Applying projective functors to arbitrary holonomic simple modules (Q6591572)

The present paper studies the action of projective functors (also known as translation functors) on holonomic simple modules over finite-dimensional complex semisimple Lie algebras \(\mathfrak{g}\). Here, a simple module is called holonomic if its Gelfand-Kirillov dimension is minimal amongst all simple modules with the same annihilator.\N\NThe main result is that if \(L\) is a holonomic simple \(\mathfrak{g}\)-module and \(V\) a finite-dimensional \(\mathfrak{g}\)-module, then \(V\otimes_\mathbb{C}L\) contains an essential semisimple submodule of finite length. In type~\(A\), this also holds for non-holonomic simple modules by \textit{C.-W. Chen} et al. [Math. Z. 297, No. 1--2, 255--281 (2021; Zbl 1486.17012)]. The proof uses the bicategory of projective functors and the combinatorics of Kazhdan-Lusztig cells to reduce the claim to a specific situation, and the main step is to show the non-existence of certain ``strange'' subquotients. The authors conjecture that \(V\otimes_\mathbb{C}L\) does not have any strange subquotients even if \(L\) is not holonomic.\N\NIn type~\(A\), the paper also describes other aspects of the structure of \(V\otimes_\mathbb{C}L\), which are conjectured to hold in general. As an application respectively motivation, the authors explain how the main result can be used to study simple supermodules over Lie superalgebras by induction of simple modules over the even part.