Sur les champs de vecteurs invariants sur l'espace tangent d'un espace symétrique réductif

A. Bouaziz ; N. Kamoun

Journal of Lie theory, Tome 25 (2015) no. 2, pp. 307-326

Résumé

\def\g{{\frak g}} \def\q{{\frak q}} \def\X{{\frak X }} Let $G$ be a real reductive and connected Lie group and $\sigma$ an involution of $G$. Let $H$ denote the identity component of the group of fixed points of $\sigma$, $\g$ the Lie algebra of $G$ and $\q$ the $-1$ eigenspace of $\sigma$ in $\g$. The group $H$ acts naturally on $\q$ via the adjoint representation. Let $C^{\infty}(\q)^H$ denote the algebra of $H$-invariant smooth functions on $\q$, and $\X(\q)^H$ the space of $H$-invariant smooth vector fields on $\q$. Any vector field $X\in \X(\q)^H$ defines naturally a derivation $D_X$ of the algebra $C^{\infty}(\q)^H$. We prove that the image of the map $X\mapsto D_X$ is the set of derivations of the algebra $C^{\infty}(\q)^H$ preserving the ideal $\Phi C^{\infty}(\q)^H$ of $C^{\infty}(\q)^H$, where $\Phi$ is a discriminant function on $\q$.

Détail

Classification : 17B20, 22F30, 22E30
Mots-clés : Lie Group, symmetric space, invariant vector field, Taylor expansion