For a connected, noncompact matrix simple Lie group $G$ so that a maximal compact subgroup $K$ has three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contains the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that a square integrable representations of the ambient group is admissible over the semisimple factor of $K.$
In this paper we consider the unitary symmetric spaces of the form X=U(p,q)/U(1)U(p,q-1) and their discrete series representations. Inspired by the work of A.Venkatesh and Y.Sekellarides on L-groups of p-adic spherical spaces we formulate and prove natural relative branching laws for the restriction to smaller subgroups of the same type and corresponding unitary spaces.We think of this as steps to formulation and proving Gan Gross Prasad conjectures for unitary spaces. Using period integral and some results of T.Kobayashi we prove an analogue of thesis conjectures.
We give a complete description of the discrete spectra in the branching law $Π|_{G'}$ with respect to the pair $(G,G')=(O(p,q), O(p',q') \times O(p'',q''))$ for irreducible unitary representations $Π$ of $G$ that are "geometric quantization" of minimal elliptic coadjoint orbits. We also construct explicitly all holographic operators and prove a closed Parseval-type formula.
In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $Π$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representati…
Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,π)$, for any irreducible representation $π$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde π} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of …
The restriction of an irreducible unitary representation $π$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $τ$ of $H$ with multiplicities $m(π,τ)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $π$, the direct integral is pointwise defined. This implies that $m(π,τ)$ is bounded above by the dimension of the space $\operatorname{Hom}_H(π^\infty|_H,τ^\infty)$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between t…
This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models.
Let $G$ be a noncompact connected simple Lie group, and $(G,G^Γ)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^Γ)$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. As an application, for $G=\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Γ)$ with $G^Γ$ n…
In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(π,n+l)$ of $\mathrm{GL}_{2n+2l}$ with respect to the block diagonal subgroup $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$ for any irreducible generic representation $π$ of $\mathrm{GL}_2$ over any $p$-adic field. We use the Shalika model of $\mathrm{Sp}(π,n)$ to construct certain zeta integrals, which were defined by Ginzburg and Kaplan independently, and study them. Finally, using these zeta integrals, we obtain a nonzero $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$-map from $\mathrm{Sp}(π,n+l)$ to $τ\boxtimesτ^\veeχ_π…
Given irreducible representations $Π$ and $π$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $θ$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(Π|_{G'}, π)\not = \{0\}. \] In the special case that both $Π$ and $π$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both …
This article is devoted to branching problems for holomorphic discrete series representations of a conformal group $G$ of a tube domain $T_Omega$ over a symmetric cone $Ω$. More precisely, we analyse restrictions of such representations to the conformal group $G'$ of a tube domain $T_{Ω'}$ holomorphically embedded in $T_Ω$. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. To do so, a stratification space for a symmetric cone is introduced. This structure put the light on a new functional model, called the \emph{s…
Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results…
We consider branching laws for the restriction of some irreducible unitary representations $Π$ of $G=O(p,q)$ to its subgroup $H=O(p-1,q)$. In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of $O(p-1,q)$ in the restriction of the unitary $Π|_{O(p-1,q) }$ are determined. By considering the restriction of packets of irreducible representations we obtain another very simple branching law, which was conjectured in Orsted-Speh (arXiv:1907.07544).
Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $Π$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $Π$ occurs inside a principal series representation of $G$, induced from the $\operatorname{O}(n)$-representation $\bigwedge\nolimits^p(\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to…
In this paper, we obtain explicit branching laws for all irreducible unitary representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The restriction turns out to be a finite direct sum of irreducible unitary representations of $P$. We also verify Duflo's conjecture for the branching law of tempered representations of $\Spin(N,1)$ with respect to a parabolic subgroup $P$. That is to show: in the framework of the orbit method, the branching law of a tempered representation is determined by the behavior of the moment map from the corresponding coadjoint orbit. A few key tools used…
In this article, we generalize Duflo's conjecture to understand the branching laws of non-discrete series. We give a unified description on the geometric side about the restriction of an irreducible unitary representation $π$ of $\mathrm{GL}_n(k)$, $k=\mathbb{R}$ or $\mathbb{C}$, to the mirabolic subgroup, where $π$ is attached to a certain kind of coadjoint orbit.
We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.
