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Geometric analysis on symmetric spaces. (English) Zbl 0809.53057
Mathematical Surveys and Monographs. 39. Providence, RI: American Mathematical Society (AMS). xiv, 611 p. (1994).
The monograph is devoted to geometric analysis on non-compact Riemannian symmetric space $$X$$. Central objects of study are the algebra $$D(X)$$ of invariant differential operators on $$X$$ and the Radon transformation between functions on $$X$$ and functions on the homogeneous space $$Y$$ of horocycles on $$X$$. Chapter 1 has introductory character. The author presents here the general notion of Radon transformation on $$X$$ into functions on $$Y$$, defined by integration along fibers of the second fibration. Two main problems are stated: (1) determine a function on the basis of its Radon transform (inversion problem) (2) determine the range and the kernel of the Radon transform (range problem). Some classical results on the solution of these problems in special cases are discussed. Chapters 2 and 4 deal with the solution of the problems for the case when $$X = G/K$$ is a Riemannian symmetric space of non-compact type, $$Y = G/MN$$ is the space of horocycles in $$X$$ and $$Z = G/M$$. Also the Plancherel formula for the Radon transformation is established here, the relations between the algebras $$D(X)$$, $$D(Y)$$ of invariant differential operators are studied in detail, and the theory of conical distributions on $$Y$$ (which are the analogs of the spherical functions on $$X$$) are developed. The subject of Chapter 3 is the Fourier transform of functions on the symmetric space $$X$$, which gives a simultaneous diagonalization of operators from $$D(X)$$. In particular, the inversion formula and Paley- Wiener type formula for the Fourier transform are obtained. Results of Chapters 1-4 are used in Chapter 5 to study some natural problems for invariant differential operators on a symmetric space $$X$$, solvability questions, the structure of the joint eigenfunctions, with the emphasis on the harmonic functions and the solutions to the invariant wave equation on $$X$$. The last Chapter 6 deals with representations of $$G$$ which naturally arise from the joint eigenspaces of the operators in the algebras $$D(X)$$ and $$D(Y)$$.

##### MSC:
 53C35 Differential geometry of symmetric spaces 58E20 Harmonic maps, etc. 43A85 Harmonic analysis on homogeneous spaces 44A12 Radon transform 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry