Berezin quantization of homogeneous bounded domains
Original Paper
First Online:
Received:
Accepted:
- 110 Downloads
- 12 Citations
Abstract
We prove that a homogeneous bounded domain admits a Berezin quantization.
Keywords
Kähler metrics Berezin quantization Bounded homogeneous domain Calabi’s diastasis functionMathematics Subject Classification (2000)
53D05 53C55 58F06Preview
Unable to display preview. Download preview PDF.
References
- 1.Arezzo C., Loi A.: Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau–Zelditch. J. Geom. Phys. 47, 87–99 (2003)MathSciNetMATHCrossRefGoogle Scholar
- 2.Arezzo C., Loi A.: Moment maps, scalar curvature and quantization of Kähler manifolds. Comm. Math. Phys. 246, 543–549 (2004)MathSciNetMATHCrossRefGoogle Scholar
- 3.Auslander L., Kostant B.: Polarization and unitary representations of solvable Lie groups. Invent. Math. 14, 255–354 (1971)MathSciNetMATHCrossRefGoogle Scholar
- 4.Berezin F.A.: Quantization. Math. USSR Izvestiya 8, 1109–1163 (1974) MR 52:16404MATHCrossRefGoogle Scholar
- 5.Calabi E.: Isometric imbeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)MathSciNetMATHCrossRefGoogle Scholar
- 6.Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization. JGP 7, 45–62 (1990)MathSciNetMATHGoogle Scholar
- 7.Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. 337, 73–98 (1993)MathSciNetMATHGoogle Scholar
- 8.Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds III. Lett. Math. Phys. 30, 291–305 (1994)MathSciNetMATHCrossRefGoogle Scholar
- 9.Cahen M., Gutt S., Rawnsley J.H.: Quantization of Kähler manifolds IV. Lett. Math. Phys. 34, 159–168 (1995)MathSciNetMATHCrossRefGoogle Scholar
- 10.Di Scala, A.J., Ishi, H., Loi, A.: Kaehler immersions of homogeneous Kaehler manifolds into complex space forms. Asian J. Math. (To appear)Google Scholar
- 11.Donaldson S.: Scalar curvature and projective embeddings, I. J. Diff. Geom. 59, 479–522 (2001)MathSciNetMATHGoogle Scholar
- 12.Dorfmeister J.: Simply transitive groups and Kähler structures on homogeneous Siegel domains. Trans. Am. Math. Soc. 288, 293–305 (1985)MathSciNetMATHGoogle Scholar
- 13.Engliš M.: Berezin quantization and reproducing kernels on complex domains. Trans. Am. Math. Soc. 348, 411–479 (1996)MATHCrossRefGoogle Scholar
- 14.Gindikin S.G.: Analysis in homogeneous domains. Russ. Math. Surv. 19, 1–89 (1964)MathSciNetCrossRefGoogle Scholar
- 15.Gramchev T., Loi A.: TYZ expansion for the Kepler manifold. Comm. Math. Phys. 289, 825–840 (2009)MathSciNetMATHCrossRefGoogle Scholar
- 16.Greco A., Loi A.: Radial balanced metrics on the unit disk. J. Geom. Phys. 60, 53–59 (2010)MathSciNetMATHCrossRefGoogle Scholar
- 17.Ishi H.: Representations of the affine transformation groups acting simply transitively on Siegel domains. J. Funct. Anal. 167(2), 425–462 (1999)MathSciNetMATHCrossRefGoogle Scholar
- 18.Ishi, H.: Unitary holomorphic multiplier representations over a homogeneous bounded domain. Adv. Pure Appl. Math. (To appear)Google Scholar
- 19.Ji S.: Inequality for distortion function of invertible shaves on Abelian varieties. Duke Math. J. 58, 657–667 (1989)MathSciNetMATHCrossRefGoogle Scholar
- 20.Kempf, G.R.: Metrics on invertible shaves on abelian varieties. Topics in algebraic geometry (Guanajuato, 1989): Aportationes Mat. Notas Investigacion 5, Soc. Mat. Mexicana, Mexico, pp. 107–108 (1992)Google Scholar
- 21.Loi A.: Regular quantizations of K manifolds and constant scalar curvature metrics. J. Geom. Phys. 53, 354–364 (2005)MathSciNetMATHCrossRefGoogle Scholar
- 22.Loi A.: Calabi’s diastasis function for Hermitian symmetric spaces. Diff. Geom. Appl. 24, 311–319 (2006)MathSciNetMATHCrossRefGoogle Scholar
- 23.Loi A., Zedda M.: K-Einstein submanifolds of the infinite dimensional projective space. Math. Ann. 350, 145–154 (2011)MathSciNetMATHCrossRefGoogle Scholar
- 24.Loi, A., Zedda, M.: Balanced metrics on Cartan and Cartan-Hartogs domain. Math. Zeitschrift (2011). (To appear)Google Scholar
- 25.Loi A., Zedda M.: Balanced metrics on Hartogs domains. Abh. Math. Sem. Univ. Hamburg 81(1), 69–77 (2011)MathSciNetMATHCrossRefGoogle Scholar
- 26.Nomura T.: Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains. J. Lie Theory 11, 185–206 (2001)MathSciNetMATHGoogle Scholar
- 27.Piatetskii-Shapiro I.I.: Automorphic Functions and the Geometry of Classical Domains. Gordon and Breach, New York, NY (1969)Google Scholar
- 28.Rawnsley J.: Coherent states and K manifolds. Q. J. Math. Oxf 2(28), 403–415 (1977)MathSciNetCrossRefGoogle Scholar
- 29.Rossi H., Vergne M.: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group. J. Funct. Anal. 13, 324–389 (1973)MathSciNetMATHCrossRefGoogle Scholar
- 30.Zhang S.: Heights and reductions of semi-stable varieties. Comp. Math. 104, 77–105 (1996)MATHGoogle Scholar
Copyright information
© Springer Science+Business Media B.V. 2012