Flexibility of affine horospherical varieties of semisimple groups

Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a=(k,+)$ the additive group of $k$. An algebraic variety $X$ is said to be flexible if the tangent space at every regular point of $X$ is generated by the tangent vectors to orbits of various regular actions of $\mathbb{G}_a$. In 1972, Vinberg and Popov introduced the class of affine $S$-varieties which are also known as affine horospherical varieties. These are varieties on which a connected algebraic group acts with an open orbit in such a way that the stationary subgroup of each point in the orbit contains a maximal unipotent subgroup of $G$. In this paper the flexibility of affine horospherical varieties of semisimple groups is proved.
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