Orbit spaces of torus actions on Hessenberg varieties

Abstract

We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to $S^{n+1} \setminus (U_1 \sqcup \ldots \sqcup U_l)$ where $S^{n+1}$ is the $(n+1)$--sphere and $U_1, \ldots, U_l$ are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.