Real reductive groups II.

*(English)*Zbl 0785.22001
Pure and Applied Mathematics, 132, Pt. 2. Boston, MA etc.: Academic Press, Inc.. xiv, 454 p. (1992).

This book finishes a two volume series. Together the two volumes give a self-contained treatment of important parts of representation theory and harmonic analysis on a real reductive Lie group \(G\).

The first volume (1988) dealt with representation theory and its analytic aspects, such as the asymptotic behaviour of matrix coefficients, the regularity theory of characters and the Harish-Chandra limit formula on the space of cusp forms (see the review in Zbl 0666.22002). The present volume deals with harmonic analysis on the group \(G\), giving a complete proof of Harish-Chandra’s Plancherel theorem, and also of the Whittaker Plancherel theorem. In my opinion the two volumes constitute an extremely valuable and important contribution to the literature of the subject.

The treatment of Harish-Chandra’s Plancherel formula is close to Harish- Chandra in spirit, but at some places differs considerably from Harish- Chandra’s original approach. In particular Harish-Chandra approached the standard intertwining operators for the generalized principal series representations via his asymptotic theory of the Eisenstein integral.

The present book starts with an independent study of these intertwining operators in Chapter 10. The main tool is a functional equation for the distribution kernel of the intertwining operator, due to the author and D. Vogan. It allows a meromorphic continuation of the intertwining operator on the space of smooth vectors. Another application of the functional equation is a generalization of L. Cohn’s determinant formula for the generalized \(C\)-function.

Chapter 11 deals with work of the author and W. Casselman on the structure of smooth completions of admissible \(({\mathfrak g},K)\)-modules. It also contains a proof of Casselman’s automatic continuity theorem.

Chapter 12 deals with the theory of the leading term of a matrix coefficient. Harish-Chandra called this the theory of the constant term because of the analogies with the behaviour of Eisenstein series in the theory of automorphic forms. The author’s approach differs from Harish- Chandra’s original approach in that he puts more emphasis on the underlying representation theory than Harish-Chandra (who concentrated on the differential equations). The present approach extends the validity of the theory from bi-\(K\)-finite matrix coefficients to matrix coefficients of \(K\)-finite vectors with \(C^ \infty\)-vectors. Via the representation theory approach it is easier to get control over the dependence of the matrix coefficients on various parameters, which is so important for the applications in the next chapter.

Chapter 13 contains the proof of Harish-Chandra’s Plancherel theorem for the space of \(K\)-finite Schwartz functions. Here the author follows the original approach of Harish-Chandra, via Eisenstein integrals, their wave packets, and the associated Harish-Chandra transforms.

In a technical sense Chapter 14 is independent of the rest of the book. It contains a nice treatment of the ‘abstract representation theory’ which provides the background for the Harish-Chandra Plancherel theorem: the theory of CCR \(C^*\)-algebras (algĂ¨bres \(C^*\) liminaires) and the desintegration of their unitary representations. This chapter ends with a discussion of a result of M. Cowling, U. Haagerup and R. Howe which makes it possible to decide a priori that the Plancherel measure for \(G\) is supported by the tempered representations.

The final Chapter 15 gives a complete treatment of the Whittaker Plancherel theorem. Although the main theorems have been announced by Harish-Chandra, most of the material of this chapter has not been published elsewhere.

The first volume (1988) dealt with representation theory and its analytic aspects, such as the asymptotic behaviour of matrix coefficients, the regularity theory of characters and the Harish-Chandra limit formula on the space of cusp forms (see the review in Zbl 0666.22002). The present volume deals with harmonic analysis on the group \(G\), giving a complete proof of Harish-Chandra’s Plancherel theorem, and also of the Whittaker Plancherel theorem. In my opinion the two volumes constitute an extremely valuable and important contribution to the literature of the subject.

The treatment of Harish-Chandra’s Plancherel formula is close to Harish- Chandra in spirit, but at some places differs considerably from Harish- Chandra’s original approach. In particular Harish-Chandra approached the standard intertwining operators for the generalized principal series representations via his asymptotic theory of the Eisenstein integral.

The present book starts with an independent study of these intertwining operators in Chapter 10. The main tool is a functional equation for the distribution kernel of the intertwining operator, due to the author and D. Vogan. It allows a meromorphic continuation of the intertwining operator on the space of smooth vectors. Another application of the functional equation is a generalization of L. Cohn’s determinant formula for the generalized \(C\)-function.

Chapter 11 deals with work of the author and W. Casselman on the structure of smooth completions of admissible \(({\mathfrak g},K)\)-modules. It also contains a proof of Casselman’s automatic continuity theorem.

Chapter 12 deals with the theory of the leading term of a matrix coefficient. Harish-Chandra called this the theory of the constant term because of the analogies with the behaviour of Eisenstein series in the theory of automorphic forms. The author’s approach differs from Harish- Chandra’s original approach in that he puts more emphasis on the underlying representation theory than Harish-Chandra (who concentrated on the differential equations). The present approach extends the validity of the theory from bi-\(K\)-finite matrix coefficients to matrix coefficients of \(K\)-finite vectors with \(C^ \infty\)-vectors. Via the representation theory approach it is easier to get control over the dependence of the matrix coefficients on various parameters, which is so important for the applications in the next chapter.

Chapter 13 contains the proof of Harish-Chandra’s Plancherel theorem for the space of \(K\)-finite Schwartz functions. Here the author follows the original approach of Harish-Chandra, via Eisenstein integrals, their wave packets, and the associated Harish-Chandra transforms.

In a technical sense Chapter 14 is independent of the rest of the book. It contains a nice treatment of the ‘abstract representation theory’ which provides the background for the Harish-Chandra Plancherel theorem: the theory of CCR \(C^*\)-algebras (algĂ¨bres \(C^*\) liminaires) and the desintegration of their unitary representations. This chapter ends with a discussion of a result of M. Cowling, U. Haagerup and R. Howe which makes it possible to decide a priori that the Plancherel measure for \(G\) is supported by the tempered representations.

The final Chapter 15 gives a complete treatment of the Whittaker Plancherel theorem. Although the main theorems have been announced by Harish-Chandra, most of the material of this chapter has not been published elsewhere.

Reviewer: E.P.van den Ban (Utrecht)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22E30 | Analysis on real and complex Lie groups |

43A80 | Analysis on other specific Lie groups |