Finite-dimensional representations of invariant differential operators, II

Abstract

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 B:=D(g)G, where G=SL3(C) and D(g)G denotes the algebra of G-invariant polynomial differential operators on the Lie algebra g of G. We also considered the representation theory of the subalgebra A of B, where A is generated by the invariant functions O(g)G⊂B and the invariant constant coefficient differential operators S(g)G⊂B. Among other things, we found that the finite-dimensional representations of A and B are completely reducible, and we could reduce the study of the finite-dimensional irreducible representations of B to those of A. Irreducible finite-dimensional representations of A 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 A and B.