We describe the branching rule from $S{p}_4$ to $S{L}_2$, where the latter is embedded via its action on binary cubic forms. We obtain both a numerical multiplicity formula, as well as a minimal system of generators for the geometric realization of the rule.

Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels de type $G_2$ sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés. Les appendices traitent de l’analogie avec les espaces symétriques réels et des espaces symétriques associés à $G_2$ réel et complexe.

Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels des type $F_4$ ou $E_6$ sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés.

Let $H_n$ be the Heisenberg group of dimension $2n+1$. Let ${\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n$ be the partial sub-Laplacians on $H_n$ and $T$ the central element of the Lie algebra of $H_n$. We prove that the kernel of the operator $m\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n,-iT\right)$ is in the Schwartz space $S\left(H_n\right)$ if $m\in S\left({\mathbb{R}}^{n+1}\right)$. We prove also that the kernel of the operator $h\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n\right)$ is in $S\left(H_n\right)$ if $h\in S\left({\mathbb{R}}^n\right)$ and that the kernel of the operator $g\left(\mathcal{L},-iT\right)$ is in $S\left(H_n\right)$ if $g\in S\left({\mathbb{R}}^2\right)$. Here $\mathcal{L}={\mathcal{L}}_1+\mathrm{\dots }+{\mathcal{L}}_n$ is the Kohn-Laplacian on $H_n$.

The universe we see gives every sign of being composed of matter. This is considered a major unsolved problem in theoretical physics. Using the mathematical modeling based on the algebra $𝐓:=𝐂\otimes 𝐇\otimes 𝐎$, an interpretation is developed that suggests that this seeable universe is not the whole universe; there is an unseeable part of the universe composed of antimatter galaxies and stuff, and an extra 6 dimensions of space (also unseeable) linking the matter side to the antimatter—at the very least.

A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is...

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ${ℝ}^d$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ${ℝ}^d$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ${\mathbb{P}}^{d-1}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...