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1. CMB Online first

Silberman, Lior
 Quantum unique ergodicity on locally symmetric spaces: the degenerate lift Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber. Keywords:quantum unique ergodicity, microlocal lift, spherical dualCategories:22E50, 43A85

2. 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 manifoldsCategories:58J53, 22D10

3. CMB 2012 (vol 56 pp. 881)

Xie, BaoHua; Wang, JieYan; Jiang, YuePing
 Free Groups Generated by Two Heisenberg Translations In this paper, we will discuss the groups generated by two Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free. Keywords:free group, Heisenberg group, complex triangle groupCategories:30F40, 22E40, 20H10

4. CMB 2012 (vol 57 pp. 424)

Sołtan, Piotr M.; Viselter, Ami
 A Note on Amenability of Locally Compact Quantum Groups In this short note we introduce a notion called quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups. Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivityCategories:20G42, 22D25, 46L89

5. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
 Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using FraÃ¯ssÃ© theory and Ramsey theory. This work generalizes some of the known results about countable structures. Keywords:universal minimal flows, ultrafilter flows, Ramsey theoryCategories:37B05, 03E02, 05D10, 22F50, 54H20

6. CMB 2012 (vol 56 pp. 647)

Valverde, Cesar
 On Induced Representations Distinguished by Orthogonal Groups Let $F$ be a local non-archimedean field of characteristic zero. We prove that a representation of $GL(n,F)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic. Keywords:distinguished representation, parabolic inductionCategory:22E50

7. CMB 2011 (vol 56 pp. 442)

Zelenyuk, Yevhen
 Closed Left Ideal Decompositions of $U(G)$ Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Äech compactification of $G$. We take the points of $Äta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U(G)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show that if $|G|$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the finest decomposition of $U(G)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U(G)$ is Hausdorff. Keywords:Stone--Äech compactification, uniform ultrafilter, closed left ideal, decompositionCategories:22A15, 54H20, 22A30, 54D80

8. CMB 2011 (vol 56 pp. 213)

Tausk, Daniel V.
 A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible It was claimed by Halmos in 1944 that if $G$ is a Hausdorff locally compact topological abelian group and if the character group of $G$ is torsion free, then $G$ is divisible. We prove that such a claim is false by presenting a family of counterexamples. While other counterexamples are known, we also present a family of stronger counterexamples, showing that even if one assumes that the character group of $G$ is both torsion free and divisible, it does not follow that $G$ is divisible. Category:22B05

9. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
 Left Invariant Einstein-Randers Metrics on Compact Lie Groups In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics. Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvatureCategories:17B20, 22E46, 53C12

10. CMB 2011 (vol 56 pp. 218)

Yang, Dilian
 Functional Equations and Fourier Analysis By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations - the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups. Keywords:functional equations, Fourier analysis, representation of compact groupsCategories:39B52, 22C05, 43A30

11. CMB 2011 (vol 56 pp. 116)

Krepski, Derek
 Central Extensions of Loop Groups and Obstruction to Pre-Quantization An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles. Keywords:loop group, central extension, prequantizationCategories:53D, 22E

12. CMB 2011 (vol 55 pp. 297)

Glasner, Eli
 The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal. Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functionsCategories:54H11, 22A05, 37B05, 54H20

13. CMB 2011 (vol 54 pp. 663)

Haas, Ruth; G. Helminck, Aloysius
 Admissible Sequences for Twisted Involutions in Weyl Groups Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$. Categories:20G15, 20G20, 22E15, 22E46, 43A85

14. CMB 2010 (vol 54 pp. 44)

Cheung, Wai-Shun; Tam, Tin-Yau
 Star-Shapedness and $K$-Orbits in Complex Semisimple Lie Algebras Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in \mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras of type $B$. Categories:22E10, 17B20

15. CMB 2010 (vol 54 pp. 126)

Jin, Yongyang; Zhang, Genkai
 Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups. Categories:22E30, 35R03, 43A80

16. CMB 2008 (vol 51 pp. 60)

Janzen, David
 F{\o}lner Nets for Semidirect Products of Amenable Groups For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided. Categories:22D05, 43A07, 22D15, 43A20

17. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
 Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$. Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10

18. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
 A KÃ¼nneth Theorem for $p$-Adic Groups Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$. Categories:22E50, 18G15, 55U25

19. CMB 2007 (vol 50 pp. 460)

Spielberg, Jack
 Weak Semiprojectivity for Purely Infinite $C^*$-Algebras We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups. Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebraCategories:46L05, 46L80, 22A22

20. CMB 2007 (vol 50 pp. 291)

Sarkar, Rudra P.; Sengupta, Jyoti
 Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space. Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principleCategories:22E30, 43A85

21. CMB 2007 (vol 50 pp. 48)

Dvorsky, Alexander
 Tensor Square of the Minimal Representation of $O(p,q)$ In this paper, we study the tensor product $\pi=\sigma^{\min}\otimes \sigma^{\min}$ of the minimal representation $\sigma^{\min}$ of $O(p,q)$ with itself, and decompose $\pi$ into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space. Category:22e46

22. CMB 2006 (vol 49 pp. 578)

Muić, Goran
 On the Structure of the Full Lift for the Howe Correspondence of $(Sp(n), O(V))$ for Rank-One Reducibilities In this paper we determine the structure of the full lift for the Howe correspondence of $(Sp(n),O(V))$ for rank-one reducibilities. Categories:22E35, 22E50, 11F70

23. 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

24. CMB 2004 (vol 47 pp. 439)

Parker, John R.
 On the Stable Basin Theorem The stable basin theorem was introduced by Basmajian and Miner as a key step in their necessary condition for the discreteness of a non-elementary group of complex hyperbolic isometries. In this paper we improve several of Basmajian and Miner's key estimates and so give a substantial improvement on the main inequality in the stable basin theorem. Categories:22E40, 20H10, 57S30

25. 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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