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Search: MSC category 22E30 ( Analysis on real and complex Lie groups [See also 33C80, 43-XX] )

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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, sigma-compactness, 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 Sobolev-type 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 Littlewood-Paley decomposition.

Keywords:Sobolev inequalities, polynomial volume growth Lie groups

3. CJM 2010 (vol 62 pp. 563)

Ishii, Taku
Whittaker Functions on Real Semisimple Lie Groups of Rank Two
We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Categories:11F70, 22E30

4. CJM 2009 (vol 61 pp. 961)

Bernon, Florent
Transfert des intégrales orbitales pour les algèbres de Lie classiques
Dans cet article, on consid\`ere un groupe semi-simple $\rmG$ classique r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un sous-groupe de Cartan compact. On d\'efinit une famille de sous-alg\`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 sous-alg\`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 Harish-Chandra 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, 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

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 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

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 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

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 non-tempered 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 Harish-Chandra 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 rank-one 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 square-integrable $\Gamma$-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 DeGeorge-Wallach 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 Segal-Bargmann Transform for Lie Groups of Compact Type
I consider a two-parameter 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 infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann 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)

Lohoué, Noël; Mustapha, Sami
Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes
Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$ and let $\nabla$ be the associated gradient. In this paper we investigate the boundness of the Riesz transform $\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with exponential volume growth and on certain homogenous spaces.

Categories:22E30, 35H05, 43A80, 43A85

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

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