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Search: MSC category 22D10 ( Unitary representations of locally compact groups )

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1. CMB 2013 (vol 57 pp. 357)

Lauret, Emilio A.
Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds
Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ are unitarily equivalent.

Keywords:representation equivalent, strongly isospectrality, compact flat manifolds
Categories:58J53, 22D10

2. CMB 2005 (vol 48 pp. 505)

Bouikhalene, Belaid
On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group
Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.
Categories:39B32, 39B42, 22D10, 22D12, 22D15

3. CMB 2004 (vol 47 pp. 215)

Jaworski, Wojciech
Countable Amenable Identity Excluding Groups
A discrete group $G$ is called \emph{identity excluding\/} if the only irreducible unitary representation of $G$ which weakly contains the $1$-dimensional identity representation is the $1$-dimensional identity representation itself. Given a unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let $P_\mu$ denote the $\mu$-average $\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu$ on an identity excluding group and every unitary representation $\pi$ there exists and orthogonal projection $E_\mu$ onto a $\pi$-invariant subspace such that $s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50

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