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51. CJM 2007 (vol 59 pp. 1301)

Furioli, Giulia; Melzi, Camillo; Veneruso, Alessandro
Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group
We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood--Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, G\'erard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

Keywords:nilpotent and solvable Lie groups, smoothness and regularity of solutions of PDEs
Categories:22E25, 35B65

52. CJM 2007 (vol 59 pp. 1050)

Raghuram, A.
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$
Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we describe its restriction to the diagonal subgroup $\mathcal{D}^* \times \mathcal{D}^*$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi$. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if $\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic case.

Categories:22E50, 22E35, 11S37

53. CJM 2007 (vol 59 pp. 917)

Currey, Bradley N.
Admissibility for a Class of Quasiregular Representations
Given a semidirect product $G = N \rtimes H$ where $N$ is%% nilpotent, connected, simply connected and normal in $G$ and where $H$ is a vector group for which $\ad(\h)$ is completely reducible and $\mathbf R$-split, let $\tau$ denote the quasiregular representation of $G$ in $L^2(N)$. An element $\psi \in L^2(N)$ is said to be admissible if the wavelet transform $f \mapsto \langle f, \tau(\cdot)\psi\rangle$ defines an isometry from $L^2(N)$ into $L^2(G)$. In this paper we give an explicit construction of admissible vectors in the case where $G$ is not unimodular and the stabilizers in $H$ of its action on $\hat N$ are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of $L^2(G)$ into $G$-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

Categories:22E27, 22E30

54. CJM 2007 (vol 59 pp. 795)

Jaworski, Wojciech; Neufang, Matthias
The Choquet--Deny Equation in a Banach Space
Let $G$ be a locally compact group and $\pi$ a representation of $G$ by weakly$^*$ continuous isometries acting in a dual Banach space $E$. Given a probability measure $\mu$ on $G$, we study the Choquet--Deny equation $\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation form the range of a projection of norm $1$ and can be represented by means of a ``Poisson formula'' on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu$. The relation between the space of solutions of the Choquet--Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the $W^*$-crossed product and coinciding precisely with the crossed product in the special case of the Choquet--Deny equation in the space $E=B(L^2(G))$ of bounded linear operators on $L^2(G)$. Other general properties of the Choquet--Deny equation in a Banach space are also discussed.

Categories:22D12, 22D20, 43A05, 60B15, 60J50

55. CJM 2007 (vol 59 pp. 148)

Muić, Goran
On Certain Classes of Unitary Representations for Split Classical Groups
In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups

Categories:22E35, 22E50, 11F70

56. CJM 2006 (vol 58 pp. 1229)

Henniart, Guy; Lemaire, Bertrand
Intégrales orbitales tordues sur $\GL(n,F)$ et corps locaux proches\,: applications
Soient $F$ un corps commutatif localement compact non archim\'edien, $G=\GL (n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\`ere de $F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour les $\kappa$-int\'egrales orbitales r\'eguli\`eres sur $G$ permettant, si $F$ est de caract\'eristique $>0$, de les relever \`a la caract\'eristique nulle. On en d\'eduit deux r\'esultats nouveaux en caract\'eristique $>0$\,: le ``lemme fondamental'' pour l'induction automorphe, et une version simple de la formule des traces tordue locale d'Arthur reliant $\kappa$-int\'egrales orbitales elliptiques et caract\`eres $\kappa$-tordus. Cette formule donne en particulier, pour une s\'erie $\kappa$-discr\`ete de $G$, les $\kappa$-int\'egrales orbitales elliptiques d'un pseudo-coefficient comme valeurs du caract\`ere $\kappa$-tordu.

Keywords:corps local, représentation lisse, intégrale orbitale tordue, induction automorphe, lemme fondamental, formule des traces locale, pseudo-coefficient
Category:22E50

57. CJM 2006 (vol 58 pp. 1203)

Heiermann, Volker
Orbites unipotentes et pôles d'ordre maximal de la fonction $\mu $ de Harish-Chandra
Dans un travail ant\'erieur, nous avions montr\'e que l'induite parabolique (normalis\'ee) d'une repr\'esentation irr\'eductible cuspidale $\sigma $ d'un sous-groupe de Levi $M$ d'un groupe $p$-adique contient un sous-quotient de carr\'e int\'egrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre \'egal au rang parabolique de $M$. L'objet de cet article est d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de Langlands.

Categories:11F70, 11F80, 22E50

58. CJM 2006 (vol 58 pp. 1095)

Sakellaridis, Yiannis
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$
The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs
Categories:22E50, 11F70, 11F85

59. CJM 2006 (vol 58 pp. 897)

Courtès, François
Distributions invariantes sur les groupes réductifs quasi-déployés
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis.

Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions
Categories:22E35, 20G40

60. CJM 2006 (vol 58 pp. 673)

Bart, Anneke; Scannell, Kevin P.
The Generalized Cuspidal Cohomology Problem
Let $\Gamma \subset \SO(3,1)$ be a lattice. The well known \emph{bending deformations}, introduced by \linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma$ into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this construction of cohomology to the context of ``branched'' totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in $S^3$ which is not infinitesimally rigid in $\SO(4,1)$. The first order deformations of this link complement are supported on a piecewise totally geodesic $2$-complex.

Categories:57M50, 22E40

61. CJM 2006 (vol 58 pp. 768)

Hu, Zhiguo; Neufang, Matthias
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
The decomposability number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\m$. In this paper, we explore the close connection between $\dec(\m)$ and the cardinal level of the Mazur property for the predual $\m_*$ of $\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra $\lone$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $\luc^*$, etc. We show that for any of these von Neumann algebras, say $\m$, the cardinal number $\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kg$ of $G$ and the least cardinality $\bg$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $\ag^{**}$.

Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre
Categories:22D05, 43A20, 43A30, 03E55, 46L10

62. CJM 2006 (vol 58 pp. 625)

Mohrdieck, Stephan
A Steinberg Cross Section for Non-Connected Affine Kac--Moody Groups
In this paper we generalise the concept of a Steinberg cross section to non-connected affine Kac--Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac--Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a ``twisted Coxeter cell'', a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a $\Cst$-action.

Category:22E67

63. CJM 2006 (vol 58 pp. 344)

Goldberg, David
Reducibility for $SU_n$ and Generic Elliptic Representations
We study reducibility of representations parabolically induced from discrete series representations of $SU_n(F)$ for $F$ a $p$-adic field of characteristic zero. We use the approach of studying the relation between $R$-groups when a reductive subgroup of a quasi-split group and the full group have the same derived group. We use restriction to show the quotient of $R$-groups is in natural bijection with a group of characters. Applying this to $SU_n(F)\subset U_n(F)$ we show the $R$ group for $SU_n$ is the semidirect product of an $R$-group for $U_n(F)$ and this group of characters. We derive results on non-abelian $R$-groups and generic elliptic representations as well.

Categories:22E50, 22E35

64. CJM 2005 (vol 57 pp. 1279)

Maad, Sara
A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains
We study the semilinear equation \begin{equation*} -\Delta_{\mathbb H} u(\eta) + u(\eta) = f(\eta, u(\eta)),\quad u \in \So(\Omega), \end{equation*} where $\Omega$ is an unbounded domain of the Heisenberg group $\mathbb H^N$, $N\ge 1$. The space $\So(\Omega)$ is the Heisenberg analogue of the Sobolev space $W_0^{1,2}(\Omega)$. The function $f\colon \overline{\Omega}\times \mathbb R\to \mathbb R$ is supposed to be odd in $u$, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator $\Delta_{\mathbb H}$ is the subelliptic Laplacian on the Heisenberg group. We give a condition on $\Omega$ which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais--Smale condition. This might be quite surprising since we deal with domains which are far from bounded. The technique we use rests on a compactness argument and the maximum principle.

Keywords:Heisenberg group, concentration compactness, Heisenberg Laplacian
Categories:22E30, 22E27

65. CJM 2005 (vol 57 pp. 1193)

Dungey, Nick
Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups
Let $K$ be a function on a unimodular locally compact group $G$, and denote by $K_n = K*K* \cdots * K$ the $n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in $L^2(G)$, we give estimates of the norms $\|K_n\|_2$ and $\|K_n\|_\infty$ for large $n$. In contrast to previous methods for estimating $\|K_n\|_\infty$, we do not need to assume that the function $K$ is a probability density or non-negative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.

Categories:22E30, 35B40, 43A99

66. CJM 2005 (vol 57 pp. 750)

Sabourin, Hervé
Sur la structure transverse à une orbite nilpotente adjointe
We are interested in Poisson structures to transverse nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $sl_n$, we construct some families of nilpotent orbits with quadratic transverse structures.

Keywords:nilpotent adjoint orbits, conormal orbits, Poisson transverse structure
Categories:22E, 53D

67. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator
Categories:43A85, 22D10, 39A70

68. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types
Categories:20G25, 22E35, 20H25

69. CJM 2005 (vol 57 pp. 616)

Muić, Goran
Reducibility of Generalized Principal Series
In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to M\oe glin and Tadi\'c.

Categories:22E35, and, 50, 11F70

70. CJM 2005 (vol 57 pp. 535)

Kim, Henry H.
On Local $L$-Functions and Normalized Intertwining Operators
In this paper we make explicit all $L$-functions in the Langlands--Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions.

Categories:11F70, 22E55

71. CJM 2005 (vol 57 pp. 159)

72. CJM 2005 (vol 57 pp. 17)

Bédos, Erik; Conti, Roberto; Tuset, Lars
On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations
We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories.

Keywords:quantum group, amenability
Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32

73. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.

Categories:20G15, 20G20, 22E15, 22E46

74. CJM 2004 (vol 56 pp. 963)

Ishiwata, Satoshi
A Berry-Esseen Type Theorem on Nilpotent Covering Graphs
We prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel.

Categories:22E25, 60J15, 58G32

75. CJM 2004 (vol 56 pp. 883)

Tandra, Haryono; Moran, William
Kirillov Theory for a Class of Discrete Nilpotent Groups
This paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups $G$. $G$ is assumed to be discrete, countable and $\pi$-radicable, with $\pi$ containing the primes less than or equal to the nilpotence class of $G$. In addition, it is assumed that all of the characters of $G$ have idempotent absolute value. Such groups are shown to be plentiful.

Category:22D10
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