  We consider BPS-counting functions $\mathcal{Z}_{N,M}$ of $N$ parallel
M5-branes probing a transverse $\mathbb{Z}_M$ orbifold geometry. These brane
web configurations can be dualised into a class of toric non-compact Calabi-Yau
threefolds which have the structure of an elliptic fibration over (affine)
$A_{N-1}$. We make this symmetry of $\mathcal{Z}_{N,M}$ manifest in particular
regions of the parameter space of the setup: we argue that for specific choices
of the deformation parameters, the supercharges of the system acquire
particular holonomy charges which lead to infinitely many cancellations among
states contributing to the partition function. The resulting (simplified)
$\mathcal{Z}_{N,M}$ can be written as a sum over weights forming a single
irreducible representation of the Lie algebra $\mathfrak{a}_{N-1}$ (or its
affine counterpart). We show this behaviour explicitly for an extensive list of
examples for specific values of $(N,M)$ and generalise the arising pattern for
generic configurations. Finally, for a particular compact M5-brane setup we use
this form of the partition function to make the duality $N\leftrightarrow M$
manifest.
