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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. 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. [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 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 [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.
[1] A. Andrianov, Quadratic forms and Hecke operators, Springer-Verlag, Heidelberg, 1987. [ Links ]
[2] P. Bayer and J. Neukirch, On automorphic forms and Hodge theory, Math. Ann. 257 (1981), 135-155. [ Links ]
[3] A. Borel, Introduction to automorphic forms, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, 1966, pp. 199-210. [ Links ]
[4] S. Eilenberg, Homology of spaces with operators. I, Trans. Amer. Math. Soc. 61 (1947), 378-417. [ Links ]
[5] H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge Univ. Press, Cambridge, 1993. [ Links ]
[6] M. Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties I, II, Univ. of Chicago, Chicago, 1963/64. [ Links ]
[7] M. Kuga, Group cohomology and Hecke operators. II. Hilbert modular surface case, Adv. Stud. Pure Math., vol. 7, North-Holland, Amsterdam, 1985, pp. 113-148. [ Links ]
[8] M. Kuga, W. Parry, and C. H. Sah, Group cohomology and Hecke operators, Progress in Math., vol. 14, Birkhäuser, Boston, 1981, pp. 223-266. [ Links ]
[9] M. Kuga and J. Sampson, A coincidence formula for locally symmetric spaces, Amer. J. Math. 94 (1972), 486-500. [ Links ]
[10] M. H. Lee, Mixed automorphic forms, torus bundles, and Jacobi forms, Lecture Notes in Math., vol. 1845, Springer-Verlag, Berlin, 2004. [ Links ]
[11] J.-S. Li and J. Schwermer, Automorphic representations and cohomology of arithmetic groups, Challenges for the 21st century (Singapore, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 102-137. [ Links ]
[12] T. Miyake, Modular forms, Springer-Verlag, Heidelberg, 1989. [ Links ]
[13] Y. H. Rhie and G. Whaples, Hecke operators in cohomology of groups, J. Math. Soc. Japan 22 (1970), 431-442. [ Links ]
[14] I. Satake, Algebraic structures of symmetric domains, Princeton Univ. Press, Princeton, 1980. [ Links ]
[15] G. Shimura, Introduction to the arithmetic theory automorphic functions, Princeton Univ. Press, Princeton, 1971. [ Links ]
[16] N. Steenrod, The topology of fiber bundles, Princeton Univ. Press, Princeton, 1951. [ Links ]
[17] B. Steinert, On annihilation of torsion in the cohomology of boundary strata of Siegel modular varieties, Bonner Mathematische Schriften, vol. 276, Universität Bonn, Bonn, 1995. [ Links ]
[18] R. O. Wells, Jr., Differential analysis on complex manifolds, Springer-Verlag, Heidelberg, 1980. [ Links ]
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