrable representation ${\pi }_0$ of $H_0$ which allows us to compute the Harish-Chandra parameter of each irreducible $H$-subrepresentation of $\pi$ as well as its multiplicity. The computation is based on the spectral analysis of the restriction of ${\pi }_0$ to a maximal compact subgroup of $H_0.$

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

We construct and give a complete classification of all the differential symmetry breaking operators ${𝔻}_{\lambda ,\nu }^m:C^{\infty }(S^3,{{𝒱}}_{\lambda }^3)\to C^{\infty }(S^2,{{ℒ}}_{m,\nu })$, between the spaces of smooth sections of a vector bundle of rank $3$ over the $3$-sphere ${{𝒱}}_{\lambda }^3\to S^3$, and a line bundle over the $2$-sphere ${{ℒ}}_{m,\nu }\to S^2$. In particular, we give necessary and sufficient conditions on the tuple of parameters $(\lambda ,\nu ,m)$ for which these operators exist.

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of ${𝔰𝔩}_2$ occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras $𝔤$.

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