1. CJM 2012 (vol 66 pp. 102)
 Birth, Lidia; Glöckner, Helge

Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$functions
and convolution of compactly supported Radon measures.
Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigmacompactness, convolution, continuity, seminorm, product estimates Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 

2. CJM 2011 (vol 64 pp. 481)
 Chamorro, Diego

Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolevtype inequalities on polynomial volume growth Lie groups.
We show in particular that improved Sobolev inequalities can be extended to this general framework
without the use of the LittlewoodPaley decomposition.
Keywords:Sobolev inequalities, polynomial volume growth Lie groups Category:22E30 

3. CJM 2010 (vol 62 pp. 563)
4. CJM 2009 (vol 61 pp. 961)
 Bernon, Florent

Transfert des intÃ©grales orbitales pour les algÃ¨bres de Lie classiques
Dans cet article, on consid\`ere un groupe semisimple $\rmG$ classique
r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un
sousgroupe de Cartan compact. On d\'efinit une famille de
sousalg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang
que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$.
Soit $\g'$ une telle sousalg\`ebre de Lie. On construit alors une
application de transfert des int\'egrales orbitales de $\g'$ dans
l'espace des int\'egrales orbitales de $\g$. On montre que cette
application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur
simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$.
Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de
rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette
application de transfert.
On utilise cette application de transfert pour obtenir une formule de
r\'eduction de l'int\'egrale de Cauchy HarishChandra pour les paires
duales d'alg\`ebres de Lie r\'eductives $\bigl( \Ug(p,q),\Ug(r,s)
\bigr)$ et $\bigl( \Sp(p,q),\Og^*(2n) \bigr)$ avec $p+q = r+s = n$.
Categories:22E30, 22E46 

5. 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 

6. 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 

7. 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 

8. 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 

9. CJM 2000 (vol 52 pp. 1192)
 Herb, Rebecca A.

Orbital Integrals on $p$Adic Lie Algebras
Let $G$ be a connected reductive $p$adic group and let $\frakg$ be its
Lie algebra. Let $\calO$ be any $G$orbit in $\frakg$. Then the orbital
integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution
on $\frakg $, and HarishChandra proved that its Fourier transform $\hat
\mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular
semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of
$\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give
a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$
sufficiently large. In the case that $\calO$ is a regular semisimple orbit,
the formula is already known by work of Waldspurger. In the case that
$\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at
infinity is already known because of its homogeneity properties. The
general case combines aspects of these two extreme cases. The formula
for $\hat\mu _{\calO}$ at infinity can be used to formulate a ``theory
of the constant term'' for the space of distributions spanned by the
Fourier transforms of orbital integrals. It can also be used to show
that the Fourier transforms of orbital integrals are ``linearly
independent at infinity.''
Categories:22E30, 22E45 

10. CJM 2000 (vol 52 pp. 412)
 Varopoulos, N. Th.

Geometric and Potential Theoretic Results on Lie Groups
The main new results in this paper are contained in the geometric
Theorems 1 and~2 of Section~0.1 below and they are related to
previous results of M.~Gromov and of myself (\cf\
\cite{1},~\cite{2}). These results are used to prove some general
potential theoretic estimates on Lie groups (\cf\ Section~0.3) that
are related to my previous work in the area (\cf\
\cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos
(\cf\ \cite{5},~\cite{21}).
Categories:22E30, 43A80, 60J60, 60J65 

11. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
squareintegrable $\Gamma$automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 

12. CJM 1999 (vol 51 pp. 816)
 Hall, Brian C.

A New Form of the SegalBargmann Transform for Lie Groups of Compact Type
I consider a twoparameter family $B_{s,t}$ of unitary transforms
mapping an $L^{2}$space over a Lie group of compact type onto a
holomorphic $L^{2}$space over the complexified group. These were
studied using infinitedimensional analysis in joint work with
B.~Driver, but are treated here by finitedimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
SegalBargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.
Categories:22E30, 81S30, 58G11 

13. CJM 1998 (vol 50 pp. 1090)
14. CJM 1998 (vol 50 pp. 356)
 Gross, Leonard

Some norms on universal enveloping algebras
The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$
supports some norms and seminorms that have arisen naturally in the
context of heat kernel analysis on Lie groups. These norms and seminorms
are investigated here from an algebraic viewpoint. It is shown
that the norms corresponding to heat kernels on the associated Lie
groups decompose as product norms under the natural isomorphism
$U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak
g_2)$. The seminorms corresponding to Green's functions are
examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It
is also shown that the algebraic dual space $U'$ is spanned by its
finite rank elements if and only if $\frak g$ is nilpotent.
Categories:17B35, 16S30, 22E30 
