Mirkovic-Vilonen cycles and polytopes for a Symmetric pair

TL;DR

This paper establishes a bijection between invariant MV cycles for a group and its fixed point subgroup under a Dynkin automorphism, providing a new proof of the twining character formula.

Contribution

It introduces a bijection between invariant MV cycles for a group and those for its symmetric subgroup, extending the understanding of MV polytopes in symmetric pairs.

Findings

Bijection between invariant MV cycles for G and G^σ

Extension of the bijection to MV cycles in irreducible representations

New proof of the twining character formula

Abstract

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 $\st$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\st$, 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.