| Title:
|
Invariant symbolic calculus for compact Lie groups (English) |
| Author:
|
Cahen, Benjamin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
139-155 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the invariant symbolic calculi associated with the unitary irreducible representations of a compact Lie group. (English) |
| Keyword:
|
compact Lie group |
| Keyword:
|
invariant symbolic calculus |
| Keyword:
|
coadjoint orbit |
| Keyword:
|
unitary representation |
| Keyword:
|
Berezin quantization |
| Keyword:
|
Weyl quantization |
| MSC:
|
22E45 |
| MSC:
|
22E46 |
| MSC:
|
81R05 |
| MSC:
|
81S10 |
| idZBL:
|
Zbl 07138659 |
| idMR:
|
MR3994322 |
| DOI:
|
10.5817/AM2019-3-139 |
| . |
| Date available:
|
2019-08-05T08:43:18Z |
| Last updated:
|
2020-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147821 |
| . |
| Reference:
|
[1] Ali, S.T., Englis, M.: Quantization methods: a guide for physicists and analysts.Rev. Math. Phys. 17 (4) (2005), 391–490. Zbl 1075.81038, MR 2151954, 10.1142/S0129055X05002376 |
| Reference:
|
[2] Arazy, J., Upmeier, H.: Weyl Calculus for Complex and Real Symmetric Domains.Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001), vol. 13, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 2002, pp. 165–181. MR 1984098 |
| Reference:
|
[3] Arazy, J., Upmeier, H.: Invariant symbolic calculi and eigenvalues of invariant operators on symmeric domains.Function spaces, interpolation theory and related topics (Lund, 2000), de Gruyter, Berlin, 2002, pp. 151–211. MR 1943284 |
| Reference:
|
[4] Arnal, D., Cahen, M., Gutt, S.: Representations of compact Lie groups and quantization by deformation.Acad. R. Belg. Bull. Cl. Sc. 3e série LXXIV 45 (1988), 123–140. MR 1027456 |
| Reference:
|
[5] Bekka, M.B., de la Harpe, P.: Irreducibility of unitary group representations and reproducing kernels Hilbert spaces.Expo. Math. 21 (2) (2003), 115–149. MR 1978060, 10.1016/S0723-0869(03)80014-2 |
| Reference:
|
[6] Berezin, F.A.: Quantization.Math. USSR Izv. 8 (5) (1974), 1109–1165. Zbl 0312.53049, MR 0395610 |
| Reference:
|
[7] Berezin, F.A.: Quantization in complex symmetric domains.Math. USSR Izv. 9 (2) (1975), 341–379. 10.1070/IM1975v009n02ABEH001480 |
| Reference:
|
[8] Brif, C., Mann, A.: Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries.Phys. Rev. A 59 (2) (1999), 971–987. MR 1679730, 10.1103/PhysRevA.59.971 |
| Reference:
|
[9] Cahen, B.: Contractions of $SU(1,n)$ and $SU(n+1)$ via Berezin quantization.J. Anal. Math. 97 (2005), 83–101. MR 2274974, 10.1007/BF02807403 |
| Reference:
|
[10] Cahen, B.: Berezin quantization on generalized flag manifolds.Math. Scand. 105 (2009), 66–84. Zbl 1183.22006, MR 2549798, 10.7146/math.scand.a-15106 |
| Reference:
|
[11] Cahen, B.: Stratonovich-Weyl correspondence for compact semisimple Lie groups.Rend. Circ. Mat. Palermo 59 (2010), 331–354. Zbl 1218.22008, MR 2745515, 10.1007/s12215-010-0026-y |
| Reference:
|
[12] Cahen, B.: Berezin quantization and holomorphic representations.Rend. Sem. Mat. Univ. Padova 129 (2013), 277–297. MR 3090642, 10.4171/RSMUP/129-16 |
| Reference:
|
[13] Cahen, B.: Berezin transform and Stratonovich-Weyl correspondence for the multi-dimensional Jacobi group.Rend. Sem. Mat. Univ. Padova 136 (2016), 69–93. MR 3593544, 10.4171/RSMUP/136-7 |
| Reference:
|
[14] 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:
|
[15] Cariñena, J.F., Gracia-Bondìa, J.M., Vàrilly, J.C.: Relativistic quantum kinematics in the Moyal representation.J. Phys. A: Math. Gen. 23 (1990), 901–933. Zbl 0706.60108, MR 1048769, 10.1088/0305-4470/23/6/015 |
| Reference:
|
[16] Englis, M.: Berezin transform on pluriharmonic Bergmann spaces.Trans. Amer. Math. Soc. 361 (2009), 1173–1188. MR 2457394, 10.1090/S0002-9947-08-04653-9 |
| Reference:
|
[17] Figueroa, H., Gracia-Bondìa, J.M., Vàrilly, J.C.: Moyal quantization with compact symmetry groups and noncommutative analysis.J. Math. Phys. 31 (1990), 2664–2671. MR 1075750, 10.1063/1.528967 |
| Reference:
|
[18] Folland, B.: Harmonic Analysis in Phase Space.Princeton Univ. Press, 1989. Zbl 0682.43001, MR 0983366 |
| Reference:
|
[19] Gracia-Bondìa, J.M.: Generalized Moyal quantization on homogeneous symplectic spaces.Deformation theory and quantum groups with applications to mathematical physics, Amherst, MA, 1990, Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992, pp. 93–114. MR 1187280 |
| Reference:
|
[20] Gracia-Bondìa, J.M., Vàrilly, J.C.: The Moyal representation for spin.Ann. Physics 190 (1989), 107–148. MR 0994048, 10.1016/0003-4916(89)90262-5 |
| Reference:
|
[21] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces.Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, Rhode Island, 2001. Zbl 0993.53002, MR 1834454, 10.1090/gsm/034 |
| Reference:
|
[22] Kirillov, A.A.: Lectures on the Orbit Method.Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, Rhode Island, 2004. MR 2069175, 10.1090/gsm/064 |
| Reference:
|
[23] Kobayashi, T.: Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs.Representation theory and automorphic forms, vol. 255, Birkhäuser Boston, Boston, MA, Progr. Math. ed., 2008, pp. 45–109. MR 2369496 |
| Reference:
|
[24] Kostant, B.: Quantization and unitary representations.Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, Heidelberg, New-York, Modern Analysis and Applications ed., 1970, pp. 87–207. Zbl 0223.53028, MR 0294568 |
| Reference:
|
[25] Neeb, K-H.: Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics.vol. 28, Walter de Gruyter, Berlin, New-York, 2000. MR 1740617 |
| Reference:
|
[26] Nomura, T.: Berezin transforms and group representations.J. Lie Theory 8 (1998), 433–440. MR 1650386 |
| Reference:
|
[27] Ørsted, B., Zhang, G.: Weyl quantization and tensor products of Fock and Bergman spaces.Indiana Univ. Math. J. 43 (2) (1994), 551–583. MR 1291529, 10.1512/iumj.1994.43.43023 |
| Reference:
|
[28] Stembridge, J.R.: Multiplicity-free products and restrictions of Weyl characters.Representation Theory 7 (2003), 404–439. MR 2017064, 10.1090/S1088-4165-03-00150-X |
| Reference:
|
[29] Stratonovich, R.L.: On distributions in representation space.Soviet Physics. JETP 4 (1957), 891–898. MR 0088173 |
| Reference:
|
[30] Unterberger, A., Upmeier, H.: Berezin transform and invariant differential operators.Commun. Math. Phys. 164 (3) (1994), 563–597. Zbl 0843.32019, MR 1291245, 10.1007/BF02101491 |
| Reference:
|
[31] Wallach, N.R.: Harmonic Analysis on Homogeneous Spaces.Marcel Dekker, Inc., 1973. MR 0498996 |
| Reference:
|
[32] Wildberger, N.J.: On the Fourier transform of a compact semi simple Lie group.J. Austral. Math. Soc. A 56 (1994), 64–116. MR 1250994, 10.1017/S1446788700034741 |
| Reference:
|
[33] Zhang, G.: Berezin transform on compact Hermitian symmetric spaces.Manuscripta Math. 97 (1998), 371–388. Zbl 0920.22008, MR 1654800, 10.1007/s002290050109 |
| . |
