Overview on the theory of double flag varieties for symmetric pairs

Let $G$ be a connected reductive algebraic group and its symmetric subgroup $K$. The variety $\text{dblFV}=K\mathrm{/}Q\times G\mathrm{/}P$ is called a double flag variety, where $Q$ and $P$ are parabolic subgroups of $K$ and $G$ respectively. In this article, we make a survey on the theory of double flag varieties for a symmetric pair $(G,K)$ and report entirely new results and theorems on this theory. Most important topic is the finiteness of $K$-orbits on $\text{dblFV}$. We summarize the classification of $\text{dblFV}$ of finite type, which are scattered in the literatures. In some respects such classifications are complete, and in some cases not. In particular, we get a classification of double flag varieties of finite type when a symmetric pair is of type AIII, using the theorems of Homma who describes ``indecomposable’’ objects of such double flag varieties. Together with these classifications, newly developed embedding theory provides double flag varieties of finite type, which are new. Other ingredients in this article are Steinberg theory, generalization of Robinson-Schensted correspondence, and orbit classification via quiver representations. We hope this article is useful for those who want to study the theory of double flag varieties.