Abstract: Let W be a reflection group in a vector space and P a rational polytope that is invariant under the W-action. On the one hand, the action of W on P induces a W-action on the toric variety X_P associated with P. On the other hand, W-action on P defines the region R of P such that each point x in R represents the W-orbit of x. Since the closure \overline{R} of R is again a rational polytope, we have the toric variety X_{\overline{R}} associated to it. It is conjectured that X_\overline{R} and the quotient variety {X_P}/W are isomorphic. In this talk, we shall see various examples of toric varieties and cohomological evidences supporting this conjecture. We also see its application to the cohomological study of regular Hessenberg varieties. This talk is based on the joint work with T. Horiguchi, M. Masuda and J. Shareshian.

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