76. 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 

77. 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 

78. 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 

79. 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 

80. 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 

81. 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 

82. 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 

83. 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 

84. 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 

85. 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 

86. CJM 2005 (vol 57 pp. 159)
87. 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 

88. 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 

89. CJM 2004 (vol 56 pp. 963)
90. 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 

91. CJM 2004 (vol 56 pp. 293)
 Khomenko, Oleksandr; Mazorchuk, Volodymyr

Structure of modules induced from simple modules with minimal annihilator
We study the structure of generalized Verma modules over a
semisimple complex finitedimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
BernsteinGelfandGelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of HarishChandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.
Keywords:parabolic induction, generalized Verma module, simple module, Ha\rish\Chand\ra bimodule, equivalent categories Categories:17B10, 22E47 

92. CJM 2004 (vol 56 pp. 168)
 Pogge, James Todd

On a Certain Residual Spectrum of $\Sp_8$
Let $G=\Sp_{2n}$ be the symplectic group defined over a number
field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental
problem in the theory of automorphic forms is to decompose the
right regular representation of $G(\mathbb{A})$ acting on the
Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main
contributions have been made by Langlands. He described, using his
theory of Eisenstein series, an orthogonal decomposition of this
space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A})
\bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A})
\bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a
cuspidal automorphic representation $\pi$ taken modulo conjugacy
(Here we normalize $\pi$ so that the action of the maximal split
torus in the center of $G$ at the archimedean places is trivial.)
and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$
is a space of residues of Eisenstein series associated to
$(M,\pi)$. In this paper, we will completely determine the space
$L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when
$M\simeq\GL_2\times\GL_2$. This is the first result on the
residual spectrum for nonmaximal, nonBorel parabolic subgroups,
other than $\GL_n$.
Categories:11F70, 22E55 

93. CJM 2003 (vol 55 pp. 1155)
 Đoković, Dragomir Ž.; Litvinov, Michael

The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$
The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$orbit (or $K^0$orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$orbit (resp.\ $K^0$orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$orbits (resp.\ $K^0$orbits) of smaller dimensions. The
description of the closure of nilpotent $K$orbits has been known for
some time, but not so for the nilpotent $K^0$orbits. A conjecture
describing the closure of nilpotent $K^0$orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 

94. CJM 2003 (vol 55 pp. 1121)
 Bettaïeb, Karem

Classification des reprÃ©sentations tempÃ©rÃ©es d'un groupe $p$adique
Soit $G$ le groupe des points d\'efinis sur un corps $p$adique d'un
groupe r\'eductif connexe. A l'aide des caract\`eres virtuels
supertemp\'er\'es de $G$, on prouve (conjectures de Clozel) que toute
repr\'esentation irr\'eductible temp\'er\'ee de $G$ est
irr\'eductiblement induite d'une essentielle d'un sousgroupe de
L\'evi de~$G$.
Category:22E 

95. CJM 2003 (vol 55 pp. 1080)
 Kellerhals, Ruth

Quaternions and Some Global Properties of Hyperbolic $5$Manifolds
We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 

96. CJM 2003 (vol 55 pp. 969)
 Glöckner, Helge

Lie Groups of Measurable Mappings
We describe new construction principles for infinitedimensional Lie
groups. In particular, given any measure space $(X,\Sigma,\mu)$ and
(possibly infinitedimensional) Lie group $G$, we construct a Lie
group $L^\infty (X,G)$, which is a Fr\'echetLie group if $G$ is so.
We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an
arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie
group, modelled on the locally convex direct sum $\bigoplus_{i\in I}
L(G_i)$.
Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 

97. CJM 2003 (vol 55 pp. 353)
 Silberger, Allan J.; Zink, ErnstWilhelm

Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$Adic Simple Algebras
Let $F$ be a $p$adic local field and let $A_i^\times$ be the unit
group of a central simple $F$algebra $A_i$ of reduced degree $n>1$
($i=1,2$). Let $\mathcal{R}^2 (A_i^\times)$ denote the set of
irreducible discrete series representations of $A_i^\times$. The
``Abstract Matching Theorem'' asserts the existence of a bijection,
the ``JacquetLanglands'' map, $\mathcal{J} \mathcal{L}_{A_2,A_1}
\colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$
which, up to known sign, preserves character values for regular
elliptic elements. This paper addresses the question of explicitly
describing the map $\mathcal{J} \mathcal{L}$, but only for ``level
zero'' representations. We prove that the restriction $\mathcal{J}
\mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to
\mathcal{R}_0^2 (A_2^\times)$ is a bijection of level zero discrete
series (Proposition~3.2) and we give a parameterization of the set of
unramified twist classes of level zero discrete series which does not
depend upon the algebra $A_i$ and is invariant under $\mathcal{J}
\mathcal{L}_{A_2,A_1}$ (Theorem~4.1).
Categories:22E50, 11R39 

98. CJM 2002 (vol 54 pp. 1100)
 Wood, Peter J.

The Operator Biprojectivity of the Fourier Algebra
In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
Keywords:locally compact group, Fourier algebra, operator space, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 

99. CJM 2002 (vol 54 pp. 769)
 Miyazaki, Takuya

Nilpotent Orbits and Whittaker Functions for Derived Functor Modules of $\Sp(2,\mathbb{R})$
We study the moderate growth generalized Whittaker functions,
associated to a unitary character $\psi$ of a unipotent subgroup,
for the nontempered cohomological representation of $G = \Sp
(2,\mathbb{R})$. Through an explicit calculation of a holonomic
system which characterizes these functions we observe that their
existence is determined by the including relation between the real
nilpotent coadjoint $G$orbit of $\psi$ in
$\mathfrak{g}_{\mathbb{R}}^\ast$ and the asymptotic support of the
cohomological representation.
Categories:22E46, 22E30 

100. CJM 2002 (vol 54 pp. 828)
 Moriyama, Tomonori

Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$
Let $\pi$ be an irreducible generalized principal series
representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic
subgroup. We show that the space of algebraic intertwining operators
from $\pi$ to the representation induced from an irreducible
admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one
dimensional. Spherical functions in the title are the images of
$K$finite vectors by this intertwining operator. We obtain an
integral expression of MellinBarnes type for the radial part of our
spherical function.
Categories:22E45, 11F70 
