Journal of Lie Theory 25 (2015), No. 2, 307--326
Copyright Heldermann Verlag 2015
Sur les champs de vecteurs invariants sur l'espace tangent d'un espace symétrique réductif
Abderrazak Bouaziz
Université de Poitiers, Laboratoire de Mathématiques et Applications, BP 30179, 86962 Futuroscope-Chasseneuil, France
bouaziz@math.univ-poitiers.fr
Nouri Kamoun
Faculté des Sciences, 5019 Monastir, Tunisia
nouri.kamoun@fsm.rnu.tn
[Abstract-pdf]
\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 $\q$.
Keywords: Lie Group, symmetric space, invariant vector field, Taylor expansion.
MSC: 17B20, 22F30, 22E30
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