Finite-dimensional representations of invariant differential operators, II

In Part I of this paper [G.W. Schwarz, Finite-dimensional representations of invariant differential operators, J. Algebra 258 (2002) 160–204] we considered the representation theory of the algebra , where and denotes the algebra of G-invariant polynomial differential operators on the Lie algebra of G. We also considered the representation theory of the subalgebra of , where is generated by the invariant functions and the invariant constant coefficient differential operators . Among other things, we found that the finite-dimensional representations of and are completely reducible, and we could reduce the study of the finite-dimensional irreducible representations of to those of . Irreducible finite-dimensional representations of are quotients of “Verma modules.” We found sufficient conditions for the irreducible quotients of Verma modules to be finite-dimensional, and we conjectured that these sufficient conditions are also necessary. In this paper we establish the conjecture, giving a complete classification of the finite-dimensional representations of and .