Related papers: Finite multiplicity theorems for induction and res…

We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full…

We prove that any simply connected non-compact semisimple Lie group $G$ admits an infinite-dimensional irreducible representation $\Pi$ with bounded multiplicity property of the restriction $\Pi|_{G'}$ for all symmetric pairs $(G, G')$. We…

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types…

We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…

Let $\Gamma$ be a finite group, let $\theta$ be an involution of $\Gamma$, and let $\rho$ be an irreducible complex representation of $\Gamma$. We bound $\dim \rho^{\Gamma^{\theta}}$ in terms of the smallest dimension of a faithful…

We study representations of the classical infinite dimensional real simple Lie groups $G$ induced from factor representations of minimal parabolic subgroups $P$. This makes strong use of the recently developed structure theory for those…

Let $P$ be a minimal parabolic subgroup of a real reductive Lie group $G$ and $H$ a closed subgroup of $G$. Then it is proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(G/H)$ contains each irreducible…

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

Let $G_{\mathbb{R}}$ be a real reductive Lie group and $G'_{\mathbb{R}}$ a reductive subgroup of $G_{\mathbb{R}}$ such that $\mathfrak{g'}$ is algebraic in $\mathfrak{g}$. In this paper, we consider restrictions of irreducible…

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of…

We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the…

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of…

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of…

Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $\Gamma$ be a finite subgroup of $G$, and let $\tau$ be a finite-dimensional representation of $K$. For $\pi$ in the unitary dual $\widehat G$ of…

Vinberg--Kimel'fel'd [Funct. Anal. Appl., 1978] established that a quasi-projective normal $G$-variety $X$ is spherical if and only if $G$-modules on the spaces $\Gamma(X, \mathcal{L})$ of global sections of $G$-equivariant line bundles are…

Let $G$ be a complex connected reductive algebraic group and let $G_{\mathbb{R}}$ be a real form of $G$. We construct a sequence of functors $L_i\mathcal{R}$ from admissible (resp. finite-length) representations of $G$ to admissible (resp.…

Let G be the real points of a simply connected, semisimple, simply laced complex Lie group, and let \tilde{G} be the nonlinear double cover of G. We discuss a set of small genuine irreducible representations of \tilde{G} which can be…

The restriction of an irreducible unitary representation $\pi$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $\tau$ of $H$ with multiplicities…

We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in…

For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is…