Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Radon transform on symmetric matrix domains

Author: Genkai Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 1351-1369
MSC (2000): Primary 22E45, 33C67, 43A85, 44A12
Published electronically: October 17, 2008
MathSciNet review: 2457402
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathbb{K}=\mathbb{R}, \mathbb{C}, \mathbb{H}$ be the field of real, complex or quaternionic numbers and $ M_{p, q}(\mathbb{K})$ the vector space of all $ p\times q$-matrices. Let $ X$ be the matrix unit ball in $ M_{n-r, r}(\mathbb{K})$ consisting of contractive matrices. As a symmetric space, $ X=G/K=O(n-r, r)/O(n-r)\times O(r)$, $ U(n-r, r)/U(n-r)\times U(r)$ and respectively $ Sp(n-r, r)/Sp(n-r)\times Sp(r)$. The matrix unit ball $ y_0$ in $ M_{r^\prime-r, r}$ with $ r^\prime \le n-1$ is a totally geodesic submanifold of $ X$ and let $ Y$ be the set of all $ G$-translations of the submanifold $ y_0$. The set $ Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $ \mathcal Rf(y)$ for functions $ f\in C_0^\infty(X)$ defined by integration of $ f$ over the subset $ y$, and the dual transform $ \mathcal R^t F(x), x\in X$ for functions $ F(y)$ on $ Y$. For $ 2r <n, 2r\le r^\prime$ with a certain evenness condition in the case $ \mathbb{K}=\mathbb{R}$, we find a $ G$-invariant differential operator $ \mathcal M$ and prove it is the right inverse of $ \mathcal R^t \mathcal R$, $ \mathcal R^t \mathcal R \mathcal M f=c f$, for $ f\in C_0^\infty(X)$, $ c\ne 0$. The operator $ f\to \mathcal R^t\mathcal Rf$ is an integration of $ f$ against a (singular) function determined by the root systems of $ X$ and $ y_0$. We study the analytic continuation of the powers of the function and we find a Bernstein-Sato type formula generalizing earlier work of the author in the set up of the Berezin transform. When $ X$ is a rank one domain of hyperbolic balls in $ \mathbb{K}^{n-1}$ and $ y_0$ is the hyperbolic ball in $ \mathbb{K}^{r^\prime -1}$, $ 1<r^\prime<n$ we obtain an inversion formula for the Radon transform, namely $ \mathcal M\mathcal R^t\mathcal R f=c f$. This generalizes earlier results of Helgason for non-compact rank one symmetric spaces for the case $ r^\prime=n-1$.

References [Enhancements On Off] (What's this?)

  • 1. J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), no. 6, 1035-1091. MR 1736928 (2001b:58038)
  • 2. T.H. Baker and P.J. Forrester, Non-symmetric Jack polynomials and intergral kernels, Duke Math. J. 95 (1998), no. 1, 1-50. MR 1646546 (2000b:33006)
  • 3. I. Cherednik, Inverse Harish-Chandra transform and difference operators, Internat. Math. Res. Notices (1997), no. 15, 733-750. MR 1470375 (99d:22018)
  • 4. J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford University Press, Oxford, 1994. MR 1446489 (98g:17031)
  • 5. R. Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 101, Springer-Verlag, Berlin, 1988. MR 954385 (89m:22015)
  • 6. I. M. Gel$ '$fand, M. I. Graev, and R. Roşu, The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds, J. Operator Theory 12 (1984), no. 2, 359-383. MR 757440 (86c:22016)
  • 7. F. B. Gonzalez and T. Kakehi, Pfaffian systems and Radon transforms on affine Grassmann manifolds, Math. Ann. 326 (2003), no. 2, 237-273. MR 1990910 (2004f:53093)
  • 8. E. Grinberg and B. Rubin, Radon inversion on Grassmannians via Gå rding-Gindikin fractional integrals, Ann. of Math. (2) 159 (2004), no. 2, 783-817. MR 2081440 (2005f:58042)
  • 9. E. L. Grinberg, Radon transforms on higher Grassmannians, J. Differential Geom. 24 (1986), no. 1, 53-68. MR 857375 (87m:22023)
  • 10. G. Heckman and H. Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16, Academic Press Inc., San Diego, CA, 1994. MR 1313912 (96j:22019)
  • 11. G. J. Heckman, Root systems and hypergeometric functions. II, Compositio Math. 64 (1987), no. 3, 353-373. MR 918417 (89b:58192b)
  • 12. G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329-352. MR 918416 (89b:58192a)
  • 13. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, London, 1978. MR 0145455 (26:2986)
  • 14. S. Helgason, The Radon transform, second ed., Progress in Mathematics, vol. 5, Birkhäuser Boston Inc., Boston, MA, 1980. MR 573446 (83f:43012)
  • 15. S. Helgason, Groups and geometric analysis, Academic Press, New York, London, 1984. MR 754767 (86c:22017)
  • 16. Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451-479. MR 0367562 (51:3804)
  • 17. T. Kakehi, Integral geometry on Grassmann manifolds and calculus of invariant differential operators, J. Funct. Anal. 168 (1999), no. 1, 1-45. MR 1717855 (2000k:53069)
  • 18. O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
  • 19. E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75-121. MR 1353018 (98f:33025)
  • 20. E. Ournycheva and B. Rubin, The Radon transform of functions of matrix argument, preprint.
  • 21. B. Rubin, Radon, cosine and sine transforms on real hyperbolic space, Adv. Math. 170 (2002), 206-223. MR 1932329 (2004b:43007)
  • 22. R. Schimming and H. Schlichtkrull, Helmholtz operators on harmonic manifolds, Acta Math. 173 (1994), no. 2, 235-258. MR 1301393 (96g:58175)
  • 23. G. Zhang, Radon transform on real, complex and quaternionic Grassmannians, Duke Math. J., 138 (2007), no. 1, 137-160. MR 2309157 (2008c:44002)
  • 24. -, Spherical transform and Jacobi polynomials on root systems of type BC, Intern. Math. Res. Notices, 2005, no. 51. 3169-3189. MR 2187504 (2006i:33014)
  • 25. -, Berezin transform on compact Hermitian symmetric spaces, Manuscripta Math. 97 (1998), no. 3, 371-388. MR 1654800 (2000c:22013)
  • 26. -, Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 195 (2002), 306-349. MR 1940358 (2004f:32026)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E45, 33C67, 43A85, 44A12

Retrieve articles in all journals with MSC (2000): 22E45, 33C67, 43A85, 44A12

Additional Information

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden

Keywords: Radon transform, inverse Radon transform, symmetric domains, Grassmannian manifolds, Lie groups, fractional integrations, Bernstein-Sato formula, Cherednik operators, invariant differential operators
Received by editor(s): January 17, 2007
Published electronically: October 17, 2008
Additional Notes: This research was supported by the Swedish Science Council (VR)
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society