Modules of Covariants

This lecture explores the decomposition of the circle of the coordinate ring of a G variety into a direct sum of simple submodules, showing that the circle is the direct sum of simple submodules of type lambda. It also demonstrates that the zero class is the subring of invariance under the action, and each OX lambda is an OX G module. The proof involves showing that the regular representation is locally finite and rational, and that the action of g on O of X preserves the algebra structure. Additionally, the lecture introduces the concept of modules of covariance, which are modules over the ring of invariance.