aa r X i v : . [ m a t h . N T ] S e p BOUNDARY AND EISENSTEIN COHOMOLOGY OF G ( Z ) JITENDRA BAJPAI AND LIFAN GUAN
Abstract.
In this article, Eisenstein cohomology of the arithmetic group G ( Z ) withcoefficients in any finite dimensional highest weight irreducible representation has beendetermined. We accomplish this by studying the cohomology of the boundary of theBorel-Serre compactification. Contents
1. Introduction 12. Preliminaries 33. Cohomology of the boundary components 84. Boundary cohomology 125. Eisenstein cohomology 26Acknowledgements 37References 381.
Introduction
Let G be a semisimple algebraic group defined over Q , K ∞ ⊂ G( R ) be a maximalcompact subgroup and S = G( R ) /K ∞ be the corresponding symmetric space. If Γ ⊂ G( Q )is an arithmetic subgroup then every representation ( ρ, M ) of G defines, in a natural way,a sheaf f M on the locally symmetric space S Γ = Γ \ S. One has the isomorphism(1) H • (Γ , M ) ∼ = H • (S Γ , f M ) , for details see Chapter 7 of [3]. Note that, throughout the paper, we use H • to representthe full cohomology group, namely, H • (Γ , M ) = ⊕ q H q (Γ , M ). On the other hand, letS Γ denote the Borel-Serre compactification of S Γ , then the inclusion i : S Γ ֒ → S Γ , whichis an homotopic equivalence, determines a sheaf i ∗ ( f M ) on S Γ and induce a canonicalisomorphism in the cohomology(2) H • (S Γ , i ∗ f M ) ∼ = H • (S Γ , f M ) , where i ∗ denotes the direct image functor defined by i . On the other hand, let ∂ S Γ = S Γ \ S Γ and j : ∂ S Γ → S Γ be the closed embedding. The following exact sequence of sheaves0 → i ! ( f M ) → i ∗ ( f M ) → j ∗ ( f M ) → Date : September 29, 2020.2010
Mathematics Subject Classification.
Key words and phrases. G , Borel-Serre compactification, Boundary and Eisenstein Cohomology. gives rise to a long exact sequence of cohomology groups associated to S Γ , · · · → H qc (S Γ , f M ) → H q (S Γ , f M ) r −→ H q ( ∂ S Γ , f M ) → · · · . Let boundary cohomology of Γ with coefficients in M denoted by H • ( ∂ S Γ , f M ) and the Eisenstein of cohomology Γ with coefficients in M denoted by H • Eis (S Γ , f M ), to be theimage of the map r . Clearly, there is an exact sequence0 → H q ! (S Γ , f M ) → H q (S Γ , f M ) r −→ H qEis (S Γ , f M ) → , where H q ! (S Γ , f M ) is the kernel of the restriction map r .The study of Eisenstein cohomology was initiated by Harder [6], and found that Eisen-stein cohomology is fundamentally related to several important topics in number theory,e.g., special values of L -functions, extension of motives, to simply mention a few. SeeHarder’s ICM report [8] for more details on the relation of Eisenstein cohomology withother topics. Interested reader is also referred to [10] for recent advances on the subject.Although lots of work have been done, our understanding of Eisenstein cohomology is stillfar from complete.The main purpose of this article is to determine the boundary and Eisenstein coho-mology of the arithmetic group G ( Z ) with coefficients in any finite dimensional highestweight representation M λ of G where λ denotes its highest weight.The Eisenstein cohomology of arithmetic subgroups of Res k/ Q G for totally real field k has been previously studied in [13]. Compared to their work, the basic setting of thispaper is more restrictive, that is we only consider cohomology of the full arithmetic group G ( Z ), but we provide complete results for the boundary and Eisenstein cohomology of G ( Z ) with coefficients in any finite dimensional highest weight representation.On the other hand, it is worth mentioning the paper [1], where the boundary andEisenstein cohomology of SL ( Z ) with coefficients in finite dimensional highest weightrepresentations is determined by using Euler characteristic, which is a fundamental caseto study among other rank two arithmetic groups. Unfortunately, their method, beingelementary but tricky and powerful, does not work here in the case of G ( Z ) for dimensionalreasons.The method we are employing here follows closely the work of Harder in [9]. In particu-lar, our method is constructive that involves the theory of Eisenstein series, intertwining op-erators, ( g , K ∞ )-cohomology and L -functions. Indeed, it is well-known that the Eisensteincohomology H • Eis (S Γ , f M ) spans a maximal isotropic subspace of the boundary cohomology H • ( ∂ S Γ , f M ) under the Poincar´e duality. Hence we are done if we manage to constructenough classes in H • Eis (S Γ , f M ). Starting from cohomology classes from H • ( ∂ S Γ , f M ), weconstruct cohomology classes in H • (S Γ , f M ) by evaluating the corresponding Eisenstein se-ries at certain special point, and then we get non-trivial Eisenstein cohomology classes byrestriction back to the boundary. Subtlety appears when the corresponding Eisenstein se-ries is not holomorphic at the special point, or equivalently, the corresponding L -functionis not holomorphic at the special point. We complete the proof by further exploringthe Hecke action and Poincar´e duality, as well as a detailed study of the corresponding( g , K ∞ )-cohomology.1.1. Main results.
Let us now give the details of the results obtained in this article.
OHOMOLOGY OF G ( Z ) 3 • Theorem 8, where the boundary cohomology with coefficients in every finite di-mensional highest weight representation is described. • Theorem 9, where we describe the Eisenstein cohomology for every finite dimen-sional highest weight representation.1.2.
Overview of the article.
We quickly summarize the content of each section ofthe article. In Section 2, we provide the details of structure of the group G : its Weylgroup, parabolic subgroups, root system and Kostant representatives, which is the firstbasic step to accomplish our goal. In Section 3, the parity conditions for the cohomologyof the boundary components, i.e. the cohomology of the parabolic subgroups have beenestablished which helps us to compute the boundary cohomology, content of Section 4,of the locally symmetric space of the arithmetic group G ( Z ). In Section 5, we achieveour main goal by completing the study of Eisenstein cohomology of G ( Z ) for every finitedimensional highest weight representation.2. Preliminaries
This section quickly review the basic properties of G and familiarize the reader withthe notations to be used throughout the article. We discuss the corresponding locally sym-metric space, Weyl group, the associated spectral sequence and Kostant representativesof the standard parabolic subgroups.2.1. Structure Theory.
Let G be the Chevalley group defined over Z of type G andΦ be the corresponding root system. Let us fix a maximal Q -split torus T and a Borelsubgroup B that contains T. The set of simple roots associated to B is denoted by∆ = { α , α } with α and α be the short and long simple roots respectively. The Weylgroup W of Φ is isomorphic to the dihedral group D . The fundamental weights associatedto this root system are given by γ = 2 α + α and γ = 3 α + 2 α .Let g denote the Lie algebra g and t ⊂ g be the Lie subalgebra associated to T. LetΦ = Φ + ∪ Φ − be the corresponding root system. We know thatΦ + = { α , α , α + α , α + α , α + α , α + 2 α } . Finally, we write ρ = P α ∈ Φ + α = 5 α + 3 α .Recall that a Q -parabolic subgroup is called standard if it contains the Borel subgroupB. Let { P , P , P } be the set of standard Q -parabolic subgroups, where P (resp. P )denotes the maximal Q -parabolic subgroup corresponding to the simple roots α (resp. α ) and P = B. Thus, p = u − α ⊕ t ⊕ α ∈ Φ + u α , and p = u − α ⊕ t ⊕ α ∈ Φ + u α , where p i denotes the Lie algebra of P i and u α denotes the root space corresponding to α . Note that the minimal Q -parabolic P is simply the group P ∩ P . Therefore, thecorresponding Levi quotients are given byM = G m , M = GL and M = GL . Let us choose and fix a maximal compact subgroup K ∞ ⊂ G ( R ). It is well-known thatit can be identified with SO ( R ). From now on throughout the article let S = G ( R ) /K ∞ ,Γ be the arithmetic group G ( Z ) and S Γ = Γ \ S. JITENDRA BAJPAI AND LIFAN GUAN -conjugacy classes of Q -Parabolic Subgroups. in this subsection, we determinethe Γ-conjugacy classes of Q -Parabolic Subgroups P Q ( G , G ( Z )), which should be well-known. But as we don’t know of a proper reference, a proof is given below. Lemma 1. P Q ( G , G ( Z )) = { P , P , P } .Proof. We have to show that for each standard parabolic P, G ( Z ) acts transitively on G / P( Q ). Since G is semisimple algebraic group split over Q , by strong approximation,we have G ( Q ) Y p G ( Z p ) = G ( A f ) , which implies that G ( Q ) /G ( Z ) = G ( A f ) / Q p G ( Z p ). Moreover, as Q is of class number1, we have P( Q ) Y p P( Z p ) = P( Q p ) . According to the Iwasawa decomposition for p -adic groups, P( Q p ) G ( Z p ) = G ( Q p ). Con-sequently, we have P( Q ) Y p G ( Z p ) = G ( A f ) . Hence P( Q ) acts transitively on G ( Q ) /G ( Z ), which implies G ( Z ) acts transitively on G ( Q ) / P( Q ).On the other hand, we have the following sequence of Galois cohomology,1 → P( Q ) → G ( Q ) → G / P( Q ) → H ( Q , P) → H ( Q , G ) → · · · . Note that P is the semi-direct product of its unipotent radical and its Levi subgroup, whichis isomorphic to either G m or GL . According to Hilbert’s Theorem 90 (see, e.g. [20,Chapter III]), we have H ( Q , P) = 1. Thus G / P( Q ) = G ( Q ) / P( Q ), from which thelemma follows. (cid:3) Irreducible Representations.
The fundamental weights associated to Φ + are givenby γ = 2 α + α and γ = 3 α + 2 α . Thus irreducible finite dimensional representationsof G are determined by their highest weights, which in this case are the linear functionalsof the form m γ + m γ with m , m non-negative integers. For any λ = m γ + m γ ,we set M λ to be the representation defined over Q with highest weight λ .2.4. Kostant Representatives.
It is known that the Weyl group W = W (Φ) is thedihedral group D given by 12 elements. They are listed in the first column of Table 1 anddescribed in the second column as a product of simple reflections s and s , associated tothe simple roots α and α respectively. Then we have s ( α ) = − α , s ( α ) = 3 α + α and s ( α ) = α + α , s ( α ) = − α . (3)In the third column we make a note of their lengths and in the last column we describethe element w · λ = w ( λ + ρ ) − ρ , where the pair ( a, b ) denotes the element aα + bα ∈ t ∗ .The Weyl group acts naturally on the set of roots. For each i ∈ { , , } , let ∆( u i )denote the set consisting of every root whose corresponding root space is contained in the OHOMOLOGY OF G ( Z ) 5 Label w ℓ ( w ) w · λw m + 3 m , m + 2 m ) w s m + 3 m − , m + 2 m ) w s m + 3 m , m + m − w s s m − , m + m − w s s m + 3 m − , m − w s s s − m − , m − w s s s m − , − m − w s s s s − m − m − , − m − w s s s s − m − , − m − m − w s s s s s − m − m − , − m − m − w s s s s s − m − m − , − m − m − w s s s s s s − m − m − , − m − m − Table 1.
The Weyl Group of G Lie algebra u i of the unipotent radical of P i . The set of Weyl representatives W P i ⊂ W associated to the parabolic subgroup P i (see [12]) is defined by W P i = (cid:8) w ∈ W : w (Φ − ) ∩ Φ + ⊂ ∆( u i ) (cid:9) . Clearly W P = W and, by using the table, one can see that W P = { , s , s s , s s s , s s s s , s s s s s } (4) W P = { , s , s s , s s s , s s s s , s s s s s } . (5)2.4.1. Kostant representatives for minimal parabolic P . w · λ = m γ + m γ w · λ = ( − m − γ + ( m + m + 1) γ w · λ = ( m + 3 m + 3) γ + ( − m − γ w · λ = ( − m − m − γ + ( m + 2 m + 2) γ w · λ = (2 m + 3 m + 4) γ + ( − m − m − γ w · λ = ( − m − m − γ + ( m + 2 m + 2) γ w · λ = (2 m + 3 m + 4) γ + ( − m − m − γ w · λ = ( − m − m − γ + ( m + m + 1) γ w · λ = ( m + 3 m + 3) γ + ( − m − m − γ w · λ = ( − m − m − γ + m γ w · λ = m γ + ( − m − m − γ w · λ = ( − m − γ + ( − m − γ Kostant representatives for maximal parabolic P . Here we take γ M = 12 α = γ − γ and κ M = 12 γ . JITENDRA BAJPAI AND LIFAN GUAN w · λ = m γ M + ( m + 2 m ) κ M w · λ = ( m + 3 m + 3) γ M + ( m + m − κ M w · λ = (2 m + 3 m + 4) γ M + ( m − κ M w · λ = (2 m + 3 m + 4) γ M + ( − m − κ M w · λ = ( m + 3 m + 3) γ M + ( − m − m − κ M w · λ = m γ M + ( − m − m − κ M Kostant representatives for maximal parabolic P . Here we take γ M = 12 α = − γ + γ and κ M = 12 γ w · λ = m γ M + (2 m + 3 m ) κ M w · λ = ( m + m + 1) γ M + ( m + 3 m − κ M w · λ = ( m + 2 m + 2) γ M + ( m − κ M w · λ = ( m + 2 m + 2) γ M + ( − m − κ M w · λ = ( m + m + 1) γ M + ( − m − m − κ M w · λ = m γ M + ( − m − m − κ M For i = 1 ,
2, we also write(6) w · λ = a i ( λ, w ) γ M i + b i ( λ, w ) κ M i . The symmetry of the coefficients can be explained by the following lemma.
Lemma 2.
Let P be a parabolic subgroup of G with Levi subgroup M , A P be the centraltorus of M and S P be the unique maximal torus of the semisimple part of M contained in T . Let N P denote the unipotent radical of P in G and w G (resp. w M ) be the longestWeyl element in W (resp. W M ). Then the following are true. (a) The map w w ′ := w M ww G defines an involution on W P and ℓ ( w ) + ℓ ( w ′ ) =dim N P . (b) w ′ ( λ + ρ ) − ρ | A P = w ( λ + ρ ) − ρ | A P . (c) w ′ ( λ + ρ ) − ρ | S P + w ′ ( λ + ρ ) − ρ | S P = − ρ | S P .Proof. The proof is the same as in Schwermer [19, Section 4.2] by noticing the fact thatthe Weyl group W is self-dual, namely w G = − (cid:3) Boundary of the Borel-Serre compactification.
In general, the boundary of theBorel-Serre compactification ∂ S Γ = S Γ \ S Γ is a finite union of the boundary components ∂ P for P ∈ P Q (G , Γ), i.e.(7) ∂ S Γ = [ P ∈P Q (G , Γ) ∂ P , where ∂ P = (Γ ∩ P( R )) \ P( R ) / (P( R ) ∩ K ∞ ) . OHOMOLOGY OF G ( Z ) 7 This decomposition (7) determines a spectral sequence in cohomology abutting to thecohomology of the boundary(8) E p,q = M prk (P)= p +1 H q ( ∂ P , f M λ ) ⇒ H p + q ( ∂ S Γ , f M λ )where prk (P) denotes the parabolic rank of P (the dimension of the maximal Q -split torusin the center of the Levi quotient M of P).In our case, as the Q -rank of G is 2, this spectral sequence is simplified to a long exactsequence (of Mayer-Vietoris) in cohomology of the following form(9) · · · → H q − ( ∂ P , f M λ ) → H q ( ∂ S Γ , f M λ ) → H q ( ∂ P , f M λ ) ⊕ H q ( ∂ P , f M λ ) → H q ( ∂ P , f M λ ) → · · · by noticing Lemma 1. We will use this long exact sequence to describe the cohomology ofthe boundary of the Borel-Serre compactification.2.6. The theorem of Kostant.
Let P = MN be the decomposition into its Levi subgroupM and unipotent radical N. Then, ∂ P is a fibre bundle overS MΓ = (Γ ∩ M( R )) \ M( R ) / (M( R ) ∩ K ∞ )with fibres isomorphic to N Γ := (Γ ∩ N( R )) \ N( R ). Hence we have H • ( ∂ P , f M λ ) = H • (S MΓ , H • (N Γ , f M λ )) , here we use H • (N Γ , f M λ ) to denote the corresponding sheaf by abuse of notation. By thetheorem of Nomizu [15], we know H • (N Γ , f M λ ) = H • ( n , M λ ) . By the theorem of Kostant [12], we get H q ( n , M λ ) = M w ∈W P : ℓ ( w )= q N w · λ , where N w · λ denotes the irreducible representation of M with highest weight w · λ . Inconclusion, we get H q ( ∂ P , f M λ ) = M w ∈W P H q − ℓ ( w ) (S MΓ , e N w · λ ) . Note that, when there is no ambiguity, we also use the old notation M w · λ to denote N w · λ .2.7. Cohomological dimension.
For any discrete group H , set the virtual cohomologicaldimension of H , denoted as vcd H , to bevcd H = min { cd H ′ : [ H : H ′ ] < ∞} , where cd H ′ refers to the cohomological dimension of H ′ . Now, using the compactificationthey had introduced, Borel and Serre showed in [2] that for any semisimple group G andits arithmetic subgroup H : vcd H = dim G − dim K − rank Q G , where K is the maximal compact subgroup of G( R ). In particular, dim G = 14, dim SO ( R ) =6 and rank Q G = 2, thus we have vcd G ( Z ) = 6. As a consequence, H q (S Γ , f M ) = 0 forall q > M of Γ. JITENDRA BAJPAI AND LIFAN GUAN
Remark 3.
It is interesting to note that, for any Q -split semisimple group G, we havevcdG( Z ) = dim N = max w ∈W ℓ ( w ) , where N is the unipotent radical of any Borel subgroup and ℓ ( w ) is the length of w . Indeed,the first equality follows from the Iwasawa decomposition, while the second one followsfrom theory of Weyl groups. Note that this does not hold in general. For example: whenG is Q -anisotropic, vcdG( Z ) equals to dim G − dim K , which is not the maximal length ofWeyl elements. 3. Cohomology of the boundary components
The cohomology of the boundary is obtained by using a spectral sequence whose termsare expressed by the cohomology of the faces associated to each standard parabolic sub-group. In this section we establish, for each standard parabolic P and irreducible represen-tation M ν of the Levi subgroup M ⊂ P with highest weight ν , a condition to be satisfiedin order to have nontrivial cohomology H • (S MΓ , f M ν ). Here f M ν is the sheaf on S MΓ givenby M ν .3.1. Minimal Parabolic Subgroup.
We analyze the parity condition imposed to theface associated to the minimal parabolic ∂ P . As mentioned, the Levi subgroup of P isthe two dimensional torus M ∼ = G m . The elements lying in Ξ := M ( Z ) ∩ K ∞ must acttrivially on the representation M ν in order to have nonzero cohomology. By using thisfact one can deduce the following Lemma 4.
Let ν be given by m ′ γ + m ′ γ . If m ′ or m ′ is odd then the correspondinglocal system f M ν in S M Γ is cohomological trivial, i.e. H • (S M Γ , f M ν ) = 0 .Proof. According to [10, Prop 4.3], we have H • (S M Γ , f M ν ) = H • ( g S M Γ , f M ν ) Ξ , where g S M Γ denotes the locally symmetric spaces associated to M . As g S M Γ is simply apoint, all the higher cohomology vanishes. It is clear that Ξ ∼ = ( Z / Z ) and for each ξ ∈ Ξ the action on the M is given by ν ( ξ ) ∈ {− , } . Thus, if there exists ξ ∈ Ξ with ν ( ξ ) = −
1, then H (S M Γ , f M ν ) = 0. On the other hand, since γ , γ forms an integralbasis for X ∗ (M ) := Hom(M , G m ), there exists ξ ∈ Ξ with ν ( ξ ) = − m ′ or m ′ is odd.This completes the proof. (cid:3) Note that every ν will be of the form w · λ for w ∈ W . We denote by W the set ofWeyl elements w such that w · λ do not satisfy the condition of Lemma 4. Remark 5.
For notational convenience, we simply use ∂ i to denote the boundary face ∂ P i associated to the parabolic subgroup P i and the arithmetic group Γ for i ∈ { , , } . OHOMOLOGY OF G ( Z ) 9 Cohomology groups of ∂ . In this case H q (S M Γ , f M w · λ ) = 0 for every q ≥
1. TheWeyl group W P = W and the lengths of its elements are between 0 and 6 as shown inthe Table 1 above. We know that H q ( ∂ , f M λ ) = M w ∈W P0 H q − ℓ ( w ) (S M Γ , f M w · λ )(10) = M w ∈W P0 : ℓ ( w )= q H (S M Γ , f M w · λ )Therefore H ( ∂ , f M λ ) = H (S M Γ , f M λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s · λ ) ⊕ H (S M Γ , f M s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s · λ ) ⊕ H (S M Γ , f M s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s · λ ) ⊕ H (S M Γ , f M s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s · λ ) ⊕ H (S M Γ , f M s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s s · λ ) ⊕ H (S M Γ , f M s s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s s s · λ )and for every q ≥
7, the cohomology groups H q ( ∂ , f M λ ) = 0.3.2. Maximal Parabolic Subgroups.
In this section we study the parity conditions forthe maximal parabolics. Let i ∈ { , } , then M i ∼ = GL and in this setting, K ∞ ∩ M i ( R ) =O ( R ) is the orthogonal group and Γ M i = GL ( Z ). ThereforeS M i Γ = GL ( Z ) \ GL ( R ) / O(2) R × > . We also consider the following double cover of S M i Γ , g S M i Γ = GL ( Z ) \ GL ( R ) / SO(2) R × > , which is isomorphic to the locally symmetric space associated to M i .Let i be 1 or 2. Recall from (6) that, for w ∈ W P i , w · λ = a i ( λ, w ) γ M i + b i ( λ, w ) κ M i . Lemma 6. If a i ( λ, w ) is odd, or equivalently, b i ( λ, w ) is odd as they are congruent modulo , we have H • (S M i Γ , f M w · λ ) = 0 . Moreover, if a i ( λ, w ) = 0 and b i ( λ, w ) / is odd, then H • (S M i Γ , f M w · λ ) = 0 .Proof. The proof here follows the proof of Lemma 4 closely. According to [10, Prop 4.3],we have H • (S M i Γ , f M w · λ ) = H • ( g S M i Γ , f M w · λ ) Ξ i , where Ξ i = M i ( Z ) ∩ K ∞ . Note that M i can be identified with GL , and K ∞ ∩ M i ( R )equals to O ( R ), hence Ξ i can be identified as GL ( Z ) ∩ O ( R ). For the element (cid:18) − − (cid:19) ∈ Ξ i , the action on H • ( g S M i Γ , f M ν ) is given by ( − a i ( λ,w ) , hence we get the first conclusion. Onthe other hand, consider the element (cid:18) − (cid:19) ∈ Ξ i , whose action on H • ( g S M i Γ , f M ν ) is given by ( − b i ( λ,w ) / when a = 0. This completes theproof. (cid:3) Note that, as a representation of M i ∼ = GL ( i = 1 , M w · λ ∼ = Sym a i ( λ,w ) V ⊗ Det bi ( λ,w ) − ai ( λ,w )2 , where V denotes the standard representation of GL . Let B ⊂ M i be a standard Borelsubgroup and N be its unipotent radical with n its Lie algebra. Then for i = 1 ,
2, we havethe following exact sequence, H (S M Γ , H ( n , M w · λ )) ֒ → H c (S M i Γ , M w · λ ) → H (S M i Γ , M w · λ ) ։ H (S M Γ , H ( n , M w · λ )) . Here, by abuse of notation, we use H ( n , M w · λ ) and H ( n , M w · λ ) to denote the corre-sponding sheaves. In view of Lemma 4 and (11), we get H (S M Γ , H ( n , M w · λ )) = 0 , if b i ( λ, w ) − a i ( λ, w )2 = 0 mod 2 ,H (S M Γ , H ( n , M w · λ )) = 0 , if b i ( λ, w ) − a i ( λ, w )2 = 0 mod 2 . Consequently, we have H (S M i Γ , M w · λ ) = H (S M i Γ , M w · λ ) , if b i ( λ, w ) − a i ( λ, w )2 = 0 mod 2 , (12) H (S M i Γ , M w · λ ) = H c (S M i Γ , M w · λ ) , if b i ( λ, w ) − a i ( λ, w )2 = 0 mod 2 . (13)This fact will be extensively used in the next section for the computation of boundarycohomology.Throughout the paper, we set(14) S m +2 = H (GL , Sym m V ) ∼ = H (GL , Sym m V ⊗ Det ) . It is well-known that H • ! (GL , M ) = H • cusp (GL , M ) for finite dimensional representation M , hence it is safe to replace “!” by “ cusp ” in (14). Consequently, by Eichler-Shimuraisomorphism, the space S m +2 can be identified with the space of holomorphic cusp formsof weight m + 2. Remark 7.
For notational convenience, in what follows we will denote the set of Weylelements for which H • (S M i Γ , f M w · λ ) = 0 by W i . In the following subsections we make note of the cohomology groups associated tothe boundary components ∂ and ∂ which will be used in the computations involved todetermine the boundary cohomology in the next section. OHOMOLOGY OF G ( Z ) 11 Cohomology of ∂ . In this case, the Levi M is isomorphic to GL and therefore H q (S M Γ , f M w · λ ) = 0 for every q ≥
2. The Weyl group W P = { e, s , s s , s s s , s s s s , s s s s s } where the length of elements are respectively 0 , , , , ,
5. Thus, H q ( ∂ , f M λ ) = M w ∈W P1 H q − ℓ ( w ) (S M Γ , f M w · λ )(15) = H q (S M Γ , f M λ ) ⊕ H q − (S M Γ , f M s · λ ) ⊕ H q − (S M , f M s s · λ ) ⊕ H q − (S M Γ , f M s s s · λ ) ⊕ H q − (S M Γ , f M s s s s · λ ) ⊕ H q − (S M Γ , f M s s s s s · λ ) . Therefore, H ( ∂ , f M λ ) = H (S M Γ , f M λ ) H ( ∂ , f M λ ) = H (S M Γ , f M λ ) ⊕ H (S M Γ , f M s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s · λ ) ⊕ H (S M Γ , f M s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s · λ ) ⊕ H (S M Γ , f M s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s · λ ) ⊕ H (S M Γ , f M s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s · λ ) ⊕ H (S M Γ , f M s s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s s · λ )and for every q ≥
7, the cohomology groups H q ( ∂ , f M λ ) = 0.3.2.2. Cohomology of ∂ . In this case, the Levi M is isomorphic to GL as well and there-fore H q (S M Γ , f M w · λ ) = 0 for every q ≥
2. The Weyl group W P = { e, s , s s , s s s , s s s s , s s s s s } where the length of elements are respectively 0 , , , , ,
5. Thus, H q ( ∂ , f M λ ) = M w ∈ W P2 H q − ℓ ( w ) (S M Γ , f M w · λ )(16) = H q (S M , f M λ ) ⊕ H q − (S M Γ , f M s · λ ) ⊕ H q − (S M Γ , f M s s · λ ) . ⊕ H q − (S M Γ , f M s s s · λ ) ⊕ H q − (S M Γ , f M s s s s · λ ) ⊕ H q − (S M Γ , f M s s s s s · λ ) . Therefore, H ( ∂ , f M λ ) = H (S M Γ , f M λ ) H ( ∂ , f M λ ) = H (S M Γ , f M λ ) ⊕ H (S M Γ , f M s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s · λ ) ⊕ H (S M Γ , f M s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s · λ ) ⊕ H (S M Γ , f M s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s · λ ) ⊕ H (S M Γ , f M s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s · λ ) ⊕ H (S M Γ , f M s s s s s · λ ) H ( ∂ , f M λ ) = H (S M Γ , f M s s s s s · λ ) and for every q ≥
7, the cohomology groups H q ( ∂ , f M λ ) = 0.4. Boundary cohomology
In this section, we discuss the cohomology of the boundary by giving complete descrip-tion of the spectral sequence. The covering of the boundary of the Borel-Serre compacti-fication defines a spectral sequence in cohomology. E p,q = M prk ( P )=( p +1) H q ( ∂ P , f M λ ) ⇒ H p + q ( ∂ S Γ , f M λ )and the nonzero terms of E p,q are for(17) ( p, q ) ∈ { ( i, n ) | ≤ i ≤ , ≤ n ≤ } . More precisely, E ,q = M i =1 H q ( ∂ i , f M λ )= M i =1 (cid:20) M w ∈W P i H q − ℓ ( w ) (S M i Γ , f M w · λ ) (cid:21) , (18) E ,q = H q ( ∂ , f M λ )= M w ∈W P0 : ℓ ( w )= q H (S M Γ , f M w · λ ) . Since G is of rank two, the spectral sequence has only two columns namely E ,q , E ,q and to study the boundary cohomology, the task reduces to analyze the following mor-phisms(19) E ,q d ,q −−→ E ,q where d ,q is the differential map and the higher differentials vanish. One has E ,q := Ker ( d ,q ) and E ,q := Coker ( d ,q ) . In addition, due to be in rank 2 situation, the spectral sequence degenerates in degree2. Therefore, we can use the fact that(20) H k ( ∂ S Γ , f M λ ) = M p + q = k E p,q . In other words, let us now consider the short exact sequence0 −→ E ,q − −→ H q ( ∂ S Γ , f M λ ) −→ E ,q −→ . (21) OHOMOLOGY OF G ( Z ) 13 Case 1 : m = 0 and m = 0 (trivial coefficient system). Following Lemma 4and Lemma 6 from Section 3, we get W = { w , w , w , w } , W = { w , w , w } and W = { w , w , w } . By using (18) we record the values of E ,q and E ,q for the distinct values of q below.Note that following (17) we know that for q ≥ E i,q = 0 for i = 1 , E ,q = H (S M Γ , f M e · λ ) ⊕ H (S M Γ , f M e · λ ) , q = 0 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 40 , otherwise , and(23) E ,q = H (S M Γ , f M e · λ ) , q = 0 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 60 , otherwise . We now make a thorough analysis of (19) to get the complete description of the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ). We begin with q = 0.4.1.1. At the level q = 0 . Observe that the short exact sequence (21) reduces to0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ . To compute E , , consider the differential d , : E , → E , . Following (22) and (23),we have d , : Q ⊕ Q −→ Q and we know that the differential d , is surjective (see [6]).Therefore E , := Ker ( d , ) = Q and E , := Coker ( d , ) = 0 . (24)Hence, we get H ( ∂ S Γ , f M λ ) = Q . At the level q = 1 . Following (24), in this case, our short exact sequence (21) reducesto 0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . Consider the differential d , : E , −→ E , and follow-ing (22) and (23), we observe that d , is map between zero spaces. Therefore, we obtain E , = 0 and E , = 0 . As a result, we get H ( ∂ S Γ , f M λ ) = 0 . At the level q = 2 . Following the similar process as in level q = 1, we get E , = 0 and E , = 0 . (25)This results into H ( ∂ S Γ , f M λ ) = 0 . At the level q = 3 . Following (25), in this case, the short exact sequence (21) reducesto 0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . In view of (13), consider the differential d , : E , −→ E , and following (22) and (23), we have E , = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , and Im ( d , ) = H Eis (S M Γ , f M w · λ ) ⊕ H Eis (S M Γ , f M w · λ ) ∼ = Q ⊕ Q . Therefore, E , = 0 and E , = 0 . (26)This gives us H ( ∂ S Γ , f M λ ) = 0 . At the level q = 4 . Following (26), in this case, the short exact sequence (21) reducesto 0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . Consider the differential d , : E , −→ E , and follow-ing (22) and (23), we have E , = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ). Since Im ( d , ) = H Eis (S M Γ , f M w · λ ) ⊕ H Eis (S M Γ , f M w · λ ) = { } by (12). Therefore, E , = 0 and E , = 0 . (27)and we get H ( ∂ S Γ , f M λ ) = 0 . At the level q = 5 . Following (27), in this case, the short exact sequence (21) reducesto 0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . Consider the differential d , : E , −→ E , and follow-ing (22) and (23), we have d , : 0 −→
0. Therefore, E , = 0 and E , = 0 . (28)and we get H ( ∂ S Γ , f M λ ) = 0 . OHOMOLOGY OF G ( Z ) 15 At the level q = 6 . Following (28), in this case, the short exact sequence (21) reducesto 0 −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . Consider the differential d , : E , −→ E , and follow-ing (22) and (23), we have d , : 0 −→ Q . Therefore, E , = 0 and E , = Q . (29)and we get H ( ∂ S Γ , f M λ ) = 0 . At the level q = 7 . Following (29), in this case, the short exact sequence (21) reducesto 0 −→ Q −→ H ( ∂ S Γ , f M λ ) −→ E , −→ , and we need to compute E , . Consider the differential d , : E , −→ E , and follow-ing (22) and (23), we have d , : 0 −→
0. Therefore, E , = 0 and E , = 0 . and we get H ( ∂ S Γ , f M λ ) = Q . Hence, we can summarize the above discussion as follows : H q ( ∂ S Γ , f M λ ) = Q , q = 0 , , otherwise . Case 2 : m = 0 , m = 0 , m even. Following Lemma 4 and Lemma 6 fromSection 3, we get W = { w , w , w , w } , W = { w , w , w } and W = { w , w , w , w } . By using (18) we record the values of E ,q and E ,q for the distinct values of q below.Note that following (17) we know that for q ≥ E i,q = 0 for i = 1 , E ,q = H (S M Γ , f M e · λ ) , q = 0 H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M e · λ ) , q = 0 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 60 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M λ ) ∼ = S m +2 , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ∼ = S m +6 ⊕ S m +4 , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ∼ = S m +6 ⊕ S m +4 , q = 4 H (S M Γ , f M w · λ ) ∼ = S m +2 , q = 60 , otherwise . Here S k is defined as in (14).We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +6 ⊕ S m +4 , q = 3 , , otherwise . Case 3 : m = 0 , m odd. Following Lemma 4 and Lemma 6 from Section 3, weget W = { w , w , w , w } , W = { w , w , w } and W = { w , w , w , w } . OHOMOLOGY OF G ( Z ) 17 By using (18) we record the values of E ,q and E ,q for the distinct values of q below. E ,q = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M w · λ ) ∼ = Q , q = 1 H (S M Γ , f M w · λ ) ∼ = Q , q = 2 H (S M Γ , f M w · λ ) ∼ = Q , q = 4 H (S M Γ , f M w · λ ) ∼ = Q , q = 50 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ Q , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ Q , q = 50 , otherwise . We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +5 ⊕ S m +3 ⊕ Q , q = 2 , S m +4 , q = 3 , , otherwise . Case 4 : m = 0 even and m = 0 . Following Lemma 4 and Lemma 6 fromSection 3, we get W = { w , w , w , w } , W = { w , w , w , w } and W = { w , w , w } . By using (18) we record the values of E ,q and E ,q for the distinct values of q below. E ,q = H (S M Γ , f M λ ) , q = 0 H (S M Γ , f M λ ) ⊕ H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M λ ) ∼ = Q , q = 0 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ∼ = Q , q = 3 H (S M Γ , f M w · λ ) ∼ = Q , q = 60 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) , q = 60 , otherwise . OHOMOLOGY OF G ( Z ) 19 We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +6 ⊕ S m +4 , q = 3 , , otherwise . Case 5 : m ( = 0) even, m ( = 0) even. Following Lemma 4 and Lemma 6 fromSection 3, we get W = { w , w , w , w } , W = { w , w , w , w } and W = { w , w , w , w } . By using (18) we record the values of E ,q and E ,q for the distinct values of q below. E ,q = H (S M Γ , f M λ ) ⊕ H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M λ ) ∼ = Q , q = 0 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ∼ = Q , q = 3 H (S M Γ , f M w · λ ) ∼ = Q , q = 60 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M λ ) ⊕ H (S M Γ , f M λ ) ⊕ Q , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ Q , q = 60 , otherwise . We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +2 ⊕ S m +2 ⊕ Q , q = 1 , S m +3 m +6 ⊕ S m +2 m +4 , q = 3 , , otherwise . Case 6 : m ( = 0) even, m odd. Following Lemma 4 and Lemma 6 from Section 3,we get W = { w , w , w , w } , W = { w , w , w , w } and W = { w , w , w , w } . By using (18) we record the values of E ,q and E ,q for the distinct values of q below. E ,q = H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M w · λ ) , q = 1 H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ∼ = Q , q = 50 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described as OHOMOLOGY OF G ( Z ) 21 follows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise . We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +3 m +5 ⊕ S m + m +3 , q = 2 , S m +2 m +4 , q = 3 , , otherwise . Case 7: m odd, m = 0 . Following Lemma 4 and Lemma 6 from Section 3, we get W = { w , w , w , w } , W = { w , w , w } and W = { w , w , w , w } . Note that this is dual to case 3. By using (18) we record the values of E ,q and E ,q forthe distinct values of q below. E ,q = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M w · λ ) ∼ = Q , q = 1 H (S M Γ , f M w · λ ) ∼ = Q , q = 2 H (S M Γ , f M w · λ ) ∼ = Q , q = 4 H (S M Γ , f M w · λ ) ∼ = Q , q = 50 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ Q , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) ⊕ Q , q = 50 , otherwise . We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +5 ⊕ S m +3 ⊕ Q , q = 2 , S m +6 , q = 3 , , otherwise . Case 8 : m odd, m ( = 0) even. Following Lemma 4 and Lemma 6 from Section 3,we get W = { w , w , w , w } , W = { w , w , w , w } and W = { w , w , w , w } . OHOMOLOGY OF G ( Z ) 23 Note that this is dual to case 6. By using (18) we record the values of E ,q and E ,q forthe distinct values of q below. E ,q = H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise , and E ,q = H (S M Γ , f M w · λ ) ∼ = Q , q = 1 H (S M Γ , f M w · λ ) ∼ = Q , q = 2 H (S M Γ , f M w · λ ) ∼ = Q , q = 4 H (S M Γ , f M w · λ ) ∼ = Q , q = 50 , otherwise . Having a thorough analysis of (19) as in previous section, we get the complete descriptionof the spaces E ,q and E ,q which will give us the cohomology H q ( ∂ S Γ , f M λ ) described asfollows : H q ( ∂ S Γ , f M λ ) = H (S M Γ , f M λ ) , q = 1 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 2 H (S M Γ , f M w · λ ) , q = 3 H (S M Γ , f M w · λ ) , q = 4 H (S M Γ , f M w · λ ) ⊕ H (S M Γ , f M w · λ ) , q = 5 H (S M Γ , f M w · λ ) , q = 60 , otherwise . We conclude the above discussion as follows : H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +3 m +5 ⊕ S m + m +3 , q = 2 , S m +3 m +6 , q = 3 , , otherwise . Case 9: m odd, m odd. By checking the parity conditions for standard parabolics,following Lemmas 4 and 6, we see that W i = ∅ for i = 0 , ,
2. This simply implies that H q ( ∂ S Γ , f M λ ) = 0 , ∀ q . A summary of the boundary cohomology of G ( Z ) . We now end this sectionby summarizing the results obtained about the boundary cohomology above in the form offollowing theorem which will be our base for further exploration on Eisenstein cohomologyin Section 5.
Theorem 8.
The boundary cohomology of the locally symmetric space S Γ of the arithmeticgroup Γ := G ( Z ) with respect to the coefficients in any highest weight representation M λ ,with λ = m λ + m λ , is described as follows. (1) Case 1 : m = 0 = m . H q ( ∂ S Γ , f M λ ) = Q , q = 0 , , otherwise . (2) Case 2 : m = 0 , m ( = 0) even. H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +6 ⊕ S m +4 , q = 3 , , otherwise . (3) Case 3 : m = 0 , m odd. H q ( ∂ S Γ , f M λ ) = S m +5 ⊕ S m +3 ⊕ Q , q = 2 , S m +4 , q = 3 , , otherwise . (4) Case 4 : m ( = 0) even, m = 0 . OHOMOLOGY OF G ( Z ) 25 H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +6 ⊕ S m +4 , q = 3 , , otherwise . (5) Case 5 : m ( = 0) even, m ( = 0) even. H q ( ∂ S Γ , f M λ ) = S m +2 ⊕ S m +2 ⊕ Q , q = 1 , S m +3 m +6 ⊕ S m +2 m +4 , q = 3 , , otherwise . (6) Case 6 : m ( = 0) even, m odd. H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +3 m +5 ⊕ S m + m +3 , q = 2 , S m +2 m +4 , q = 3 , , otherwise . (7) Case 7 : m odd, m = 0 . H q ( ∂ S Γ , f M λ ) = S m +5 ⊕ S m +3 ⊕ Q , q = 2 , S m +6 , q = 3 , , otherwise . (8) Case 8 : m odd, m ( = 0) even. H q ( ∂ S Γ , f M λ ) = S m +2 , q = 1 , S m +3 m +5 ⊕ S m + m +3 , q = 2 , S m +3 m +6 , q = 3 , , otherwise . (9) Case 9 : m odd, m odd. H q ( ∂ S Γ , f M λ ) = 0 , ∀ q . Eisenstein cohomology
In this section, by using the information obtained about boundary cohomology of Γ := G ( Z ), we will determine the Eisenstein cohomology with coefficients in M λ . Let us recallthat, at any degree q , the Eisenstein cohomology H qEis (S Γ , f M λ ) is, by definition, the theimage of the restriction map r : H q (S Γ , f M λ ) −→ H q ( ∂ S Γ , f M λ ).5.1. Main result on the Eisenstein cohomology of G ( Z ) . The following is one ofthe main result of this article, that gives both the dimension of the Eisenstein cohomologytogether with its sources - the corresponding parabolic subgroups.Indeed, it is clear from the definition that the Eisenstein cohomology H • Eis (S Γ , f M λ ) isdefined over Q as f M λ is defined over Q . But in the theorem stated below, we will consider H • Eis (S Γ , f M λ ⊗ C ) instead. The reason is that, our method yields a basis of H • Eis (S Γ , f M λ ⊗ C ), but since the method is transcendental, the basis we get is not necessarily defined over Q . Let Σ k be the canonical basis of normalized eigenfunctions of S k . For i = 1 ,
2, weset k i ( λ, w ) = a i ( λ, w ) + 2, where a i ( λ, w ) is the constant defined in (6). Then by theEichler-Shimura isomorphism, for i = 1 ,
2, we have H (S M i Γ , f M w · λ ⊗ C ) = M ψ ∈ Σ ki ( λ,w ) H (S M i Γ , f M w · λ ⊗ C )( ψ ) , where the C -vector spaces H (S M i Γ , f M w · λ ⊗ C )( ψ ) are of dimension 1. Set Z k = { ψ ∈ Σ k : L (1 / , π ψ ) = 0 } and Y k = { ψ ∈ Σ k : L (1 / , Sym π ψ ) = 0 } . Theorem 9. (1)
Case 1 : m = 0 = m . H qEis (S Γ , f M λ ⊗ C ) = C for q = 00 otherwise . (2) Case 2 : m = 0 , m ( = 0) even. H qEis (S Γ , f M λ ⊗ C ) = L ψ ∈Y m C ψ , q = 3 S m +6 ⊕ L ψ / ∈Y m C ψ , q = 4 S m +2 , q = 60 , otherwise . (3) Case 3 : m = 0 , m odd. OHOMOLOGY OF G ( Z ) 27 H qEis (S Γ , f M λ ⊗ C ) = L ψ ∈Y m C ψ , q = 3 L ψ / ∈Y m C ψ , q = 4 S m +5 ⊕ S m +3 ⊕ C , q = 50 , otherwise . (4) Case 4 : m ( = 0) even, m = 0 . H qEis (S Γ , f M λ ⊗ C ) = L ψ ∈Z m C ψ , q = 3 L ψ / ∈Z m C ψ ⊕ S m +4 , q = 4 S m +2 , q = 60 , otherwise . (5) Case 5 : m ( = 0) even, m ( = 0) even. H qEis (S Γ , f M λ ⊗ C ) = S m +3 m +6 ⊕ S m +2 m +4 , q = 4 S m +2 ⊕ S m +2 ⊕ C , q = 60 , otherwise . (6) Case 6 : m ( = 0) even, m odd. H qEis (S Γ , f M λ ⊗ C ) = S m +2 m +4 , q = 4 S m +3 m +5 ⊕ S m + m +3 , q = 5 S m +2 , q = 60 , otherwise . (7) Case 7 : m odd, m = 0 . H qEis (S Γ , f M λ ⊗ C ) = L ψ ∈Z m C ψ , q = 3 L ψ / ∈Z m C ψ , q = 4 S m +5 ⊕ S m +3 ⊕ C , q = 50 , otherwise . (8) Case 8 : m odd, m ( = 0) even. H qEis (S Γ , f M λ ⊗ C ) = S m +3 m +6 , q = 4 S m +3 m +5 ⊕ S m + m +3 , q = 5 S m +2 , q = 60 , otherwise . (9) Case 9 : m odd, m odd. H qEis (S Γ , f M λ ) = 0 , ∀ q . Now, the proof of Theorem 9 will occupy the rest of the paper, and will follow in severalsteps. The proof closely follow the strategy developed in [9] (see also [13], [16] and [17]).5.2.
General Strategy.
We now briefly describe the strategy which will be detailed outin the rest of this section. As aforementioned, our approach relies on the fact that theEisenstein cohomology spans a maximal isotropic subspace of the boundary cohomologywith respect to the Poincar´e dual pairing (see Theorem 10). Indeed, certain cohomologyclasses are constructed using the theory of Eisenstein series, we are done if the cohomologyclasses thus constructed spans a maximal isotropic subspace.More precisely, let ω be a harmonic differential form that represents certain cohomologyclass in the boundary cohomology H q ! ( ∂ i , f M λ )( i = 0 , , E ( ω, θ ) on S Γ , where θ is acertain parameter, will be discussed in Section 5.4. If the differential forms E ( ω, θ ) isholomorphic at certain θ ω , we get a non-trivial harmonic form E ( ω, θ ω ) that representscertain cohomology class in H q (S Γ , f M λ ) on S Γ . By restricting the harmonic form back tothe boundary, we get a non-trivial Eisenstein cohomology class in H qEis ( ∂ S Γ , f M λ ). Thisgeometric formulation is closely related to the theory of ( g , K ∞ )-cohomology and theclassical Eisenstein series. In particular, the restriction of the cohomology class E ( ω, θ ω )to the boundary can be computed using the constant term of Eisenstein series. On theother hand, if the the differential form E ( ω, θ ) has a simple pole at θ ω (or along somehyperplane that contains θ ω ), by taking residue, we still get a differential form E ′ ( ω, θ ω ),whose restriction to the boundary also gives certain Eisenstein cohomology class. OHOMOLOGY OF G ( Z ) 29 The study of Eisenstein cohomology is basically divided into two parts. In the firstpart, we study the Eisenstein cohomology classes that comes from maximal boundarycomponents, that is, those constructed from cohomology class in H • ! ( ∂ i , f M λ ) with i =1 ,
2. In the second part, we study the Eisenstein cohomology classes that comes fromthe minimal boundary component, that is, those constructed from cohomology class in H • ! ( ∂ , f M λ ).5.3. Poincar´e duality.
For simplicity, we write H q ! ( ∂ S Γ , f M λ ) := H q ! ( ∂ , f M λ ) ⊕ H q ! ( ∂ , f M λ ) ⊂ H q ( ∂ S Γ , f M λ ) , and H q ! ,Eis ( ∂ S Γ , f M λ ) := H qEis (S Γ , f M λ ) ∩ H q ! ( ∂ S Γ , f M λ ) . We shall need the following theorem from [10].
Theorem 10.
Under the Poincar´e dual pairing h· , ·i H q ( ∂ S Γ , f M λ ) × H − q ( ∂ S Γ , f M λ ) → Q , H q ! ( ∂ S Γ , f M λ ) × H − q ! ( ∂ S Γ , f M λ ) → Q , we have H qEis ( ∂ S Γ , f M λ ) = H − qEis ( ∂ S Γ , f M λ ) ⊥ , H q ! ,Eis ( ∂ S Γ , f M λ ) = H − q ! ,Eis ( ∂ S Γ , f M λ ) ⊥ . In particular, the Eisenstein cohomology is a maximal isotropic subspace of the boundarycohomology under the Poincar´e duality.
Let A (resp. A f ) be the ring of adeles (resp. finite adeles) of Q and K f = Q p G ( Z p ).It is clear that the Poincar´e dual pairings are Hecke equivariant, hence Theorem 10 canbe further refined by considering the Hecke action. Now let i = 1 ,
2, the inner cohomology H • ! ( ∂ i , f M λ ), considered as a G ( A f ) module, can be decomposed as H • ! ( ∂ i , f M λ ) = M w ∈W P i H ℓ ( w ) ( S M i Γ , f M w · λ )= M w ∈W P i M π = π ∞ ⊗ π f m ( π ) H ℓ ( w ) ( m i , K M i ∞ , π ∞ ⊗ M w · λ )( π f ) . Here π denotes a cuspidal automorphic representation of M i ( A ) with unramified π f and m ( π ) denotes the multiplicity of π . A cohomology class in H • ! ( ∂ i , f M λ ) is said to be oftype ( π, w ) if it comes from the summand H ℓ ( w ) ( m i , K M i , π ∞ ⊗ M w · λ )( π f ) in the abovedecomposition.Now let β , β ∈ H • ! ( ∂ i , f M λ ) be cohomology classes of type ( π , w ) and ( π , w ) respec-tively. Recall that the Poincare dual pairing is also G ( A f ) equivariant, hence h β , β i = 0if π ,f = π ,f , which implies π = π be strong multiplicity one. On the other hand, fordimension reasons, h β , β i = 0 only when ℓ ( w ) + ℓ ( w ) = 5, which is equivalent to sayingthat w is mapped to w under the involution introduced in Lemma 2. In conclusion, weget Lemma 11.
Let i = 1 , and β , β ∈ H • ! ( ∂ i , f M λ ) be cohomology classes of type ( π , w ) and ( π , w ) respectively. Then h β , β i is nonzero only if π = π and w = w ′ . Let W P i > = { w ∈ W P i : ℓ ( w ) ≥ ℓ ( w ′ ) } . In view of this lemma, we may regroup H • ! ( ∂ i , f M λ ) using parameters w ∈ W P i > as follows: H • ! ( ∂ i , f M λ ) = M w ∈W P i> M ψ ∈ Σ ki ( λ,w ) H • ! ( ∂ i , f M λ )( π ψf , w ) , with H • ! ( ∂ i , f M λ )( π ψf , w ) := H ℓ ( w ′ )! (S M i Γ , f M w ′ · λ )( π ψf ) ⊕ H ℓ ( w )! (S M i Γ , f M w · λ )( π ψf ) , where π ψ denotes the automorphic representation associated to the Hecke eigenform ψ .By the multiplicity one theorem, dim H ℓ ( w )! (S M , f M w · λ )( π ψf ) = 1 for any ( π ψ , w ). Hence,by combining Theorem 10 and Lemma 11, we get Proposition 12.
The Eisenstein cohomology H • ! ,Eis ( ∂ S Γ , f M λ ) decomposes as H • ! ,Eis ( ∂ S Γ , f M λ ) = H • ! ,Eis ( ∂ , f M λ ) ⊕ H • ! ,Eis ( ∂ , f M λ ) , where H • ! ,Eis ( ∂ i , f M λ ) = M w ∈W P i> M ψ ∈ Σ ki ( λ,w ) H ! ,Eis ( ∂ i , f M λ )( π ψ , w ) , with H • ! ,Eis ( ∂ i , f M λ )( π ψ , w ) equals either H ℓ ( w ′ )! (S M i Γ , f M w ′ · λ )( π ψf ) or H ℓ ( w )! (S M i Γ , f M w · λ )( π ψf ) . Eisenstein forms.
Let ω ∈ Ω ∗ ( ∂ i , f M λ )( i = 0 , ,
2) be a differential form on theboundary component ∂ i and e ω ∈ Ω ∗ (Γ P i \ S , f M λ ) be the pull-back of ω along the projectionof Γ P \ S = ∂ i × A P i to the first factor. For any θ ∈ b i := a ∗ P i ⊗ C , set ω θ = e ω × a θ + ρ i ∈ Ω ∗ (Γ P i \ S , f M λ ) , where ρ i = ρ | b i Then we define the corresponding Eisenstein form as E ( ω, θ ) = X γ ∈ Γ / Γ P i ω θ ◦ γ ∈ Ω ∗ (S Γ , f M λ ) . It is well-known that the series E ( ω, θ ) converges in certain region and admits an analyticcontinuation to a meromorphic function on b i .5.5. Constant terms and intertwining operators.
To proceed, we consider the rep-resentation theoretic reformulation of the Eisenstein forms. The complex of smooth formsΩ ∗ (S Γ , f M λ ) can be computed asΩ ∗ (S Γ , f M λ ) = C ∗ ( g , K ∞ ; C ∞ (Γ \ G ( R )) ⊗ M λ )= C ∗ ( g , K ∞ ; C ∞ ( G ( Q ) \ G ( A )) ⊗ M λ ) K f . Consequently, we have H • (S Γ , f M λ ) = H • ( g , K ∞ ; C ∞ ( G ( Q ) \ G ( A )) ⊗ M λ ) K f Now let P i be a standard parabolic, π = π ∞ ⊗ π f be a cuspidal representation of M i ( A )and let θ ∈ b i be a parameter. For ψ θ ∈ V ( θ, π ) := Ind G P i π ⊗ C θ + ρ i , where C θ + ρ i denotes OHOMOLOGY OF G ( Z ) 31 the one-dimensional representation of P i ( A ) given by the character θ + ρ i . Define theEisenstein series as E P i ( θ, π, ψ θ )( g ) = X γ ∈ P i ( Q ) \ G ( Q ) ψ θ ( γg ) . For θ from certain region, the Eisenstein series defined above converges absolutely, hencedefines an intertwining operator E θ : V ( θ, π ) → C ∞ ( G ( Q ) \ G ( A )) , in this region. The Eisenstein series has a meromorphic continuation to b i , hence de-fines a meromorphic continuation of the corresponding intertwining operator. When theEisenstein series has a simple pole at θ , by taking the derivative, we get an intertwiningoperator E ′ θ : V ( θ, π ) → C ∞ ( G ( Q ) \ G ( A )) . It is clear that, for a closed form ω ∈ Ω ∗ ( ∂ i , f M λ ) that represents certain cohomology classof type ( π, w ), the intertwining operator E is holomorphic (resp. have a simple pole) at θ if and only if the corresponding Eisenstein form E ( ω, θ ) is holomorphic (resp. have asimple pole) at θ .To determine whether the Eisenstein series is holomorphic or not, it suffices to look atthe constant terms. First, we consider the case when the parabolic subgroup is maximal.Let Q = M Q N Q be another standard parabolic subgroup. Then the constant term alongQ is defined as E Q ( θ, π, ψ θ )( g ) = Z N Q ( Q ) \ N Q ( A ) E ( θ, π, ψ θ )( ng ) dn. Since the maximal parabolic subgroups of G are self-conjugate, the constant term isnon-trivial only when Q = P i , where we have E P i ( θ, π, ψ θ )( g ) = ψ θ ( g ) + M ( θ, π, w P i ) ψ θ ( g ) , where w P i is the longest element in W P i and M ( θ, π, w P i ) is a global intertwining operatorfrom V ( θ, π ) to V ( − θ, π ). The global intertwining operator M ( θ, π, w P i ) is a product oflocal intertwining operator M ( θ, π, w P i ) = A ( θ, π ∞ , w P i ) ⊗ A ( θ, π f , w P i ) , where A ( θ, π f , w P i ) = ⊗ p A ( θ, π p , w P i ) . Note that if π p is unramified, the induced representation V ( θ, π p ) is also unramified. Worksof Langlands and Gindikin-Karpelevich, see, for example [4], give a description of the localintertwining operator on G ( Z p )-invariant vectors. As a consequence, we have the followingdescription of the global intertwining operator. Lemma 13. (1)
Let i = 1 and θ = zγ ∈ b . Then, M ( θ, π, w P ) = c ( θ, π ) A ( θ, π ∞ , w P ) ⊗ A ′ ( θ, π f , w P ) where A ′ ( θ, π f , w P i ) is the intertwining operator from V ( θ, π f ) := ⊗ p V ( θ, π p ) to V ( − θ, π f ) that sends a normalized K -invariant vector in V ( θ, π f ) to a normalized K -invariant vector in V ( − θ, π f ) and c ( θ, π ) = L ( z, π ) L ( z + 1 , π ) ζ (2 z ) ζ (2 z + 1) L (3 z, π ) L (3 z + 1 , π ) . (2) Let i = 2 and θ = zγ ∈ b . Then, M ( θ, π, w P ) = c ( θ, π ) A ( θ, π ∞ , w P ) ⊗ A ′ ( θ, π f , w P ) where A ′ ( θ, π f , w P ) is the intertwining operator from V ( θ, π f ) to V ( − θ, π f ) thatsends a normalized K -invariant vector in V ( θ, π f ) to a normalized K -invariantvector in V ( − θ, π f ) and c ( θ, π ) = L ( z, Sym π ) L ( z + 1 , Sym π ) ζ (2 z ) ζ (2 z + 1) . Proof.
Recall that the factor of the intertwining operator for G is given by the actionof the adjoint action of the L-group of the Levi component of the uniponent radical.For maximal parabolic subgroup, the Levi is isomorphic to GL . Hence the L-group isisomorphic to GL ( C ). Let V ∼ = C be the standard representation of GL ( C ) of dimension2. As determined in [4], the adjoint action of GL ( C ) on L n and the adjoint action ofGL ( C ) on L n decompose as L n = V ⊗ ∧ V ⊕ V ⊕ ∧ V, L n = Sym V ⊗ ( ∧ V ) − ⊕ ∧ V. Since π is unramified, the corresponding automorphic representation of ∧ π f is trivial.Hence we have the factor c ( z, π ) = L ( z, π ) L ( z + 1 , π ) ζ (2 z ) ζ (2 z + 1) L (3 z, π ) L (3 z + 1 , π ) . and the factor for P is given by c ( z, π ) = L ( z, Sym π ) L ( z + 1 , Sym π ) ζ (2 z ) ζ (2 z + 1) . (cid:3) Now let i = 0. Here we consider the special case when π = C is the trivial representation.For θ = z γ + z γ ∈ b , set V ( θ ) = V ( θ, C ). Then the constant term can be computedas E P ( θ, ψ θ )( g ) = X w ∈ W M ( θ, w )( ψ θ ) , where M ( θ, w ) denotes a global intertwining functor from V ( θ ) to V ( w · θ ). Again, theglobal intertwining operator is a product of local intertwining operators M ( θ, π, w ) = A ( θ, π ∞ , w ) ⊗ A ( θ, π f , w ) , where A ( θ, π f , w ) = ⊗ p A ( θ, π p , w ) . Lemma 14.
Let i = 0 and θ = z γ + z γ . Then M ( θ, π, w ) = c ( θ, w ) A ( θ, π ∞ , w ) ⊗ A ′ ( θ, π f , w ) where A ′ ( θ, π f , w ) is the intertwining operator from V ( θ, π f ) := ⊗ p V ( θ, π p ) to V ( w · θ, π f ) that sends a normalized K -invariant vector in V ( θ, π f ) to a normalized K -invariant vectorin V ( w · θ, π f ) and c ( θ, w ) = Y α ∈ Φ + w − α ∈− Φ + ζ ( h α, γ i ( z + 1) + h α, γ i ( z + 1) − ζ ( h α, γ i ( z + 1) + h α, γ i ( z + 1)) . OHOMOLOGY OF G ( Z ) 33 Proof.
This follows from direct computation, for a quick reference see [9, p. 159] and [7,Section 1.2.4] for the details. (cid:3)
The inner part of the Eisenstein cohomology.
Now we are ready to determinethe space H • ! ,Eis ( ∂ S Γ , f M λ ). We begin with the following lemma. Lemma 15.
Let λ = m γ + m γ and set θ iλ,w := − w ( λ + ρ ) | b i ( i = 1 , . (1) The constant term c ( θ, π ) has a simple pole at θ λ,w if w = w , m = 0 and L (1 / , π ) is non-zero and is holomorphic at θ λ,w otherwise. (2) The constant term c ( θ, π ) has a simple pole at θ λ,w if w = w , m = 0 and L (1 / , Sym π ) is non-zero and is holomorphic at θ λ,w otherwise.Proof. As ρ = γ M + 3 κ M , we have − w ( λ + ρ ) | b = −
12 ( t i ( w, λ ) + 3) γ . Note that, for θ = zγ , c ( θ, π ) = L ( z, π ) L ( z + 1 , π ) ζ (2 z ) ζ (2 z + 1) L (3 z, π ) L (3 z + 1 , π ) . It is well-known that L ( z, π ) appearing here are holomorphic and non-zero at z when ℜ z >
1. Hence in view of Section 2.5, for c ( θ, π ) to have pole, it is necessary to have w = w , m = 0. The possible pole comes from the simple pole of the zeta function at2 z = 1. But the simple pole may be canceled by a possible zero of L ( z, π ) at z = 1 / ρ = γ M + 5 κ M , we have − w ( λ + ρ ) | b = − ( t i ( w, λ ) + 5) γ . Note that for θ = zγ , c ( θ, π ) = L ( z, Sym π ) L ( z + 1 , Sym π ) ζ (2 z ) ζ (2 z + 1) . Since the automorphic representations π considered here are all unramified, the corre-sponding central character is trivial, hence π is not monomial. Then, according to [11],the L -function L ( z, Sym π ) is entire. Hence, in view of Section 2.5, for c ( θ, π ) to havepole it is necessary to have w = w , m = 0. The possible pole comes from the simplepole of the zeta function at 2 z = 1. But the simple pole may be canceled by a possiblezero of L ( z, Sym π ) at z = 1 /
2. This shows the (2). (cid:3)
We shall need the following theorem.
Theorem 16.
Let i = 1 , , π be a cuspidal automorphic representation of M i ( A ) and w ∈W P i > . Let β ∈ H ℓ ( w )! ( ∂ i , f M λ ) be a cohomology class of type ( π, w ) and ω ∈ Ω ∗ ( ∂ i , f M λ ) be a closed harmonic form the represents β . (1) If the Eisenstein series E ( ω, θ ) is holomorphic at θ iλ,w , then E ( ω, θ iλ,w ) ∈ Ω ∗ (S Γ , f M λ ) is a closed form such that the restriction of its cohomology class to the boundary r ([ E ( ω, θ iλ,w )]) ∈ H ℓ ( w )! ( ∂ i , f M λ ) is non-trivial and of type ( π, w ) . (2) If the Eisenstein series E ( ω, θ ) has a simple pole at θ iλ,w , then the residue E ′ ( ω, θ iλ,w ) ∈ Ω ∗ (S Γ , f M λ ) is a closed form such that its restriction to the boundary r ([ E ′ ( ω, θ iλ,w )]) ∈ H ℓ ( w ′ )! ( ∂ i , f M λ ) is non-trivial and of type ( π, w ′ ) . Proof.
When the group G is of Q -rank 1, the corresponding statement are proved in [5].The same proof works when the cohomology classes come from the maximal boundary,hence works for this theorem as well. See also [16]. (cid:3) The subspace of H • ! ( ∂ i , f M λ ) spanned by the cohomology classes of form r ([ E ( ω, θ iλ,w )])appearing in (1) (resp. r ([ E ′ ( ω, θ iλ,w )]) appearing in (2)) is denoted as H • reg ( ∂ i , f M λ ) (resp, H • res ( ∂ i , f M λ )). Then we have the natural inculsion(30) M i =1 , (cid:16) H • reg ( ∂ i , f M λ ) ⊕ H • res ( ∂ i , f M λ ) (cid:17) ⊂ H • ! ,Eis ( ∂ S Γ , f M λ ) . By combining Proposition 12, Lemma 15 and Theorem 16, the above inclusion (30) isindeed an equality. Moreover, we conclude that
Proposition 17.
Let λ = m γ + m γ , w ∈ W P i > ( i = 1 , and ψ ∈ Σ k i ( λ,w ) . Then (1) H • ! ,Eis ( ∂ i , f M λ )( π ψf , w ) = H ℓ ( w ′ )! (S M Γ , f M w ′ · λ )( π ψf ) if w = w , m = 0 , and L (1 / , π ) = 0 ,H ℓ ( w )! (S M Γ , f M w · λ )( π ψf ) otherwise . (2) H • ! ,Eis ( ∂ i , f M λ )( π ψf , w ) = H ℓ ( w ′ )! (S M Γ , f M w ′ · λ )( π ψf ) if w = w , m = 0 , and L (1 / , Sym π ) = 0 ,H ℓ ( w )! (S M Γ , f M w · λ )( π ψf ) otherwise . Note that as the inculsion (30) is an equality, in case H • ( ∂ S Γ , f M λ ) = H • ! ( ∂ S Γ , f M λ ), wehave already determined H • Eis (S Γ , f M λ ). Hence we are left to treat the cases 1, 3, 5 and 7of Theorem 8.5.7. The boundary part of the Eisenstein cohomology.
In this section, we deter-mine the Eisenstein cohomology classes that come from the minimal boundary ∂ . As aconsequence, we determine H • Eis (S Γ , f M λ ) for all the cases left. Throughout this subsec-tion, we assume that H • ( ∂ S Γ , f M λ ) = H • ! ( ∂ S Γ , f M λ ), or equivalently, we are consideringthe cases 1, 3, 5 and 7 of Theorem 8.Let β be a cohomology class in H ( e ∂ , f M λ ) , and ω ∈ Ω ( e ∂ , f M λ ) be a closed harmonicform that represents β . Recall that, as a T-module, we have H ( e ∂ , f M λ ) = H ( g S M Γ , H (N , f M λ )) = C − λ − ρ . The overall idea for construction of Eisenstein cohomology classes is the same as before. Ifthe Eisenstein form E ( ω, θ ) is holomorphic at θ λ := − λ − ρ , then E ( ω, θ λ ) ∈ Ω (S Γ , f M λ ) isa closed form such that the restriction of its cohomology class to the boundary is non-trivial,see [18, Theorem 7.2]. Otherwise, we need to take residues of the Eisenstein form andcompute their restriction to the boundary using the constant term. As before, we denotethe subspace of H • ( ∂ S Γ , f M λ ) spanned the Eisenstein cohomology classes that comes from Here e ∂ denotes the cover of ∂ , which is easily seen to be isomorphic to the unipotent radical N. OHOMOLOGY OF G ( Z ) 35 restriction of Eisenstein form (resp. residues of the Eisenstein form) as H • B,reg ( ∂ S Γ , f M λ )(resp, H • B,res ( ∂ S Γ , f M λ )).For simplicity, set H • B,Eis ( ∂ S Γ , f M λ ) = H • B,reg ( ∂ S Γ , f M λ ) ⊕ H • B,res ( ∂ S Γ , f M λ ) . Then we have the natrual inclusion(31) H • B,Eis ( ∂ S Γ , f M λ ) ⊕ H • ! ,Eis ( ∂ S Γ , f M λ ) ⊂ H • Eis ( ∂ S Γ , f M λ ) . According to Theorem 10, the Eisenstein cohomology H • Eis ( ∂ S Γ , f M λ ) is a maximal isotropicsubspace of boundary cohomology under the Poincar´e duality. In particular,dim H • Eis ( ∂ S Γ , f M λ ) = 12 dim H • ( ∂ S Γ , f M λ ) . On the other hand, according to Theorem 8, we always havedim H • ( ∂ S Γ , f M λ ) − dim H • ! ( ∂ S Γ , f M λ ) = 2 , for the cases studied in this subsection. Hence to determine H • Eis ( ∂ S Γ , f M λ ), it suffices tofill H • B,Eis ( ∂ S Γ , f M λ ) with one non-trivial cohomology class.The following proposition is the main result of this subsection Proposition 18.
Let notations be as in Theorem 8. (1)
In case 1, we have H • Eis ( ∂ S Γ , f M λ ) = H B,res ( ∂ S Γ , f M λ ) = Q . (2) In case 3 and 7, we have H • B,Eis ( ∂ S Γ , f M λ ) = H B,res ( ∂ S Γ , f M λ ) = Q . (3) In case 5, we have H • B,Eis ( ∂ S Γ , f M λ ) = H B,reg ( ∂ S Γ , f M λ ) = Q .Proof. The proof will be based on the computation of ( g , K ∞ )-cohomology of certaininduced modules. To begin, let us start with the proof of part (2). Without loss ofgenerality, we may assume that λ = m γ for some m > c ( θ, w ) = Y α ∈ Φ + w − α ∈− Φ + ζ ( h α, γ i ( z + 1) + h α, γ i ( z + 1) − ζ ( h α, γ i ( z + 1) + h α, γ i ( z + 1)) . Hence, in case of w = s the constant term c ( θ, w ) has a simple pole along z = 0. Hence,the corresponding Eisenstein series has a simple pole along the line z = 0. By taking theresidue along the line z = 0, we get an intertwining operator E ′ : Ind G P C − λ − ρ → C ∞ ( G ( Q ) \ G ( A )) , which factors through the Langlands quotient J λ . More precisely, we have the followingdiagram. Ind G P C − λ − ρ Ind G P C − w ( λ + ρ ) C ∞ ( G ( Q ) \ G ( A )) J λ E ′ ρ A ( − λ − ρ,π ∞ ,s ) φ Note that, J λ = Ind G P C − w ( λ + ρ ) , and the intertwining operator ρ is simply the induction of the intertwining operatorInd M B C − λ − ρ → C − w ( λ + ρ ) , where B denotes the corresponding Borel subgroup of M . Namely, ρ is the mapInd G P Ind M B C − λ − ρ → Ind G P C − w ( λ + ρ ) . By taking cohomology of the map φ , we get H ( g , K ∞ , J λ ⊗ M λ ) φ −→ H (S Γ , f M λ ⊗ C ) . Moreover, the map φ fits into the following diagram. H ( g , K ∞ , J λ ⊗ M λ ) H ( ∂ S Γ , f M λ ⊗ C ) H (S M Γ , f M w · λ ⊗ C ) H (S Γ , f M λ ⊗ C ) Φ φ φ ′ φ ′′ r Here the map φ ′′ is given as in Section 4.3. In view of the constant term expansion, themap Φ is easily seen to be an isomorphism. This completes the proof of part (2).For the proof of part (1) and part (3), the same strategy applies and the proofs areindeed easier. In part (1), the constant term, hence the Eisenstein series, has a doublepole at z = 0 , z = 0. By taking successive residues, the Langlands quotient we getis the constant representation, which provides nontrivial Eisenstein cohomology classes H B,res ( ∂ S Γ , f M λ ). While in case 5, the corresponding Eisenstein series is holomorphic atthe special point θ λ . Hence part (3) can be proved by just taking cohomology of the map E , see [18] for more general cases. (cid:3) OHOMOLOGY OF G ( Z ) 37 Proof of Theorem 9.
Now to finish the proof, we only need to interpret Proposition17 using more concrete terms and then combine it with Proposition 18.According to Proposition 17, when m , m >
0, we always have H • ! ,Eis ( ∂ S Γ , f M λ ) = M q =4 H q ! ( ∂ S Γ , f M λ ) . Indeed, this also follows directly from the the main results of [14] by noticing that rank G − rank K ∞ = 0.5.8.1. When m > , m = 0, special attention needs to be paid on H q ( ∂ S Γ , f M λ )( π ψ , w )in case w = w , for which we have ℓ ( w ) = 3 and ℓ ( w ′ ) = ℓ ( w ) = 2. Thus, we have H qEis (S Γ , f M λ ⊗ C ) = H q ( ∂ S Γ , f M λ ⊗ C ) q = 5 , , , q = 0 , , , , , and H ,Eis (S Γ , f M λ ⊗ C )( π ψ , w ) = C ψ, if L (1 / , π ψ ) = 0 ,H ,Eis (S Γ , f M λ ⊗ C )( π ψ , w ) = C ψ, if L (1 / , π ψ ) = 0 . m = 0 , m >
0. A similar argument shows that H qEis (S Γ , f M λ ⊗ C ) = H q ( ∂ S Γ , f M λ ⊗ C ) q = 5 , , , q = 0 , , , , , and H ,Eis (S Γ , f M λ ⊗ C )( π ψ , w ) = C ψ, if L (1 / , Sym π ψ ) = 0 ,H ,Eis (S Γ , f M λ ⊗ C )( π ψ , w ) = C ψ, if L (1 / , Sym π ψ ) = 0 . By combing above disscusion with Proposition 18, Theorem 9 can be verified througha case-by-case study. (cid:3)
Acknowledgements
The authors would like to thank the Georg-August Universit¨at G¨ottingen, TechnischeUniversit¨at Dresden, and Max Planck Institue f¨ur Mathematics, Bonn, Germany, wheremuch of the discussion and work on this project was accomplished, for its hospitality andwonderful working environment. In addition, authors would like to extend their thanksto G¨unter Harder for several discussions on the subject and the encouragement duringthe writing of this article. This work is financially supported by ERC Consolidator grants648329 and 681207.
References [1] J. Bajpai, G. Harder, I. Horozov, and M. V. Moya Giusti. Boundary and Eisenstein cohomology ofSL ( Z ). Math. Ann. , 377(1-2):199–247, 2020. 2[2] A. Borel and J.-P. Serre. Corners and arithmetic groups.
Comment. Math. Helv. , 48:436–491, 1973.Avec un appendice: Arrondissement des vari´et´es `a coins, par A. Douady et L. H´erault. 7[3] A. Borel and N. Wallach.
Continuous cohomology, discrete subgroups, and representations of reduc-tive groups , volume 67 of
Mathematical Surveys and Monographs . American Mathematical Society,Providence, RI, second edition, 2000. 1[4] J. W. Cogdell, H. H. Kim, and M. R. Murty.
Lectures on automorphic L -functions , volume 20 of FieldsInstitute Monographs . American Mathematical Society, Providence, RI, 2004. 31, 32[5] G. Harder. On the cohomology of discrete arithmetically defined groups. In
Discrete subgroups of Liegroups and applications to moduli (Internat. Colloq., Bombay, 1973) , pages 129–160. Oxford Univ.Press, Bombay, 1975. 34[6] G. Harder. Eisenstein cohomology of arithmetic groups. The case GL . Invent. Math. , 89(1):37–118,1987. 2, 13[7] G. Harder. Some results on the Eisenstein cohomology of arithmetic subgroups of GL n . In Cohomologyof arithmetic groups and automorphic forms (Luminy-Marseille, 1989) , volume 1447 of
Lecture Notesin Math. , pages 85–153. Springer, Berlin, 1990. 33[8] G. Harder. Eisenstein cohomology of arithmetic groups and its applications to number theory. In
Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) , pages 779–790.Math. Soc. Japan, Tokyo, 1991. 2[9] G. Harder. The Eisenstein motive for the cohomology of GSp ( Z ). In Geometry and arithmetic , EMSSer. Congr. Rep., pages 143–164. Eur. Math. Soc., Z¨urich, 2012. 2, 28, 33[10] G. Harder and A. Raghuram.
Eisenstein cohomology for GL N and the special values of Rankin-Selberg L -functions , volume 203 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ,2020. 2, 8, 9, 29[11] H. H. Kim and F. Shahidi. Symmetric cube L -functions for GL are entire. Ann. of Math. (2) ,150(2):645–662, 1999. 33[12] B. Kostant. Lie algebra cohomology and the generalized Borel-Weil theorem.
Ann. of Math. (2) ,74:329–387, 1961. 5, 7[13] J.-S. Li and J. Schwermer. Constructions of automorphic forms and related cohomology classes forarithmetic subgroups of G . Compositio Math. , 87(1):45–78, 1993. 2, 28[14] J.-S. Li and J. Schwermer. On the Eisenstein cohomology of arithmetic groups.
Duke Math. J. ,123(1):141–169, 2004. 37[15] K. Nomizu. On the cohomology of compact homogeneous spaces of nilpotent Lie groups.
Ann. ofMath. (2) , 59:531–538, 1954. 7[16] J. Schwermer. On arithmetic quotients of the Siegel upper half space of degree two.
Compositio Math. ,58(2):233–258, 1986. 28, 34[17] J. Schwermer. Cohomology of arithmetic groups, automorphic forms and L -functions. In Cohomologyof arithmetic groups and automorphic forms (Luminy-Marseille, 1989) , volume 1447 of
Lecture Notesin Math. , pages 1–29. Springer, Berlin, 1990. 28[18] J. Schwermer. Eisenstein series and cohomology of arithmetic groups: the generic case.
Invent. Math. ,116(1-3):481–511, 1994. 34, 36[19] J. Schwermer. On Euler products and residual Eisenstein cohomology classes for Siegel modular vari-eties.
Forum Math. , 7(1):1–28, 1995. 6[20] J.-P. Serre.
Galois cohomology . Springer Monographs in Mathematics. Springer-Verlag, Berlin, englishedition, 2002. Translated from the French by Patrick Ion and revised by the author. 4
Mathematisches Institut, Georg-August Universit¨at G¨ottingen, D-37073 Germany.
E-mail address : [email protected] Mathematisches Institut, Georg-August Universit¨at G¨ottingen, D-37073 Germany.
E-mail address ::