Soit $G_{\mathbf{C}}$ un groupe de Lie complexe et $G_{\mathbf{R}}$ une forme réelle fermée de $G_{\mathbf{C}}$. Un couple $(G_{\mathbf{C}},G_{\mathbf{R}})$ est dit pseudo-convexe, s’il existe sur $G_{\mathbf{C}}$ une fonction régulière, strictement p.s.h., invariante par l’action de $G_{\mathbf{R}}$ et d’exhaustion sur $G_{\mathbf{C}}/G_{\mathbf{R}}$. On dit que $G_{\mathbf{R}}$ est à spectre imaginaire pur, si pour tout $X$ de Lie$\left(G_{\mathbf{R}}\right)$, les valeurs propres de ad$X$ sont imaginaires pures. Pour $G_{\mathbf{C}}$ à radical simplement connexe, cette dernière propriété équivaut à la pseudo-convexité de $(G_{\mathbf{C}},G_{\mathbf{R}})$. Pour $(G_{\mathbf{C}},G_{\mathbf{R}})$ pseudo-convexe et sous une hypothèse de sous-groupe discret,...

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.

Let $X$ be a del Pezzo surface of degree $1$, and let $G$ be the simple Lie group of type ${\text{E}}_8$. We construct a locally closed embedding of a universal torsor over $X$ into the $G$-orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Néron-Severi torus $T$ of $X$ identified with a maximal torus of $G$ extended by the group of scalars. Moreover, the $T$-invariant hyperplane sections of the torsor defined by the roots of $G$ are the inverse images...

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra . Furthermore, assume that the linear adjoint action of on is diagonalizable with non-purely imaginary eigenvalues. Let $\tau =In{d}_H^{N⋊H}1$. We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (+)*, and a...

We define partial spectral integrals $S_R$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $log(2+L_{\alpha })f\in L²$, where $L_{\alpha }$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R}f\left(x\right)$ converges a.e. to f(x) as R → ∞.

Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_{R}f:={\int }_0^{R}d{E}_{\lambda }f,R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_0^{\infty }\lambda d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that $S_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log(2+ℒ)f\in L^2\left(G\right).$