We determine the dimensions of $\textrm{Ext}$-groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category $\mathcal{O}$ for complex semisimple Lie algebras and affine Kac-Moody algebras.

The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\text{S}{{\text{L}}_{2}}(\mathbb{Z})$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_{r}^{\left( 1 \right)}$ and $D_{r}^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the $B_{r}^{(1)},D_{r}^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_{2}^{(1)}\cong C_{2}^{(1)}$ and $D_{7}^{(1)}$. The ${{B}_{2,3}}$ and ${{D}_{7,3}}$ exceptionals are cousins of the ${{\varepsilon }_{6}}$-exceptional and ${{\varepsilon }_{8}}$-exceptional, respectively, in the $\text{A-D-E}$ classification for $A_{1}^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine $u(1)$.