26. 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 

27. 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
nonelementary 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 

28. 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 

29. CMB 2003 (vol 46 pp. 332)
 Đoković, Dragomir Z.; Tam, TinYau

Some Questions about Semisimple Lie Groups Originating in Matrix Theory
We generalize the wellknown result that a square traceless complex
matrix is unitarily similar to a matrix with zero diagonal to
arbitrary connected semisimple complex Lie groups $G$ and their Lie
algebras $\mathfrak{g}$ under the action of a maximal compact subgroup
$K$ of $G$. We also introduce a natural partial order on
$\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for
all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This
partial order is $K$invariant and induces a partial order on the
orbit space $\mathfrak{g}/K$. We prove that, under some restrictions
on $\mathfrak{g}$, the set $f(K\cdot x)$ is starshaped with respect
to the origin.
Categories:15A45, 20G20, 22E60 

30. CMB 2002 (vol 45 pp. 466)
 Arthur, James

A Note on the Automorphic Langlands Group
Langlands has conjectured the existence of a universal group, an
extension of the absolute Galois group, which would play a fundamental
role in the classification of automorphic representations. We shall
describe a possible candidate for this group. We shall also describe
a possible candidate for the complexification of Grothendieck's
motivic Galois group.
Categories:11R39, 22E55 

31. CMB 2002 (vol 45 pp. 364)
 Deitmar, Anton

Mellin Transforms of Whittaker Functions
In this note we show that for an arbitrary reductive Lie group
and any admissible irreducible Banach representation the Mellin
transforms of Whittaker functions extend to meromorphic functions.
We locate the possible poles and show that they always lie along
translates of walls of Weyl chambers.
Categories:11F30, 22E30, 11F70, 22E45 

32. CMB 2002 (vol 45 pp. 436)
33. CMB 2002 (vol 45 pp. 220)
 Hakim, Jeffrey; Murnaghan, Fiona

Globalization of Distinguished Supercuspidal Representations of $\GL(n)$
An irreducible supercuspidal representation $\pi$ of $G=
\GL(n,F)$, where $F$ is a nonarchimedean local field of
characteristic zero, is said to be ``distinguished'' by a
subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if
$\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue
of this notion for and irreducible, automorphic, cuspidal
representation associated to $\GL(n)$. Under certain general
hypotheses, it is shown in this paper that every distinguished,
irreducible, supercuspidal representation may be realized as a
local component of a distinguished, irreducible automorphic,
cuspidal representation. Applications to the theory of
distinguished supercuspidal representations are provided.
Categories:22E50, 22E35, 11F70 

34. CMB 2001 (vol 44 pp. 408)
35. CMB 2001 (vol 44 pp. 491)
 Wang, Weiqiang

Resolution of Singularities of Null Cones
We give canonical resolutions of singularities of several cone
varieties arising from invariant theory. We establish a connection
between our resolutions and resolutions of singularities of closure of
conjugacy classes in classical Lie algebras.
Categories:14L35, 22G 

36. CMB 2001 (vol 44 pp. 482)
37. CMB 2001 (vol 44 pp. 429)
 Henniger, J. P.

Ergodic Rotations of Nilmanifolds Conjugate to Their Inverses
In answer to a question posed in \cite{G}, we give sufficient
conditions on a Lie nilmanifold so that any ergodic rotation of the
nilmanifold is metrically conjugate to its inverse. The condition is
that the Lie algebra be what we call quasigraded, and is weaker than
the property of being graded. Furthermore, the conjugating map can be
chosen to be an involution. It is shown that for a special class of
groups, the condition of quasigraded is also necessary. In certain
examples there is a continuum of conjugacies.
Categories:28Dxx, 22E25 

38. CMB 2001 (vol 44 pp. 298)
39. CMB 2000 (vol 43 pp. 459)
 Ndogmo, J. C.

Properties of the Invariants of Solvable Lie Algebras
We generalize to a field of characteristic zero certain properties of
the invariant functions of the coadjoint representation of solvable
Lie algebras with abelian nilradicals, previously obtained over the
base field $\bbC$ of complex numbers. In particular we determine
their number and the restricted type of variables on which they
depend. We also determine an upper bound on the maximal number of
functionally independent invariants for certain families of solvable
Lie algebras with arbitrary nilradicals.
Categories:17B30, 22E70 

40. CMB 2000 (vol 43 pp. 380)
41. CMB 2000 (vol 43 pp. 90)
42. CMB 2000 (vol 43 pp. 47)
 Božičević, Mladen

A Property of Lie Group Orbits
Let $G$ be a real Lie group and $X$ a real analytic manifold.
Suppose that $G$ acts analytically on $X$ with finitely many
orbits. Then the orbits are subanalytic in $X$. As a consequence
we show that the microsupport of a $G$equivariant sheaf on $X$ is
contained in the conormal variety of the $G$action.
Categories:32B20, 22E15 

43. CMB 1999 (vol 42 pp. 393)
 Savin, Gordan

A Class of Supercuspidal Representations of $G_2(k)$
Let $H$ be an exceptional, adjoint group of type $E_6$ and split
rank 2, over a $p$adic field $k$. In this article we discuss the
restriction of the minimal representation of $H$ to a dual pair
$\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of
dimension 9 over $k$. In particular, we discover an interesting
class of supercuspidal representations of $G_2(k)$.
Categories:22E35, 22E50, 11F70 

44. CMB 1998 (vol 41 pp. 463)
 Moran, Alan

The right regular representation of a compact right topological group
We show that for certain compact right topological groups,
$\overline{r(G)}$, the strong operator topology closure of
the image of the right regular representation of $G$ in
${\cal L}({\cal H})$, where ${\cal H} = \L2$, is a compact
topological group and introduce a class of representations,
${\cal R}$, which effectively transfers the representation
theory of $\overline{r(G)}$ over to $G$. Amongst the groups
for which this holds is the class of equicontinuous groups
which have been studied by Ruppert in [10]. We use familiar
examples to illustrate these features of the theory and to
provide a counterexample. Finally we remark that every
equicontinuous group which is at the same time a Borel group
is in fact a topological group.
Category:22D99 

45. CMB 1998 (vol 41 pp. 368)
46. CMB 1997 (vol 40 pp. 376)
47. CMB 1997 (vol 40 pp. 183)
 Kepert, Andrew G.

The range of group algebra homomorphisms
A characterisation of the range of a homomorphism between two
commutative group algebras is presented which implies, among other
things, that this range is closed. The work relies mainly on the
characterisation of such homomorphisms achieved by P.~J.~Cohen.
Categories:43A22, 22B10, 46J99 

48. CMB 1997 (vol 40 pp. 72)