##
**Hyperfunctions and harmonic analysis on symmetric spaces.**
*(English)*
Zbl 0555.43002

Progress in Mathematics, Vol. 49. Boston - Basel - Stuttgart: BirkhĂ¤user. XIV, 185 p. SFr. 48.00; DM 58.00 (1984).

Ever since M. Kashiwara, A. Kowata, K. Minomura, K. Okamoto, T. Oshima and M. Tanaka in [Ann. Math., II. Ser. 107, 1-39 (1978; Zbl 0377.43012)] have demonstrated the strength of an approach based on the use of the theory of hyperfunctions to problems in harmonic analysis by proving Helgason’s conjecture, and appreciation for that technique has been growing among harmonic analysis. The book reviewed is apparently the first monograph devoted specifically to applications of the theory of hyperfunctions in harmonic analysis. The applications considered are based on results concerning systems of PDEs with regular singularity and used for solving two problems in the analysis on symmetric spaces. The first is that of representing eigenfunctions of invariant differential operators on symmetric space as Poisson transforms of their hyperfunction boundary values. This was originally posed by S. Helgason [Adv. Math. 5, 1-154 (1970; Zbl 0209.254)] in the context of Riemannian symmetric spaces and in its full generality remained open untill 1978, when it was positively answered in the remarkable work of Kashiwara et al. quoted above. The solution has been preceded by some partial results by Helgason and several teams composed of certain of the future coauthors of the final and conclusive version. The book of Schlichtkrull brings (with some modifications) an original proof of Kashiwara’s et al. solution to the Helgason conjecture in an almost complete form - almost referring to the omission of a certain set of singular eigenvalues, what would have required yet more refined machinery, along with some extensions due to the author.

The book is split into 8 chapters each concluded with a section of historical and bibliographical notes - these are: 1) Hyperfunctions and microlocal analysis - an introduction, 2) Differential equations with regular singularities, 3) Riemannian symmetric spaces and invariant differential operators- preliminaries, 4) A compact imbedding, 5) Boundary values and Poisson integral representations, 6) Boundary values on the full boundary, 7) Semisimple symmetric spaces, 8) Construction of functions with integrable square.

The first two chapters are a rapid introduction to hyperfunctions, microlocal analysis and partial differential equations with regular singularities. There the proofs are missing or replaced by a study of a typical example. The notions from sheaf theory, which are needed are also discussed. The short Chapter 3 gives a summary of mostly standard notions and results about Riemannian symmetric spaces and parabolic subgroups of real reducitve groups - no proofs are given. The real work begins in Ch. 4 which gives an exposition of Oshima’s construction of an imbedding of a Riemannian symmetric space \(X=G/K\) into a compact real analytic G-space \(\tilde X,\) which is larger then the usual Satake-Furstenberg compactification and contains boundary components different from the usual maximal Furstenberg boundary. They play a role later (Chapter 6). The idea is that the invariant differential operators on X extend to G- invariant differential operators on \(\tilde X\) with regular singularities along the boundary. Then the general results from Chapter 2 may be applied to yield the existence of boundary values of eigenfunctions and to show that the Poisson transform of these gives the original eigenfunctions. This is done following Kashiwara et al. in Ch. 5 for the Furstenberg boundary whereas in Ch. 6 resuls of the author for the other boundary components are given.

The two last Chapters are devoted to the second major theme - construction of the Flensted-Jensen discrete series for a semisimple nonriemannian symmetric space.

The reviewer has found reading the book a real pleasure.

The book is split into 8 chapters each concluded with a section of historical and bibliographical notes - these are: 1) Hyperfunctions and microlocal analysis - an introduction, 2) Differential equations with regular singularities, 3) Riemannian symmetric spaces and invariant differential operators- preliminaries, 4) A compact imbedding, 5) Boundary values and Poisson integral representations, 6) Boundary values on the full boundary, 7) Semisimple symmetric spaces, 8) Construction of functions with integrable square.

The first two chapters are a rapid introduction to hyperfunctions, microlocal analysis and partial differential equations with regular singularities. There the proofs are missing or replaced by a study of a typical example. The notions from sheaf theory, which are needed are also discussed. The short Chapter 3 gives a summary of mostly standard notions and results about Riemannian symmetric spaces and parabolic subgroups of real reducitve groups - no proofs are given. The real work begins in Ch. 4 which gives an exposition of Oshima’s construction of an imbedding of a Riemannian symmetric space \(X=G/K\) into a compact real analytic G-space \(\tilde X,\) which is larger then the usual Satake-Furstenberg compactification and contains boundary components different from the usual maximal Furstenberg boundary. They play a role later (Chapter 6). The idea is that the invariant differential operators on X extend to G- invariant differential operators on \(\tilde X\) with regular singularities along the boundary. Then the general results from Chapter 2 may be applied to yield the existence of boundary values of eigenfunctions and to show that the Poisson transform of these gives the original eigenfunctions. This is done following Kashiwara et al. in Ch. 5 for the Furstenberg boundary whereas in Ch. 6 resuls of the author for the other boundary components are given.

The two last Chapters are devoted to the second major theme - construction of the Flensted-Jensen discrete series for a semisimple nonriemannian symmetric space.

The reviewer has found reading the book a real pleasure.

Reviewer: A.Strasburger

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

43A85 | Harmonic analysis on homogeneous spaces |

46F15 | Hyperfunctions, analytic functionals |

22E46 | Semisimple Lie groups and their representations |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

58J15 | Relations of PDEs on manifolds with hyperfunctions |