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

Abstract

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

arXiv comment18 pages, in French