Coherent states over Grassmann manifolds and the WKB-exactness in path integral

$\Un{N}$ 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.