Enveloping algebras of Lie groups with discrete series (Q1209673)

Let \({\mathfrak g}\) be the Lie algebra and \(U({\mathfrak g})\) the enveloping algebra of the real connected unimodular Lie group \(G\) with a discrete series. This paper proves that \(U({\mathfrak g})\), when localized at one central element, becomes a tensor product of a Weyl algebra over a ring of Laurent polynomials in one variable and the enveloping algebra of a reductive Lie algebra. In particular it settles the Gelfand-Kirillov conjecture in the affirmative for unimodular solvable groups with a discrete series. The proof uses an earlier structural result of the first author [Ann. Math., II. Ser. 104, 431-458 (1976; Zbl 0359.22007)] that any such group is the semidirect product of a semisimple group and a \(H\)- group. The methods are classical and largely inspired by [\textit{J. Dixmier}, Algèbre enveloppantes (1974; Zbl 0308.17007)] and the calculations are very explicit.