Finite-dimensional representations of invariant differential operators

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. Set , the algebra of G-invariant polynomial differential operators on V. Let be restriction, where denotes the differential operators on . Much attention of late has been given to the study of Imρ and Kerρ. Less well studied are properties of itself. For example: • What is the representation theory of ? What are the primitive ideals of ? • Does have finite-dimensional representations? In particular, is an FCR algebra? Little is known about these questions when G is noncommutative. We give answers for the adjoint representation of , already an interesting and difficult case.