Print version ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 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. 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 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. , , ) 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. ). Such automorphic forms are also linked to families of abelian varieties parametrized by the locally symmetric space (cf. , , ). 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. ). Hecke operators are certain averaging operators acting on the space of automorphic forms (cf. , , ), 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. , , , ). Hecke operators on the cohomology of more general groups were also investigated by Rhie and Whaples in . 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  (see also ). 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 , 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 ,  and .
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
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 .
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 .
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
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
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
then we define its operation on by
for all .
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
with , we set
where the summation is over the set of representatives of the double cosets contained in and
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
Thus by using (2.9) again, we obtain
hence it follows that
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 .
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
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
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
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
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 .
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 .
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.
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 . 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 .
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
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.
for all and .
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
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
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
Since is generated by the -cells, a -cochain is uniquely determined by its values for the -cells . The coboundary operator
on is defined by
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
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
of . Then the quotient group
is the equivariant -th cohomology group of over .
We denote by the subgroup of consisting of the cochains with an equivariant coboundary, that is,
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. ) that there is an exact sequence of the form
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
for . We define the left action of the group acts on by
for all and . Thus, if acts on an abelian group on the left, then we can consider the equivariant cohomology groups of over .
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
for each and each generator of . By (3.9) the coboundary map is given by
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
for some . Since for each , we have
for all . Thus for , we see that
for some element , where is a permutation of . For each and we have
Comparing this with , we see that
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
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
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 .
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).
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. ).
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.  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
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
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.
for all .
If the coboundary homomorphism is denoted by for with , then (4.3) means that
In particular we have
(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 .
for all .
Proof. See [18, Theorem 3.11].
is said to be acyclic if for all and .
Let be a sheaf of abelian groups over , and let
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].
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 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.
Let be a representation of in a finite-dimensional real vector space , and define an action of on by
for all and . We equip the real vector space with the Euclidean topology and denote by
the quotient of with respect to the -action in (4.5). Then the natural projection map induces a surjective map such that the diagram
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
We define the map by
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
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
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
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
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
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
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
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 .
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
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
for some elements . Given a -form , we denote by the -form defined by
Proof. Given an element satisfying (4.19) and , let be an element of such that
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
on . Thus, using the canonical isomorphism (4.17), we obtain the operator
for each , which is a Hecke operator on determined by .
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.
(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 .
Proof. For each the fiber of over is isomorphic to the discrete additive group . There exist an open covering of and a homeomorphism
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
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
of is a path in from to and yields an isomorphism of onto .
Given and a path from to , we define an isomorphism by
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
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 .
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
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 .
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
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
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
for all and . In order to discuss Hecke operators on we now consider a pair of smooth -sheeted covering maps .
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
If , we shall use the same symbol to denote either the map
sending to or the map
which it induces.
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
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.
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 .
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
Thus maps the singular simplexes in into cells of and induces a homomorphism
of groups of -chains. We also see that
for all .
(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.
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
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 .
satisfying the relations
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.
for , and
where the homomorphism is from the exact sequence (3.14) for the complex .
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.
for each .
for all and . Thus it follows that is equivariant, and therefore the map determines an isomorphism (6.12).
for each , where is regarded as a -module via the representation .
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
we define the map by
for all and . Since clearly commutes with the boundary operator for the complex , it induces the map
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.
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 . Thus it follows that .
Given , we denote by the subgroup of defined by
We assume that and that
with . Then by Lemma 2.4 we have
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
on the -th equivariant cohomology group with coefficients in . We denote by
the canonical isomorphism (6.12).
for all , where denotes the cohomology class of in .
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
for all , where is as in (6.4).
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 .
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
hence the lemma follows.
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
for all . We now fix a point and choose the set of representatives of -orbits in such a way that
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.
Recibido: 13 de marzo de 2008
Aceptado: 28 de mayo de 2009