Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Abstract

We study three related homological properties of modules in the BGG category $ \mathcal {O}$ for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category $ \mathcal {O}$ with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category $ \mathcal {O}$ for $ \mathfrak{gl}(m\vert n)$ can contain infinitely many non-equivalent blocks, which we work out explicitly for $ \mathfrak{gl}(3\vert 1)$. All of this is in sharp contrast with category $ \mathcal {O}$ for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.