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
LittlewoodPaley
decomposition related to the spectral resolution of the full Laplacian.
This requires a careful
analysis due also to the nonhomogeneous 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 multiplicityfree by proving an appropriate theorem on invariant
distributions; this then proves a multiplicityone 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, multiplicityfree subspaces each of which
is the image of a wavelet transform . We also show that, with the
assumption of (almosteverywhere) trivial stabilizers,
nonunimodularity is necessary for the existence of admissible
vectors.
Categories:22E27, 22E30 

54. CJM 2007 (vol 59 pp. 795)
 Jaworski, Wojciech; Neufang, Matthias

The ChoquetDeny 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 ChoquetDeny 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 ChoquetDeny 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 ChoquetDeny equation in
the space $E=B(L^2(G))$ of bounded linear operators on $L^2(G)$. Other
general properties of the ChoquetDeny equation in a Banach space are also
discussed.
Categories:22D12, 22D20, 43A05, 60B15, 60J50 

55. CJM 2007 (vol 59 pp. 148)
56. CJM 2006 (vol 58 pp. 1229)
 Henniart, Guy; Lemaire, Bertrand

IntÃ©grales 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 pseudocoefficient comme valeurs du caract\`ere
$\kappa$tordu.
Keywords:corps local, reprÃ©sentation lisse, intÃ©grale orbitale tordue, induction automorphe, lemme fondamental, formule des traces locale, pseudocoefficient Category:22E50 

57. CJM 2006 (vol 58 pp. 1203)
 Heiermann, Volker

Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de HarishChandra
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
sousgroupe de Levi $M$ d'un groupe $p$adique contient un
sousquotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de HarishChandra 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 CasselmanShalika Formula for the Shalika Model of $\operatorname{GL}_n$
The CasselmanShalika 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.,
multiplicityfree 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:CasselmanShalika, 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 rÃ©ductifs quasidÃ©ployÃ©s
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des
$F$points d'un groupe r\'eductif connexe quasid\'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 sousgroupe 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 sousespace de
dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la
r\'eunion des sousgroupes 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 semisimples, 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 nontrivial 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 nonzero 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 nontrivial 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 nonzero
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 NonConnected Affine KacMoody Groups
In this paper we generalise the concept of a Steinberg
cross section to nonconnected affine KacMoody 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 KacMoody 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 nonvoid.)
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 quasisplit 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
nonabelian $R$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 PalaisSmale
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 nonnegative.
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 Ã une orbite nilpotente adjointe
We are interested in Poisson structures to
transverse nilpotent adjoint orbits in a complex semisimple 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
Kirillovtype 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 nonunitary 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
LanglandsShahidi 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
nonvaninishing 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 CoAmenability 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 coamenability 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)
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
torsionfree 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 
