  \(\Un{N}\) coherent states over Grassmann manifold,
\(\grsmn{N}{n}\simeq\Un{N}/ (\Un{n}\times \Un{N-n})\), are formulated to be
able to argue the WKB-exactness, so called the localization of
Duistermaat-Heckman, in the path integral representation of a character
formula. The exponent in the path integral formula is proportional to an
integer \(k\) that labels the \(\Un{N}\) representation. Thus when \(k
\rightarrow\infty\) a usual semiclassical approximation, by regarding \(k \sim
1 / \hbar\), can be performed yielding to a desired conclusion. The mechanism
of the localization is uncovered by means of a view from an extended space
realized by the Schwinger boson technique.
