Explicit quantization of the Kepler manifold

Blattner, Robert J.; Wolf, Joseph A.

doi:10.1090/S0002-9939-1979-0539648-1

Explicit quantization of the Kepler manifold
HTML articles powered by AMS MathViewer

by Robert J. Blattner and Joseph A. Wolf PDF

Proc. Amer. Math. Soc. 77 (1979), 145-149 Request permission

Abstract:

Any representation $\pi$ of ${\text {SO}}(2,4)$ quantizing the Kepler manifold has the same lowest highest weight as the representation ${\nu _0}$ in the Sternberg-Wolf description of the $U(2,2)$-restriction of the metaplectic representation of ${\text {Sp}}(4;{\mathbf {R}})$. Hence, modulo covering groups, $\pi$ is unitarily equivalent to ${\nu _0}$.

References

Robert J. Blattner, Intertwining operators and the half-density pairing, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975, pp. 1–12. MR 0390130
Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965
C. Itzykson, Remarks on boson commutation rules, Comm. Math. Phys. 4 (1967), 92–122. MR 211692
D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), no. 4, 481–507. MR 478225, DOI 10.1002/cpa.3160310405
Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR 0294568
G. Mack and I. T. Todorov, Irreducibility of the ladder representations of $U(2,\,2)$ when restricted to the Poincaré subgroup, J. Mathematical Phys. 10 (1969), 2078–2085. MR 250582, DOI 10.1063/1.1664804
Enrico Onofri, Dynamical quantization of the Kepler manifold, J. Mathematical Phys. 17 (1976), no. 3, 401–408. MR 395530, DOI 10.1063/1.522907
J. H. Rawnsley, On the pairing of polarizations, Comm. Math. Phys. 58 (1978), no. 1, 1–8. MR 468930
Shlomo Sternberg and Joseph A. Wolf, Hermitian Lie algebras and metaplectic representations. I, Trans. Amer. Math. Soc. 238 (1978), 1–43. MR 486325, DOI 10.1090/S0002-9947-1978-0486325-5

Discrete series of hermitian representations of the Lie algebra of

Joseph A. Wolf, Conformal group, quantization, and the Kepler problem, Group theoretical methods in physics (Fourth Internat. Colloq., Nijmegen, 1975) Lecture Notes in Phys., Vol. 50, Springer, Berlin, 1976, pp. 217–222. MR 0499018
Joseph A. Wolf, Representations associated to minimal co-adjoint orbits, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977) Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 329–349. MR 519619