Cohomology of the universal centralizers I: the adjoint group case

We compute the rational cohomology of the universal centralizer $J_G$ (also known as the Toda system or BFM space) for a complex (connected) semisimple group $G$ of adjoint form. While $J_G$ exhibits interesting and increasingly complex topology as the rank of $G$ rises, its rational cohomology is surprisingly simple--it coincides with that of a point. In a subsequent work [Jin2], we will extend this analysis to the case of $J_G$ for general semisimple $G$. In particular, we will show that its rational cohomology has pure Hodge structure.