Representations of Solvable Lie Algebras. IV: An Elementary Proof of The (U/P) E -Structure Theorem

Abstract

In this paper we give a shorter and much more elementary proof of a theorem which describes the structure of certain localisations of the enveloping algebra of a completely solvable Lie algebra. Such a localisation is shown to be a twisted group algebra where the group is free abelian of finite rank and the coefficient ring is a polynomial extension of a Weyl algebra. Introduction. Let g be a completely solvable Lie algebra over a field k of characteristic zero, P a prime ideal of the enveloping algebra U = U(g), E the semicentre of U/P (see ? 1 below) and (U/P )E the corresponding quotient ring. (The interest of (U/P)E is that any simple U-module with annihilator P is naturally a (U/P )E-module and (U/P )E is a simple algebra.) If g is nilpotent then E is the centre of U/P and (U/P)E is a Weyl algebra An, n > 0, where An = K[yl, * .. * yng a/ ayl X .. a/ ayn 1 the "ring of differential operators with polynomial coefficients". In [4] it was shown that if g is completely solvable then (U/P)E may be regarded as a ring of differential operators in which the multiplication has been altered by a 2cocycle. In [5] the cohomology group involved was determined and it followed readily that such a "twisted" ring of differential operators had a much more elementary presentation as a "group algebra" of a free abelian group in which the group elements induce automorphisms on the coefficient ring. This group algebra is constructed (see ? 1) from the data (V, 6, G) (and is denoted by ?(V, 6, G)) where V is a finite-dimensional vector space, 8 is an alternating bilinear form on V and G is a finitely generated subgroup of the additive group of the dual space V*. The proof that (U/P)E is isomorphic to 6T(V, 6, G) as given in [4] and [5] is rather complicated. In particular, the proofs in [4] depend on results on smash products from [3]. In this paper we give an elementary proof that (U/P )E (V, 8, G ), which is completely independent of [3]. In order to make the whole argument intelligible we briefly sketch those Received by the editors September 8, 1976. AMS (MOS) subject classifications (1970). Primary 17B35; Secondary 16A08.