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## Revista de la Unión Matemática Argentina

*versão impressa* ISSN 0041-6932

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

**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. Introduction**This 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 satisfying a certain transformation formula with respect to the linear fractional action of a discrete subgroup of , and they are closely linked to the geometry of the associated Riemann surface . For example, modular forms for can be interpreted as holomorphic sections of a line bundle over , and the space of such modular forms of a given weight corresponds to a certain cohomology group of with local coefficients or with some cohomology group of the discrete group (cf. [1], [2], [5]) with coefficients in some -module. Modular forms can be extended to automorphic forms of several variables by using holomorphic functions either on the Cartesian product of copies of for Hilbert modular forms or on the Siegel upper half space of degree for Siegel modular forms. More general automorphic forms can also be considered by using semisimple Lie groups. Indeed, given a semisimple Lie group of Hermitian type and a discrete subgroup of , we can consider automorphic forms for defined on the quotient of by a maximal compact subgroup of . The space has the structure of a Hermitian symmetric domain, and automorphic forms on for are holomorphic functions on satisfying an appropriate transformation formula with respect to the natural action of on (cf. [3]). Such automorphic forms are also linked to families of abelian varieties parametrized by the locally symmetric space (cf. [6], [10], [14]). Close connections between automorphic forms for the discrete group and the group cohomology of or the de Rham cohomology of 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 is an automorphic form on a Hermitian symmetric domain for a discrete subgroup of described above, then can be interpreted as an algebraic correspondence on the quotient space , which has the structure of a complex manifold, assuming that is torsion-free. Such a correspondence is determined by a pair of holomorphic maps , where is another discrete subgroup of . The maps and can be used to construct a Hecke operator on the de Rham cohomology of . 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 algebras**In 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 , consider double cosets determined by two commensurable subgroups of , 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 be a group. Two subgroups and are said to be *commensurable* (or is said to be commensurable with ) if

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

*Proof.* The relation is clearly reflexive and symmetric. Let , and be subgroups of with and . We consider the map

| (2.1) |

sending the left coset to the left coset for each . If with , then ; hence we see that . Thus the map (2.1) is injective, and therefore we have

which implies that

Similarly, it can be shown that

and hence we obtain

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

Given a subgroup of , we set

which will be called the *commensurator* of in .

**Lemma 2.2.** *The commensurator* *is a subgroup of* *containing* *.*

*Proof.* Given , since , we see that

is commensurable with . However, the commensurability implies that ; hence we have

Thus , and therefore is a subgroup of . Since clearly contains , the proof of the lemma is complete.

*Proof.* If and , then we have

hence , which shows that . On the other hand, if , then we have

hence . Thus we have . Similarly, it can be shown that , and therefore we obtain .

**Proposition 2.4.** *Let* *, and let* *. Then the double coset* *can be decomposed into disjoint unions of the form*

| (2.2) |

*for some positive integers* *and* *, where* *and* *are complete sets of coset representatives of* *and* *, respectively.*

*Proof.* We note first that a right coset of contained in can be written in the form for some . If with is another subset of , we see that if and only if , which is equivalent to the condition that

Since , the index is finite. Thus, if is a set of representatives of , each determines a unique coset contained in ; hence we have

Similarly, it can be shown that

where .

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

Given and a commutative ring be with identity, we denote by the free -module generated by the double cosets with . Thus an element of can be written in the form

where the coefficients are zero except for a finite number of . We denote by the number of right cosets contained in . Thus, if is as in (2.2), then . If is an element of given by , then we set

| (2.3) |

and refer to it as the *degree* of .

We now consider an -module and assume that the subsemigroup acts on on the right by

for . Thus we have

for all and . Given , let denote the submodule of consisting of the -invariant elements of , that is,

If the double coset with and has a decomposition of the form

| (2.4) |

then we define its operation on by

| (2.5) |

for all .

**Lemma 2.5.** *The operation of* *on* *in* (2.15) *is* *independent of the choice of the representatives* *of the right cosets* *of* *in* (2.4) *and*

*for all* *.*

*Proof.* If are subsets of with , then for some . Thus we see that for all ; hence is independent of the choice of the representatives . On the other hand, if has a decomposition as in (2.4), then we see that

for all . Thus we have

hence it follows that .

We see easily that the map given by (2.15) is in fact a homomorphism of -modules. We now extend this by defining an -module homomorphism associated to each element of by

for and .

Given elements and double cosets of the form

| (2.6) |

with , we set

| (2.7) |

where the summation is over the set of representatives of the double cosets contained in and

| (2.8) |

is the number of pairs with and such that . Since except for a finitely many double cosets , the sum on the right hand side of (2.7) is a finite sum.

Let denote the free -module generated by the right cosets with . Then acts on by right multiplication. On the other hand, there is a natural injective map sending to . By using this injection we may regard as an -submodule of , and under this identification we see easily that

| (2.9) |

If the double cosets and are as in (2.6), using (2.15) and (2.8), we have

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

Thus by using (2.9) again, we obtain

hence it follows that

| (2.10) |

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

**Lemma 2.6.** *Let* *and* *be as in* (2.6)*, and let* *with* *be as in* (2.8)*. Then we have*

*for each* *.*

*Proof.* We assume that has a decomposition of the form

Then the relation holds if and only if for exactly one . Thus, if is as in (2.8), we see that

hence the lemma follows from this and the fact that .

**Lemma 2.7.** *If* *and* *, then* *we have*

*Proof.* Let and be as in (2.7). Then, using (2.3) and Lemma 2.6, we have

However, by (2.8) the right hand side of this relation is equal to the number of pairs with and and therefore is equal to . Thus the lemma follows by extending this result linearly.

2.3. **Hecke algebras.** Given , the operation in (2.7) induces a bilinear map

defined by

| (2.11) |

Using (2.10), we see that the operation of on coincides with the multiplication operation in (2.11), that is,

| (2.12) |

for all and .

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

for all , and . From this and (2.12) we obtain

| (2.13) |

for all , and .

Given , we set

Then by (2.13) the multiplication operation on is associative and is an algebra over with identity . When , we shall simply write

**Definition 2.8.** Given , the algebra is called the *Hecke algebra over* of with respect to . If , then is simply called the *Hecke algebra* of with respect to .

Let and be two subsemigroups of with . Then certainly is a subset of . If with are regarded as elements of , their product can be written in the form

| (2.14) |

where the summation is over the set of representatives of the double cosets contained in . However, we have ; hence the product in (2.14) coincides with the product of and in . Thus we see that is a subalgebra of .

**Proposition 2.9.** *Let* *, and assume that* *. Then the quotients* *and* *have* *a common set of coset representatives.*

*Proof.* We assume that can be decomposed as

Then it can be shown that is nonempty for all and . Indeed, if and are disjoint for some and , then , and therefore we have

which is a contradiction. Thus, in particular, we have for each . If for each , then we see that and . Hence we have

and is a common set of coset representatives.

We now discuss the commutativity of the Hecke algebra . Note that an involution on is a map satisfying

for all .

**Theorem 2.10.** *Let* *be an involution on* *, and assume that* *an element* *satisfies*

| (2.15) |

*for all* *. Then the associated Hecke algebra* *is* *commutative.*

*Proof.* Given with , using (2.15), we have

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

for some . Similarly, if is another element of , we have

for some positive integer and for . We now assume that

where and are as in (2.8). Then we have

where we used the fact that

Hence it follows that is a commutative algebra.

**Example 2.11.** Let for some positive integer , and consider the subgroup and the subsemigroup

of . Then we see that the transposition is an involution satisfying

Given , by the elementary divisor theorem the corresponding double coset can be written as

for some diagonal matrix , where the diagonal entries are positive integers satisfying for each . Hence we see that

Thus by Theorem 2.10 the Hecke algebra is commutative.

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 with coefficients in a -module by using both homogeneous and nonhomogeneous cochains is given in Section 3.1. Given a complex on which a group acts on the left and a left -module , in Section 3.2 we construct the associated equivariant cohomology of with coefficients in following Eilenberg [4]. We also obtain an isomorphism between this equivariant cohomology and the cohomology of 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 be a group, and let be a left -module. Thus is an abelian group on which acts on the left. Then the cohomology of with coefficients in can be described by using either homogeneous or nonhomogeneous cochains.

Given a nonnegative integer , let denote the group consisting of the -valued functions on the -fold Cartesian product of , called *nonhomogeneous* *-cochains.* We then consider the map defined by

for all and . Then is the coboundary map for nonhomogeneous -cochains satisfying . The associated -th cohomology group of with coefficients in is given by

where is the kernel of and is the image .

For each we also consider the group of *homogeneous* *-cochains* consisting of the maps satisfying

for all . We then define the map by

| (3.2) |

for all and , which is the coboundary map for nonhomogeneous -cochains satisfying . Then the corresponding -th cohomology group of in is given by

where is the kernel of and is the image .

We can establish a correspondence between homogeneous and nonhomogeneous cochains as follows. Given and , we consider the elements and given by

for all . Then we see that

for all and . Thus, by extending linearly we obtain the linear maps

such that and are identity maps on and , 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* *, we have*

*for all* *and* *.*

*Proof.* Given elements and , using (3.1), (3.2) and (3.3), we have

On the other hand, if , by using (3.1), (3.2) and (3.4) we see that

hence the lemma follows.

From Lemma 3.1 we see that the diagram

is commutative, which implies that there is a canonical isomorphism

for each .

3.2. **Equivariant cohomology.** Let be a complex, which can be described as follows. The elements of the complex are called *cells*, and there is a nonnegative integer associated to each cell called the *dimension* of the cell. A cell of dimension is referred to as a -cell, and the *incidence number* associated to the a -cell and a -cell is an integer that is nonzero only for a finite number of -cells and satisfies

| (3.5) |

for . Given , we denote by the free abelian group generated by the -cells, and the elements of are called *-chains*. The boundary operator on is the homomorphism

of abelian groups given by

| (3.6) |

for each generator of , where the summation is over the generators of . Then it can be shown that satisfies .

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

| (3.7) |

Since is generated by the -cells, a -cochain is uniquely determined by its values for the -cells . The coboundary operator

| (3.8) |

on is defined by

| (3.9) |

for all and , and the condition implies . Then the *-th cohomology group of the complex* *over* is given by the quotient

where is the kernel of and is the image of .

We now assume that a group acts on and on , both on the left. Given , an element is said to be an *equivariant* *-cochain* if it satisfies

| (3.10) |

for all and , where is as in (3.7). We denote by the subgroup of consisting of the equivariant cochains. If is the coboundary map in (3.8) and if is an equivariant -cochain, then we have

for all , which shows that is an equivariant -cochain. We define an *equivariant* *-cocycle* to be an element of the group

and an *equivariant* *-coboundary* an element of the subgroup

| (3.11) |

of . Then the quotient group

| (3.12) |

is the *equivariant* *-th cohomology group* of over .

We denote by the subgroup of consisting of the cochains with an equivariant coboundary, that is,

| (3.13) |

An element of is called a *residual* *-cocycle*. A *residual* *-coboundary*, on the other hand, is an element of the group

If and , then by (3.11) the element satisfies

hence by (3.13) the group is a subgroup of . The corresponding quotient group

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

| (3.14) |

where the homomorphisms and are induced by the inclusions

and the map is given by the coboundary map on .

We now consider the complex defined as follows. The -cells in are ordered -tuples of elements of , so that is the free abelian group generated by the -fold Cartesian product of . Given a -cell and a -cell , we define the incidence number to be if and zero otherwise, where means deleting the entry . Then it can be shown that the integer satisfies (3.5), so that is indeed a complex. By (3.6) its boundary operator on is given by

| (3.15) |

for . We define the left action of the group acts on by

| (3.16) |

for all and . Thus, if acts on an abelian group on the left, then we can consider the equivariant cohomology groups of over .

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

| (3.17) |

*for each* *.*

*Proof.* For each the group of -cochains over associated to the complex is given by

Thus consists of maps satisfying

where is a -cell in and for each . Therefore may be regarded as the free abelian group generated by the maps of the form

By (3.10) and (3.16) an element is equivariant if

| (3.18) |

for each and each generator of . By (3.9) the coboundary map is given by

for all , where we used (3.15). Thus we see that the space of equivariant elements of coincides with the space of homogeneous -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 be a fixed group. If is a subgroup of , as in Section 2.2 we denote by its commensurator. Given a subsemigroup of , recall that is the set of mutually commensurable subgroups of such that

We choose an element and denote by the associated Hecke algebra described in Section 2.3. Thus is the -algebra generated by double cosets with .

Given a subgroup of , we consider the Hecke algebra associated to the subsemigroup of . Let with be an element of that has a decomposition of the form

| (3.20) |

for some . Since for each , we have

for all . Thus for , we see that

| (3.21) |

for some element , where is a permutation of . For each and we have

Comparing this with , we see that

| (3.22) |

for all .

Given a nonnegative integer and a -module , let be the group of homogeneous -cochains described in Section 3.1. For an element and a double coset with that has a decomposition as in (3.20), we consider the map given by

where the maps are determined by (3.21). Then it is known that is an element of (see [13]). Thus each double coset with determines the -linear map

| (3.23) |

defined by

| (3.24) |

for , where and each is as in (3.21). Then the map is independent of the choice of representatives of the coset decomposition of modulo . Furthermore, it can be shown that

| (3.25) |

for each , where and are coboundary maps on and , respectively. Thus the map in (3.23) induces a homomorphism

which is the *Hecke operator* on corresponding to .

The Hecke operators can also be described by using nonhomogeneous cochains as follows. For each we denote by the group of nonhomogeneous -cochains over as in Section 3.1. Given and with as in (3.20), we set

for all .

**Proposition 3.3.** *Given* *, the map* *is an* *element of* *and satisfies*

*for all* *, where the operators*

*Proof.* Given , by (3.3) we have

for all . Thus for , using (3.24), we obtain

Hence by using (3.4) we have

for all . However, it follows from (3.22) that

for . Hence we obtain

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

Let and be the coboundary maps for nonhomogeneous cochains. Then, using Lemma 3.1 and (3.25), we have

for all ; hence it follows that

for each . Therefore the map also induces the Hecke operator

on that is compatible with .

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 singular cohomology of differentiable manifolds with coefficients in a real vector space. If is a fundamental group of a manifold and is a representation of in a finite-dimensional real vector space, we can consider the associated vector bundle over . In Section 4.2 we construct the de Rham cohomology of with coefficients in . This cohomology is identified, in Section 4.3, with certain cohomology of the universal covering space of associated to the representation of . We use this identification to introduce Hecke operators on the de Rham cohomology of with coefficients in (cf. [6]).

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

which we also write as

in terms of the graded sheaf over .

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

is exact, where denotes the usual coboundary operator for singular cochains. We denote by the sheaf over generated by the presheaf . Then the previous exact sequence induces the exact sequence

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

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

| (4.1) |

where is the exterior differentiation operator and 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 .

(iii) Let be a complex manifold of complex dimension , and let the sheaf of -forms on . Given with , we consider the sequence

| (4.2) |

where denotes the sheaf of holomorphic -forms on that is the kernel of morphism . Then the Poincaré lemma implies the sequence (4.2) is exact and therefore is a resolution of the sheaf .

Given a sheaf over a topological space , in order to define the cohomology of with coefficients in we now construct a particular resolution of . Let together with a local homeomorphism be the associated *étale space*, which means that is a topological space such that is isomorphic to the sheaf of sections of . Let be the presheaf defined by

for each open subset . Then is in fact a sheaf and is known as the *sheaf of discontinuous sections of* *over* , and the natural map determines an injective morphism of sheaves. We set

and define inductively

for . Then the natural morphisms determine short exact sequences of sheaves over of the form

for . These sequences induce the long exact sequence

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

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

so that the collection becomes a cochain complex.

**Definition 4.2.** Given a sheaf over , the -th cohomology group of the cochain complex is called the -th *cohomology group* *of* *with coefficients in* and is denoted by , that is,

| (4.3) |

for all .

If the coboundary homomorphism is denoted by for with , then (4.3) means that

In particular we have

**Definition 4.3.** (i) A sheaf over a topological space is *flabby* if for any open set the restriction map is surjective.

(ii) A sheaf over a topological space is *soft* if for any closed set the restriction map is surjective.

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

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

*for all* *.*

*Proof.* See [18, Theorem 3.11].

**Definition 4.5.** A resolution of a sheaf over of the form

is said to be *acyclic* if for all and .

Let be a sheaf of abelian groups over , and let

| (4.4) |

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

Thus we can consider the cohomology groups of the cochain complex .

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

*for all* *.*

*Proof.* See [18, Theorem 3.13].

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

*Proof.* Let be a closed subset of , and consider an element . Then can be extended to a neighborhood of . Define an element satisfying for and for . Since is soft, can be extended to an element . Then is an extension of .

Let be a vector space over , and let with be the sheaf of -valued -forms on a differentiable manifold . Let be the sheaf obtained by modifying in Example 4.1(i) by using and singular -cochains. We consider the corresponding graded sequences and of sheaves over . Then the -th *singular cohomology group* and the -th *de Rham cohomology group* with coefficients in are defined by

for each . On the other hand, if with as in Example 4.1(iii), then the *Dolbeault cohomology group* of of type is defined by

for .

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

*for all* *, where* *denotes the* *-th cohomology group* *of* *with coefficients in the constant sheaf* *.*

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

*for all* *with* *, where* *is the sheaf of* *holomorphic* *-forms on* *.*

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

Using the argument of the partition of unity, it can be shown that and are soft sheaves. Since the sheaf is a module over for each , it follows from Lemma 4.7 that is soft. Thus, using Theorem 4.4 and Theorem 4.6, we see that

Similarly, each is soft; hence we have

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

4.2. **De Rham cohomology and vector bundles.** Let be a manifold, and let be the universal covering space of . Let be the fundamental group of , so that can be identified with the quotient space .

Let be a representation of in a finite-dimensional real vector space , and define an action of on by

| (4.5) |

for all and . We equip the real vector space with the Euclidean topology and denote by

| (4.6) |

the quotient of with respect to the -action in (4.5). Then the natural projection map induces a surjective map such that the diagram

| (4.7) |

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

**Proposition 4.9.** *The set* *has the structure of a locally constant* *vector bundle over* *with fiber* *whose fibration is the map* *in* (4.7).

*Proof.* Let be an open cover of such that the inverse image of each under is homeomorphic to . By taking smaller open sets if necessary we may assume that is either connected or empty for all . For each we select a connected component of . If , then there exists a unique element such that

| (4.8) |

We define the map by

| (4.9) |

for all , where is the element of with . Then we see easily that is a bijection. We shall now introduce a vector space structure on each fiber with . Given , we define the map by

| (4.10) |

for all . Then is bijective, and therefore we can define a vector space structure on by transporting the one on via the map . We need to show that such a structure is independent of . Let . If and are the elements with . Then from (4.8) we see that . Using this and the relations (4.5), (4.9) and (4.10), we obtain

for all . Hence we see that the diagram

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

can be used as a local trivialization for each .

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

| (4.11) |

where denotes the tangent space of at and is the fiber of at . We then define, for each , the function on which associates to each an -valued alternating -linear map given by

| (4.12) |

where .

**Definition 4.10.** A *-valued* *-form* on is a function on which assigns to each an alternating -linear map of the form (4.11) such that the function in (4.12) is differentiable.

Let be an open cover of . Noting that is locally constant by Proposition 4.9, we denote by the constant transition function on for . Then a -valued -form on can be regarded as a collection of -valued -forms on satisfying

on for all with . Since each is constant, we have

hence the collection determines a -valued -form on . Thus, if denotes the space of all -valued -forms on , the map determines an operator

| (4.13) |

with for each . Then the *de Rham cohomology of* *with coefficients in* is the cohomology of the cochain complex with the coboundary operator (4.13). Thus the quotient

| (4.14) |

for is the -th de Rham cohomology of with coefficients in .

4.3. **Hecke operators on de Rham cohomology.** Let , , , and the representation be as in Section 4.2. Given , the space of all -valued -forms on is spanned by the elements of the form with and . By setting

we obtain the map with ; hence we can consider the associated cochain complex whose cohomology is the de Rham cohomology of with coefficients in . By Theorem 4.8 there is a canonical isomorphism

for each . This isomorphism can be described more explicitly as follows. Given , the group of -cochains considered in Theorem 4.8 can be written as

where is the group of -chains. Thus each element of is a finite sum of the form with , where each is a map from a -simplex in a Euclidean space to . To each -form we set

| (4.15) |

for . If with , the Stokes theorem implies that

Thus the map is well-defined map on the set of -cycles in and therefore is an element of . On the other hand, if with , then we have

hence the map is a well-defined map from to , and according to Theorem 4.8 this map is an isomorphism.

For each , we set

| (4.16) |

Then we see that

hence we obtain the cochain complex . If the -th cohomology group for this complex is denoted by , then the next proposition shows that it can be identified with the -th de Rham cohomology group with coefficients in .

**Proposition 4.11.** *There is a canonical isomorphism*

| (4.17) |

*for each* *, where* *is as in* (4.14)*.*

*Proof.* Let and be the canonical projection maps as in the commutative diagram (4.7). Given , we define the map by

for all . Then for and we have

hence we see that

| (4.18) |

If , we define the element by

for all and . Using this and (4.18), we have

for all , which implies that . Now we see easily that the map determines an isomorphism between and ; hence the lemma follows.

We now want to introduce Hecke operators on , which by Proposition 4.11 may be regarded as Hecke operators on . Let denote the commensurator of as in Section 3.3, and consider an element such that the double coset has a decomposition of the form

| (4.19) |

for some elements . Given a -form , we denote by the -form defined by

| (4.20) |

**Lemma 4.12.** *If* *, then* *for* *each* *.*

*Proof.* Given an element satisfying (4.19) and , let be an element of such that

for some element as in (3.21), so that the set is a permutation of . If , then by (3.21), (4.16) and (4.20) the -form satisfies

for all ; hence it follows that .

By Lemma 4.12 for each there is a linear map

However, since commutes with , the same operator induces the operator

| (4.21) |

on . Thus, using the canonical isomorphism (4.17), we obtain the operator

for each , which is a *Hecke operator* on determined by .

**5. Cohomology with local coefficients**

In this section we discuss the cohomology of a topological space with coefficients in a system of local groups as well as Hecke operators acting on such cohomology. Section 5.1 contains the description of a system of local groups associated to a representation of the fundamental group of in a finite-dimensional real vector space. When is a differentiable manifold, we show that the cohomology of with coefficients in the sheaf of sections of is canonically isomorphic to the de Rham cohomology of the universal covering space of associated to introduced in Section 4.3. In Section 5.2 we discuss the homology and cohomology of with coefficients in a general system of local groups. We introduce Hecke operators in Section 5.3 acting on de Rham cohomology of with coefficients in the vector bundle considered in Section 4.2.

5.1. **Local systems.** Let be an arcwise connected topological space with fundamental group , and let be its universal covering space. Thus can be identified with the quotient space . Given , we denote by the homotopy class of curves from to . The homotopy class containing the inverse of a curve belonging to is denoted by , and the symbol denotes the homotopy class obtained by traversing first a path in the class followed by a path in the class . We fix a base point , and denote the class simply by . We also use to denote the class of closed paths.

**Definition 5.1.** A *system of local groups* on is a collection of groups for satisfying the following conditions:

(i) For each class of paths in there is an isomorphism .

(ii) If the transform of under the isomorphism in (i) is denoted by , then we have for all and .

The group , where is the base point of , will be denoted simply by . Then each element determines an endomorphism of ; hence acts on on the right.

Let be a representation of in a finite-dimensional real vector space . We denote by the vector space equipped with the discrete topology, and set

where the quotient is taken with respect to the action in (4.5) with replaced with . Then the natural projection map induces a surjective map .

**Proposition 5.2.** *For each* *, let* *be the fiber* *of* *over* *. Then the space* *, regarded as the collection* *of its fibers is a system of local groups on* *.*

*Proof.* For each the fiber of over is isomorphic to the discrete additive group . There exist an open covering of and a homeomorphism

| (5.1) |

for each such that for all and induces an isomorphism for each . If , since is totally disconnected, any curve from to determines uniquely an isomorphism which depends only on the homotopy class of (see [16, Section 13]). Thus the collection is a system of local groups on .

We now assume that is a differentiable manifold and denote by the vector bundle over given by (4.6), where is equipped with the Euclidean topology. We denote by the sheaf of germs of -valued -forms on . If denotes the space of sections of , we obtain the cochain complex whose coboundary map

is induced by the exterior differentiation map. Since the natural isomorphism

commutes with , it determines a canonical isomorphism

| (5.2) |

for each .

**Proposition 5.3.** *Let* *be the sheaf of germs of continuous sections* *of the local system* *in Proposition* *5.2**. Then for each* *there* *are canonical isomorphisms*

*between the* *-th cohomology group* *of the complex* *and the* *-th cohomology group* *of* *with* *coefficients in* *.*

*Proof.* The second isomorphism was proved in Proposition 4.11. As for the first isomorphism, by using the Poincaré lemma it can be shown that the sheaf is locally constant and that the sequence

is exact. Hence by Theorem 4.6 there is a canonical isomorphism

Thus the lemma follows from this and (5.2).

5.2. **Homology and cohomology with local coefficients.** Let , and be as in Section 5.1, so that can be identified with the quotient space . We consider a local system on . If is a Euclidean simplex and is a singular -simplex in , we set . Since the leading vertex of the -th face for coincides with that of , we see that . For , however, the 0-th face has as its leading vertex and is not connected with . In this case the leading edge

| (5.3) |

of is a path in from to and yields an isomorphism of onto .

Given and a path from to , we define an isomorphism by

| (5.4) |

for all . We assume that the groups are topological and that the isomorphisms of onto are continuous.

We now introduce a cochain complex defined as follows. Given , a -cochain on over belonging to is a function which assigns an element to each singular -simplex in . We define the homomorphism by

| (5.5) |

for each -simplex and , where is as in (5.3). Then it can be shown that the homomorphism satisfies and therefore is a coboundary map for the cochain complex . Thus we obtain the associated -th cohomology group

of with coefficients in .

Let be a base point, so that the fundamental group of can be written as , and set . Then is an abelian group, and by (5.4) the group acts on on the left. Let be the singular complex in , and for each let denote the group of singular -chains in . Then is the free abelian group generated by the singular -simplexes in , and there is a boundary map given by

for a singular -simplex in associated to a Euclidean singular -simplex . Then the group of singular -cochains with coefficients in is given by

and its coboundary operator is defined by

for all and . By (3.10) a -cochain is equivariant with respect to if

for all and . We denote by the subgroup of consisting of the equivariant cochains. If is the coboundary map, by (3.12) the equivariant -th singular cohomology group of over is given by

where denotes the kernel of the map .

**Theorem 5.4.** *There is a canonical isomorphism*

*for each* *.*

*Proof.* In this proof we shall regard an element as the homotopy class of paths in joining the base point with , where is the natural projection map. Then for each the element is well-defined, and we have

for each . If , we define the cochain by

| (5.6) |

for all -simplex in , where is the leading vertex of . Since , the element belongs to , and therefore is a cochain belonging to . Thus (5.6) determines a homomorphism . If denotes the coboundary map for , then by using (5.6) we have

where is the -th face of and is the leading vertex of . Since is the leading vertex of , for we see that . For , however, we have , where is the leading edge of the simplex in . Hence we obtain

where we used (5.5) and (5.6). Thus we see that . We shall now show that the map is an isomorphism. First, if is a nonzero element of , then for some -simplex in . Hence there is a simplex in such that and , and therefore is injective. To consider the surjectivity of we note that, if is a -simplex in with leading vertex and if , then is the leading vertex of and

hence the cochain is equivariant. Now, let be an equivariant -cochain in over . Given a -simplex in , we choose a -simplex in with and consider the element of , where is the leading vertex of . If is replaced by with , then we have

where we used the fact that is equivariant. Hence the element of is independent of the choice of . We now define a cochain by

Then we see that

and therefore , which implies the surjectivity of . We have thus shown that is an isomorphic mapping of the group of cochains onto the group of equivariant cochains. Since in addition , it follows that and are mapped isomorphically onto the groups and , respectively. This proves that the map determines the isomorphism .

5.3. **Hecke operators.** Let , and be as in Section 5.1, and let be the vector bundle over given by (4.6) associated to a representation of in a finite-dimensional vector space over .

We consider another manifold , where is the fundamental group and is the universal covering space of . Let be a group homomorphism, and let be a map that is equivariant with respect to , which means that

for all and . Then induces a map . We now define an action of on by

for all , and . Then the corresponding quotient space

| (5.7) |

is a vector bundle over with fiber , whose fibration is induced by the natural projection map . The next lemma shows that this bundle is essentially the same as the vector bundle over obtained by pulling back via .

**Lemma 5.5.** *The bundle* *over* *in* (5.7) *is canonically* *isomorphic to the pullback bundle* *.*

*Proof.* We note that the pullback bundle over is given by

| (5.8) |

where is the fibration for the bundle . We introduce the notations

for the respective natural projection maps. Then (5.8) can be written in the form

Since and , the condition is equivalent to the relation for some . Using this and the fact that , we see that

Thus we may define a map by

for all and . If , we have

hence is a well-defined surjective map. To verify the injectivity of we consider elements and satisfying

Then we have

for some . Thus we obtain

and therefore is injective and the proof of the lemma is complete.

Let be the -th cohomology group for the cochain complex for each considered in Section 4.2. If denotes the dual space of , then we have the natural identification

where denotes the contragredient of .

We assume that is a smooth -sheeted covering map for some positive integer . Then the associated pull-back map determines the homomorphism

of cohomology groups. On the other hand, according to the Poincaré duality, there are canonical isomorphisms

where denotes the dual space of . Then the *Gysin map* associated to is the linear map

such that the diagram

is commutative; here is the dual of the linear map

Thus the Gysin map is characterized by the condition

| (5.9) |

for all and . In order to discuss Hecke operators on we now consider a pair of smooth -sheeted covering maps .

**Definition 5.6.** For the *Hecke operator on* associated to the pair is the map

given by

| (5.10) |

We now consider the case where is equal to and is a subgroup of with . Let be the set of coset representatives of in , so that we have

| (5.11) |

If , we shall use the same symbol to denote either the map

sending to or the map

which it induces.

**Theorem 5.7.** *Let* *be covering maps, and assume* *that* *is induced by the identity map on* *. Then for* *the associated Hecke operator on* *is given by*

*for all* *.*

*Proof.* First, we shall determine for , where is the inclusion map. By (5.9) we have

for all . Let and with be the differential forms on corresponding to and , respectively. If and are fundamental domains of and , respectively, then by (5.11) the domain can be written as a disjoint union of the form

Using this and the fact that the lifting of is the identity map on , we have

Thus, if denotes the -form on defined by

| (5.12) |

then we have

We now need to show that is an element of . Given and a positive integer , using (5.11), we see that

for some . Thus there is an element such that

Using this and (5.12), we see that

which shows that belongs to . Hence it follows that is the element of corresponding to under the canonical isomorphism (4.17). Therefore we obtain

for all , and hence the proof of the theorem is complete.

**6. Compatibility of Hecke operators**

The goal of this section is to establish the compatibility among the Hecke operators acting on various types of cohomology groups. Given a discrete group acting on a Riemannian symmetric space and a representation of in a finite-dimensional vector space , Section 6.1 describes the canonical isomorphisms among the de Rham cohomology of associated to , the cohomology of with coefficients in , and the equivariant singular cohomology of . The compatibility between Hecke operators on singular cohomology and the ones on de Rham cohomology is discussed in Section 6.2. In Section 6.3 it is shown that the Hecke operators on the de Rham cohomology and those on the group cohomology are compatible under the canonical isomorphism obtained in Section 6.1.

6.1. **De Rham, singular and group cohomology.** Let be a Riemannian symmetric space, and let be a discrete group acting on properly discontinuously. We regard the associated quotient space as a subset of consisting of the set of representatives of the orbits of in . We shall review relations between singular and group cohomology discussed by Eilenberg in [4].

Let be a singular -simplex in , where is a Euclidean simplex with ordered vertices . Then the vertices of in can be written uniquely in the form

| (6.1) |

for some . Let be the singular complex in as in Section 5.2, and let be the complex for the cohomology of the group considered in Section 3.2. If is as in (6.1), we define to be the -cell of given by

Thus maps the singular simplexes in into cells of and induces a homomorphism

| (6.2) |

of groups of -chains. We also see that

| (6.3) |

for all .

**Definition 6.1.** (i) A -cell in is said to be *basic* if its first vertex is the identity element of , that is, if for some .

(ii) A simplex in is called *basic* if its leading vertex is one of the points in , where is regarded as a subset of consisting of the set of representatives of -orbits in as above.

**Lemma 6.2.** *If* *, then for each integer* *with* *there is a homomorphism*

| (6.4) |

*satisfying*

| (6.5) |

*for all* *with* *.*

*Proof.* First, we choose a point , and define by

for all , where is the basic 0-cell. Then we see that

for ; hence satisfies (6.5). In order to define for by induction, we assume that the maps have been defined for all with and they satisfy

| (6.6) |

for all . Given a basic cell in , its image under is an integral chain in of dimension , and therefore by the first condition in (6.6) it is a cycle in , that is,

Since the space is acyclic in dimensions less than and , there is an element of , which we denote by , such that

We now obtain the homomorphism

by extending the map to all the -chains belonging to .

**Lemma 6.3.** *Given integers* *and* *with* *and* *, there exist homomorphisms*

*satisfying the relations*

| (6.7) |

| (6.8) |

*for all* *,* *and* *.*

*Proof.* Given a basic 0-cell in , we consider the 0-cycle . Since is a point in and since is pathwise connected, there is an integral 1-chain such that . We extend this to the nonbasic 0-cells by

for all , and use induction for general as follows. Assume that has been defined for all -cells with and that the relations in (6.7) hold. Given a basic -simplex of , consider the -chain

in . Then we have

and hence the chain is an -cycle in . Since , there is an -chain such that . We now extend the map to all the -chains in including nonbasic ones by using the first condition in (6.7). The construction of can be obtained in a similar manner.

**Theorem 6.4.** *Let* *be a topological space that is acyclic in dimensions* *less than* *, and let* *be a group acting on* *without fixed points. If* *is a left* *-module, then we have*

| (6.9) |

*for* *, and*

| (6.10) |

*where the homomorphism* *is from the exact sequence* (3.14) *for the* *complex* *.*

*Proof.* Since the map in (6.2) satisfies (6.3), it induces the homomorphism

| (6.11) |

for each . From (6.7) and (6.8) we see that the maps and are chain homotopic to the identity maps and , respectively. Since the maps , and are equivariant, for the homomorphisms in (6.11) are isomorphisms. Hence we obtain (6.9) by combining the isomorphism with the relation (3.17). In order to prove (6.10) we consider the commutative diagram

induced by and the exact sequence (3.14). Since (see [4, p. 47]), the map is an isomorphism. On the other hand, is injective because . Using the relation and the fact that both and are isomorphisms, we see that is injective and has the same image as . However, we have

hence we obtain (6.10).

Let be a representation of in a finite-dimensional vector space over as in Section 5.3, so that can be regarded as a left -module.

**Proposition 6.5.** *There is a canonical isomorphism*

| (6.12) |

*for each* *.*

*Proof.* If with is as in (4.15), then by Theorem 4.8 the map determines an isomorphism between and . On the other hand, if , we have

for all and . Thus it follows that is equivariant, and therefore the map determines an isomorphism (6.12).

**Corollary 6.6.** *If* *is contractible, there is a canonical isomorphism*

*for each* *, where* *is regarded as a* *-module via the* *representation* *.*

*Proof.* This follows from the isomorphisms (6.9) and (6.12).

6.2. **Singular and de Rham cohomology.** Let be a semisimple Lie group, and let be the associated symmetric space, which can be identified with the quotient of by a maximal compact subgroup. Let be a discrete subgroup of , and let be the associated locally symmetric space. Let be a representation of in a finite-dimensional real vector space .

Given , the group of -cochains in with coefficients in can be written as

where denotes the group of -chains in . If with

| (6.13) |

we define the map by

| (6.14) |

for all and . Since clearly commutes with the boundary operator for the complex , it induces the map

| (6.15) |

which is the *Hecke operator* on the -th singular cohomology group with coefficients in .

**Lemma 6.7.** *The map* *given by* (6.14) *sends* *-equivariant* *-cochains to* *-equivariant* *-cochains.*

*Proof.* Let be an element of such that the corresponding double coset has a decomposition given by (6.13). Then, as in (3.21), for each and there are elements and such that

| (6.16) |

Furthermore, the set is a permutation of for each . Let and , where is the subspace of consisting of -equivariant cochains. Then, since is -equivariant, we have

for all . Using this, (6.14) and (6.16), we obtain

for all . Thus it follows that .

Given , we denote by the subgroup of defined by

| (6.17) |

and set

We assume that and that

| (6.18) |

with . Then by Lemma 2.4 we have

| (6.19) |

If , then ; hence for each we have

Thus it follows that , and therefore the map , induces a map . However, since , there is another map induced by the identity map on . Thus the maps and are -sheeted covering maps of , and by Definition 5.6 they determine the Hecke operator on for each . By identifying with using the canonical isomorphism (4.17) we obtain the Hecke operator

for each . On the other hand, by Lemma 6.7 the Hecke operator (6.15) induces the Hecke operator

on the -th equivariant cohomology group with coefficients in . We denote by

the canonical isomorphism (6.12).

**Theorem 6.8.** *Given* *and* *, we have*

*for all* *, where* *denotes the cohomology class of* *in* *.*

*Proof.* Let , and let be an element of satisfying (6.13) and (6.19). Then, using (6.14) and Theorem 5.7, we have

for all ; hence the theorem follows.

6.3. **De Rham and group cohomology.** Let , , and the representation of a discrete subgroup of in be as in Section 6.2.

Let be the group of singular -chains as in Section 5.2, and let be a singular -simplex belonging to , where is a Euclidean simplex with ordered vertices . Then, as in Section 6.1, the vertices of can be written uniquely in the form

for some , where is regarded as a subset of consisting of representatives of the orbits of . Given a -form on , we define the associated map by

| (6.20) |

for all , where is as in (6.4).

| (6.21) |

*for all* *.*

*Proof.* Given , using the construction of in the proof of Lemma (4.8), we see easily that

for all . Hence by using (4.16) we obtain

for all .

By Lemma 6.9 the map given by (6.20) is a homogeneous -cochain for the cohomology of described in Section 3.1. Thus we have

for all , where is regarded as a -module via the representation . We denote by and the coboundary maps for the complexes and , respectively.

**Lemma 6.10.** *The map* *given by* (6.20) *satisfies*

*for all* *.*

*Proof.* Given and , using (6.20), we see that

where we used the relation from (6.5). However, since the boundary operator and the coboundary operator are given by (3.15) and (3.2), respectively, we have

hence the lemma follows.

By Lemma 6.10 the map given by (6.20) induces the canonical isomorphism

| (6.22) |

for each .

**Lemma 6.11.** *Let* *, and let* *be a* *-cycle such that* *, where* *is as in* (6.2)*. Then we have*

*for all closed* *-forms* *.*

*Proof.* If , by using (6.7) we see that

where we used the fact that is a cycle. Thus the formula (6.20) can be written in the form

since is a closed form.

Let be a reductive group containing , and let be the commensurability group of . Given and , let be the Hecke operator on group cohomology described in Section 3.3. Let be the Hecke operator in (4.21), which may be regarded as a Hecke operator on by using the canonical isomorphism

considered in (4.17).

**Theorem 6.12.** *Let* *be the isomorphism in* (6.22)*. Then we have*

*for all* *.*

*Proof.* Assume that the double coset containing has a decomposition of the form

for some elements . If and , as was described in (3.21), we have

for some element , where is a permutation of . Since is a -module via the representation , the formula (3.24) can be written in the form

for each -cocycle and . Thus we have

| (6.23) |

for all . We now fix a point and choose the set of representatives of -orbits in such a way that

| (6.24) |

Then, if is as in (6.2), we see that

for . From this and Lemma 6.11, we obtain

Using this, (6.23), and the relation for , we have

where for each . However, since is a permutation of , the condition (6.24) implies that for each . Hence we have

and therefore it follows that

Thus we obtain , and the proof of the theorem is complete.

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**Min Ho Lee**** ** Department of Mathematics,

University of Northern Iowa,

Cedar Falls, IA 50614, U.S.A.

lee@math.uni.edu

**Recibido**: 13 de marzo de 2008

**Aceptado**: 28 de mayo de 2009