versión impresa 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. 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