The α-Cosine Transform and Intertwining Integrals on Real Grassmannians

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

In this paper we describe the range of the α-cosine transform between real Grassmannians in terms of the decomposition under the action of the special orthogonal group. As one of the steps in the proof we show that the image of certain intertwining operators between maximally degenerate principal series representations is irreducible.

Keywords

Line Bundle Parabolic Subgroup Full Subcategory Composition Series Perverse Sheave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We express our gratitude to T. Braden for the explanations of the results of [7], and to B. Rubin for communication us Proposition 2.2 and important remarks on the first version of the paper. We are grateful to A. Beilinson, J. Bernstein, A. Braverman, and D. Vogan for very useful discussions. We thank A. Koldobsky for useful discussions and some references.

References

  1. 1.
    S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11(2), 244–272 (2001)Google Scholar
  2. 2.
    A. Alesker, J. Bernstein, Range characterization of the cosine transform on higher Grassmannians. Adv. Math. 184(2), 367–379 (2004)Google Scholar
  3. 3.
    D. Barbasch, S. Sahi, B. Speh, Degenerate series representations for GL(2n, R) and Fourier analysis. Symposia Mathematica, Rome, 1988. Sympos. Math., vol. XXXI (Academic, London, 1990), pp. 45–69Google Scholar
  4. 4.
    A. Beilinson, J. Bernstein, Localisation de g-modules (French). C. R. Acad. Sci. Paris Sèr. I Math. 292(1), 15–18 (1981)Google Scholar
  5. 5.
    F. Bien, D-Modules and Spherical Representations. Mathematical Notes, vol. 39 (Princeton University Press, Princeton, 1990)Google Scholar
  6. 6.
    W. Borho, J.-L. Brylinski, Differential operators on homogeneous spaces, III, Characteristic varieties of Harish-Chandra modules and primitive ideals. Invent. Math. 80(1), 1–68 (1985)Google Scholar
  7. 7.
    T. Braden, M. Grinberg, Perverse sheaves on rank stratifications. Duke Math. J. 96(2), 317–362 (1999)Google Scholar
  8. 8.
    W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G. Canad. J. Math. 41(3), 385–438 (1989)Google Scholar
  9. 9.
    T. Fujimura, On some degenerate principal series representations of O(p, 2). J. Lie Theor. 11(1), 23–55 (2001)Google Scholar
  10. 10.
    I.M. Gelfand, M.I. Graev, R.I. Roşu, The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. J. Operat. Theor. 12(2), 359–383 (1984)Google Scholar
  11. 11.
    P. Goodey, R. Howard, Processes of flats induced by higher-dimensional processes. Adv. Math. 80(1), 92–109 (1990)Google Scholar
  12. 12.
    P. Goodey, R. Howard, in Processes of Flats Induced by Higher-Dimensional Processes. II. Integral Geometry and Tomography (Arcata, CA, 1989), Contemp. Math., vol. 113 (Am. Math. Soc., Providence, 1990), pp. 111–119Google Scholar
  13. 13.
    P. Goodey, R. Howard, M. Reeder, Processes of flats induced by higher-dimensional processes. III. Geom. Dedicata 61(3), 257–269 (1996)Google Scholar
  14. 14.
    E.L. Grinberg, Radon transforms on higher Grassmannians. J. Diff. Geom. 24(1), 53–68 (1986)Google Scholar
  15. 15.
    R. Howe, S.T. Lee, Degenerate principal series representations of GLn(C) and GLn(R). J. Funct. Anal. 166(2), 244–309 (1999)Google Scholar
  16. 16.
    R. Howe, E.C. Tan, Homogeneous functions on light cones: The infinitesimal structure of some degenerate principal series representations. Bull. Am. Math. Soc. (N.S.) 28(1), 1–74 (1993)Google Scholar
  17. 17.
    K.D. Johnson, Degenerate principal series and compact groups. Math. Ann. 287(4), 703–718 (1990)Google Scholar
  18. 18.
    M. Kashiwara, Representation theory and D-modules on flag varieties. Orbites unipotentes et représentations, III. Astérisque 173–174(9), 55–109 (1989)Google Scholar
  19. 19.
    A. Koldobsky, Inverse formula for the Blaschke-Levy representation. Houston J. Math. 23, 95–108 (1997)Google Scholar
  20. 20.
    H. Kraft, C. Procesi, On the geometry of conjugacy classes in classical groups. Comment. Math. Helv. 57(4), 539–602 (1982)Google Scholar
  21. 21.
    S.S. Kudla, S. Rallis, Degenerate principal series and invariant distributions. Israel J. Math. 69(1), 25–45 (1990)Google Scholar
  22. 22.
    S.T. Lee, Degenerate principal series representations of Sp(2n, R). Compos. Math. 103(2), 123–151 (1996)Google Scholar
  23. 23.
    S.T. Lee, H.Y. Loke, Degenerate principal series representations of U(p, q) and Spin 0(p, q). Compos. Math. 132(3), 311–348 (2002)Google Scholar
  24. 24.
    G. Matheron, Un théorème d’unicité pour les hyperplans poissoniens (French). J. Appl. Probab. 11, 184–189 (1974)Google Scholar
  25. 25.
    G. Matheron, in Random Sets and Integral Geometry. Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1975)Google Scholar
  26. 26.
    S.T. Lee, On some degenerate principal series representations of U(n, n). J. Funct. Anal. 126(2), 305–366 (1994)Google Scholar
  27. 27.
    B. Rubin, The Calderón reproducing formula, windowed X-ray transforms and Radon transforms in L p-spaces. J. Fourier Anal. Appl. 4, 175–197 (1998)Google Scholar
  28. 28.
    B. Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform. Adv. Appl. Math. 29, 471–497 (2002)Google Scholar
  29. 29.
    S. Sahi, Jordan algebras and degenerate principal series. J. Reine Angew. Math. 462, 1–18 (1995)Google Scholar
  30. 30.
    R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper (German). Math. Nachr. 44, 55–75 (1970)Google Scholar
  31. 31.
    E. Spodarev, Cauchy-Kubote type integral formula for generalized cosine transforms. IZV. Nats. Akad. Nauk Armenii Mat. 37(1), 52–69 (2002) translation in J. Contemp. Math. Anal. 37(1), 47–63 (2002)Google Scholar
  32. 32.
    M. Sugiura, Representations of compact groups realized by spherical functions on symmetric spaces. Proc. Jpn. Acad. 38, 111–113 (1962)Google Scholar
  33. 33.
    M. Takeuchi, Modern Spherical Functions. Translated from the 1975 Japanese original by Toshinobu Nagura. Translations of Mathematical Monographs, vol. 135 (American Mathematical Society, Providence, 1994)Google Scholar
  34. 34.
    D. Vogan, Gelfand–Kirillov dimension of Harish-Chandra modules. Inv. Math. 48, 75–98 (1978)Google Scholar
  35. 35.
    N.R. Wallach, in Real Reductive Groups. I. Pure and Applied Mathematics, vol. 132 (Academic, Boston, 1988)Google Scholar
  36. 36.
    N.R. Wallach, in Real Reductive Groups. II. Pure and Applied Mathematics, vol. 132-II (Academic, Boston, 1992)Google Scholar
  37. 37.
    D.P. Želobenko, Compact Lie Groups and Their Representations. Translated from the Russian by Israel Program for Scientific Translations. Translations of Mathematical Monographs, vol. 40 (American Mathematical Society, Providence, 1973)Google Scholar
  38. 38.
    G.K. Zhang, Jordan algebras and generalized principal series representations. Math. Ann. 302(4), 773–786 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Sackler Faculty of Exact Sciences, Department of MathematicsTel Aviv UniversityTel AvivIsrael

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