| Title:
|
Weyl quantization for the semidirect product of a compact Lie group and a vector space (English) |
| Author:
|
Cahen, Benjamin |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
325-347 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be the semidirect product $V\rtimes K$ where $K$ is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\mathcal O$ be a coadjoint orbit of $G$ associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation $\pi$ of $G$. We consider the case when the corresponding little group $H$ is the centralizer of a torus of $K$. By dequantizing a suitable realization of $\pi$ on a Hilbert space of functions on ${\mathbb C}^n$ where $n=\dim (K/H)$, we construct a symplectomorphism between a dense open subset of ${\mathcal O}$ and the symplectic product ${\mathbb C}^{2n}\times {\mathcal O}'$ where ${\mathcal O}'$ is a coadjoint orbit of $H$. This allows us to obtain a Weyl correspondence on ${\mathcal O}$ which is adapted to the representation $\pi$ in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C.R. Acad. Sci. Paris t. 325, série I (1997), 803--806]. (English) |
| Keyword:
|
Weyl quantization |
| Keyword:
|
Berezin quantization |
| Keyword:
|
semidirect product |
| Keyword:
|
coadjoint orbits |
| Keyword:
|
unitary representations |
| MSC:
|
22E46 |
| MSC:
|
22E99 |
| MSC:
|
32M10 |
| MSC:
|
81S10 |
| idZBL:
|
Zbl 1212.81015 |
| idMR:
|
MR2573408 |
| . |
| Date available:
|
2009-09-23T21:34:15Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134907 |
| . |
| Reference:
|
[1] Ali S.T., Englis M.: Quantization methods: a guide for physicists and analysts.Rev. Math. Phys. 17 (2005), no. 4, 391--490. Zbl 1075.81038, MR 2151954, 10.1142/S0129055X05002376 |
| Reference:
|
[2] 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:
|
[3] Baguis P.: Semidirect products and the Pukansky condition.J. Geom. Phys. 25 (1998), 245--270. MR 1619845, 10.1016/S0393-0440(97)00028-4 |
| Reference:
|
[4] Berezin F.A.: Quantization.Math. USSR Izv. 8, 5 (1974), 1109--1165. Zbl 0976.83531, 10.1070/IM1974v008n05ABEH002140 |
| Reference:
|
[5] Cahen B.: Deformation program for principal series representations.Lett. Math. Phys. 36 (1996), 65--75. Zbl 0843.22020, MR 1371298, 10.1007/BF00403252 |
| Reference:
|
[6] Cahen B.: Quantification d'une orbite massive d'un groupe de Poincaré généralisé.C.R. Acad. Sci. Paris Sér. I Math. 325 (1997), 803--806. Zbl 0883.22016, MR 1483721, 10.1016/S0764-4442(97)80063-8 |
| Reference:
|
[7] Cahen B.: Quantification d'orbites coadjointes et théorie des contractions.J. Lie Theory 11 (2001), 257--272. Zbl 0973.22009, MR 1851792 |
| Reference:
|
[8] Cahen B.: Contractions of $SU(1,n)$ and $SU(n+1)$ via Berezin quantization.J. Anal. Math. 97 (2005), 83--102. MR 2274974, 10.1007/BF02807403 |
| Reference:
|
[9] Cahen B.: Weyl quantization for semidirect products.Differential Geom. Appl. 25 (2007), 177--190. Zbl 1117.81087, MR 2311733, 10.1016/j.difgeo.2006.08.005 |
| Reference:
|
[10] Cahen B.: Weyl quantization for principal series.Beiträge Algebra Geom. 48 (2007), no. 1, 175--190. Zbl 1134.22010, MR 2326408 |
| Reference:
|
[11] Cahen B.: Berezin quantization for discrete series.preprint Univ. Metz (2008), to appear in Beiträge Algebra Geom. MR 2682458 |
| Reference:
|
[12] Cahen B.: Berezin quantization on generalized flag manifolds.preprint Univ. Metz (2008), to appear in Math. Scand. MR 2549798 |
| Reference:
|
[13] 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:
|
[14] Cotton P., Dooley A.H.: Contraction of an adapted functional calculus.J. Lie Theory 7 (1997), 147--164. Zbl 0882.22015, MR 1473162 |
| Reference:
|
[15] Folland B.: Harmonic Analysis in Phase Space.Princeton Univ. Press, Princeton, 1989. Zbl 0682.43001, MR 0983366 |
| Reference:
|
[16] Gotay M.: Obstructions to quantization.in Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, Springer, NewYork, 2000, pp. 271--316. Zbl 1041.53507, MR 1766362 |
| Reference:
|
[17] 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 |
| Reference:
|
[18] Hörmander L.: The Analysis of Linear Partial Differential Operators III. Pseudodifferential Operators.Grundlehren der Mathematischen Wissenschaften, 274, Springer, Berlin-Heidelberg-NewYork, 1985. MR 0781536 |
| Reference:
|
[19] Kirillov A.A.: Elements of the Theory of Representations.Grundlehren der Mathematischen Wissenschaften, 220, Springer, Berlin-Heidelberg-New York, 1976. Zbl 0342.22001, MR 0412321, 10.1007/978-3-642-66243-0 |
| Reference:
|
[20] Kirillov A.A.: Lectures on the Orbit Method.Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, Providence, Rhode Island, 2004. MR 2069175 |
| Reference:
|
[21] Kostant B.: Quantization and unitary representations.in Modern Analysis and Applications, Lecture Notes in Mathematics 170, Springer, Berlin-Heidelberg-New York, 1970, pp. 87--207. Zbl 0249.53016, MR 0294568, 10.1007/BFb0079068 |
| Reference:
|
[22] Neeb K.-H.: Holomorphy and Convexity in Lie Theory.de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter, Berlin, New York, 2000. Zbl 0936.22001, MR 1740617 |
| Reference:
|
[23] Rawnsley J.H.: Representations of a semi direct product by quantization.Math. Proc. Cambridge Philos. Soc. 78 (1975), 345--350. Zbl 0313.22014, MR 0387499, 10.1017/S0305004100051793 |
| Reference:
|
[24] Simms D.J.: Lie Groups and Quantum Mechanics.Lecture Notes in Mathematics, 52, Springer, Berlin-Heidelberg-New York, 1968. Zbl 0161.24002, MR 0232579, 10.1007/BFb0069914 |
| Reference:
|
[25] Taylor M.E.: Noncommutative Harmonis Analysis.Mathematical Surveys and Monographs, Vol. 22, American Mathematical Society, Providence, Rhode Island, 1986. MR 0852988 |
| Reference:
|
[26] Voros A.: An algebra of pseudo differential operators and the asymptotics of quantum mechanics.J. Funct. Anal. 29 (1978), 104--132. MR 0496088, 10.1016/0022-1236(78)90049-6 |
| Reference:
|
[27] Wallach N.R.: Harmonic Analysis on Homogeneous Spaces.Pure and Applied Mathematics, Vol. 19, Marcel Dekker, New York, 1973. Zbl 0265.22022, MR 0498996 |
| Reference:
|
[28] Wildberger N.J.: Convexity and unitary representations of a nilpotent Lie group.Invent. Math. 89 (1989), 281--292. MR 1016265, 10.1007/BF01388854 |
| Reference:
|
[29] Wildberger N.J.: On the Fourier transform of a compact semisimple Lie group.J. Austral. Math. Soc. A 56 (1994), 64--116. Zbl 0842.22015, MR 1250994, 10.1017/S1446788700034741 |
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