2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71382The exceptional configuration of the minimal resolution $\hat{S}_G $ of a Kleinian quotient surface $S_G (:= \CZ^2/G)$ is depicted by a $A$-$D$-$E$ Coxeter-Dynkin diagram. In this article, we show that branching indices of the affine $A$-$D$-$E$ diagram is geometrically characterized by a certain special function $F$ of $S_G$ as the multiplicities of its divisor components in $\hat{S}_G$, a version parallel to the elliptic fibration near certain types of simple singular fibers in Kodaira's elliptic surface theory. We further obtain the uniqueness property of the function $F$ (modular local units) among all local functions in $S_G$ near the singular point whose divisors in $\hat{S}_G $ display the affine $A$-$D$-$E$ diagram configuration.Latex 12 pages; Typos fixed, Improved version of Theorem 1 with more discussions addedAlgebraic GeometryRepresentation Theory14J17, 14L30, 20C30, 32S25text