Essential dimension of reductive groups

Abstract

In this paper, we present the essential dimension of reductive groups. We propose a method to construct generically free representations of split reductive groups with finite or connected center, which gives upper bounds on the essential dimension of those groups. Combining our upper bound with previously known lower bound, we give the exact value of the essential dimension of some types of reductive groups. As an application, we determine the essential dimension of a semisimple group of classical type or E6, and its strict reductive envelope under certain conditions on its center. Among these, the result on groups of type B and D has an application to quadratic forms. We also give the essential dimension of some semisimple groups of type B for which we cannot apply our method using generically free representation.