##
**Real reductive groups I.**
*(English)*
Zbl 0666.22002

Pure and Applied Mathematics (Academic Press) 132-1. Boston, MA etc.: Academic Press, Inc. (ISBN 0-12-732960-9). xix, 412 p. (1988).

This book deals with the representation theory of real reductive Lie groups. Its emphasis is on analytic aspects of the theory. As the author announces it will be followed by a second volume dealing with Harish- Chandra’s Plancherel theorem.

The present book deals with an enormous wealth of material containing such highlights as the subquotient theorem, the subrepresentation theorem, Jacquet modules, asymptotics of matrix coefficients, Zuckerman induction, orbital integrals, character theory, the discrete series, and (\({\mathfrak g},K)\)-cohomology. Nevertheless the author has succeeded to keep the treatment completely self-contained; only familiarity with the representation theory of compact groups is presupposed. Tools whose introduction would have interrupted the flow of the argumentation are dealt with in numerous appendices. Notable examples of this are Zuckerman’s translation principle, the theory of radial components of differential operators, Kostant’s theorem on nilpotent orbits, results of de Rham and Gelfand-Shilov on fundamental solutions and Kostant’s theorem on \({\mathfrak n}\)-cohomology.

His own contributions to the subject have allowed Wallach to cut his own path through the existing theory, presenting many shortcuts in and between original expositions along the way. In the following we shall indicate the author’s route to give an impression of this.

Chapters 1 and 2 deal with elementary representation theory and the structure theory of a real reductive Lie group G. Then in Chapter 3 the basic theory of \(({\mathfrak g},K)\)-modules including the admissibility results of Harish-Chandra is treated. The sub-quotient theorem of Harish- Chandra and Lepowsky for irreducible \(({\mathfrak g},K)\)-modules is proved. In particular it implies that such modules always do come from Hilbert space representations hence admit K-finite matrix coefficients. Using growth estimates on these, Casselman’s subrepresentation theorem is then rapidly obtained.

Chapter 4 deals with the asymptotic behaviour of matrix coefficients of admissible representations. Harish-Chandra’s original approach was to consider matrix coefficients of K-finite vectors: from the system of differential equations satisfied by these coefficients he obtained convergent series expansions for them. This approach was later clarified and refined by Casselman and Miličić who showed that the system of differential equations has regular singularities at infinity. On the other hand, for the completion of his proof of the Plancherel formula Harish-Chandra did not need full expansions: it sufficed to have the top order part (called the ‘constant term’) in the asymptotics of K-finite matrix coefficients of tempered representations together with good estimates on the remainder terms. Harish-Chandra achieved this by what he called ‘the method of the constant term’.

In the present book Wallach follows a different approach, involving asymptotic expansions for a bigger class of matrix coefficients; the expansions do not converge in general, but are genuine asymptotic expansions. The approach was developed by Casselman and himself: it was motivated by the p-adic theory of Jacquet and by the above mentioned theory of the constant term. We shall briefly indicate the type of asymptotics that is dealt with here.

Let \(\pi\) be an admissible representation of finite length of G in a Hilbert space H. Let \(H^{\infty}\) denote the (Fréchet) space of \(C^{\infty}\) vectors and let \(H^*_ K\) denote the contragredient of the Harish-Chandra module \(H_ K\). Let \(\sigma \in H^*_ K\) be fixed, and consider its matrix coefficient \(m_ v(g)=\sigma(\pi(g)v)\) \((g\in G)\) with a \(C^{\infty}\) vector \(v\in H^{\infty}\). Then for \(X\in a^+\) (the positive Weyl chamber associated with a minimal parabolic subgroup), one has an asymptotic expansion \[ m_ v(\exp tX)\sim \sum_{\mu \in E}e^{t\mu (X)}\sum_{\eta \in L^+}e^{-t\eta (X)}p_{\mu,\eta}(tX,v)\quad (t\to \infty). \] Here E is a finite subset of \(a^*_ c\), \(L^+\) is the semi-lattice spanned by the positive a- roots, and the \(p_{\mu,\eta}(X,v)\) are polynomials in X: they are obtained via Jacquet module theory. Similar expansions are obtained along the walls of \(a^+\). The \(p_{\mu,\eta}\) and the remainder terms satisfy estimates that are locally uniform in X and uniform in v (in terms of continuous seminorms of \(H^{\infty})\). This uniformity makes the asymptotics into a powerful tool for analysis. The author announces that in the second volume the theory of the constant term will be obtained as a consequence of the presented method.

Chapter 5 deals with the Langlands classification of irreducible \(({\mathfrak g},K)\)-modules in terms of tempered irreducible \(({\mathfrak g},K)\)-modules.

Chapters 6-8 deal with the theory of the discrete series. The approach followed here deviates from Harish-Chandra’s original approach in that it starts, in Chapter 6, with a construction of a series of (\({\mathfrak g},K)\)- modules (called the ‘fundamental series’) by a method equivalent to Zuckerman’s derived functor construction. By the author’s elementary proof of Vogan’s unitarity result the modules of this fundamental series are shown to be unitary. Moreover by the Jacquet module approach to the asymptotics of matrix coefficients the fundamental series are seen to be tempered, and square integrable when G has a compact Cartan group. After this it remains to be shown that the constructed series exhaust the discrete series. This is achieved in Chapters 7, 8.

Chapter 7 deals with analysis on the space of cusp forms, which is quickly shown to contain the space of K-finite matrix coefficients of the discrete series (eventually the latter space is shown to be dense). The author has found a way of treating Harish-Chandra’s theory of orbital integrals on the Schwartz space before developing character theory. Another deviation of Harish-Chandra’s original approach is the simpler proof of a restricted version of Harish-Chandra’s limit formula: the restricted version holds for a compact Cartan subgroup, and on the space of cusp forms. It turns out to be sufficient for the determination of the discrete series. At this stage it is shown that G has a discrete series if and only if there is a compact Cartan subgroup.

In Chapter 8 the local integrability theorem for characters on G is obtained. The proof given follows the lines of Harish-Chandra’s original proof. The main difference is that the theorem which asserts that analytic G-invariant differential operators which annihilate the G- invariant smooth functions on \({\mathfrak g}\) also annihilate the G-invariant distributions is replaced by a stronger result, using an idea of Duistermaat (unpublished). Chapter 8 finishes with the proof that the fundamental series exhaust the discrete series, when G has a compact Cartan subgroup.

The final Chapter 9 stands somewhat apart: depending only on the first six chapters, it gives an introduction to \(({\mathfrak g},K)\)-cohomology, treats vanishing theorems of Kumaresan, Enright, Vogan-Zuckerman and the calculation of \(({\mathfrak g},K)\)-cohomology with respect to the tensor product of a finite dimensional and an irreducible unitary representation (due to Vogan and Zuckerman).

The present book deals with an enormous wealth of material containing such highlights as the subquotient theorem, the subrepresentation theorem, Jacquet modules, asymptotics of matrix coefficients, Zuckerman induction, orbital integrals, character theory, the discrete series, and (\({\mathfrak g},K)\)-cohomology. Nevertheless the author has succeeded to keep the treatment completely self-contained; only familiarity with the representation theory of compact groups is presupposed. Tools whose introduction would have interrupted the flow of the argumentation are dealt with in numerous appendices. Notable examples of this are Zuckerman’s translation principle, the theory of radial components of differential operators, Kostant’s theorem on nilpotent orbits, results of de Rham and Gelfand-Shilov on fundamental solutions and Kostant’s theorem on \({\mathfrak n}\)-cohomology.

His own contributions to the subject have allowed Wallach to cut his own path through the existing theory, presenting many shortcuts in and between original expositions along the way. In the following we shall indicate the author’s route to give an impression of this.

Chapters 1 and 2 deal with elementary representation theory and the structure theory of a real reductive Lie group G. Then in Chapter 3 the basic theory of \(({\mathfrak g},K)\)-modules including the admissibility results of Harish-Chandra is treated. The sub-quotient theorem of Harish- Chandra and Lepowsky for irreducible \(({\mathfrak g},K)\)-modules is proved. In particular it implies that such modules always do come from Hilbert space representations hence admit K-finite matrix coefficients. Using growth estimates on these, Casselman’s subrepresentation theorem is then rapidly obtained.

Chapter 4 deals with the asymptotic behaviour of matrix coefficients of admissible representations. Harish-Chandra’s original approach was to consider matrix coefficients of K-finite vectors: from the system of differential equations satisfied by these coefficients he obtained convergent series expansions for them. This approach was later clarified and refined by Casselman and Miličić who showed that the system of differential equations has regular singularities at infinity. On the other hand, for the completion of his proof of the Plancherel formula Harish-Chandra did not need full expansions: it sufficed to have the top order part (called the ‘constant term’) in the asymptotics of K-finite matrix coefficients of tempered representations together with good estimates on the remainder terms. Harish-Chandra achieved this by what he called ‘the method of the constant term’.

In the present book Wallach follows a different approach, involving asymptotic expansions for a bigger class of matrix coefficients; the expansions do not converge in general, but are genuine asymptotic expansions. The approach was developed by Casselman and himself: it was motivated by the p-adic theory of Jacquet and by the above mentioned theory of the constant term. We shall briefly indicate the type of asymptotics that is dealt with here.

Let \(\pi\) be an admissible representation of finite length of G in a Hilbert space H. Let \(H^{\infty}\) denote the (Fréchet) space of \(C^{\infty}\) vectors and let \(H^*_ K\) denote the contragredient of the Harish-Chandra module \(H_ K\). Let \(\sigma \in H^*_ K\) be fixed, and consider its matrix coefficient \(m_ v(g)=\sigma(\pi(g)v)\) \((g\in G)\) with a \(C^{\infty}\) vector \(v\in H^{\infty}\). Then for \(X\in a^+\) (the positive Weyl chamber associated with a minimal parabolic subgroup), one has an asymptotic expansion \[ m_ v(\exp tX)\sim \sum_{\mu \in E}e^{t\mu (X)}\sum_{\eta \in L^+}e^{-t\eta (X)}p_{\mu,\eta}(tX,v)\quad (t\to \infty). \] Here E is a finite subset of \(a^*_ c\), \(L^+\) is the semi-lattice spanned by the positive a- roots, and the \(p_{\mu,\eta}(X,v)\) are polynomials in X: they are obtained via Jacquet module theory. Similar expansions are obtained along the walls of \(a^+\). The \(p_{\mu,\eta}\) and the remainder terms satisfy estimates that are locally uniform in X and uniform in v (in terms of continuous seminorms of \(H^{\infty})\). This uniformity makes the asymptotics into a powerful tool for analysis. The author announces that in the second volume the theory of the constant term will be obtained as a consequence of the presented method.

Chapter 5 deals with the Langlands classification of irreducible \(({\mathfrak g},K)\)-modules in terms of tempered irreducible \(({\mathfrak g},K)\)-modules.

Chapters 6-8 deal with the theory of the discrete series. The approach followed here deviates from Harish-Chandra’s original approach in that it starts, in Chapter 6, with a construction of a series of (\({\mathfrak g},K)\)- modules (called the ‘fundamental series’) by a method equivalent to Zuckerman’s derived functor construction. By the author’s elementary proof of Vogan’s unitarity result the modules of this fundamental series are shown to be unitary. Moreover by the Jacquet module approach to the asymptotics of matrix coefficients the fundamental series are seen to be tempered, and square integrable when G has a compact Cartan group. After this it remains to be shown that the constructed series exhaust the discrete series. This is achieved in Chapters 7, 8.

Chapter 7 deals with analysis on the space of cusp forms, which is quickly shown to contain the space of K-finite matrix coefficients of the discrete series (eventually the latter space is shown to be dense). The author has found a way of treating Harish-Chandra’s theory of orbital integrals on the Schwartz space before developing character theory. Another deviation of Harish-Chandra’s original approach is the simpler proof of a restricted version of Harish-Chandra’s limit formula: the restricted version holds for a compact Cartan subgroup, and on the space of cusp forms. It turns out to be sufficient for the determination of the discrete series. At this stage it is shown that G has a discrete series if and only if there is a compact Cartan subgroup.

In Chapter 8 the local integrability theorem for characters on G is obtained. The proof given follows the lines of Harish-Chandra’s original proof. The main difference is that the theorem which asserts that analytic G-invariant differential operators which annihilate the G- invariant smooth functions on \({\mathfrak g}\) also annihilate the G-invariant distributions is replaced by a stronger result, using an idea of Duistermaat (unpublished). Chapter 8 finishes with the proof that the fundamental series exhaust the discrete series, when G has a compact Cartan subgroup.

The final Chapter 9 stands somewhat apart: depending only on the first six chapters, it gives an introduction to \(({\mathfrak g},K)\)-cohomology, treats vanishing theorems of Kumaresan, Enright, Vogan-Zuckerman and the calculation of \(({\mathfrak g},K)\)-cohomology with respect to the tensor product of a finite dimensional and an irreducible unitary representation (due to Vogan and Zuckerman).

Reviewer: E.P.van den Ban

### MSC:

22E46 | Semisimple Lie groups and their representations |

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

57T10 | Homology and cohomology of Lie groups |