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Using the generalized coherent states we argue that the path integral formulae for SU (2) and SU (1, 1) (in the discrete series) are WKB exact, if the starting point is expressed as the trace of e −iTĤ withĤ being given by a linear combination of generators. In our case, WKB approximation is achieved by taking a large "spin" limit: J, K → ∞. The result is obtained directly by knowing that the each coefficient vanishes under the J −1 (K −1 ) expansion and is examined by another method to be legitimated. We also point out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression leads us to a wrong result. Therefore a great care must be taken when some geometrical action would be adopted,

U(N ) coherent states over Grassmann manifold, G N,n ≃ U(N )/(U(n) × U(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 U(N ) representation. Thus when k → ∞ a usual semiclassical approximation, by regarding k ∼ 1/h, 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.

Analysis of the WKB exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart D N,1 is studied. U(N + 1) or U(N, 1) based on the Schwinger bosons leads us to CP N or D N,1 path integral expression for the quantity, tre −iHT , with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.

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