Cohomology of the universal centralizers I: the adjoint group case

TL;DR

This paper computes the rational cohomology of the universal centralizer for complex adjoint semisimple groups, revealing a surprisingly simple structure despite complex topology, and sets the stage for broader future analysis.

Contribution

It provides the first explicit computation of the rational cohomology of the universal centralizer in the adjoint case, showing it is trivial and paving the way for generalizations.

Findings

Rational cohomology of $J_G$ is trivial, matching that of a point.

Topology of $J_G$ becomes more complex with higher rank.

Future work will analyze the cohomology for general semisimple groups.

Abstract

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.