Complexity of modules over classical Lie superalgebras

The complexity of the simple and the Kac modules over the general linear Lie superalgebra $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$ of type $A$ was computed by Boe, Kujawa, and Nakano in 2012. A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\mathrm{\mid }2n)$ of type $C$. The two Lie superalgebras are both of Type I which will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. We also compute a categorical invariant, z-complexity, introduced in Boe et al., and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the z-complexity of the simple modules over the Type II Lie superalgebras $\mathfrak{o}\mathfrak{s}\mathfrak{p}(3\mathrm{\mid }2)$, $D(2,1;\alpha )$, $G(3)$, and $F(4)$.