Quantization of the Geodesic flow on Quaternion Projective Spaces

Abstract

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.