Finite-dimensional representations of invariant differential operators, II

✍ Scribed by Gerald W. Schwarz

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 97 KB
• Volume: 266
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00281-3

✦ Synopsis

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 = SL 3 (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 finitedimensional 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.