On torus orbit closures in flag varieties
Let $G$ be a semisimple Lie group, $T$ a maximal torus of $G$, and $B$ a Borel subgroup of $G$ containing $T$. For a parabolic subgroup $P$ containing $B$, the homogeneous space $G/P$ is a smooth projective algebraic variety, called a flag variety. If $P$ is minimal, that is, $P=B$, then we call $G/B$ a full flag variety. When $P$ is maximal, we call $G/P$ a Grassmannian variety. The left multiplication of $T$ on $G$ induces that on $G/P$. Considering $T$-orbit closures, we obtain lots of toric varieties including toric Schubert varieties and toric Richardson varieties. In this talk, we study the topological and geometric properties of these toric varieties. Especially, we consider toric Schubert varieties in $G/B$ and their cohomological rigidity. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.