Mirković-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 $G$. Then $(G,G^\sigma)$ is a symmetric pair. In this paper, we get a bijection between the set of $\sigma$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\sigma$, 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.