Spherical harmonics and integral geometry on projective spaces

Author:
Eric L. Grinberg

Journal:
Trans. Amer. Math. Soc. **279** (1983), 187-203

MSC:
Primary 53C65; Secondary 43A90, 58G15

DOI:
https://doi.org/10.1090/S0002-9947-1983-0704609-1

MathSciNet review:
704609

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Abstract: The Radon transform on associates to a point function the hyperplane function by integration over the hyperplane . If is the dual transform, we can invert by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason [**7**] with very simple computations.

We view the Radon transform as a -invariant map between representations of the group of isometries on function spaces attached to . Pulling back to a sphere via a suitable Hopf fibration and using the theory of spherical harmonics, we can decompose these representations into irreducibles. The scalar by which acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for . The action of is immediately related to the spectrum of . This shows that can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact -point homogeneous spaces: , , , as well as spheres.

**[1]**M. Berger et al.,*Le spectre d'une variété Riemannienne*. Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971.**[2]**Jean Dieudonné,*Special functions and linear representations of Lie groups*, CBMS Regional Conference Series in Mathematics, vol. 42, American Mathematical Society, Providence, R.I., 1980. Expository lectures from the CBMS Regional Conference held at East Carolina University, Greenville, North Carolina, March 5–9, 1979. MR**557540****[3]**I. M. Gel′fand, M. I. Graev, and Z. Ja. Šapiro,*Differential forms and integral geometry*, Funkcional. Anal. i Priložen.**3**(1969), no. 2, 24–40 (Russian). MR**0244919****[4]**I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin,*Generalized functions. Vol. 5*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. MR**0435835****[5]**Victor Guillemin,*The Radon transform on Zoll surfaces*, Advances in Math.**22**(1976), no. 1, 85–119. MR**0426063**, https://doi.org/10.1016/0001-8708(76)90139-0**[6]**V. Guillemin and D. Schaeffer,*Fourier integral operators from the Radon transform point of view*, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 297–300. MR**0380520****[7]**SigurÄ‘ur Helgason,*The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds*, Acta Math.**113**(1965), 153–180. MR**0172311**, https://doi.org/10.1007/BF02391776**[8]**-,*The Radon transform*, Progress in Math., Birkhäuser, Basel, 1980.**[9]**S. Kobayashi and K. Nomizu,*Foundations of differential geometry*, vols. 1,2, Wiley (Interscience), New York. 1969.**[10]**Ian R. Porteous,*Topological geometry*, 2nd ed., Cambridge University Press, Cambridge-New York, 1981. MR**606198****[11]**R. T. Smith,*The spherical representations of groups transitive on 𝑆ⁿ*, Indiana Univ. Math. J.**24**(1974/75), 307–325. MR**0364557**, https://doi.org/10.1512/iumj.1974.24.24028**[12]**Hermann Weyl,*The classical groups*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR**1488158****[13]**George W. Whitehead,*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508****[14]**Robert S. Strichartz,*𝐿^{𝑝} estimates for Radon transforms in Euclidean and non-Euclidean spaces*, Duke Math. J.**48**(1981), no. 4, 699–727. MR**782573**, https://doi.org/10.1215/S0012-7094-81-04839-0

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0704609-1

Keywords:
Integral geometry,
Radon transform,
spherical harmonics,
projective spaces

Article copyright:
© Copyright 1983
American Mathematical Society