  We determine, in an inductive framework, the vertices of the polytope
$P(s,K)$ controlling the conjugacy classes of elements which product to one in
the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This
extends earlier work of the authors in related contexts. One feature of this
work is the use of Kontsevich compactifications of the moduli of $P$-bundles
(replacing the use of quot schemes in type A) which are related to
semi-infinite geometry. We also obtain a quantum generalization of Fulton's
conjecture valid for all $G$.
