Group Actions: Differential of Orbit Map

This lecture explores the differential of group actions on vector spaces, focusing on the representation of a linear algebraic group on a finite dimensional vector space. The differential of the action is shown to be related to the Lie algebra of the group and the tangent space of the vector space. The lecture also delves into the orbit map, discussing its differential, image, and kernel, providing insights into the behavior of linear actions and stabilizers. Additionally, a lemma is presented regarding the surjectivity of differentials in dominant morphisms between irreducible affine varieties. The proof involves detailed algebraic geometric reasoning and concludes with a practical example involving the action of a general linear group on square matrices.