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Keywords:
Weyl quantization; Berezin quantization; semidirect product; coadjoint orbits; unitary representations
Weyl quantization; Berezin quantization; semidirect product; coadjoint orbits; unitary representations
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].
References:
[1] Ali S.T., Englis M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17 (2005), no. 4, 391--490. DOI 10.1142/S0129055X05002376 | MR 2151954 | Zbl 1075.81038
[2] Arnal D., Cortet J.-C.: Nilpotent Fourier Transform and Applications. Lett. Math. Phys. 9 (1985), 25--34. DOI 10.1007/BF00398548 | MR 0774736 | Zbl 0616.46041
[3] Baguis P.: Semidirect products and the Pukansky condition. J. Geom. Phys. 25 (1998), 245--270. DOI 10.1016/S0393-0440(97)00028-4 | MR 1619845
[4] Berezin F.A.: Quantization. Math. USSR Izv. 8, 5 (1974), 1109--1165. DOI 10.1070/IM1974v008n05ABEH002140 | Zbl 0976.83531
[5] Cahen B.: Deformation program for principal series representations. Lett. Math. Phys. 36 (1996), 65--75. DOI 10.1007/BF00403252 | MR 1371298 | Zbl 0843.22020
[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. DOI 10.1016/S0764-4442(97)80063-8 | MR 1483721 | Zbl 0883.22016
[7] Cahen B.: Quantification d'orbites coadjointes et théorie des contractions. J. Lie Theory 11 (2001), 257--272. MR 1851792 | Zbl 0973.22009
[8] Cahen B.: Contractions of $SU(1,n)$ and $SU(n+1)$ via Berezin quantization. J. Anal. Math. 97 (2005), 83--102. DOI 10.1007/BF02807403 | MR 2274974
[9] Cahen B.: Weyl quantization for semidirect products. Differential Geom. Appl. 25 (2007), 177--190. DOI 10.1016/j.difgeo.2006.08.005 | MR 2311733 | Zbl 1117.81087
[10] Cahen B.: Weyl quantization for principal series. Beiträge Algebra Geom. 48 (2007), no. 1, 175--190. MR 2326408 | Zbl 1134.22010
[11] Cahen B.: Berezin quantization for discrete series. preprint Univ. Metz (2008), to appear in Beiträge Algebra Geom. MR 2682458
[12] Cahen B.: Berezin quantization on generalized flag manifolds. preprint Univ. Metz (2008), to appear in Math. Scand. MR 2549798
[13] Cahen M., Gutt S., Rawnsley J.: Quantization on Kähler manifolds I, Geometric interpretation of Berezin quantization. J. Geom. Phys. 7 (1990), 45--62. DOI 10.1016/0393-0440(90)90019-Y | MR 1094730
[14] Cotton P., Dooley A.H.: Contraction of an adapted functional calculus. J. Lie Theory 7 (1997), 147--164. MR 1473162 | Zbl 0882.22015
[15] Folland B.: Harmonic Analysis in Phase Space. Princeton Univ. Press, Princeton, 1989. MR 0983366 | Zbl 0682.43001
[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. MR 1766362 | Zbl 1041.53507
[17] Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, Rhode Island, 2001. MR 1834454 | Zbl 0993.53002
[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
[19] Kirillov A.A.: Elements of the Theory of Representations. Grundlehren der Mathematischen Wissenschaften, 220, Springer, Berlin-Heidelberg-New York, 1976. DOI 10.1007/978-3-642-66243-0 | MR 0412321 | Zbl 0342.22001
[20] Kirillov A.A.: Lectures on the Orbit Method. Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, Providence, Rhode Island, 2004. MR 2069175
[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. DOI 10.1007/BFb0079068 | MR 0294568 | Zbl 0249.53016
[22] 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 | Zbl 0936.22001
[23] Rawnsley J.H.: Representations of a semi direct product by quantization. Math. Proc. Cambridge Philos. Soc. 78 (1975), 345--350. DOI 10.1017/S0305004100051793 | MR 0387499 | Zbl 0313.22014
[24] Simms D.J.: Lie Groups and Quantum Mechanics. Lecture Notes in Mathematics, 52, Springer, Berlin-Heidelberg-New York, 1968. DOI 10.1007/BFb0069914 | MR 0232579 | Zbl 0161.24002
[25] Taylor M.E.: Noncommutative Harmonis Analysis. Mathematical Surveys and Monographs, Vol. 22, American Mathematical Society, Providence, Rhode Island, 1986. MR 0852988
[26] Voros A.: An algebra of pseudo differential operators and the asymptotics of quantum mechanics. J. Funct. Anal. 29 (1978), 104--132. DOI 10.1016/0022-1236(78)90049-6 | MR 0496088
[27] Wallach N.R.: Harmonic Analysis on Homogeneous Spaces. Pure and Applied Mathematics, Vol. 19, Marcel Dekker, New York, 1973. MR 0498996 | Zbl 0265.22022
[28] Wildberger N.J.: Convexity and unitary representations of a nilpotent Lie group. Invent. Math. 89 (1989), 281--292. DOI 10.1007/BF01388854 | MR 1016265
[29] Wildberger N.J.: On the Fourier transform of a compact semisimple Lie group. J. Austral. Math. Soc. A 56 (1994), 64--116. DOI 10.1017/S1446788700034741 | MR 1250994 | Zbl 0842.22015
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