Geometric construction of Gelfand-Tsetlin modules over simple Lie algebras (Q2001431)

The authors consider the Gelfand-Tsetlin modules of the following type. Let \(\theta\) be a the maximal root of a complex simple Lie algebra \(\mathfrak g\). Let \(s_{\theta}\) be a subalgebra generated by the corresponding \(\mathfrak{sl}_2\)-triple. Let \(Z(s_{\theta})\subset U(s_{\theta})\) be the center of its universal enveloping and let \(\Gamma_{\theta}\) be a subalgebra in \(U(\mathfrak g)\) generated by the Cartan subalgebra \(\mathfrak h\subset \mathfrak g\) and \(Z(s_{\theta})\). The \(\theta\)-Gelfand-Tsetlin module is a Gelfand-Tsetlin module associated with the commutative subalgebra \(\Gamma_{\theta}\subset U(\mathfrak g)\). When \(\mathfrak g = \mathfrak{sl}_n\) these modules form a subclass of Gelfand-Tsetlin modules with infinite-dimensional weight subspaces. In the paper under review for the \(\theta\)-Gelfand-Tsetlin module a geometric realization is given, the generators of the algebra are interpreted as differential operators.