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 \mathrm{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 \mathrm{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 Poincaré 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 \mathrm{Hess}_{\Theta}(N,I) and the W_{\Theta} -invariant subring of the cohomology ring of the regular nilpotent Hessenberg variety \mathrm{Hess}(N,I) . Furthermore, we provide a description of the cohomology ring for a regular nilpotent partial Hessenberg variety \mathrm{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.