On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras (Q2706607)

Let \(G\) be a connected semisimple real Lie group with finite center, let \(K\) be a maximal compact subgroup of \(G\), and let \(Q\) be a minimal parabolic subgroup of \(G\) with the corresponding Langlands decomposition \(Q = MAN\). If \(\mathfrak g\) and \(\mathfrak n\) denote the complexification of the Lie algebra of \(G\) resp. \(N\) and \(\mathcal A\) denotes the augmentation ideal of the universal enveloping algebra of \(\mathfrak n\), then an important result of \textit{W. Casselman} says that \({\mathcal A}V\neq V\) for every admissible finitely generated \(({\mathfrak g},K)\)-module \(V\). Although the statement is purely algebraic, Casselman's original proof is analytic in nature. The first algebraic proof is due to \textit{A. Beilinson} and \textit{J. Bernstein} [in: Representation Theory of Reductive Groups, Prog. Math. 40, 35-52 (1983; Zbl 0526.22013)] being a consequence of a more general result proved via \(\mathcal D\)-module theory techniques. Moreover, \textit{W. Casselman} and \textit{M. S. Osborne} [Math. Ann. 233, 193-198 (1978; Zbl 0355.20041)] gave an algebraic proof under the strong restriction that the nilpotent radical \(\mathfrak n\) is abelian.NEWLINENEWLINENEWLINEThe main purpose of the paper under review is to study the structure of prime ideals related to finitely generated modules over universal enveloping algebras of finite-dimensional nilpotent Lie algebras over an algebraically closed field of characteristic zero. More generally, let \(\mathcal K\) be the class of all Noetherian rings such that every ideal has a centralizing sequence of generators and every prime ideal is completely prime. Then it is well-known that for every prime ideal \(\mathcal P\) of \({\mathcal R}\in {\mathcal K}\) the set \({\mathcal R}\setminus {\mathcal P}\) is a (left and right) denominator set, i.e., one can localize \(\mathcal R\) and any \(\mathcal R\)-module with respect to this set. Then one can define the support \(\text{Supp}(M)\) of an \(\mathcal R\)-module \(M\) completely analogously to the commutative case. Set also \({\mathcal V}(M) := \{{\mathcal P}\in \text{Spec}({\mathcal R})\mid{\mathcal P}\supseteq\text{Ann}_{\mathcal R}(M)\}\). The first main result says that for every \({\mathcal R}\in{\mathcal K}\) and every finitely generated \(\mathcal R\)-module \(M\) the support of \(M\) is contained in \({\mathcal V}(M)\) and \({\mathcal P} M\neq{\mathcal P}\) for every \({\mathcal P}\in\text{Supp}(M)\). Contrary to the commutative case there is no equality in general. The support can even be empty for a faithful module, as the author shows for the universal enveloping algebra of the three-dimensional Heisenberg algebra. This indicates already why the support is the main obstacle in extending the algebraic proof of Casselman and Osborne to non-abelian nilpotent radicals \(\mathfrak n\). Nevertheless, the author is able to prove a stability result of \textit{W. Casselman} and \textit{M. S. Osborne} [loc. cit.] in the non-commutative situation. This enables him to further pursue the Casselman-Osborne approach for those \(\mathfrak g\)-modules \(V\) which as \(\mathfrak n\)-modules are finitely generated and satisfy \(\text{Supp}(V) = {\mathcal V}(V)\). The author speculates that the latter might be true for every admissible finitely generated \(({\mathfrak g},K)\)-module \(V\). This would be a considerable generalization of Casselman's theorem since then there were (infinitely) many maximal ideals \(\mathcal M\) with \({\mathcal M}V\neq V\).