Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 078, 19 pages      arXiv:1108.3357      http://dx.doi.org/10.3842/SIGMA.2011.078
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Harmonic Analysis on Quantum Complex Hyperbolic Spaces

Olga Bershtein and Yevgen Kolisnyk
Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103, Kharkov, Ukraine

Received April 30, 2011, in final form August 10, 2011; Published online August 18, 2011

Abstract
In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order q-difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.

Key words: quantum groups, harmonic analysis on quantum symmetric spaces; q-difference operators; Al-Salam-Chihara polynomials; Plancherel formula.

pdf (455 kb)   tex (20 kb)

References

  1. Akhiezer N.I., Glazman I.M., Theory of linear operators in Hilbert space, Dover Publications, New York, 1993.
  2. Bershtein O., Kolisnyk Ye., Plancherel measure for the quantum matrix ball. I, J. Math. Phys. Anal. Geom. 5 (2009), 315-346, arXiv:0903.4068.
  3. Bershtein O., Sinelshchikov S., Function theory on a q-analog of complex hyperbolic space, arXiv:1009.6063.
  4. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  5. Faraut J., Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. 58 (1979), 369-444.
  6. Gaspar G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.
  7. Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996.
  8. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  9. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998, http://aw.twi.tudelft.nl/~koekoek/askey/.
  10. Koelink E., Stokman J.V., Fourier transforms on the quantum SU(1,1) group, with an appendix by M. Rahman, Publ. Res. Inst. Math. Sci. 37 (2001), 621-715, math.QA/9911163.
  11. Molchanov V.F., Spherical functions on hyperboloids, Mat. Sb. 99 (1976), no. 2, 139-161 (English transl.: Math. USSR Sb. 28 (1976), no. 2, 119-139).
  12. Molchanov V.F., Harmonic analysis on homogeneous spaces, Itogi Nauki i Tekhniki, Vol. 59, VINITI, Moscow, 1990, 5-144 (English transl.: Encycl. Math. Sci., Vol. 59, Springer, Berlin, 1995, 1-135).
  13. Onishchik A.L., Vinberg E.B., Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990.
  14. Reshetikhin N.Yu., Takhtadzhyan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), 178-206 (English transl.: Leningrad Math. J. 1 (1990), 193-225).
  15. Shklyarov D., Sinel'shchikov S., Vaksman L., Fock representations and quantum matrices, Internat. J. Math. 15 (2004), 855-894, math.QA/0410605.
  16. Shklyarov D., Sinel'shchikov S., Vaksman L., On function theory on quantum disc: q-differential equations and Fourier transform, math.QA/9809002.
  17. Shklyarov D., Sinel'shchikov S., Stolin A., Vaksman L., On a q-analogue of the Penrose transform, Ukr. Phys. J. 47 (2003), 288-292.
  18. Ueno K., Spectral analysis for the Casimir operator on the quantum group SU(1,1), Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 2, 42-44.
  19. Vaksman L.L., Quantum bounded symmetric domains, translated by O. Bershtein and S. Sinel'shchikov, Translations of Mathematical Monographs, Vol. 238, American Mathematical Society, Providence, RI, 2010.
  20. Vaksman L.L., Korogodskii L.I., Spherical functions on the quantum group SU(1,1) and a q-analogue of the Mehler-Fock formula, Funktsional. Anal. i Prilozhen. 25 (1991), no. 1, 60-62 (English transl.: Funct. Anal. Appl. 25 (1991), no. 1, 48-49).
  21. van Dijk G., Sharshov Yu.A., The Plancherel formula for line bundles on complex hyperbolic spaces, J. Math. Pures Appl. 79 (2000), 451-473.

Previous article   Next article   Contents of Volume 7 (2011)