Automorphism groups of toroidal horospherical varieties

We establish a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties, that is, toric fibrations over rational homogeneous spaces. The key ingredient is a characterization of the Demazure roots of toric fibers that extend to the total space. In particular, we provide a criterion for the reductivity of the connected automorphism groups of such varieties. As an application, we prove the K-unstability of certain $\mathbb{P}^1$-bundles over rational homogeneous spaces.