Abstract
We study a problem of the geometric quantization for the quaternionprojective space. First we explain a Kähler structure on the punctured cotangent bundleof the quaternion projective space, whose Kähler form coincides withthe natural symplectic form on the cotangent bundle and show thatthe canonical line bundle of this complex structure is holomorphicallytrivial by explicitly constructing a nowhere vanishing holomorphicglobal section. Then we construct a Hilbert space consisting of acertain class of holomorphic functions on the punctured cotangentbundle by the method ofpairing polarization and incidentally we construct an operatorfrom this Hilbert space to the L 2 space of the quaternionprojective space. Also we construct a similar operator between thesetwo Hilbert spaces through the Hopf fiberation.We prove that these operators quantizethe geodesic flow of the quaternion projective space tothe one parameter group of the unitary Fourier integral operatorsgenerated by the square root of the Laplacian plus suitable constant.Finally we remark that the Hilbert space above has the reproducing kernel.
Similar content being viewed by others
References
Besse, A. L.: Manifolds All of Whose Geodesics Are Closed, Springer-Verlag, Berlin, 1978.
Furutani, K. and Tanaka, R.: A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to geometric quantization I, J. Math. Kyoto Univ. 34 (1994), 719-737.
Furutani, K. and Yoshizawa, S.: A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to geometric quantization II, Japan. J. Math. 21 (1995), 355-392.
Furutani, K.: A Kähler structure on the punctured cotangent bundle of the Cayley projective plane, in: de Gosson (ed.), Proceedings of the Conference in the Honor of Jean Leray, Karlskrona, Kluwer Acad. Publ., Dordrecht, to appear.
Ii, K.: On a Bargmann-type transform and a Hilbert space of holomorphic functions, Tôhoku Math. J. (1) 38 (1986), 57-69.
Ii, K. and Morikawa, T.: Kähler structures on the tangent bundle of Riemannian manifolds of constant positive curvature, Bull. Yamagata Univ. Natur. Sci. 14 (1999), 141-154.
Lichtenstein, W.: A system of quadrics describing the orbit of the highest weight vector, Proc. Amer. Math. Soc. 84 (1982), 605-608.
Murakami, S.: Exceptional Simple Lie Groups and Related Topics in Recent Differential Geometry, in Differential Geometry and Topology, Springer Lecture Notes 1369, Springer, New York, 1987.
Rawnsley, J. H.: Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. 28 (1977), 403-415.
Rawnsley, J. H.: A non-unitary pairing of polarization for the Kepler problem, Trans. Amer. Math. Soc. 250 (1979), 167-180.
Souriau, J. M.: Sur la variété de Kepler, Symposia Math. 14 (1974), 343-360.
Sz?ke, R.: Complex structures on the tangent bundle of Riemannian manifolds, Math. Ann. 291 (1991), 409-428.
Sz?ke, R.: Adapted complex structures and geometric quantization, Nagoya J. Math. 154 (1999), 171-183.
Sz?ke, R.: Involutive structures on tangent bundles of symmetric spaces, Math. Ann. 319 (2001), 319-348.
Woodhouse, N. M. J.: Geometric Quantization, 2nd edn., Oxford Mathematical Monographs, Clarendon Press, Oxford, 1997.
Yokota, I.: Realizations of involutive automorphisms ? and G ? of exceptional linear Lie Groups G. Part I, G = G 2, F 4 and E 6, Tsukuba J. Math. 14 (1990), 185-223.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Furutani, K. Quantization of the Geodesic Flow on Quaternion Projective Spaces. Annals of Global Analysis and Geometry 22, 1–27 (2002). https://doi.org/10.1023/A:1016202322387
Issue Date:
DOI: https://doi.org/10.1023/A:1016202322387