The support of an irreducible Lie algebra representation
Let \(\mathbb{G}\) be a simple complex finite-dimensional Lie algebra and \(\mathbb{H}\) be a Cartan subalgebra of \(\mathbb{G}\). The well-known result by S. Fernando and V. Futorny states that any simple weight (with respect to \(\mathbb{H}\)) \(\mathbb{G}\)-module is either dense (i.e. has the maximal possible support, which coincides with a coset in \(\mathbb{H}^*\) with respect to the root lattice) or is a unique simple quotient of some generalized Verma module, associated with a parabolic subalgebra \(P\) of \(\mathbb{G}\) and with a simple dense module over the semi-simple part of \(P\). In the last case the support of our module is bounded in the weight space by a hyperplane and can be easily constructed. In the paper under review, the authors generalize this result to the following situation. Let \(\mathbb{G}\) be any finite-dimensional complex Lie algebra and \(\mathbb{H}\) be a nilpotent subalgebra of \(\mathbb{G}\). Then in a natural sense one can consider the notion of a weight (with respect to \(\mathbb{H}\)) \(\mathbb{G}\)-module. In the first main result of the paper, the authors describe in a simple way the support of any simple \(\mathbb{H}\)-weight \(\mathbb{G}\)-module. In the second main result, they consider the case when \(\mathbb{H}\) contains a Cartan subalgebra for the semi-simple part of \(\mathbb{G}\). In this case they show how one can reconstruct the support of a simple \(\mathbb{H}\)-weight \(\mathbb{G}\)-module in terms of its convex hull and of a certain sublattice of the root lattice. The authors also explain their results with three examples, which include \(sl(2)\), \(sl(3)\) and the direct sum of \(sl(2)\) with a Heisenberg algebra.
scientific article; zbMATH DE number 1234245