  We evaluate the twisted partition function of four-dimensional $\mathcal{N} =
1$ supersymmetric Yang--Mills theory reduced to a point for all simple gauge
groups. The partition function is expressed as a sum of residues. The types of
residues that appear are classified by distinguished nilpotent orbits.
Surprisingly, the multiplicity of residues of each type is proportional to
their common value. The sum over residues has the same form as the Plancherel
measure for Yang's system of particles. Intriguingly, the precise constants
appearing in the Plancherel measure also appear in the formal degrees of
unipotent discrete series representations of $p$-adic groups.
