1 4 Fe b 20 04 Dixmier ’ s Problem 6 for the Weyl Algebra ( the Generic Type Problem )

In the fundamental paper [11], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problem 3 was solved by Joseph and Stein [15] (using results of McConnel and Robson [20]), problem 5 was solved by the author in [7]. Using a (difficult) polarization theorem for the Weyl algebra A1 Joseph [15] solved problem 6. Problems 1, 2, and 4 are still open. Note that these problems make sense for non-commutative algebras of Gelfand-Kirillov < 3, and some of them (after minor modifications) make sense for an arbitrary noncommutative algebra. In this paper a short proof is given to Dixmier’s problem 6 for many noncommutative algebras A of Gelfand-Kirillov < 3 (a typical example is the ring of differential operators D(X) on a smooth irreducible algebraic curve X). An affirmative answer to this problem leads to clarification of the structure of maximal commutative subalgebras of these algebras A (a typical example of the algebra A is any noncommutative subalgebra of Gelfand-Kirillov < 3 of the division ring Q(D(X)) for the algebra D(X)), and the result is rather surprising: for a given maximal commutative subalgebra C of the algebra A, (almost) all non-central elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent, (ii) weakly nilpotent, (iii) generic, (iv) generic except for a subset Ka + Z(A) of strongly semi-simple elements, (iv) generic except for a subset Ka+Z(A) of weakly semi-simple elements, where K := K\{0} and Z(A) is the centre of the algebra A. For an arbitrary algebra A, Dixmier’s problem 6 is essentially a question: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], degt(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (eg, rings of differential operators D(Y ) on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra U(G) of a completely solvable algebraic Lie algebra G).