## Duke Mathematical Journal

### Radon transform on real, complex, and quaternionic Grassmannians

Genkai Zhang

#### Abstract

Let $G_{n,k}({\mathbb K})$ be the Grassmannian manifold of $k$-dimensional ${\mathbb K}$-subspaces in ${\mathbb K}^n$, where ${\mathbb K}={\mathbb R}, {\mathbb C}, {\mathbb H}$ is the field of real, complex, or quaternionic numbers. For $1\le k \lt k^\prime \le n-1$, we define the Radon transform $({\mathcal R}f)(\eta)$, $\eta \in G_{n,k^{\prime}}({\mathbb K})$, for functions $f(\xi)$ on $G_{n,k}({\mathbb K})$ as an integration over all $\xi \subset \eta$. When $k+k^\prime \le n$, we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive ($k\times k$)-matrices over ${\mathbb K}$. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians

#### Article information

Source
Duke Math. J. Volume 138, Number 1 (2007), 137-160.

Dates
First available in Project Euclid: 9 May 2007

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1178738562

Digital Object Identifier
doi:10.1215/S0012-7094-07-13814-6

Mathematical Reviews number (MathSciNet)
MR2309157

Zentralblatt MATH identifier
1125.26013

#### Citation

Zhang, Genkai. Radon transform on real, complex, and quaternionic Grassmannians. Duke Math. J. 138 (2007), no. 1, 137--160. doi:10.1215/S0012-7094-07-13814-6. http://projecteuclid.org/euclid.dmj/1178738562.

#### References

• J. Faraut and A. KoráNyi, Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford Univ. Press, New York, 1994.
• J. Faraut and G. Travaglini, Bessel functions associated with representations of formally real Jordan algebras, J. Funct. Anal. 71 (1987), 123--141.
• E. L. Grinberg, Radon transforms on higher Grassmannians, J. Differential Geom. 24 (1986), 53--68.
• E. L. Grinberg and B. Rubin, Radon inversion on Grassmannians via Gå rding-Gindikin fractional integrals, Ann. of Math. (2) 159 (2004), 783--817.
• S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978.
• —, Groups and Geometric Analysis, Pure Appl. Math. 113, Academic Press, Orlando, Fla., 1984.
• —, The Radon Transform, 2nd ed., Progr. Math. 5, Birkhäuser, Boston, 1999.
• —, Remarks on B. Rubin: Radon, cosine and sine transforms on real hyperbolic space,'' Adv. Math. 192 (2005), 225.
• C. S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474--523.
• T. Kakehi, Integral geometry on Grassmann manifolds and calculus of invariant differential operators, J. Funct. Anal. 168 (1999), 1--45.
• E. Ournycheva and B. Rubin, The Radon transform of functions of matrix argument, preprint,\arxivmath/0406573v1[math.FA]
• B. Rubin, Radon, cosine and sine transforms on real hyperbolic space, Adv. Math. 170 (2002), 206--223.
• R. S. Strichartz, Harmonic analysis on Grassmannian bundles, Trans. Amer. Math. Soc. 296 (1986), 387--409.