Continuous cohomology, discrete subgroups, and representations of reductive groups. 2nd ed.

*(English)*Zbl 0980.22015
Mathematical Surveys and Monographs. 67. Providence, RI: American Mathematical Society (AMS). xvii, 260 p. (2000).

There are many properties of topological groups that can be put into the framework of cohomology. We mention two of them: the amenability and the property (T) of Kazhdan. We recall these notions and the links with the cohomology of groups.

A locally compact group \(G\) is called amenable if there exists a nontrivial invariant mean on the space \(C_{\text{b}}(G)\) of continuous and bounded functions on \(G\). Obviously, any compact group is amenable. It is proved that \(G\) is amenable if and only if \(H^1(G,E)=0\) for any dual Banach \(G\)-module \(E\). The locally compact group \(G\) has the property (T) of Kazhdan if the identity representation is an isolated point in \(\hat G\) (the unital dual of \(G\)). This property is equivalent to \(H^1(G,E)=\{0\}\) for any unital \(G\)-module \(E\). The cohomology of locally compact groups or of Lie groups is a domain that requires a solid background in the representation theory of groups (in particular, the machinery of induction).

This monograph, written by two important specialists in the domain, is the development of a seminar on the cohomology of discrete subgroups of Lie groups, held at the Institute for Advanced Studies in Princeton.

The first chapter starts with recalling the relative cohomology (in the spirit of Cartan-Eilenberg) \(H^*({\mathfrak g},{\mathfrak k};V)\) of a \(\mathfrak g\)-module \(V\), where \({\mathfrak k}\) is a subalgebra of the Lie algebra \(\mathfrak g\). If \(G\) is a Lie group and \(K\) a closed connected subgroup of \(G\), then \(H^*({\mathfrak g},{\mathfrak k};V)\) is isomorphic with the cohomology of the module of differential forms, \(G\)-invariant on \(G/K\) and valued in \(V\). In particular, if \(G\) is compact and connected, and \(V\) is a trivial \(G\)-module of finite dimension, then by using de Rham’s theorem, we obtain: \(H^*({\mathfrak g},{\mathfrak k};V)\simeq H^*(G/K;V)\). Also, we have to mention, in this chapter, the Poincaré duality (Proposition 7.6): If \(G\) is a connected and reductive Lie group, \(K\) is a maximal subgroup, \(m=\dim G/K\), \(E\) is a \(G\)-module of finite dimension, and \(V\) is an irreducible and admissible \(({\mathfrak g},K)\)-module, then \(H^q(g,k;E\otimes V)\) is isomorphic to the dual of \(H^{m-q}({\mathfrak g},{\mathfrak k};E\otimes V)\).

In the second chapter, the vanishing theorem of Matsushima type (Theorem 8.3) is proved. The Langlands classification of admissible and irreducible representations of Lie groups is presented in the fourth chapter. It is proved that any irreducible and uniformly bounded representation is either tempered or is a Langlands quotient. These results are used to prove the vanishing of the cohomology of \(({\mathfrak g},K)\)-modules.

A new proof is given for the Delorme theorem: Let \(G\) be a connected, semisimple Lie group with finite center. If the identity representation is not an isolated point in the unital dual, then there exists a \(({\mathfrak g},K)\)-module \(H\) such that \(H^1({\mathfrak g},{\mathfrak k};H)\neq \{0\}\). In the seventh chapter the isomorphisms are proved between the cohomologies \(H^*(\Gamma,E)\) and \(H^*({\mathfrak g},{\mathfrak k};I^\infty(E))\), where \(\Gamma\) is a discrete subgroup of the Lie group \(G\) having a finite number of connected components, \(E\) is a \(\Gamma\)-module of finite dimension and \(I^\infty(E)\) is the differentiable induced module. The eighth chapter contains the construction of a nontrivial unitary representation \(V\) of \(SU(p,q)\) \((p\geq q>0)\) such that \(H^*({\mathfrak g},{\mathfrak k};V)\neq \{0\}\).

The nineth chapter is an excellent introduction into the continuous (respectively, differentiable) cohomology of locally compact (respectively, real) Lie groups, in such a gentle manner that even a beginner in the study of the cohomology of locally compact groups can take it as a good start. In the second section of this chapter the classical lemma of Shapiro is presented: If \(H\) is a closed subgroup of the group \(G\), such that there exists a continuous section of \(G/H\) in \(G\), \(U\) is a continuous \(H\)-module, and \(I(U)\) is the induced module, then \(H^q_{\text{ct}}(H,U)\) is isomorphic to \(H^q_{\text{ct}}(H,I(U))\) \((q\in\mathbb{N})\). This result remains true in the differentiable case, when there always exist local sections. Under fairly general assumptions, a result is proved obtained by Hochschild and Mostow: If \(G\) is a Lie group countable at infinity and \(E\) is a differentiable \(G\)-module, then \(H^q_{\text{ct}}(G,E)\) is isomorphic to \(H^q_{\text{ct}}(G,E)\) (the differentiable cohomology). At the end of the chapter the theorem of Künneth is proved.

The differentiable cohomology of a semisimple Lie group can be calculated by using differential forms on a symmetric space \(G/K\), where \(K\) is a maximal compact subgroup (chapters II and IX). In chapter XI it is shown that in the \(p\)-adic case the cohomology can be calculated by cochains on the Bruhat-Tits building. In the last paragraph of this chapter, Künneth type formulas are proven for \(p\)-adic groups. In the same chapter, the analog of Langlands classification (see chapter IV) is presented. The third paragraph introduces the class \(\Pi_\infty(G)\) of admissible and irreducible representations of a \(p\)-adic reductive group \(G\) and it is proven that this class contains admissible and irreducible representations with compact kernel. A generalization of R. Howe concerning the behaviour of the functions \(a\mapsto \langle \pi(a),y\rangle\), where \((\pi,E)\in\Pi_\infty(G)\), is also presented.

Since the publication of the first edition of this monograph other results on the isomorphism of the continuous cohomology and other types of cohomologies have been obtained. For example, \(H^q_{\text{ct}}(G,E)\) is isomorphic to \(H^q_{L^p_{\text{loc}}}(G,E)\) [see Ph. Blanc, Ann. Sci. Éc. Norm. Supér. 12, 137-167 (1979; Zbl 0429.57012)]. If \(G\) is a nilpotent, connected and simply connected group, then \(H^q_L(G,E)\) is isomorphic to the tempered cohomology \(H^q_{\text{temp}}(G,E)\), cf. F. du Cloux [J. Funct. Anal. 85, 420-457 (1989; Zbl 0401.22004)].

If \(G_1\) is a \(p\)-adic reductive group (respectively, a Lie group) and \(G_2\) is a totally disconnected group, the cohomology of the product \(G_1\times G_2\) is presented in chapter X (respectively, chapter XII) and Künneth type formulas are proved (see X.6.1 and XII.3.1).

If \(\Gamma\) is a compact discrete subgroup of a product group \(G_1\times G_2\), where \(G_1\), \(G_2\) are as above, and \(E\) is a unitary \(\Gamma\)-module, of finite dimension, then (see Proposition XIII.1.3) \(H^*(\Gamma;E)=H^*_{\text{ct}}(G,I^G_{\Gamma,2}(E))\), where \(I^G_{\Gamma,2}\) is the induced unitary \(G\)-module. If \(G\) is the product of a finite number of connected and reductive groups \(G_s\), on a local field \(k_s\), then any irreducible unitary representation of \(G\) can be decomposed into a Hilbertian tensor product of irreducible unitary representations of \(G_s\). Consequently, the decomposition \(H^*_{\text{ct}}(G,H_\pi)=\bigotimes_{s\in S}H^*_{\text{ct}}(G_s,H_{\pi_s})\) holds (see Proposition XII.2.2).

In the last chapter (added to the second edition) it is shown how one can generalize the results proven in the case of cocompact subgroups (cf. chapters VII and XVII) to the case of \(S\)-arithmetic subgroups, and, in particular, arithmetic ones.

This monograph contains many original results that are presented in a systematic fashion.

A locally compact group \(G\) is called amenable if there exists a nontrivial invariant mean on the space \(C_{\text{b}}(G)\) of continuous and bounded functions on \(G\). Obviously, any compact group is amenable. It is proved that \(G\) is amenable if and only if \(H^1(G,E)=0\) for any dual Banach \(G\)-module \(E\). The locally compact group \(G\) has the property (T) of Kazhdan if the identity representation is an isolated point in \(\hat G\) (the unital dual of \(G\)). This property is equivalent to \(H^1(G,E)=\{0\}\) for any unital \(G\)-module \(E\). The cohomology of locally compact groups or of Lie groups is a domain that requires a solid background in the representation theory of groups (in particular, the machinery of induction).

This monograph, written by two important specialists in the domain, is the development of a seminar on the cohomology of discrete subgroups of Lie groups, held at the Institute for Advanced Studies in Princeton.

The first chapter starts with recalling the relative cohomology (in the spirit of Cartan-Eilenberg) \(H^*({\mathfrak g},{\mathfrak k};V)\) of a \(\mathfrak g\)-module \(V\), where \({\mathfrak k}\) is a subalgebra of the Lie algebra \(\mathfrak g\). If \(G\) is a Lie group and \(K\) a closed connected subgroup of \(G\), then \(H^*({\mathfrak g},{\mathfrak k};V)\) is isomorphic with the cohomology of the module of differential forms, \(G\)-invariant on \(G/K\) and valued in \(V\). In particular, if \(G\) is compact and connected, and \(V\) is a trivial \(G\)-module of finite dimension, then by using de Rham’s theorem, we obtain: \(H^*({\mathfrak g},{\mathfrak k};V)\simeq H^*(G/K;V)\). Also, we have to mention, in this chapter, the Poincaré duality (Proposition 7.6): If \(G\) is a connected and reductive Lie group, \(K\) is a maximal subgroup, \(m=\dim G/K\), \(E\) is a \(G\)-module of finite dimension, and \(V\) is an irreducible and admissible \(({\mathfrak g},K)\)-module, then \(H^q(g,k;E\otimes V)\) is isomorphic to the dual of \(H^{m-q}({\mathfrak g},{\mathfrak k};E\otimes V)\).

In the second chapter, the vanishing theorem of Matsushima type (Theorem 8.3) is proved. The Langlands classification of admissible and irreducible representations of Lie groups is presented in the fourth chapter. It is proved that any irreducible and uniformly bounded representation is either tempered or is a Langlands quotient. These results are used to prove the vanishing of the cohomology of \(({\mathfrak g},K)\)-modules.

A new proof is given for the Delorme theorem: Let \(G\) be a connected, semisimple Lie group with finite center. If the identity representation is not an isolated point in the unital dual, then there exists a \(({\mathfrak g},K)\)-module \(H\) such that \(H^1({\mathfrak g},{\mathfrak k};H)\neq \{0\}\). In the seventh chapter the isomorphisms are proved between the cohomologies \(H^*(\Gamma,E)\) and \(H^*({\mathfrak g},{\mathfrak k};I^\infty(E))\), where \(\Gamma\) is a discrete subgroup of the Lie group \(G\) having a finite number of connected components, \(E\) is a \(\Gamma\)-module of finite dimension and \(I^\infty(E)\) is the differentiable induced module. The eighth chapter contains the construction of a nontrivial unitary representation \(V\) of \(SU(p,q)\) \((p\geq q>0)\) such that \(H^*({\mathfrak g},{\mathfrak k};V)\neq \{0\}\).

The nineth chapter is an excellent introduction into the continuous (respectively, differentiable) cohomology of locally compact (respectively, real) Lie groups, in such a gentle manner that even a beginner in the study of the cohomology of locally compact groups can take it as a good start. In the second section of this chapter the classical lemma of Shapiro is presented: If \(H\) is a closed subgroup of the group \(G\), such that there exists a continuous section of \(G/H\) in \(G\), \(U\) is a continuous \(H\)-module, and \(I(U)\) is the induced module, then \(H^q_{\text{ct}}(H,U)\) is isomorphic to \(H^q_{\text{ct}}(H,I(U))\) \((q\in\mathbb{N})\). This result remains true in the differentiable case, when there always exist local sections. Under fairly general assumptions, a result is proved obtained by Hochschild and Mostow: If \(G\) is a Lie group countable at infinity and \(E\) is a differentiable \(G\)-module, then \(H^q_{\text{ct}}(G,E)\) is isomorphic to \(H^q_{\text{ct}}(G,E)\) (the differentiable cohomology). At the end of the chapter the theorem of Künneth is proved.

The differentiable cohomology of a semisimple Lie group can be calculated by using differential forms on a symmetric space \(G/K\), where \(K\) is a maximal compact subgroup (chapters II and IX). In chapter XI it is shown that in the \(p\)-adic case the cohomology can be calculated by cochains on the Bruhat-Tits building. In the last paragraph of this chapter, Künneth type formulas are proven for \(p\)-adic groups. In the same chapter, the analog of Langlands classification (see chapter IV) is presented. The third paragraph introduces the class \(\Pi_\infty(G)\) of admissible and irreducible representations of a \(p\)-adic reductive group \(G\) and it is proven that this class contains admissible and irreducible representations with compact kernel. A generalization of R. Howe concerning the behaviour of the functions \(a\mapsto \langle \pi(a),y\rangle\), where \((\pi,E)\in\Pi_\infty(G)\), is also presented.

Since the publication of the first edition of this monograph other results on the isomorphism of the continuous cohomology and other types of cohomologies have been obtained. For example, \(H^q_{\text{ct}}(G,E)\) is isomorphic to \(H^q_{L^p_{\text{loc}}}(G,E)\) [see Ph. Blanc, Ann. Sci. Éc. Norm. Supér. 12, 137-167 (1979; Zbl 0429.57012)]. If \(G\) is a nilpotent, connected and simply connected group, then \(H^q_L(G,E)\) is isomorphic to the tempered cohomology \(H^q_{\text{temp}}(G,E)\), cf. F. du Cloux [J. Funct. Anal. 85, 420-457 (1989; Zbl 0401.22004)].

If \(G_1\) is a \(p\)-adic reductive group (respectively, a Lie group) and \(G_2\) is a totally disconnected group, the cohomology of the product \(G_1\times G_2\) is presented in chapter X (respectively, chapter XII) and Künneth type formulas are proved (see X.6.1 and XII.3.1).

If \(\Gamma\) is a compact discrete subgroup of a product group \(G_1\times G_2\), where \(G_1\), \(G_2\) are as above, and \(E\) is a unitary \(\Gamma\)-module, of finite dimension, then (see Proposition XIII.1.3) \(H^*(\Gamma;E)=H^*_{\text{ct}}(G,I^G_{\Gamma,2}(E))\), where \(I^G_{\Gamma,2}\) is the induced unitary \(G\)-module. If \(G\) is the product of a finite number of connected and reductive groups \(G_s\), on a local field \(k_s\), then any irreducible unitary representation of \(G\) can be decomposed into a Hilbertian tensor product of irreducible unitary representations of \(G_s\). Consequently, the decomposition \(H^*_{\text{ct}}(G,H_\pi)=\bigotimes_{s\in S}H^*_{\text{ct}}(G_s,H_{\pi_s})\) holds (see Proposition XII.2.2).

In the last chapter (added to the second edition) it is shown how one can generalize the results proven in the case of cocompact subgroups (cf. chapters VII and XVII) to the case of \(S\)-arithmetic subgroups, and, in particular, arithmetic ones.

This monograph contains many original results that are presented in a systematic fashion.

Reviewer: Ion Colojoara (Bucureşti)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22Exx | Lie groups |

22E40 | Discrete subgroups of Lie groups |

22E35 | Analysis on \(p\)-adic Lie groups |

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

17B55 | Homological methods in Lie (super)algebras |

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