Adjoint vector fields on the tangent space of semisimple symmetric spaces

Abstract: Let $\frak{g}$ be a semisimple complex Lie algebra and $\vartheta \in \operatorname{Aut} \frak{g}$ be an involution. If $\frak{g} = \frak{k} \oplus \frak{p}$ is the decomposition associated to $\vartheta$, define a Lie subalgebra of $\operatorname{End} \frak{p}$ by $\tilde{\frak{k}} = \{X : \forall f \in S(\p^*)^\frak{k},  X.f = 0\}$. We prove that $\operatorname{ad}_\frak{p}(\frak{k}) = \tilde{\frak{k}}$ if and only if each irreducible factor of rank one of the symmetric pair $(\frak{g},\frak{k})$ is isomorphic to $(\frak{so}(q+1), \frak{so}(q))$.

Keywords: complex Lie algebras, semisimple Lie algebras, Lie subalgebras, symmetric pairs

Classification (MSC91): 53C35; 17B20

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