Sur les champs de vecteurs invariants sur l'espace tangent d'un espace sym\'etrique

TL;DR

This paper studies invariant vector fields on the tangent space of a symmetric space, characterizing derivations of invariant smooth functions that preserve a specific ideal related to a discriminant function.

Contribution

It provides a description of derivations induced by invariant vector fields on the algebra of invariant functions, linking them to the preservation of a discriminant ideal.

Findings

The image of the map from invariant vector fields to derivations matches those preserving the discriminant ideal.

Invariant vector fields induce derivations that preserve a specific ideal in the algebra of invariant functions.

The work characterizes the structure of derivations compatible with the symmetry and invariance properties.

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 $\mathfrak q$.