Regular nilpotent partial Hessenberg varieties

Let $G$ be a complex semisimple linear algebraic group. Fix a subset $\Theta$ of simple roots. Given a lower ideal $I$ in positive roots, one can define the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$ in the full flag variety $G/B$. For a $\Theta$-ideal $I$ (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ in the partial flag variety $G/P$. In this manuscript we first provide a summand formula and a product formula for the Poincar\'e polynomial of regular nilpotent partial Hessenberg varieties. It is a well-known result from Bernstein-Gelfand-Gelfand that the cohomology ring of the partial flag variety $G/P$ is isomorphic to the invariants in the cohomology ring of the full flag variety $G/B$ under an action of the parabolic Weyl group $W_\Theta$ generated by $\Theta$. We generalize this result to regular nilpotent partial Hessenberg varieties. More concretely, we give an isomorphism between the cohomology ring of a regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ and the $W_\Theta$-invariant subring of the cohomology ring of the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$. Furthermore, we provide a description of the cohomology ring for a regular nilpotent partial Hessenberg variety $\mbox{Hess}_\Theta(N,I)$ in terms of the $W_\Theta$-invariants in the logarithmic derivation module of the ideal arrangement $\mathcal{A}_I$, which is a generalization of the result by Abe-Masuda-Murai-Sato with the author.