K-Stability of Fano spherical varieties

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Sz'ekelyhidi, it yields a criterion for the existence of a K"ahler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of K"ahler-Ricci solitons.