# Invariant differential operators on the tangent space of some symmetric spaces

• Volume: 49, Issue: 6, page 1711-1741
• ISSN: 0373-0956

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## Abstract

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Let $𝔤$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta$ and set $𝔨=\mathrm{Ker}\left(\vartheta -I\right)$, $𝔭=\mathrm{Ker}\left(\vartheta +I\right)$. We consider the differential operators, $𝒟\left(𝔭{\right)}^{K}$, on $𝔭$ that are invariant under the action of the adjoint group $K$ of $𝔨$. Write $\tau :𝔨\to \mathrm{Der}\phantom{\rule{0.166667em}{0ex}}𝒪\left(𝔭\right)$ for the differential of this action. Then we prove, for the class of symmetric pairs $\left(𝔤,𝔨\right)$ considered by Sekiguchi, that $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪\left(𝔭{\right)}^{K}\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. An immediate consequence of this equality is the following result of Sekiguchi: Let $\left({𝔤}_{0},{𝔨}_{0}\right)$ be a real form of one of these symmetric pairs $\left(𝔤,𝔨\right)$, and suppose that $T$ is a ${K}_{0}$-invariant eigendistribution on ${𝔭}_{0}$ that is supported on the singular set. Then, $T=0$. In the diagonal case $\left(𝔤,𝔨\right)=\left({𝔤}^{\text{'}}\oplus {𝔤}^{\text{'}},{𝔤}^{\text{'}}\right)$ this is a well-known result due to Harish-Chandra.

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