Quantization of the 4-Dimensional Nilpotent Orbit of Sl(3, R)

Brylinski, Ranee

Canadian journal of mathematics, Tome 49 (1997) no. 5, pp. 916-943

Résumé

We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit OR of SL(3, R). The space of quantization is the space of holomorphic functions on c 2 - {0}) which are square integrable with respect to a signed measure defined by a Meijer G-function. We construct the quantization out a non-flat Kaehler structure on c 2 - {0}) (the universal cover of OR) with Kaehler potential ρ |z|4.

Détail

DOI : 10.4153/CJM-1997-048-0

Mots-clés : 81S10, 32C17, 22E70

Bibliographie

Biedenharn, L.C., Cusson, R.Y., Han, M.Y., Weaver, O.L., Hadronic Regge sequences as primitive realizations of SL(3, ℝ) symmetry. Physics Letters (1972), 257–260. and Errata, Physics Letters 73B(1978), 503. Google Scholar

Brylinski, R., Holomorphic Realizations of Minimal Unitary Representations of E6, E7 and E8. Preprint 1995. Google Scholar

Brylinski, R., Instantons and Kaehler geometry of nilpotent orbits I and II. Google Scholar

Brylinski, R., Geometric quantization and half-form models of unitary minimal representations II. In preparation. Google Scholar

Brylinski, R. and B.Kostant, Minimal representations, geometric quantization and unitarity. Proc.Natl. Acad. Sci. USA 91(1994), 6026–6029. Google Scholar

Brylinski, R., Differential operators on conical Lagrangian manifolds. In: Lie Theory and Geometry: In Honor of B. Kostant (Eds. Brylinski, J.-L., Brylinski, R., Guillemin, V., Kac, V.). Progr. Math., Vol. 123, Birkhauser, Boston, 1994. 65–96. Google Scholar

Brylinski, R., Lagrangian models of minimal representations of E6, E. and E8. In: Functional Analysis on the Eve of the 21th Century: Festschrift in Honor of the Eightieth Birthday of I. M. Gelfand (Eds. Gindikin, S., Lepowsky, J., Wilson, R.). Progr. Math., Vol. 131, Birkhauser, Boston, 1995. 13–63. Google Scholar

Brylinski, R., Geometric quantization and half-form models of unitary minimal representations I. Preprint 1997. Google Scholar

Faraut, J. and Koranyi, A., Analysis on Symmetric Cones. Oxford University Press, Oxford, 1994. Google Scholar

Folland, G.B., Harmonic Analysis on Phase Space. Ann. of Math. Studies 122(1989). Google Scholar

Joseph, D.W., Representations of the algebra of SL(3, ℝ) with j = 2. Preprint 1969. Google Scholar

Kirillov, A.A., Geometric quantization. In: Dynamical Systems IV (Eds. Arnold, V.I. and Novikov, S.P.). Encyclopaedia of Mathematical Sciences, Vol. 4, Springer Verlag, 1990. 137–172. Google Scholar

Mathai, A.M., A handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford University Press, New York, 1993. Google Scholar

Paris, R.B. and Wood, A.D., Asymptotics of high order differential equations. Pitman Research Notes in Math Series, Vol. 129, Longman Scientific and Technical, Harlow, 1986. Google Scholar

Rawnsley, J. and Sternberg, S., On representations associated to the minimal nilpotent coadjoint orbit of SL(3, ℝ). Amer. J. Math. 104(1982), 1153–1180. Google Scholar

Sijacki, D., The unitary irreducible representations of S¯ L(3, ℝ). J. Math. Phys. 16(1975), 298–311. Google Scholar

Torasso, P.. Quantification geometrique et representations de SL3(ℝ). Acta Mathematica 150(1983). 153–242. Google Scholar

Whitaker, E.T. and Watson, G.N., A Course in Modern Analysis. Cambridge University Press, Cambridge, 1935. Google Scholar