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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

 

Hecke operators on cohomology

Min Ho Lee

Abstract. Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham cohomology of locally symmetric spaces, and the cohomology of symmetric spaces with coefficients in a system of local groups. We construct canonical isomorphisms among such cohomology groups and discuss the compatibility of the Hecke operators with respect to those canonical isomorphisms. 2000 Mathematics Subject Classification. 11F60, 20J06, 55N25, 57T10.

1. IntroductionThis paper is mainly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham cohomology of locally symmetric spaces, and the cohomology of symmetric spaces with coefficients in a system of local groups. We construct canonical isomorphisms among such cohomology groups and discuss the compatibility of the Hecke operators with respect to those canonical isomorphisms. Automorphic forms play a major role in number theory, and they are closely related to many other areas of mathematics. Modular forms, or automorphic forms of one variable, are holomorphic functions on the Poincaré upper half plane H satisfying a certain transformation formula with respect to the linear fractional action of a discrete subgroup Γ of SL (2,ℝ ) , and they are closely linked to the geometry of the associated Riemann surface X = Γ \H . For example, modular forms for Γ can be interpreted as holomorphic sections of a line bundle over X , and the space of such modular forms of a given weight corresponds to a certain cohomology group of X with local coefficients or with some cohomology group of the discrete group Γ (cf. [1], [2], [5]) with coefficients in some Γ -module. Modular forms can be extended to automorphic forms of several variables by using holomorphic functions either on the Cartesian product  n H of n copies of H for Hilbert modular forms or on the Siegel upper half space Hn of degree n for Siegel modular forms. More general automorphic forms can also be considered by using semisimple Lie groups. Indeed, given a semisimple Lie group G of Hermitian type and a discrete subgroup Γ of G , we can consider automorphic forms for Γ defined on the quotient D = G ∕K of G by a maximal compact subgroup K of G . The space D has the structure of a Hermitian symmetric domain, and automorphic forms on D for Γ are holomorphic functions on D satisfying an appropriate transformation formula with respect to the natural action of Γ on D (cf. [3]). Such automorphic forms are also linked to families of abelian varieties parametrized by the locally symmetric space Γ \D (cf. [6], [10], [14]). Close connections between automorphic forms for the discrete group Γ ⊂ G and the group cohomology of Γ or the de Rham cohomology of D with certain coefficients have also been studied in numerous papers over the years (see e.g. [11]). Hecke operators are certain averaging operators acting on the space of automorphic forms (cf. [1], [12], [15]), and they are an important component of the theory of automorphic forms. For example, they are used to obtain Euler products associated to modular forms which lead to some multiplicative properties of Fourier coefficients of those automorphic forms. In light of the fact that automorphic forms are closely related to the cohomology of the corresponding discrete subgroups of a semisimple Lie group, it would be natural to study the Hecke operators on the cohomology of the discrete groups associated to automorphic forms as was done in a number of papers (see e.g. [6], [8], [7], [17]). Hecke operators on the cohomology of more general groups were also investigated by Rhie and Whaples in [13]. On the other hand, if f is an automorphic form on a Hermitian symmetric domain D = G ∕K for a discrete subgroup Γ of G described above, then f can be interpreted as an algebraic correspondence on the quotient space Γ \D , which has the structure of a complex manifold, assuming that Γ is torsion-free. Such a correspondence is determined by a pair of holomorphic maps  ′ λ, μ : Γ \D → Γ \D , where  ′ Γ is another discrete subgroup of G . The maps λ and μ can be used to construct a Hecke operator on the de Rham cohomology of Γ \D . The idea of Hecke operators on cohomology of complex manifolds of the kind described above was suggested, for example, by Kuga and Sampson in [9] (see also [7]). The goal of this paper is to discuss relations among different types of cohomology described above and establish the compatibility of the Hecke operators acting on those cohomology groups. The organization of the paper is as follows. In Section 2 we review Hecke algebras associated to subgroups of a given group, whose examples include the algebras of Hecke operators considered in the subsequent sections. In Section 3 we describe the cohomology of groups as well as Hecke operators acting on such cohomology. We also discuss equivariant cohomology and its relation with group cohomology. The de Rham cohomology of a locally symmetric space with coefficients in a vector bundle is discussed in Section 4 by using the language of sheaves, and then Hecke operators are introduced on Rham cohomology groups. In Section 5 we study the cohomology of a locally symmetric space with coefficients in a local system of groups in connection with other types of cohomology. Hecke operators are also considered for this cohomology. Section 6 is concerned with compatibility of Hecke operators. We discuss canonical isomorphisms among de Rham, singular, and group cohomology and show that the Hecke operators acting on those cohomology groups are compatible with one another under those canonical isomorphisms.

2. Hecke algebrasIn this section we review some of the basic properties of Hecke algebras. In Section 2.1 we discuss the commensurability relation on the set of subgroups of a given group G , consider double cosets determined by two commensurable subgroups of G , and describe decompositions of such double cosets in terms of left or right cosets of one of those two subgroups. We introduce a binary operation on the set of double cosets in Section 2.2, which is used in Section 2.3 to construct the structure of an algebra, known as a Hecke algebra, on the set of double cosets determined by a single subgroup of the given group. More details and some additional properties of Hecke algebras can be found, for example, in [6], [12] and [15].

2.1. Double cosets. Let G be a group. Two subgroups Γ and  ′ Γ are said to be commensurable (or Γ is said to be commensurable with Γ ′ ) if

 ′ ′ ′ [Γ : Γ ∩ Γ ] < ∞, [Γ : Γ ∩ Γ ] < ∞,

that is, if Γ ∩ Γ ′ has finite index in both Γ and Γ ′ . We shall write  ′ Γ ~ Γ when Γ is commensurable with  ′ Γ . If H is a subgroup of G and if K is a subset of G containing H , then we shall denote by K ∕H (resp. H \K ) the set of left (resp. right) cosets of H in K . Lemma 2.1. The commensurability relation ~ is an equivalence relation.

Proof. The relation ~ is clearly reflexive and symmetric. Let Γ 1 , Γ 2 and Γ 3 be subgroups of G with Γ 1 ~ Γ 2 and Γ 2 ~ Γ 3 . We consider the map

Γ 1 ∩ Γ 2∕Γ 1 ∩ Γ 2 ∩ Γ 3 → Γ 2∕Γ 2 ∩ Γ 3
(2.1)

sending the left coset γ(Γ 1 ∩ Γ 2 ∩ Γ 3) to the left coset γ(Γ 2 ∩ Γ 3) for each γ ∈ Γ 1 ∩ Γ 2 . If  ′ γ (Γ 2 ∩ Γ 3) = γ (Γ 2 ∩ Γ 3) with  ′ γ,γ ∈ Γ 1 ∩ Γ 2 , then  -1 ′ γ γ ∈ Γ 1 ∩ Γ 2 ∩ Γ 3 ; hence we see that  ′ γ(Γ 1 ∩ Γ 2 ∩ Γ 3) = γ (Γ 1 ∩ Γ 2 ∩ Γ 3) . Thus the map (2.1) is injective, and therefore we have

[Γ 1 ∩ Γ 2 : Γ 1 ∩ Γ 2 ∩ Γ 3] ≤ [Γ 2 : Γ 2 ∩ Γ 3] < ∞,

which implies that

[Γ : Γ ∩ Γ ∩ Γ ] = [Γ : Γ ∩ Γ ][Γ ∩ Γ : Γ ∩ Γ ∩ Γ ] < ∞. 1 1 2 3 1 1 2 1 2 1 2 3

Similarly, it can be shown that

[Γ 3 : Γ 1 ∩ Γ 2 ∩ Γ 3] < ∞,

and hence we obtain

pict

Thus the relation is transitive, and therefore the lemma follows.

Given a subgroup Γ of G , we set

^ -1 Γ = {α ∈ G | α Γ α ~ Γ },

which will be called the commensurator of Γ in G .

Lemma 2.2. The commensurator ^Γ is a subgroup of G containing Γ .

Proof. Given  ^ α,β ∈ Γ , since  - 1 α Γ α ~ Γ , we see that

(α β- 1)- 1Γ (αβ -1) = β(α- 1Γ α)β -1

is commensurable with βΓ β-1 . However, the commensurability β -1Γ β ~ Γ implies that Γ ~ β Γ β- 1 ; hence we have

 -1 -1 -1 (αβ ) Γ (α β ) ~ Γ .

Thus  -1 ^ α β ∈ Γ , and therefore ^ Γ is a subgroup of G . Since ^ Γ clearly contains Γ , the proof of the lemma is complete.

Lemma 2.3. If Γ ~ Γ ′ , then ^Γ ′ = ^Γ .

Proof. If Γ ~ Γ ′ and α ∈ Γ ′ , then we have

 - 1 -1 ′ ′ α Γ α ~ α Γ α = Γ ~ Γ ;

hence  ^ α ∈ Γ , which shows that  ′ ^ Γ ⊂ Γ . On the other hand, if  ′ β ∈ ^Γ , then we have

β- 1Γ β ~ β -1Γ ′β ~ Γ ′ ~ Γ ;

hence β ∈ ^Γ . Thus we have ^Γ ′ ⊂ ^Γ . Similarly, it can be shown that ^Γ ⊂ ^Γ ′ , and therefore we obtain Γ^′ = ^Γ .

Proposition 2.4. Let Γ ~ Γ ′ , and let α ∈ ^Γ . Then the double coset Γ αΓ ′ can be decomposed into disjoint unions of the form

 ′ ∐r ∐s ′ Γ αΓ = Γ α γi = δjα Γ i=1 j=1
(2.2)

for some positive integers r and s , where  r {γi}i=1 and  s {δj}j=1 are complete sets of coset representatives of  ′ -1 ′ (Γ ∩ α Γ α )\ Γ and Γ ∕(Γ ∩ α-1Γ ′α ) , respectively.

Proof. We note first that a right coset of Γ contained in Γ αΓ ′ can be written in the form Γ αγ for some γ ∈ Γ ′ . If Γ α γ′ with γ′ ∈ Γ ′ is another subset of Γ α Γ ′ , we see that Γ αγ = Γ αγ ′ if and only if  ′- 1 ′ -1 γ γ ∈ Γ ∩ α Γ α , which is equivalent to the condition that

(Γ ′ ∩ α -1Γ α )γ = (Γ ′ ∩ α -1Γ α )γ′.

Since α -1Γ α ~ Γ ~ Γ ′ , the index [Γ ′ : Γ ′ ∩ α- 1Γ α] is finite. Thus, if {γi}ri=1 is a set of representatives of (Γ ′ ∩ α-1Γ α)\Γ ′ , each γi determines a unique coset Γ αγ i contained in Γ α Γ ′ ; hence we have

 r ′ ∐ Γ α Γ = Γ α γi. i=1

Similarly, it can be shown that

 s ′ ∐ ′ Γ α Γ = δjαΓ , j=1

where s = [Γ : Γ ∩ α -1Γ ′α] .

2.2. Operations on double cosets. Let G be the group considered in Section 2.1, and fix a subsemigroup Δ of G . We denote by C (Δ ) the collection of subgroups Γ of G that are mutually commensurable and satisfy

Γ ⊂ Δ ⊂ ^Γ .

Given  ′ Γ ,Γ ∈ C(Δ ) and a commutative ring R be with identity, we denote by  ′ HR (Γ ,Γ ;Δ ) the free R -module generated by the double cosets Γ αΓ ′ with α ∈ Δ . Thus an element of HR (Γ ,Γ ′;Δ ) can be written in the form

 ∑ ′ cαΓ αΓ , α ∈Δ

where the coefficients cα ∈ R are zero except for a finite number of α . We denote by  ′ deg(Γ α Γ ) the number of right cosets Γ γ contained in  ′ Γ αΓ . Thus, if  ′ Γ αΓ is as in (2.2), then  ′ deg(Γ α Γ ) = r . If η is an element of HR (Γ ,Γ ′;Δ ) given by  ∑ η = α∈Δ cαΓ αΓ ′ , then we set

 ∑ deg η = cαdeg (Γ αΓ ′) α∈Δ
(2.3)

and refer to it as the degree of η .

We now consider an R -module M and assume that the subsemigroup Δ ⊂ G acts on M on the right by

(m, δ) ↦→ m ⋅ δ ∈ M

for (m, δ) ∈ M × Δ . Thus we have

m ⋅ 1 = m, m ⋅ (δδ′) = (m ⋅ δ) ⋅ δ′

for all m ∈ M and  ′ δ,δ ∈ Δ . Given Γ ∈ C(Δ ) , let  Γ M denote the submodule of M consisting of the Γ -invariant elements of M , that is,

 Γ M = {m ∈ M | m ⋅ γ = m for all γ ∈ Γ }.

If the double coset  ′ Γ αΓ with α ∈ Δ and  ′ Γ ,Γ ∈ C (Δ ) has a decomposition of the form

 ∐d Γ αΓ ′ = Γ α , i i=1
(2.4)

then we define its operation on M Γ by

 ∑ d m | Γ αΓ ′ = m ⋅ αi i=1
(2.5)

for all  Γ m ∈ M .

Lemma 2.5. The operation of Γ αΓ ′ on M Γ in (2.15) is independent of the choice of the representatives α i of the right cosets of Γ in (2.4) and

m | Γ αΓ ′ ∈ M Γ ′

for all m ∈ M Γ .

Proof. If Γ α ,Γ α ′ i i are subsets of  ′ Γ αΓ with  ′ Γ αi = Γ αi , then  ′ αi = γαi for some γ ∈ Γ . Thus we see that  ′ m ⋅ αi = (m ⋅ γ ) ⋅ αi = m ⋅ αi for all  Γ m ∈ M ; hence  ′ m | Γ αΓ is independent of the choice of the representatives αi . On the other hand, if Γ αΓ ′ has a decomposition as in (2.4), then we see that

 ∐d Γ α Γ ′ = Γ α γ′ i i=1

for all γ′ ∈ Γ ′ . Thus we have

 d d ′ ′ ∑ ′ ∑ ′ (m | Γ α Γ) ⋅ γ = m ⋅ (αiγ ) = m ⋅ αi = m | Γ α Γ ; i=1 i=1

hence it follows that m | Γ αΓ ′ ∈ M Γ ′ .

We see easily that the map m ↦→ (m | Γ α Γ ′) given by (2.15) is in fact a homomorphism of R -modules. We now extend this by defining an R -module homomorphism associated to each element of HR (Γ ,Γ ′;Δ) by

 ∑ m | η = cα(m | Γ αΓ ′) α

for  Γ m ∈ M and  ∑ ′ ′ η = αcαΓ αΓ ∈ HR (Γ ,Γ ;Δ) .

Given elements Γ 1,Γ 2,Γ 3 ∈ C(Δ ) and double cosets of the form

 ∐r ∐s Γ 1α Γ 2 = Γ 1αi, Γ 2βΓ 3 = Γ 2βj i=1 j=1
(2.6)

with α, β ∈ Δ , we set

 ∑ (Γ 1α Γ 2) ⋅ (Γ 2βΓ 3) = cγΓ 1γΓ 3, γ
(2.7)

where the summation is over the set of representatives γ ∈ Δ of the double cosets Γ 1γ Γ 3 contained in Δ and

cγ = # {(i,j ) | Γ 1αiβj = Γ 1γ}
(2.8)

is the number of pairs (i,j) with 1 ≤ i ≤ r and 1 ≤ j ≤ s such that Γ 1αiβj = Γ 1γ . Since cγ = 0 except for a finitely many double cosets Γ 1γΓ 3 , the sum on the right hand side of (2.7) is a finite sum.

Let R[Γ 1\Δ ] denote the free R -module generated by the right cosets Γ 1α with α ∈ Δ . Then Δ acts on R[Γ 1\Δ ] by right multiplication. On the other hand, there is a natural injective map H (Γ ,Γ ;Δ ) → R [Γ \Δ ] R 1 2 1 sending  ∐ Γ 1αΓ 2 = iΓ 1αi to ∑ iΓ 1αi . By using this injection we may regard HR (Γ 1,Γ 2;Δ ) as an R -submodule of R [Γ 1\Δ ] , and under this identification we see easily that

 Γ HR (Γ 1,Γ 2;Δ ) = R [Γ 1\Δ ]2.
(2.9)

If the double cosets Γ 1α Γ 2 and Γ 2βΓ 3 are as in (2.6), using (2.15) and (2.8), we have

 ∑r ∑ r ∑s ∑ (Γ 1α Γ 2) | (Γ 2βΓ 3) = Γ 1αi | (Γ 2βΓ 3) = Γ 1αiβj = cγΓ 1γ. i=1 i=1 j=1 γ

Using Lemma 2.5 and the identification (2.9) with Γ ,Γ ′ replaced by Γ 2,Γ 3 , we see that

∑ Γ cγΓ 1γ ∈ R [Γ 1\ Δ] 3. γ

Thus by using (2.9) again, we obtain

∑ ∑ c Γ γ = c Γ γΓ ; γ 1 γ 1 3 γ γ

hence it follows that

(Γ 1α Γ 2) ⋅ (Γ 2βΓ 3) = (Γ 1α Γ 2) | (Γ 2βΓ 3).
(2.10)

From this and Lemma 2.5 we see that the operation in (2.7) is independent of the choice of the representatives αi , βj and γ .

Lemma 2.6. Let Γ 1αΓ 2 and Γ 2β Γ 3 be as in (2.6), and let cγ with Γ 1γΓ 3 ⊂ Δ be as in (2.8). Then we have

cγ deg (Γ 1γΓ 3) = # {(i,j) | Γ 1αiβjΓ 3 = Γ 1γΓ 3}

for each γ ∈ Δ .

Proof. We assume that Γ 1γΓ 3 ⊂ Δ has a decomposition of the form

 ∐ t Γ 1γΓ 3 = Γ 1γk. k=1

Then the relation Γ 1αiβjΓ 3 = Γ 1γΓ 3 holds if and only if Γ 1αiβj = Γ 1γk for exactly one k ∈ {1, ...,t} . Thus, if cγ is as in (2.8), we see that

 ∑t #{ (i,j) | Γ 1αiβj Γ 3 = Γ 1γΓ 3} = #{ (i,j) | Γ 1αiβj = Γ 1γ} = cγt; k=1

hence the lemma follows from this and the fact that deg(Γ 1γ Γ 2) = t .

Lemma 2.7. If η1 ∈ HR (Γ 1,Γ 2;Δ) and η2 ∈ HR (Γ 2,Γ 3;Δ ) , then we have

deg(η1 ⋅ η2) = deg (η1)deg(η2).

Proof. Let Γ 1α Γ 2 ∈ HR (Γ 1,Γ 2;Δ) and Γ 2βΓ 3 ∈ HR (Γ 2,Γ 3;Δ ) be as in (2.7). Then, using (2.3) and Lemma 2.6, we have

deg[(Γ α Γ ) ⋅ (Γ βΓ )] = ∑ c deg(Γ γ Γ ). 1 2 2 3 γ 1 3 γ

However, by (2.8) the right hand side of this relation is equal to the number of pairs (i,j) with 1 ≤ i ≤ r and 1 ≤ j ≤ s and therefore is equal to rs = deg(Γ 1αΓ 2) ⋅ deg(Γ 2α Γ 3) . Thus the lemma follows by extending this result linearly.

2.3. Hecke algebras. Given Γ 1,Γ 2,Γ 3 ∈ C(Δ ) , the operation in (2.7) induces a bilinear map

HR (Γ 1,Γ 2;Δ) × HR (Γ 2,Γ 3;Δ ) → HR (Γ 1,Γ 3;Δ )

defined by

(∑ ) (∑ ) ∑ aαΓ 1αΓ 2 ⋅ bβΓ 2β Γ 3 = aαbβ(Γ 1αΓ 2) ⋅ (Γ 2β Γ 3). α β α,β
(2.11)

Using (2.10), we see that the operation of HR (Γ 2,Γ 3;Δ ) on HR (Γ 1,Γ 2;Δ ) = R[Γ 1\ Δ]Γ 2 coincides with the multiplication operation in (2.11), that is,

η1 ⋅ η2 = η1 | η2
(2.12)

for all η1 ∈ HR (Γ 1,Γ 2;Δ ) and η2 ∈ HR (Γ 2,Γ 3;Δ ) .

If M is an R -module on which Δ acts on the right, then it follows easily from the definition that

(m | η1) | η2 = m | (η1 ⋅ η2)

for all  Γ 1 m ∈ M , η1 ∈ HR (Γ 1,Γ 2;Δ ) and η2 ∈ HR (Γ 2,Γ 3;Δ) . From this and (2.12) we obtain

(η1 ⋅ η2) ⋅ η3 = η1 ⋅ (η2 ⋅ η3)
(2.13)

for all η1 ∈ HR (Γ 1,Γ 2;Δ ) , η2 ∈ HR (Γ 2,Γ 3;Δ ) and η3 ∈ HR (Γ 3,Γ 4;Δ ) .

Given Γ ∈ C (Δ ) , we set

HR (Γ ;Δ ) = HR (Γ ,Γ ;Δ ).

Then by (2.13) the multiplication operation on HR (Γ ;Δ ) is associative and HR (Γ ;Δ ) is an algebra over R with identity Γ . When R = ℤ , we shall simply write

H (Γ ;Δ ) = H ℤ(Γ ;Δ ) = H ℤ(Γ ,Γ ;Δ ).

Definition 2.8. Given Γ ∈ C (Δ ) , the algebra HR (Γ ;Δ ) is called the Hecke algebra over R of Γ with respect to Δ . If R = ℤ , then H (Γ ;Δ ) = H (Γ ;Δ) ℤ is simply called the Hecke algebra of Γ with respect to Δ .

Let Δ and  ′ Δ be two subsemigroups of G with  ′ Δ ⊂ Δ . Then certainly HR (Γ ;Δ ) is a subset of  ′ HR (Γ ;Δ ) . If Γ αΓ ,Γ βΓ ∈ HR (Γ ;Δ ) with α, β ∈ Δ are regarded as elements of  ′ HR (Γ ;Δ ) , their product can be written in the form

 ∑ ′ (Γ αΓ ) ⋅ (Γ βΓ ) = cγΓ γ Γ , γ
(2.14)

where the summation is over the set of representatives γ of the double cosets Γ γΓ contained in Δ ′ . However, we have Γ γ Γ ⊂ Γ αΓ β Γ ⊂ Δ ; hence the product in (2.14) coincides with the product of Γ αΓ and Γ βΓ in HR (Γ ;Δ ) . Thus we see that HR (Γ ;Δ ) is a subalgebra of HR (Γ ;Δ ′) .

Proposition 2.9. Let α ∈ ^Γ , and assume that |Γ \Γ αΓ | = |Γ α Γ ∕Γ | . Then the quotients Γ \Γ αΓ and Γ αΓ ∕Γ have a common set of coset representatives.

Proof. We assume that Γ αΓ can be decomposed as

 ∐d ∐d Γ α Γ = Γ α = β Γ . i i i=1 i=1

Then it can be shown that Γ αi ∩ βjΓ is nonempty for all i and j . Indeed, if Γ αi and βiΓ are disjoint for some i and j , then Γ α ⊂ ⋃ β Γ i k⁄=j k , and therefore we have

 ⋃ Γ α Γ = Γ αi Γ = βkΓ , k⁄=j

which is a contradiction. Thus, in particular, we have Γ αi ∩ βiΓ ⁄= ∅ for each i . If δi ∈ Γ αi ∩ βiΓ for each i , then we see that Γ αi = Γ δi and δiΓ = βiΓ . Hence we have

 d d ∐ ∐ Γ αΓ = Γ δi = δiΓ , i=1 i=1

and {δ }d i i=1 is a common set of coset representatives.

We now discuss the commutativity of the Hecke algebra HR (Γ ;Δ ) . Note that an involution on Δ is a map ι : Δ → Δ satisfying

(αβ )ι = β ια ι, (αι)ι

for all α,β ∈ Δ .

Theorem 2.10. Let ι : Δ → Δ be an involution on Δ , and assume that an element Γ ∈ C(Δ ) satisfies

Γ ι = Γ , Γ α ιΓ = Γ α Γ
(2.15)

for all α ∈ Δ . Then the associated Hecke algebra H (Γ ;Δ ) R is commutative.

Proof. Given α ∈ Δ with  ∐ Γ α Γ = d Γ αi i=1 , using (2.15), we have

 d ι ι ∐ ι Γ α Γ = Γ α Γ = (Γ α Γ ) = α iΓ . i=1

Hence by Lemma 2.9 the sets Γ \ Γ αΓ and Γ αΓ ∕Γ have a common set of coset representatives. Thus we may write

 ∐d ∐d Γ αΓ = Γ αi = αiΓ i=1 i=1

for some α1, ...,αd ∈ Δ . Similarly, if β is another element of Δ , we have

 ∐ s ∐s Γ βΓ = Γ βj = βjΓ i=1 j=1

for some positive integer s and βj ∈ Δ for 1 ≤ j ≤ s . We now assume that

 ∑ ∑ (Γ α Γ ) ⋅ (Γ βΓ ) = cγ(Γ γΓ ), (Γ βΓ ) ⋅ (Γ α Γ ) = c′γ(Γ γΓ ), γ γ

where cγ and c′γ ∈ Δ are as in (2.8). Then we have

pict

where we used the fact that

 ι ι ∑ ′ ∑ ′ ι (Γ β Γ ) ⋅ (Γ α Γ ) = (Γ β Γ ) ⋅ (Γ αΓ ) = cγ(Γ γΓ ) = cγ(Γ γ Γ ). γ γ

Hence it follows that HR (Γ ;Δ) is a commutative algebra.

Example 2.11. Let G = GL (n,ℚ ) for some positive integer n , and consider the subgroup Γ = SL (n,ℤ ) and the subsemigroup

Δ = {α ∈ M (n,ℤ) | det α > 0}

of G . Then we see that the transposition α ↦→ tα is an involution satisfying

t ^ Γ = Γ , Γ ⊂ Δ ⊂ Γ .

Given α ∈ Δ , by the elementary divisor theorem the corresponding double coset Γ α Γ can be written as

Γ α Γ = Γ αdΓ

for some diagonal matrix αd = diag(d1,...,dn) , where the diagonal entries d1,...,dn are positive integers satisfying di | di+1 for each i . Hence we see that

 t t Γ α Γ = Γ αdΓ = Γ αdΓ = Γ αΓ .

Thus by Theorem 2.10 the Hecke algebra H (Γ ;Δ ) = H (Γ ;Δ ) ℤ is commutative.

 

3. Group cohomology

In this section we review group cohomology and its relation with equivariant cohomology as well as Hecke operators acting on group cohomology. The description of the cohomology of a group G with coefficients in a G -module by using both homogeneous and nonhomogeneous cochains is given in Section 3.1. Given a complex K on which a group Γ acts on the left and a left Γ -module A , in Section 3.2 we construct the associated equivariant cohomology of K with coefficients in A following Eilenberg [4]. We also obtain an isomorphism between this equivariant cohomology and the cohomology of Γ with the same coefficients. We then discuss Hecke operators acting on group cohomology in Section 3.3 introduced by Rhie and Whaples [13].

3.1. Cohomology of groups. Let G be a group, and let M be a left G -module. Thus M is an abelian group on which G acts on the left. Then the cohomology of G with coefficients in M can be described by using either homogeneous or nonhomogeneous cochains.

Given a nonnegative integer q , let  q C (G,M ) denote the group consisting of the M -valued functions f : Gq → M on the q -fold Cartesian product Gq = G × ⋅⋅⋅ × G of G , called nonhomogeneous q -cochains. We then consider the map ∂ : Cq (G, M ) → Cq+1 (G,M ) defined by

pict

for all f ∈ Cq (G, M ) and (σ1,...,σq+1 ) ∈ Gq+1 . Then ∂ is the coboundary map for nonhomogeneous q -cochains satisfying ∂2 = 0 . The associated q -th cohomology group of G with coefficients in M is given by

Hq (G, M ) = Zq (G,M )∕Bq(G, M ),

where Zq (G, M ) is the kernel of ∂ : Cq(G, M ) → Cq+1(G, M ) and Bq (G, M ) is the image ∂ : Cq- 1(G, M ) → Cq (G,M ) .

For each q ≥ 0 we also consider the group ℭq(G,M ) of homogeneous q -cochains consisting of the maps  q+1 ϕ : G → M satisfying

ϕ (σ σ0,...,σσq) = σ ϕ(σ0,...,σq)

for all σ, σ0,...,σq ∈ G . We then define the map δ : ℭq(G, M ) → ℭq+1(G, M ) by

 q∑+1 (δϕ )(σ0, ...,σq+1) = (- 1)iϕ (σ0,...,σi-1,σi+1,...,σq+1) i=0
(3.2)

for all  q ϕ ∈ ℭ (G, M ) and  n+2 (σ0, ...,σq+1) ∈ G , which is the coboundary map for nonhomogeneous q -cochains satisfying δ2 = 0 . Then the corresponding q -th cohomology group of G in M is given by

 q q q ℌ (G, M ) = ℨ (G, M )∕𝔅 (G, M ),

where  q ℨ (G,M ) is the kernel of  q q+1 ∂ : ℭ (G,M ) → ℭ (G, M ) and  q 𝔅 (G, M ) is the image  q-1 q ∂ : ℭ (G, M ) → ℭ (G, M ) .

We can establish a correspondence between homogeneous and nonhomogeneous cochains as follows. Given f ∈ Cq (G,M ) and ϕ ∈ ℭq (G,M ) , we consider the elements fH ∈ ℭq(G, M ) and ϕ ∈ Cq (G, M ) N given by

pict

for all σ0,σ1,...,σq ∈ G . Then we see that

pict

for all  q f ∈ C (G,M ) and  q ϕ ∈ ℭ (G,M ) . Thus, by extending linearly we obtain the linear maps

 q q q q (⋅)H : C (G, M ) → ℭ (G, M ), (⋅)N : ℭ (G,M ) → C (G, M )

such that (⋅)H ∘ (⋅)N and (⋅)N ∘ (⋅)H are identity maps on  q ℭ (G, M ) and Cq (G, M ) , respectively. The next lemma shows that this correspondence between homogeneous and nonhomogeneous cochains is compatible with the coboundary maps.

Lemma 3.1. Given a nonnegative integer q , we have

(∂f)H = δfH, (δϕ)N = ∂ ϕN

for all  q f ∈ C (G, M ) and  q ϕ ∈ ℭ (G,M ) .

Proof. Given elements σ0,σ1,...,σq+1 ∈ G and f ∈ Cq(G, M ) , using (3.1), (3.2) and (3.3), we have

pict

On the other hand, if ϕ ∈ ℭq+1(G, M ) , by using (3.1), (3.2) and (3.4) we see that

pict

hence the lemma follows.

From Lemma 3.1 we see that the diagram

 (⋅)H (⋅)N Cq (G,M ) ---→ ℭq(G,M ) ---→ Cq(G, M ) || || || ∂↓ δ↓ ↓∂ (⋅) (⋅) Cq+1 (G,M ) ---H→ ℭq+1(G, M ) ---N→ Cq+1(G, M )

is commutative, which implies that there is a canonical isomorphism

 q ~ q H (G, M )= ℌ (G, M )

for each q ≥ 0 .

3.2. Equivariant cohomology. Let K be a complex, which can be described as follows. The elements of the complex K are called cells, and there is a nonnegative integer associated to each cell called the dimension of the cell. A cell σq ∈ K of dimension q ≥ 0 is referred to as a q -cell, and the incidence number [σ : σ ] q+1 q associated to the a q -cell σ q and a (q + 1 ) -cell σ q+1 is an integer that is nonzero only for a finite number of q -cells σq and satisfies

∑ [σq+1 : σq][σq : σq- 1] = 0 σq
(3.5)

for q ≥ 1 . Given q ≥ 0 , we denote by C (K ) q the free abelian group generated by the q -cells, and the elements of Cq(K ) are called q -chains. The boundary operator on Cq (K ) is the homomorphism

∂ : C (K ) → C (K ) q q-1

of abelian groups given by

 ∑ ∂ σq = [σq : σq-1]σq-1 σ q-1
(3.6)

for each generator σq of Cq(K ) , where the summation is over the generators σq-1 of Cq- 1(K ) . Then it can be shown that ∂ satisfies ∂2 = ∂ ∘ ∂ = 0 .

Given an abelian group A , we consider the associated group of q -cochains given by

Cq (K, A) = Hom (C (K ),A ). q
(3.7)

Since Cq (K ) is generated by the q -cells, a q -cochain f is uniquely determined by its values f(σq) for the q -cells σq . The coboundary operator

 q q+1 δ : C (K, A ) → C (K, A )
(3.8)

on Cq (K, A) is defined by

(δf )(c) = f (∂c)
(3.9)

for all f ∈ Cq(K, A ) and c ∈ Cq+1 (K ) , and the condition ∂2 = 0 implies δ2 = 0 . Then the q -th cohomology group of the complex K over A is given by the quotient

Hq (K, A ) = Zq (K, A)∕Bq (K, A ),

where  q Z (K, A ) is the kernel of  q q+1 δ : C (K,A ) → C (K, A ) and Bq (K, A) is the image Bq (K, A) of δ : Cq- 1(K, A ) → Cq(K, A ) .

We now assume that a group Γ acts on K and on A , both on the left. Given q ≥ 0 , an element f ∈ Cq(K, A ) is said to be an equivariant q -cochain if it satisfies

f (γc) = γf(c)
(3.10)

for all γ ∈ Γ and c ∈ Cq (K ) , where  q C (K,A ) is as in (3.7). We denote by  q C E(K, A ) the subgroup of  q C (K,A ) consisting of the equivariant cochains. If δ is the coboundary map in (3.8) and if f is an equivariant q -cochain, then we have

δf(γcq+1) = f(∂ γcq+1) = f(γ∂cq+1) = γf (∂cq+1) = γ[δf(cq+1)]

for all γ ∈ Γ , which shows that δf is an equivariant (q + 1) -cochain. We define an equivariant q -cocycle to be an element of the group

ZqE (K, A ) = Zq(K, A ) ∩ CqE(K, A )

and an equivariant q -coboundary an element of the subgroup

 q q-1 B E(K, A) = δC E (K, A )
(3.11)

of Bq(K, A ) . Then the quotient group

 q q q H E(K, A ) = ZE(K, A )∕B E(K, A)
(3.12)

is the equivariant q -th cohomology group of K over A .

We denote by ZqR (K, A ) the subgroup of Cq (K, A) consisting of the cochains with an equivariant coboundary, that is,

 q q q+1 Z R(K, A ) = {c ∈ C (K, A ) | δc ∈ B E (K, A)}.
(3.13)

An element of  q ZR (K,A ) is called a residual q -cocycle. A residual q -coboundary, on the other hand, is an element of the group

 q q q B R(K, A ) = B (K, A ) + C E(K, A ).

If  q b ∈ B (K, A) and  q c ∈ CE (K, A ) , then by (3.11) the element  q (b + c) ∈ B R(K, A ) satisfies

δ(b + c) = δc ∈ δCq (K, A ) = Bq+1(K, A ); E E

hence by (3.13) the group BqR(K, A) is a subgroup of ZqR (K, A) . The corresponding quotient group

 q q q H R(K, A ) = ZR(K, A )∕B R(K, A)

is the residual q -th cohomology group of K over A . Then it can be shown (cf. [4]) that there is an exact sequence of the form

⋅⋅⋅-→δ Hq (K, A )-→ξ Hq (K, A )-→η Hq (K, A )-→δ Hq+1 (K, A ) → ⋅⋅⋅ , E R E
(3.14)

where the homomorphisms ξ and η are induced by the inclusions

Zq (K, A) ⊂ Zq (K, A) ⊂ Zq (K, A ), E R

 q q q B E(K, A ) ⊂ B (K, A ) ⊂ BR (K, A)

and the map δ is given by the coboundary map on  q C (K, A ) .

We now consider the complex K Γ defined as follows. The q -cells in K Γ are ordered (q + 1) -tuples (γ0,...,γq) of elements of Γ , so that Cq (KΓ ) is the free abelian group generated by the (q + 1) -fold Cartesian product Γ q+1 of Γ . Given a q -cell ^γ = (γ0,...,γq) and a (q - 1) -cell ^α = (α0,...,αq- 1) , we define the incidence number [^γ : ^α ] to be  i (- 1) if ^α = (γ0,..., ^γi,...,γq) and zero otherwise, where ^γi means deleting the entry γi . Then it can be shown that the integer [^γ : ^α] satisfies (3.5), so that K Γ is indeed a complex. By (3.6) its boundary operator on Cq (KΓ ) is given by

 ∑q i ∂ (γ0,...,γq) = (- 1)(γ0,...,^γi,...,γq) ∈ Cq-1(K Γ ) i=0
(3.15)

for γ ,...,γ ∈ Γ 0 q . We define the left action of the group Γ acts on Cq (KΓ ) by

γ(γ0,...,γq) = (γγ0,...,γ γq)
(3.16)

for all γ ∈ Γ and  q+1 (γ0,...,γq) ∈ Γ . Thus, if Γ acts on an abelian group A on the left, then we can consider the equivariant cohomology groups HqE (KΓ ,A) of K Γ over A .

Proposition 3.2. Given a left Γ -module A , there is a canonical isomorphism

 q ~ q H (Γ ,A )= H E(K Γ ,A )
(3.17)

for each q ≥ 0 .

Proof. For each q ≥ 0 the group of q -cochains over A associated to the complex Kq is given by

 q q C (KΓ ,A) = Hom (C (K Γ ),A ).

Thus Cq(K Γ ,A ) consists of maps f : Cq(K Γ ) → A satisfying

 ( ∑ ) ∑ f mi ^γi = mif (^γi) i i

where ^γi is a q -cell in K Γ and mi ∈ A for each i . Therefore Cq (K Γ ,A) may be regarded as the free abelian group generated by the maps of the form

h : Γ q+1 → A.

By (3.10) and (3.16) an element f ∈ Cq(K ,A ) Γ is equivariant if

γf (γ0,...,γq) = f(γ(γ0,...,γq)) = f(γ γ0,...,γγq)
(3.18)

for each γ ∈ Γ and each generator (γ0,...,γq) ∈ Γ q+1 of Cq (K Γ ) . By (3.9) the coboundary map δ : Cq(K Γ ,A ) → Cq+1 (K Γ ,A ) is given by

pict

for all f ∈ Cq (K Γ ,A) , where we used (3.15). Thus we see that the space of equivariant elements of Cq (K ,A ) Γ coincides with the space ℭq (Γ ,A ) of homogeneous q -cochains considered in Section 3.1; hence the proposition follows.

3.3. Hecke operators on group cohomology. In this section, we discuss Hecke operators acting on the group cohomology. Let G be a fixed group. If Γ is a subgroup of G , as in Section 2.2 we denote by ^Γ its commensurator. Given a subsemigroup Δ of G , recall that C(Δ ) is the set of mutually commensurable subgroups Γ of G such that

Γ ⊂ Δ ⊂ ^Γ .

We choose an element Γ ∈ C (Δ) and denote by H (Γ ;Δ ) the associated Hecke algebra described in Section 2.3. Thus H (Γ ;Δ ) is the ℤ -algebra generated by double cosets Γ αΓ with α ∈ Δ .

Given a subgroup Γ of G , we consider the Hecke algebra H (Γ ;^Γ ) associated to the subsemigroup Δ = ^Γ of G . Let Γ α Γ with α ∈ ^Γ be an element of  ^ H (Γ ;Γ ) that has a decomposition of the form

 ∐d Γ αΓ = Γ α i i=1
(3.20)

for some α1, ...,αd ∈ ^Γ . Since Γ α Γ γ = Γ αΓ for each γ ∈ Γ , we have

 d d Γ αΓ = ∐ Γ α = ∐ Γ α γ i i i=1 i=1

for all γ ∈ Γ . Thus for 1 ≤ i ≤ d , we see that

α γ = ξ (γ) ⋅ α i i i(γ)
(3.21)

for some element ξi(γ) ∈ Γ , where (α1 (γ),...,αd(γ)) is a permutation of (α1,...,αd ) . For each i and γ,γ ′ ∈ Γ we have

 ′ ′ ′ (αiγ)γ = ξi(γ ) ⋅ αi(γ)γ = ξi(γ) ⋅ ξi(γ)(γ ) ⋅ αi(γ)(γ′).

Comparing this with α (γγ ′) = ξ (γ γ′)α ′ i i i(γγ) , we see that

i(γγ′) = i(γ )(γ ′), ξi(γγ′) = ξi(γ) ⋅ ξi(γ)(γ′)
(3.22)

for all γ,γ′ ∈ Γ .

Given a nonnegative integer p and a Γ -module M , let  p ℭ (Γ ,M ) be the group of homogeneous p -cochains described in Section 3.1. For an element ϕ ∈ ℭp(Γ ,M ) and a double coset Γ αΓ with α ∈ ^Γ that has a decomposition as in (3.20), we consider the map ϕ ′ : Γ p+1 → M given by

 d ′ ∑ -1 ϕ (γ0,...,γp) = α i ⋅ ϕ(ξi(γ0),...,ξi(γp)), i=1

where the maps ξ : Γ → Γ i are determined by (3.21). Then it is known that  ′ ϕ is an element of  p ℭ (Γ ,M ) (see [13]). Thus each double coset Γ αΓ with α ∈ ^Γ determines the ℂ -linear map

𝔗 (α) : ℭp(Γ ,M ) → ℭp (Γ ,M )
(3.23)

defined by

 d ∑ -1 (𝔗 (α)ϕ)(γ0,...,γp) = αi ⋅ ϕ(ξi(γ0),...,ξi(γp)) i=1
(3.24)

for ϕ ∈ ℭp(Γ ,M ) , where Γ α Γ = ∐ Γ αi 1≤i≤d and each ξi is as in (3.21). Then the map 𝔗(α ) is independent of the choice of representatives of the coset decomposition of Γ αΓ modulo Γ . Furthermore, it can be shown that

𝔗(α ) ∘ δp = δp ∘ 𝔗(α )
(3.25)

for each p ≥ 1 , where δp and δp+1 are coboundary maps on ℭp(Γ ,M ) and ℭp+1(Γ ,M ) , respectively. Thus the map 𝔗 (α) in (3.23) induces a homomorphism

 p p 𝔗 (α) : H (Γ ,M ) → H (Γ ,M ),

which is the Hecke operator on  p H (Γ ,M ) corresponding to α .

The Hecke operators can also be described by using nonhomogeneous cochains as follows. For each q ≥ 0 we denote by Cq (Γ ,M ) the group of nonhomogeneous q -cochains over M as in Section 3.1. Given f ∈ Cq (Γ ,M ) and α ∈ ^Γ with Γ αΓ as in (3.20), we set

pict

for all γ1,...,γq ∈ Γ .

Proposition 3.3. Given α ∈ ^Γ , the map T(α)f : Γ q → M is an element of Cq (Γ ,M ) and satisfies

T (α)f = (𝔗 (α)fH )N

for all  q f ∈ C (Γ ,M ) , where the operators

 q q q q (⋅)H : C (G, M ) → ℭ (G,M ), (⋅)N : ℭ (G, M ) → C (G, M )

are as in (3.3) and (3.4).

Proof. Given  q f ∈ C (Γ ,M ) , by (3.3) we have

 -1 -1 -1 fH (σ0, σ1,...,σq) = σ0 ⋅ f(σ0 σ1, σ1 σ2,...,σq- 1σq )

for all σ ,σ ,...,σ ∈ Γ 0 1 q . Thus for α ∈ ^Γ , using (3.24), we obtain

pict

Hence by using (3.4) we have

pict

for all γ1,...,γq ∈ Γ . However, it follows from (3.22) that

 -1 ξi(γ1 ⋅⋅⋅γk-1) ξi(γ1⋅⋅⋅γk-1γk) = ξi(γ1⋅⋅⋅γk-1)(γk )

for 2 ≤ k ≤ q . Hence we obtain

 d (𝔗(α )f ) (γ ,...,γ ) = ∑ α- 1f(ξ(γ ),ξ (γ ),ξ (γ ),...,ξ (γ )), H N 1 q i i 1 i(γ1) 2 i(γ1γ2) 3 i(γ1⋅⋅⋅γq-1) q i=1

and therefore the proposition follows from this and (3.26).

Let ∂q : Cq (Γ ,M ) → Cq+1 (Γ ,M ) and ∂q+1 : Cq+1 (Γ ,M ) → Cq+2(Γ ,M ) be the coboundary maps for nonhomogeneous cochains. Then, using Lemma 3.1 and (3.25), we have

pict

for all  q f ∈ C (Γ ,M ) ; hence it follows that

T(α ) ∘ ∂q = ∂q+1 ∘ T (α )

for each q ≥ 0 . Therefore the map T (α) : Cq (Γ ,M ) → Cq(Γ ,M ) also induces the Hecke operator

 q q T(α ) : H (Γ ,M ) → H (Γ ,M )

on  q H (Γ ,M ) that is compatible with 𝔗 (α) .

 

4. De Rham cohomology

The focus of this section is on the de Rham cohomology of differentiable manifolds with coefficients in a vector bundle and Hecke operators on such cohomology. In Section 4.1 we review basic properties of the sheaf cohomology including the sheaf-theoretic interpretation of the de Rham and C ∞ singular cohomology of differentiable manifolds with coefficients in a real vector space. If Γ is a fundamental group of a manifold X and ρ is a representation of Γ in a finite-dimensional real vector space, we can consider the associated vector bundle Vρ over X . In Section 4.2 we construct the de Rham cohomology of X with coefficients in Vρ . This cohomology is identified, in Section 4.3, with certain cohomology of the universal covering space of X associated to the representation ρ of Γ . We use this identification to introduce Hecke operators on the de Rham cohomology of X with coefficients in V ρ (cf. [6]).

4.1. Cohomology of sheaves. Let X be a topological space, and let S be a sheaf over X of certain algebraic objects, such as abelian groups, rings, and modules (see e.g. [18] for the definition and basic properties of sheaves). If U is an open subset of X , we denote by Γ (X, S) or S(U ) the space of sections of S over U . Then a resolution of S is an exact sequence of morphisms of sheaves of the form

 0 1 2 0 → S → F → F → F → ⋅⋅⋅ ,

which we also write as

 ∙ 0 → S → F

in terms of the graded sheaf F ∙ = {F i}i≥0 over X .

Example 4.1. (i) Let A be an abelian group regarded as a constant sheaf over a topological space X . Given an open set U ⊂ X , let  p S (U, A) denote the group of singular p -cochains in U with coefficients in A . If U is a unit ball in a Euclidean space, then its cohomology group is zero. Hence the sequence

 p- 1 δ p δ p+1 ⋅⋅⋅ → S (U, A )-→ S (U, A )-→ S (U, A )

is exact, where δ denotes the usual coboundary operator for singular cochains. We denote by  p S (A ) the sheaf over X generated by the presheaf U ↦→ Sp (U, A ) . Then the previous exact sequence induces the exact sequence

 d d 0 → A → S0(A )-→ S1 (A ) -→ ⋅⋅⋅ ,

of sheaves, which is a resolution of the sheaf A over X .

(ii) Let ℝ be the constant sheaf of real numbers, and let X be a differentiable manifold of real dimension n . We denote by Ep the sheaf of real-valued p -forms on X . Then we have a sequence of the form

 ι d d d 0 → ℝ -→ E 0-→ E 1-→ ⋅⋅⋅-→ E n → 0,
(4.1)

where d is the exterior differentiation operator and ι is the natural inclusion map. Using the Poincaré lemma, it can be shown that the sequence (6.4) is exact and therefore is a resolution of the sheaf ℝ .

(iii) Let X be a complex manifold of complex dimension n , and let E p,q the sheaf of (p,q) -forms on X . Given p with 0 ≤ p ≤ n , we consider the sequence

 - - - 0 → Ωp → Ep,0 -→∂ Ep,1-→∂ ⋅⋅⋅-→∂ E p,n → 0,
(4.2)

where  p Ω denotes the sheaf of holomorphic p -forms on X that is the kernel of morphism -- ∂ : Ep,0 → Ep,1 . Then the -- ∂ Poincaré lemma implies the sequence (4.2) is exact and therefore is a resolution of the sheaf Ωp .

Given a sheaf S over a topological space X , in order to define the cohomology of X with coefficients in S we now construct a particular resolution of S . Let  E S together with a local homeomorphism  E ϖ : S → X be the associated étale space, which means that  E S is a topological space such that S is isomorphic to the sheaf of sections of ϖ . Let ℭ0 (S ) be the presheaf defined by

 0 E ℭ (S )(U) = {s : U → S | ϖ ∘ s = 1U}

for each open subset U ⊂ X . Then ℭ0(S ) is in fact a sheaf and is known as the sheaf of discontinuous sections of S over X , and the natural map  0 S (U) → ℭ (S )(U ) determines an injective morphism  0 S → ℭ (S ) of sheaves. We set

 1 0 1 0 1 F (S) = ℭ (S )∕S, ℭ (S ) = ℭ (F (S )),

and define inductively

F i(S ) = ℭi-1(S)∕F i-1(S), ℭi(S ) = ℭ0(F i(S ))

for i ≥ 2 . Then the natural morphisms determine short exact sequences of sheaves over X of the form

0 → S → ℭ0(S ) → ℭ1(S ) → 0,

 i i i+1 0 → F (S) → ℭ (S) → ℭ (S) → 0

for i ≥ 2 . These sequences induce the long exact sequence

0 → S → ℭ0 (S ) → ℭ1 (S) → ℭ2(S) → ⋅⋅⋅ ,

which is called the canonical resolution of S . By taking the global section of each term of this exact sequence we obtain a sequence of the form

0 → Γ (X, S) → Γ (X, ℭ0(S )) → Γ (X,ℭ1 (S)) → ⋅⋅ ⋅ ,

which is in fact a cochain complex. For each i ≥ 0 we set

 i i C (X,S ) = Γ (X, ℭ (S )),

so that the collection C ∙(X, S) = {Ci (X, S)}i≥0 becomes a cochain complex.

Definition 4.2. Given a sheaf S over X , the q -th cohomology group of the cochain complex C ∙(X,S ) is called the q -th cohomology group of X with coefficients in S and is denoted by Hq (X,S ) , that is,

Hq (X, S ) = Hq (C ∙(X, S))
(4.3)

for all q ≥ 0 .

If the coboundary homomorphism  i i+1 C (X, S ) → C (X, S ) is denoted by  i δ for i ≥ - 1 with  -1 C (X, S ) = 0 , then (4.3) means that

 q i i-1 H (X,S ) = Ker δ ∕Im δ .

In particular we have

H0 (X, S ) = Ker δ0 = Γ (X, S).

Definition 4.3. (i) A sheaf F over a topological space X is flabby if for any open set U ⊂ X the restriction map F (X ) → F (U ) is surjective.

(ii) A sheaf F over a topological space X is soft if for any closed set U ⊂ X the restriction map F (X ) → F(U ) is surjective.

(iii) A sheaf F of abelian groups over a paracompact Hausdorff space X is fine if for any disjoint subsets Y1 and Y2 of X there is an automorphism ϕ : F → F which induces the zero map on a neighborhood of Y1 and the identity map on a neighborhood of Y2 .

Theorem 4.4. Let S be a sheaf over a paracompact Hausdorff space X . If S is soft, then

 q H (X, S ) = 0

for all q ≥ 1 .

Proof. See [18, Theorem 3.11].

Definition 4.5. A resolution of a sheaf S over X of the form

 0 1 2 0 → S → A → A → A → ⋅⋅⋅

is said to be acyclic if Hj (X, Ai) = 0 for all i ≥ 0 and j ≥ 1 .

Let S be a sheaf of abelian groups over X , and let

 0 1 2 0 → S → A → A → A → ⋅⋅⋅
(4.4)

be a resolution of S . By taking the global section of each term of this exact sequence we obtain a cochain complex of the form

0 → Γ (X, S ) → Γ (X, A0) → Γ (X, A1 ) → Γ (X, A2) → ⋅⋅⋅ .

Thus we can consider the cohomology groups Hq (Γ (X, A ∙)) of the cochain complex Γ (X,A ∙) = {Γ (X, Ai)}i≥0 .

Theorem 4.6. If the resolution (4.4) of the sheaf S over X is acyclic, then there is a canonical isomorphism

 q q ∙ H (X, S ) = H (Γ (X, A ))

for all q ≥ 0 .

Proof. See [18, Theorem 3.13].

Lemma 4.7. Let R be a sheaf of rings over X , and let M be a sheaf of modules over R . If R is soft, then M is soft.

Proof. Let K be a closed subset of X , and consider an element s ∈ M (K ) . Then s can be extended to a neighborhood U of K . Define an element h ∈ R (K ∪ (X - U )) satisfying h(x) = 1 for x ∈ K and h(x) = 0 for x ∈ X - U . Since R is soft, h can be extended to an element ^ h ∈ R (X ) . Then ^ h ⋅ s ∈ M (X ) is an extension of s .

Let F be a vector space over ℝ , and let  q E (F ) with q ≥ 0 be the sheaf of F -valued q -forms on a differentiable manifold X . Let  q S ∞(F ) be the sheaf obtained by modifying Sq(A ) in Example 4.1(i) by using A = F and C ∞ singular q -cochains. We consider the corresponding graded sequences E∙ = {E i} i≥0 and S∙ (F) = {Si (F )} ∞ ∞ i≥0 of sheaves over X . Then the q -th  ∞ C singular cohomology group  q H ∞ (X, F) and the q -th de Rham cohomology group  q H DR(X, F ) with coefficients in F are defined by

 q Hq∞ (X, F ) = Hq (Γ (X, S∙∞ (F)), H DR (X, F) = Hq (Γ (X, E ∙(F)))

for each q ≥ 0 . On the other hand, if  p,∙ p,q E = {E }q≥0 with  p,q E as in Example 4.1(iii), then the Dolbeault cohomology group of X of type (p,q) is defined by

 (p,q) q p,∙ H (X ) = H (Γ (X, E ))

for p, q ≥ 0 .

Theorem 4.8. (i) Let F be a vector space over ℝ . If X is a differentiable manifold, then there are canonical isomorphisms

Hq (X, F) ~= Hq∞ (X, F ) ~= HqDR (X, F )

for all q ≥ 0 , where Hq (X, F ) denotes the q -th cohomology group of X with coefficients in the constant sheaf F .

(ii) If X is a complex manifold of complex dimension n , then there is a canonical isomorphism

H (p,q)(X ) ~= Hq (X, Ωp),

for all p,q ≥ 0 with p + q = 2n , where  p Ω is the sheaf of holomorphic p -forms on X .

Proof. Given a manifold X , there are resolutions of the constant sheaf F of the form

0 → F → E∙(F ), 0 → F → S ∙(F ). ∞

Using the argument of the partition of unity, it can be shown that S0∞(F ) and E0(F ) are soft sheaves. Since the sheaf Sq∞ (F ) is a module over S0 (F ) ∞ for each q ≥ 0 , it follows from Lemma 4.7 that  q S ∞(F ) is soft. Thus, using Theorem 4.4 and Theorem 4.6, we see that

Hq (X, F) ~= Hq (Γ (X, S ∙∞(F )) = Hq∞(X, F ).

Similarly, each Eq(X ) is soft; hence we have

 q ~ q ∙ q H (X,F ) = H (Γ (X, E (F )) = H DR(X, F ),

which proves (i). As for (ii), we consider the resolution (4.2) of Ωp and use the fact that the sheaves E p,q are soft.

4.2. De Rham cohomology and vector bundles. Let X be a manifold, and let D be the universal covering space of X . Let Γ = π1(X ) be the fundamental group of X , so that X can be identified with the quotient space Γ \D .

Let ρ be a representation of Γ in a finite-dimensional real vector space F , and define an action of Γ on D × F by

γ ⋅ (z,v ) = (γz, ρ(γ)v)
(4.5)

for all γ ∈ Γ and (z,v) ∈ D × F . We equip the real vector space F with the Euclidean topology and denote by

Vρ = Γ \D × F
(4.6)

the quotient of D × F with respect to the Γ -action in (4.5). Then the natural projection map pr1 : D × F → D induces a surjective map π : V ρ → X such that the diagram

 ϖ^ D × F -- -→ V ρ || || pr1↓ ↓π ϖ D -- -→ X
(4.7)

is commutative, where ^ϖ and ϖ denote the canonical projection maps. The surjective map π determines the structure of a vector bundle over X on V ρ as can be seen in the following proposition.

Proposition 4.9. The set Vρ has the structure of a locally constant vector bundle over X = Γ \D with fiber F whose fibration is the map π : V ρ → X in (4.7).

Proof. Let {U } α α∈I be an open cover of X such that the inverse image  -1 π (Uα) of each U α under ϖ is homeomorphic to U α . By taking smaller open sets if necessary we may assume that U α ∩ Uβ is either connected or empty for all α,β ∈ I . For each α ∈ I we select a connected component ^Uα of π- 1(U α) . If U α ∩ U β ⁄= ∅ , then there exists a unique element γα,β ∈ Γ such that

γ ^U ∩ ^U ⁄= ∅. α,β α β
(4.8)

We define the map  -1 ψ α : Uα × F → π (U α) by

ψ α(x,y) = ^ϖ (^x,y)
(4.9)

for all (x,y) ∈ Uα × F , where ^x is the element of U^α with ϖ (^x) = x . Then we see easily that ψ α is a bijection. We shall now introduce a vector space structure on each fiber  - 1 V ρ,x = π (x ) with x ∈ X . Given x ∈ U α ⊂ X , we define the map ψα,x : F → Vρ,x by

ψ (v) = ψ (x,v) α,x α
(4.10)

for all v ∈ F . Then ψ α,x is bijective, and therefore we can define a vector space structure on Vρ,x by transporting the one on F via the map ψα,x . We need to show that such a structure is independent of x . Let x ∈ U ∩ U α β . If ^x ∈ ^U α α and ^x ∈ ^U β β are the elements with ϖ (^xα) = x = ϖ (^x α) . Then from (4.8) we see that x^β = γα,β ^xα . Using this and the relations (4.5), (4.9) and (4.10), we obtain

pict

for all v ∈ F . Hence we see that the diagram

 ρ(γα,β) F ----→ F || || ψα(x,v)↓ ↓ψβ(x,v) Vρ,x ------ Vρ,x

is commutative, which shows that the vector space structure on Vρ,x is independent of x . Finally, we note that the map

ϕ = ψ- 1: π -1(U ) → U × F α α α α

can be used as a local trivialization for each α ∈ I .

Given a positive integer p , we first define a function which assigns to each x ∈ X an alternating p -linear map

ξx : Tx(X ) × ⋅⋅⋅ × Tx (X ) → Vρ,x,
(4.11)

where Tx(X ) denotes the tangent space of X at x ∈ X and Vρ,x is the fiber of Vρ at x . We then define, for each α ∈ I , the function ξα on U α which associates to each x ∈ Uα an F -valued alternating p -linear map ξ (x) α given by

ξα(x) = ϕα,x ∘ ξx,
(4.12)

where ϕα,x = ϕα |Vρ,x .

Definition 4.10. A Vρ -valued p -form on X is a function ξ on X which assigns to each x ∈ X an alternating p -linear map ξx of the form (4.11) such that the function ξ α in (4.12) is differentiable.

Let {Uα} α∈I be an open cover of X . Noting that Vρ is locally constant by Proposition 4.9, we denote by C α,β ∈ GL (F ) the constant transition function on U α ∩ U β for α,β ∈ I . Then a Vρ -valued p -form on X can be regarded as a collection { ωα}α∈I of F -valued p -forms ω α on U α satisfying

ωβ = Cα,βωα

on U α ∩ U β for all α,β ∈ I with Uα ∩ Uβ ⁄= ∅ . Since each C α,β is constant, we have

dω β = d(C α,βω α) = Cα,βdωα;

hence the collection {dω } α α∈I determines a V (ρ) -valued (p + 1) -form on X . Thus, if  p E (X, Vρ) denotes the space of all V ρ -valued p -forms on X , the map {ωα }α∈I ↦→ {d ωα}α∈I determines an operator

d : Ep(X, Vρ) → E p+1(X, Vρ)
(4.13)

with d2 = 0 for each p ≥ 0 . Then the de Rham cohomology of X with coefficients in V ρ is the cohomology of the cochain complex  ∙ p E (X, V ρ) = {E (V(ρ ))}p≥0 with the coboundary operator (4.13). Thus the quotient

 Ker (d : Eq(X, Vρ) → Eq+1(X, Vρ)) Hq (X, Vρ) = -------------q-1----------------- dE (X, Vρ)
(4.14)

for q ≥ 0 is the q -th de Rham cohomology of X with coefficients in V ρ .

4.3. Hecke operators on de Rham cohomology. Let D , Γ , X = Γ \D , and the representation ρ : Γ → GL (F) be as in Section 4.2. Given p ≥ 0 , the space Ep(D, F) of all F -valued p -forms on D is spanned by the elements of the form ω ⊗ v with ω ∈ Ep(D ) and v ∈ F . By setting

d(ω ⊗ v) = (dω) ⊗ v

we obtain the map  p p+1 d : E (D, F ) → E (D,F ) with  2 d = 0 ; hence we can consider the associated cochain complex E ∙(D,F ) = {E p(D, F )}p≥0 whose cohomology is the de Rham cohomology H *DR(D, F ) of D with coefficients in F . By Theorem 4.8 there is a canonical isomorphism

HqDR (D,F ) ~= Hq∞ (D,F )

for each q ≥ 0 . This isomorphism can be described more explicitly as follows. Given q ≥ 0 , the group Sq∞(D, F ) of C∞ q -cochains considered in Theorem 4.8 can be written as

 q ∞ S ∞ (D, F ) = Hom (S q ,F),

where  ∞ S q is the group of  ∞ C q -chains. Thus each element of  ∞ S q is a finite sum of the form  ∑ c = iaiΞi with ai ∈ ℤ , where each Ξi : s → D is a C ∞ map from a q -simplex in a Euclidean space to D . To each q -form ω ∈ E q(D, F ) we set

 ∫ ∑ ∫ fω(c) = cω = ai Ξ *iω i s
(4.15)

for  ∑ ∞ c = iaiΞi ∈ Sq . If  ′ ′′ c = c + ∂c with  ′′ ∞ c ∈ Sq+1 , the Stokes theorem implies that

 ∫ ∫ ∫ ∫ ∫ ∫ fω(c′) = ω = ω + ω = ω + d ω = ω = fω(c). c+∂c′′ c ∂c′′ c c′′ c

Thus the map c ↦→ fω(c) is well-defined map on the set of q -cycles in S ∞q and therefore is an element of Sq∞ (D,F ) . On the other hand, if