| Title:
|
Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group (English) |
| Author:
|
Cahen, Benjamin |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
35-48 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a quasi-Hermitian Lie group with Lie algebra $\mathfrak g$ and $K$ be a compactly embedded subgroup of $G$. Let $\xi _0$ be a regular element of ${\mathfrak g}^{\ast }$ which is fixed by $K$. We give an explicit $G$-equivariant diffeomorphism from a complex domain onto the coadjoint orbit $\mathcal {O}({\xi _0})$ of $\xi _0$. This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where ${\mathcal O}({\xi _0})$ is associated with a unitary irreducible representation of $G$ which is holomorphically induced from a unitary character of $K$. In particular, we consider the case $G=SU(p,q)$ and the case where $G$ is the Jacobi group. (English) |
| Keyword:
|
quasi-Hermitian Lie group |
| Keyword:
|
coadjoint orbit |
| Keyword:
|
stereographic projection |
| Keyword:
|
Berezin quantization |
| Keyword:
|
unitary holomorphic representation |
| Keyword:
|
unitary group |
| Keyword:
|
Jacobi group |
| MSC:
|
22E10 |
| MSC:
|
22E15 |
| MSC:
|
22E45 |
| MSC:
|
32M05 |
| MSC:
|
32M10 |
| MSC:
|
32M15 |
| MSC:
|
81S10 |
| idZBL:
|
Zbl 06285752 |
| idMR:
|
MR3202747 |
| . |
| Date available:
|
2013-08-02T07:53:29Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143389 |
| . |
| Reference:
|
[1] Arnal, D., Cortet, J.-C.: Nilpotent Fourier transform and applications. Lett. Math. Phys. 9 (1985), 25–34. Zbl 0616.46041, MR 0774736, 10.1007/BF00398548 |
| Reference:
|
[2] Bar-Moshe, D.: A method for weight multiplicity computation based on Berezin quantization. SIGMA 5 (2009), 091, 1–12. Zbl 1188.22009, MR 2559670 |
| Reference:
|
[3] Bar-Moshe, D., Marinov, M. S.: Realization of compact Lie algebras in Kähler manifolds. J. Phys. A: Math. Gen. 27 (1994), 6287–6298. Zbl 0843.58056, MR 1306179, 10.1088/0305-4470/27/18/035 |
| Reference:
|
[4] Berezin, F. A.: Quantization. Math. USSR Izv. 8, 5 (1974), 1109–1165. Zbl 0312.53049, 10.1070/IM1974v008n05ABEH002140 |
| Reference:
|
[5] Berezin, F. A.: Quantization in complex symmetric domains. Math. USSR Izv. 9, 2 (1975), 341–379. 10.1070/IM1975v009n02ABEH001480 |
| Reference:
|
[6] Berceanu, S.: A holomorphic representation of the Jacobi algebra. Rev. Math. Phys. 18 (2006), 163–199. Zbl 1099.81036, MR 2228923, 10.1142/S0129055X06002619 |
| Reference:
|
[7] Berceanu, S., Gheorghe, A.: On the geometry of Siegel–Jacobi domains. Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783–1798. Zbl 1250.22010, MR 2876095, 10.1142/S0219887811005920 |
| Reference:
|
[8] Bernatska, J., Holod, P.: Geometry and topology of coadjoint orbits of semisimple Lie groups. Mladenov, I. M., de León, M. (eds) Proceedings of the 9th international conference on ’Geometry, Integrability and Quantization’, June 8–13, 2007, Varna, Bulgarian Academy of Sciences, Sofia, 2008, 146–166. Zbl 1208.22009, MR 2436268 |
| Reference:
|
[9] Berndt, R., Böcherer, S.: Jacobi forms and discrete series representations of the Jacobi group. Math. Z. 204 (1990), 13–44. Zbl 0695.10024, MR 1048065, 10.1007/BF02570858 |
| Reference:
|
[10] Berndt, R., Schmidt, R.: Elements of the Representation Theory of the Jacobi Group. Progress in Mathematics 163, Birkhäuser Verlag, Basel, 1988. MR 1634977 |
| Reference:
|
[11] Cahen, B.: Deformation program for principal series representations. Lett. Math. Phys. 36 (1996), 65–75. Zbl 0843.22020, MR 1371298, 10.1007/BF00403252 |
| Reference:
|
[12] Cahen, B.: Contraction de $SU(1,1)$ vers le groupe de Heisenberg. In: Mathematical works, Part XV, Séminaire de Mathématique Université du Luxembourg, Luxembourg, (2004), 19–43. Zbl 1074.22005, MR 2143420 |
| Reference:
|
[13] Cahen, B.: Weyl quantization for semidirect products. Diff. Geom. Appl. 25 (2007), 177–190. Zbl 1117.81087, MR 2311733, 10.1016/j.difgeo.2006.08.005 |
| Reference:
|
[14] Cahen, B.: Multiplicities of compact Lie group representations via Berezin quantization. Mat. Vesnik 60 (2008), 295–309. Zbl 1199.22016, MR 2465811 |
| Reference:
|
[15] Cahen, B.: Contraction of compact semisimple Lie groups via Berezin quantization. Illinois J. Math. 53, 1 (2009), 265–288. Zbl 1185.22008, MR 2584946 |
| Reference:
|
[16] Cahen, B.: Berezin quantization on generalized flag manifolds. Math. Scand. 105 (2009), 66–84. Zbl 1183.22006, MR 2549798 |
| Reference:
|
[17] Cahen, B.: Contraction of discrete series via Berezin quantization. J. Lie Theory 19 (2009), 291–310. Zbl 1185.22007, MR 2572131 |
| Reference:
|
[18] Cahen, B.: Berezin quantization for discrete series. Beiträge Algebra Geom. 51 (2010), 301–311. MR 2682458 |
| Reference:
|
[19] Cahen, B.: Stratonovich-Weyl correspondence for discrete series representations. Arch. Math. (Brno) 47 (2011), 41–58. Zbl 1240.22011, MR 2813546 |
| Reference:
|
[20] Cahen, B.: Berezin quantization and holomorphic representations. Rend. Sem. Mat. Univ. Padova, to appear. |
| Reference:
|
[21] Cahen, M., Gutt, S., Rawnsley, J.: Quantization on Kähler manifolds I, Geometric interpretation of Berezin quantization. J. Geom. Phys. 7 (1990), 45–62. MR 1094730, 10.1016/0393-0440(90)90019-Y |
| Reference:
|
[22] Cahen, M., Gutt, S., Rawnsley, J.: Quantization on Kähler manifolds III. Lett. Math. Phys. 30 (1994), 291–305. MR 1271090, 10.1007/BF00751065 |
| Reference:
|
[23] Cotton, P., Dooley, A. H.: Contraction of an adapted functional calculus. J. Lie Theory 7 (1997), 147–164. Zbl 0882.22015, MR 1473162 |
| Reference:
|
[24] Folland, B.: Harmonic Analysis in Phase Space., Princeton Univ. Press, Princeton, 1989. Zbl 0682.43001, MR 0983366 |
| Reference:
|
[25] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Graduate Studies in Mathematics 34, American Mathematical Society, Providence, Rhode Island, 2001. Zbl 0993.53002, MR 1834454 |
| Reference:
|
[26] Kirillov, A. A.: Lectures on the Orbit Method, Graduate Studies in Mathematics. 64, American Mathematical Society, Providence, Rhode Island, 2004. MR 2069175 |
| Reference:
|
[27] Kostant, B.: Quantization and unitary representations. In: Modern Analysis and Applications, Lecture Notes in Mathematics 170, Springer-Verlag, Berlin, Heidelberg, New York, (1970), 87–207. Zbl 0223.53028, MR 0294568 |
| Reference:
|
[28] Neeb, K-H.: Holomorphy and Convexity in Lie Theory. de Gruyter Expositions in Mathematics 28, Walter de Gruyter, Berlin, New York, 2000. MR 1740617 |
| Reference:
|
[29] Satake, I: Algebraic Structures of Symmetric Domains. Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1971. MR 0591460 |
| Reference:
|
[30] Skrypnik, T. V.: Coadjoint orbits of compact Lie groups and generalized stereographic projection. Ukr. Math. J. 51 (1999), 1939–1944. MR 1752044, 10.1007/BF02525136 |
| Reference:
|
[31] Varadarajan, V. S.: Lie groups, Lie Algebras and Their Representations. Graduate Texts in Mathematics 102, Springer-Verlag, Berlin, 1984. Zbl 0955.22500, MR 0746308 |
| . |
