  We study a problem of the geometric quantization for the quaternion
projective space. First we explain a Kaehler structure on the punctured
cotangent bundle of the quaternion projective space, whose Kaehler form
coincides with the natural symplectic form on the cotangent bundle and show
that the canonical line bundle of this complex structure is holomorphically
trivial by explicitly constructing a nowhere vanishing holomorphic global
section. Then we construct a Hilbert space consisting of a certain class of
holomorphic functions on the punctured cotangent bundle by the method of
pairing polarization and incidentally we construct an operator from this
Hilbert space to the $L_2$ space of the quaternion projective space. Also we
construct a similar operator between these two Hilbert spaces through the Hopf
fiberation. We prove that these operators quantize the geodesic flow of the
quaternion projective space to the one parameter group of the unitary Fourier
integral operators generated by the square root of the Laplacian plus suitable
constant. Finally we remark that the Hilbert space above has the reproducing
kernel.
