Globality in semisimple Lie groups

Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 493-536.

Dans la première section de cet article nous caractérisons les cônes convexes fermés de $W$ de l’algèbre de Lie $sl(2,\mathbf{R}{)}^n$, qui sont invariants sous l’action d’un groupe compact maximal du groupe adjoint et qui sont contrôlables dans le groupe $Sl(2,\mathbf{R}{)}^{n^{\sim }}$, c’est-à-dire tels que l’image exponentielle de $W$ engendre le groupe tout entier (Theorem 1.3). Dans la section 2 nous développons des instruments algébriques concernant le système de racines réelles relatives à une sous-algèbre de Cartan compacte plongée et les cônes invariants dans les algèbres de Lie semi-simples. Dans la section 3 nous utilisons ces instruments, en combinaison avec des résultats de la section 1, pour caractériser les cônes invariants dans une algèbres de Lie semi-simples qui sont contrôlables dans le groupe simplement connexe associé. Si $L$ est simple nous obtenons une caractérisation des cônes invariants $W\subseteq L$ qui sont globaux, c’est-à-dire pour lesquels il existe un semi-groupe fermé $S\subseteq G$ avec $L\left(S\right)=W$.

In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,\mathbf{R}{)}^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,\mathbf{R}{)}^{n^{\sim }}$, i.e., for which the subsemigroup $S=(expW)$ generated by the exponential image of $W$ agrees with the whole group $G$ (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly embedded Cartan algebras and invariant cones in semisimple Lie algebras. In Section 3 these tools, combined with the results from Section 1, yield a characterization of those invariant cones in a semisimple Lie algebra $L$ which are controllable in the associated simply connected Lie group $G$. If $L$ is simple, we even get a characterization of those invariant wedges $W\subseteq L$ which are global in $G$, i.e., for which there exists a closed subsemigroup $S\subseteq G$ having $W$ as its tangent wedge $L\left(S\right)$.

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