Mirkovic-Vilonen cycles and polytopes for a Symmetric pair

Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\sigma$ be a Dynkin automorphism on $G$. In this paper, we get a bijection between the set of $\text{st}$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^{\text{st}}$, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula.