The Regular Representation

This lecture introduces the regular representation, a crucial tool in proving results for a linear algebraic group acting on a variety. The regular representation maps functions to linear maps, forming an abstract representation from the group to a vector space. The lecture explains the concepts of locally finite and rational representations, highlighting their importance in approximating homomorphisms between algebraic groups. The regular representation on an affine variety is shown to be locally finite and rational, providing a powerful tool for studying group actions on varieties. The proof demonstrates how the regular representation behaves under group actions, showcasing its utility in analyzing properties of group actions through representations.