Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group

Benjamin Cahen

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2013)

  • Volume: 52, Issue: 1, page 35-48
  • ISSN: 0231-9721

Abstract

top
Let G be a quasi-Hermitian Lie group with Lie algebra 𝔤 and K be a compactly embedded subgroup of G . Let ξ 0 be a regular element of 𝔤 * which is fixed by K . We give an explicit G -equivariant diffeomorphism from a complex domain onto the coadjoint orbit 𝒪 ( ξ 0 ) of ξ 0 . This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where 𝒪 ( ξ 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 = S U ( p , q ) and the case where G is the Jacobi group.

How to cite

top

Cahen, Benjamin. "Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.1 (2013): 35-48. <http://eudml.org/doc/260687>.

@article{Cahen2013,
abstract = {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.},
author = {Cahen, Benjamin},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group; quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group},
language = {eng},
number = {1},
pages = {35-48},
publisher = {Palacký University Olomouc},
title = {Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group},
url = {http://eudml.org/doc/260687},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Cahen, Benjamin
TI - Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 1
SP - 35
EP - 48
AB - 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.
LA - eng
KW - quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group; quasi-Hermitian Lie group; coadjoint orbit; stereographic projection; Berezin quantization; unitary holomorphic representation; unitary group; Jacobi group
UR - http://eudml.org/doc/260687
ER -

References

top
  1. Arnal, D., Cortet, J.-C., 10.1007/BF00398548, Lett. Math. Phys. 9 (1985), 25–34. (1985) Zbl0616.46041MR0774736DOI10.1007/BF00398548
  2. Bar-Moshe, D., A method for weight multiplicity computation based on Berezin quantization, SIGMA 5 (2009), 091, 1–12. (2009) Zbl1188.22009MR2559670
  3. Bar-Moshe, D., Marinov, M. S., 10.1088/0305-4470/27/18/035, J. Phys. A: Math. Gen. 27 (1994), 6287–6298. (1994) Zbl0843.58056MR1306179DOI10.1088/0305-4470/27/18/035
  4. Berezin, F. A., 10.1070/IM1974v008n05ABEH002140, Math. USSR Izv. 8, 5 (1974), 1109–1165. (1974) Zbl0312.53049DOI10.1070/IM1974v008n05ABEH002140
  5. Berezin, F. A., 10.1070/IM1975v009n02ABEH001480, Math. USSR Izv. 9, 2 (1975), 341–379. (1975) DOI10.1070/IM1975v009n02ABEH001480
  6. Berceanu, S., 10.1142/S0129055X06002619, Rev. Math. Phys. 18 (2006), 163–199. (2006) Zbl1099.81036MR2228923DOI10.1142/S0129055X06002619
  7. Berceanu, S., Gheorghe, A., 10.1142/S0219887811005920, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783–1798. (2011) Zbl1250.22010MR2876095DOI10.1142/S0219887811005920
  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. (2008) Zbl1208.22009MR2436268
  9. Berndt, R., Böcherer, S., 10.1007/BF02570858, Math. Z. 204 (1990), 13–44. (1990) Zbl0695.10024MR1048065DOI10.1007/BF02570858
  10. Berndt, R., Schmidt, R., Elements of the Representation Theory of the Jacobi Group, Progress in Mathematics 163, Birkhäuser Verlag, Basel, 1988. (1988) MR1634977
  11. Cahen, B., 10.1007/BF00403252, Lett. Math. Phys. 36 (1996), 65–75. (1996) Zbl0843.22020MR1371298DOI10.1007/BF00403252
  12. Cahen, B., Contraction de S U ( 1 , 1 ) vers le groupe de Heisenberg, In: Mathematical works, Part XV, Séminaire de Mathématique Université du Luxembourg, Luxembourg, (2004), 19–43. (2004) Zbl1074.22005MR2143420
  13. Cahen, B., 10.1016/j.difgeo.2006.08.005, Diff. Geom. Appl. 25 (2007), 177–190. (2007) Zbl1117.81087MR2311733DOI10.1016/j.difgeo.2006.08.005
  14. Cahen, B., Multiplicities of compact Lie group representations via Berezin quantization, Mat. Vesnik 60 (2008), 295–309. (2008) Zbl1199.22016MR2465811
  15. Cahen, B., Contraction of compact semisimple Lie groups via Berezin quantization, Illinois J. Math. 53, 1 (2009), 265–288. (2009) Zbl1185.22008MR2584946
  16. Cahen, B., Berezin quantization on generalized flag manifolds, Math. Scand. 105 (2009), 66–84. (2009) Zbl1183.22006MR2549798
  17. Cahen, B., Contraction of discrete series via Berezin quantization, J. Lie Theory 19 (2009), 291–310. (2009) Zbl1185.22007MR2572131
  18. Cahen, B., Berezin quantization for discrete series, Beiträge Algebra Geom. 51 (2010), 301–311. (2010) MR2682458
  19. Cahen, B., Stratonovich-Weyl correspondence for discrete series representations, Arch. Math. (Brno) 47 (2011), 41–58. (2011) Zbl1240.22011MR2813546
  20. Cahen, B., Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear. 
  21. Cahen, M., Gutt, S., Rawnsley, J., 10.1016/0393-0440(90)90019-Y, J. Geom. Phys. 7 (1990), 45–62. (1990) MR1094730DOI10.1016/0393-0440(90)90019-Y
  22. Cahen, M., Gutt, S., Rawnsley, J., 10.1007/BF00751065, Lett. Math. Phys. 30 (1994), 291–305. (1994) MR1271090DOI10.1007/BF00751065
  23. Cotton, P., Dooley, A. H., Contraction of an adapted functional calculus, J. Lie Theory 7 (1997), 147–164. (1997) Zbl0882.22015MR1473162
  24. Folland, B., Harmonic Analysis in Phase Space, , Princeton Univ. Press, Princeton, 1989. (1989) Zbl0682.43001MR0983366
  25. Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics 34, American Mathematical Society, Providence, Rhode Island, 2001. (2001) Zbl0993.53002MR1834454
  26. Kirillov, A. A., Lectures on the Orbit Method, Graduate Studies in Mathematics, 64, American Mathematical Society, Providence, Rhode Island, 2004. (2004) MR2069175
  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. (1970) Zbl0223.53028MR0294568
  28. Neeb, K-H., Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics 28, Walter de Gruyter, Berlin, New York, 2000. (2000) MR1740617
  29. Satake, I, Algebraic Structures of Symmetric Domains, Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1971. (1971) MR0591460
  30. Skrypnik, T. V., 10.1007/BF02525136, Ukr. Math. J. 51 (1999), 1939–1944. (1999) MR1752044DOI10.1007/BF02525136
  31. Varadarajan, V. S., Lie groups, Lie Algebras and Their Representations, Graduate Texts in Mathematics 102, Springer-Verlag, Berlin, 1984. (1984) Zbl0955.22500MR0746308

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.