# 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 $\it{\Phi}C^{\infty}(\mathfrak q)^H$ of $C^{\infty}(\mathfrak q)^H$, where $\it{\Phi}$ is a discriminant function on $\mathfrak q$.
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https://hal.archives-ouvertes.fr/hal-00853349
Contributor : Abderrazak Bouaziz <>
Submitted on : Thursday, August 22, 2013 - 2:39:35 PM
Last modification on : Wednesday, September 5, 2018 - 1:30:08 PM
Long-term archiving on : Saturday, November 23, 2013 - 4:22:14 AM

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• HAL Id : hal-00853349, version 1
• ARXIV : 1308.5108

### Citation

Abderrazak Bouaziz, Nouri Kamoun. Sur les champs de vecteurs invariants sur l'espace tangent d'un espace symétrique. 2013. ⟨hal-00853349⟩

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