Eisenstein series and the top degree cohomology of arithmetic subgroups of SL n = Q

The cohomology H .;E/ of a torsion-free arithmetic subgroup  of the special linear Q-group G D SLn may be interpreted in terms of the automorphic spectrum of  . Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes 1Po of associate proper parabolic Q-subgroups of G. Each summandH 1Po.;E/ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in 1Po. The cohomology H .;E/ vanishes above the degree given by the cohomological dimension cd./ D 1 2 n.n 1/. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes 1Po for which the corresponding summandH cd./ 1Po .;E/ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H cd./ 1Qo .;C/. Finally, in the case of a principal congruence subgroup .q/, q D p > 5, p 3 a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes 1Qo, there is a precise formula for their dimension.

1. Introduction 1.1. Given a connected semi-simple algebraic Q-group G of rk Q G > 0, the group of real-valued points G D G.R/ is a real Lie group. Let K be a maximal compact subgroup of G. The homogeneous space G=K D X, the symmetric space associated to G, is diffeomorphic to R d.G/ , where d.G/ D dim G dim K, hence X is contractible.
An arithmetic subgroup of G is a discrete subgroup of G, and acts properly discontinuously on X. If is torsion-free, the action is free, and the quotient space nX is a smooth manifold of dimension d.G/. Since the underlying algebraic Q-group G has positive Q-rank, nX is non-compact but of finite volume. It can be viewed as the interior of a natural compactification with boundary, due to Borel and Serre [2]. This compactification is obtained as the quotient under of a G.Q/-equivariant partial compactification X.
The inclusion nX ! nX is a homotopy equivalence. The boundary .nX / is glued together out of faces e 0 .Q/, where Q ¤ G ranges over a set of representatives for the finitely many -conjugacy classes of proper parabolic Q-subgroups of G. The codimension of a given face in nX is the parabolic Q-rank of Q. The closures of the faces e 0 .Q/ form a closed cover of the boundary .nX / whose nerve is the quotient under the natural action of of the Tits building T G of proper parabolic Q-subgroups of G (cf. [2,Section 8.4.]).
Let . ; E/ be a finite-dimensional irreducible representation of the real Lie group G on a complex vector space. We assume that this representation originates from an algebraic representation of the algebraic Q-group G. Then the cohomology H .; E/ is isomorphic to the cohomology H .nX; E/ [computed, for example, via the de Rham complex of E-valued -invariant differential forms on X ]. Interpreting these latter groups in terms of the automorphic spectrum of , by [5] resp. [6], there is a decomposition of the cohomology into the cuspidal cohomology (i.e. classes represented by cuspidal automorphic forms for G with respect to ) and the Eisenstein cohomology (i.e. classes represented by forms arising from Eisenstein series for G with respect to ). The latter space is decomposed according to the classes ¹Pº of associate proper parabolic Q-subgroups of G, that is, eventually we obtain Each summand indexed by ¹Pº is built up by Eisenstein series (appropriate derivatives or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in ¹Pº. Note that an associate class ¹Pº falls into finitely many G.Q/-conjugacy classes. The role of the theory of Eisenstein series, especially their residues, in describing the structure of the cohomology H .nX; E/ was dealt with by various authors, among them, Harder [9], Franke [5], J.-S. Li [21], and the author of this paper in [27], [29], [31], or, jointly with Grbac, in [7], [8]. Nevertheless, the full elucidation of the structure of the Eisenstein cohomology of such arithmetic quotients is still a major outstanding challenge.
The cohomology H .nX; E/ vanishes above the degree given by the cohomological dimension cd./. Therefore it is natural, possibly even technically necessary, to start with this extreme non-trivial case. It is the aim of this paper, in the case of a congruence subgroup of the Q-group G D SL n , to unfold the automorphic view on the cohomology group H cd./ .nX; E/ and to gain some insight into the internal structure of the cohomology in this top degree. This includes some quantitative results regarding the size of these cohomology spaces. Note that, by [2,Corollary 11.4.3], cd./ D dim X rk.G/. Therefore, in the case at issue, we obtain cd./ D n.n 1/ 2 , because dim X D n.nC1/ 2 1 and rk.G/ D n 1.

1.2.
With regard to associated classes ¹Pº, we use that the G.Q/-conjugacy classes of parabolic Q-subgroups of G are in bijection with the subsets J Q of the set of simple roots determined by the Levi decomposition P 0 D L 0 N 0 of the parabolic subgroup of upper triangular matrices where the maximal split torus L 0 is the group of diagonal matrices in G with determinant one. If e i is the projection of L 0 to its i th component, then generated by regular Eisenstein cohomology classes as constructed for each of the faces e 0 .Q/, Q 2 ¹Pº, up to -conjugacy, is non-trivial and its dimension is given by ¹Pº .nXI C/ D conj G OE¹Pº conj .P/ dim C H cd./ cusp .e 0 .P/I C/; where conj G OE¹Pº D n r r and conj .P/ denotes the number of -conjugacy classes of P.
Given a prime power q D p > 2, the number conj .q/ .P/ is determined in Lemma 5.7. The remaining cases of index sets in J cd are dealt with in a similar way to a certain extent in Section 5.2. Finally, in the case of a principal congruence subgroup .q/, q D p > 5, p 3 a prime, we give lower bounds for the size of these spaces. For certain associate classes ¹Qº there is a precise formula for their dimension.
If ¹Pº D ¹P 0 º is the associate class of minimal parabolic Q-subgroups of SL n =Q, represented by the standard parabolic of upper triangular matrices, we have already given a nonvanishing result for the space H cd./ ¹P 0 º .nX; E/ in the thesis work [25], [26] in the case of the trivial coefficient system, and in [29,Section 7] in the generic case. This was completed by giving a lower bound for the dimension of this space. For the sake of completeness, we briefly review this result in the case E D C in Section 6.
In Section 7, we compare our approach via automorphic forms to the structure of the top degree cohomology group H cd./ .nX; E/ with prior work where one uses the relation with the Steinberg module and its realization as the reduced cohomology of the Tits building attached to SL n =Q. Finally, we indicate how our results can possibly be extended to the special linear group over a number field or even to other groups than SL n .

Notation and conventions.
(1) Let Q be the field of rational numbers. We denote by V the set of places of Q, and by V f the set of non-archimedean places. The archimedean place is denoted by v D 1. Let Q v be the completion of Q at v, and Z v the ring of integers of Q v for v 2 V f . Let A (resp. I) be the ring of adeles (resp. the group of ideles) of Q. We denote by A f the finite adeles.
(2) The algebraic groups considered are linear. If H is an algebraic group defined over a field k, and k 0 is a commutative k-algebra, we denote by H.k 0 / the group of k 0 -valued points of H. If H is a connected Q-group, the group of real points H.R/, endowed with the topology associated to the one of R, is a real Lie group. We denote its Lie algebra by the corresponding small gothic letter h.
(3) Given a connected algebraic Q-group G, we put where X Q .G/ denotes the group of Q-morphisms G ! GL 1 . The group is defined over Q and normal in G. Any character is trivial on the unipotent radical R u .G/ of G, and, given any Levi The real Lie group 0 G.R/ contains each compact subgroup of G.R/ and each arithmetic subgroup of G.

Arithmetic quotients and their cohomology
Let be an arithmetically defined subgroup of a connected semi-simple algebraic Q-group G of positive Q-rank. Then it can be viewed as a discrete subgroup of the real Lie group G D G.R/. Given a maximal compact subgroup K G, the associated symmetric space is X D G=K on which G, and thus , acts properly. The arithmetically defined quotient space nX is non-compact but has finite volume; it may be interpreted as the interior of a compact orbifold with corners. These finitely many corners are parametrized by the -conjugacy classes of proper parabolic Q-subgroups of G. We briefly review this construction and its role within investigations regarding the cohomology of nX. In particular, we describe the structure of such a corner e 0 .P / as a fibre bundle and its cohomology. The cuspidal cohomology of e 0 .P / is the source for the eventual construction of cohomology classes for nX which are represented by Eisenstein series or residues of such.
2.1. Parabolic subgroups, Levi subgroups, and roots. Let G be a connected semisimple algebraic group defined over Q. We assume that G has Q-rank greater than zero. Fix a minimal parabolic subgroup P 0 of G defined over Q and a Levi subgroup L 0 of P 0 defined over Q. By definition, a standard parabolic subgroup P of G is a parabolic subgroup P of G defined over Q that contains P 0 . Analogously, a standard Levi subgroup L of G is a Levi subgroup of any standard parabolic subgroup P of G such that L contains L 0 . A given standard parabolic subgroup P of G has a unique standard Levi subgroup L. We denote by P D L P N P the corresponding Levi decomposition of P over Q. When the dependency on the parabolic subgroup is clear from the context we suppress the subscript P from the notation.
Let P be a parabolic Q-subgroup of G, and let N be the unipotent radical of P. We denote by Ä W P ! P=N DW M the canonical projection of P on the reductive Q-group M. Let S P be the maximal central Q-split torus of M, and let S P be denote the identity component of S P .R/. A subgroup A of a Levi subgroup of P D P.R/ such that A is mapped under Ä isomorphically on S P is called a split component of P . Note that a Levi subgroup of P is isomorphic via Ä to the group of real points M.R/. Two split components of P are conjugate under N D N.R/.
Fix a maximal compact subgroup K of the real Lie group G D G.R/. We denote by A P the uniquely determined split component of P which is stable under the Cartan involution ‚ K attached to K. Let M D Z G .A P / defined to be the centralizer of A P in G; it is the uniquely determined ‚-stable Levi subgroup of P . Then we have the decomposition P D M Ë N as a semi-direct product. Analogously we have P D A P 0 P .
The projection Ä induces a canonical isomorphism W M ! M.R/; we denote by 0 M the inverse image under of 0 M.R/. Thus, we obtain M D 0 M A P . Since M is ‚ K -stable, we have K \ P D K \ 0 M , and K \ P is a maximal compact subgroup of 0 M; M , and P . By definition, the parabolic rank prk(P ) of P is the dimension of A P .
A parabolic pair is defined to be a pair .P; A/ which is given by the group of real points P of a parabolic subgroup P of G together with a split component A of P . Having fixed a minimal parabolic Q-subgroup P 0 of G, a parabolic pair .P; Let h be a Cartan subalgebra of g which contains a 0 . We denote by H the corresponding Cartan subgroup Z G .h/ of G. LetˆDˆ.g C ; h C / be the set of roots of g C with respect to h C , and letˆR DˆR.g C ; a 0;C / be the set of R-roots, that is, the set of roots of g C with respect to a 0;C . Occasionally we may view the elements ofˆ(resp.ˆR/ as roots of G.C/ with respect to H (resp. A 0 ). Given a parabolic pair .P; A/, we denote byˆ.P; A/ the set of roots of P with respect to A, and by .P; A/ the set of simple roots inˆ.P; A/. Similarly, we will make no distinction between a character of A and its differential.
Since we have fixed a minimal parabolic Q-subgroup P 0 of G, together with the unique split component A 0 of P 0 which is ‚ K -stable, positivity onˆR is determined by the condition that the set of positive rootsˆC R is equal toˆ.P 0 ; A 0 /. We choose an order onˆwhich is compatible with this order onˆR, that is, the restriction of a positive element is positive. Given this order onˆ, we denote byˆC (resp. by ) the set of positive roots (resp. simple roots).
Let .P; A/ be a semi-standard parabolic pair. Then b WD h \ 0 m is a Cartan subalgebra of the Lie algebra 0 m of 0 M , and one has a direct sum decomposition h D b˚a. We may (and will) identify a C (resp. b C ) with the space of linear forms on h which vanish on b (resp. a). Thus, there is a canonical isomorphism LetˆM Dˆ.m C ; h C / Dˆ. 0 m C ; b C / denote the set of roots of m C with respect to h C . Then we can identifyˆM with the set of roots which vanish on a, and M D \ˆM is the set of simple roots with respect to the order induced by the one onˆ.
Given a parabolic pair .P; A/, the element P 2 a is defined by where n denotes the Lie algebra of the unipotent radical N of P . As usual, we put Since the parabolic pair .P; A/ is semi-standard, we see ja 0 D P 0 and jb D 0 M . If .P; A/ is standard, we have ja D P .
2.2. Arithmetic quotients and the adjunction of corners. Given a connected semisimple algebraic Q-group G of rk Q G > 0, the group of real-valued points G D G.R/ is a real Lie group. Let K be a maximal compact subgroup of G. Since any two of these are conjugate to one another by an inner automorphism, the homogeneous space G=K D X may be viewed as the space of maximal compact subgroups of G. The space X is diffeomorphic to An arithmetic subgroup of G is a discrete subgroup of G, and acts properly discontinuously on X. If is torsion-free, the action is free, and the quotient space nX is a smooth manifold of dimension d.G/. Since the underlying algebraic Q-group G has positive Q-rank, nX is non-compact but of finite volume. It can be viewed as the interior of a natural compactification with boundary, the adjunction of corners, due to Borel and Serre [2].
This compactification is obtained as the quotient under of a G.Q/-equivariant partial compactification X . The inclusion nX ! nX is a homotopy equivalence. The boundary .nX / is glued together out of faces e 0 .Q/, where Q ranges over a set of representatives for the -conjugacy classes of proper parabolic Q-subgroups of G.
A single face is described by the fibration (induced from the projection Ä W P ! P=N) 2.3. Cohomology. Let . ; E/ be a finite-dimensional irreducible representation of the real Lie group G on a real or complex vector space; we assume that this representation originates from an algebraic representation of the algebraic Q-group G. Let . .X; E/; d / be the complex of smooth E-valued differential forms on X. Given an arithmetic torsion-free subgroup of G, the singular cohomology H .nX; Q E/ of the manifold with coefficients in the local system defined by . ; E/ is canonically isomorphic to the cohomology H . .X; E/ /, the de Rham cohomology.
These latter cohomology groups may be interpreted in terms of relative Lie algebra cohomology groups. Indeed, let C 1 .G/ denote the space of C 1 -functions on G. We endow C 1 .G/˝E with the G-module structure given as the tensor product of the right regular representation l of G on C 1 .G/ and of . ; E/. The space C 1 .G/ K of all C 1 -vectors f for which f l.K/ is a finite-dimensional subspace of C 1 .G/ is preserved under the action of the Lie algebra g, obtained by differentiation of l, and compatible with the action of K. Therefore, since C 1 .G/ K is locally finite as a K-module, the space of K-finite vectors C 1 .G/ K is .g; K/-module. Taking into account the action of the discrete torsion-free group , there is an isomorphism of .G=K; E/ onto the complex C .g; K; C 1 .nG/ K˝E /, hence there is an isomorphism H .nX; Q E/ Š H . .G=K; E/ / ! H .g; K; C 1 .nG/ K˝E /: Let L 2 .nG/ be the space of square-integrable functions (modulo the center) on nG, viewed as usual as a unitary G-module via right-translations. The theory of Eisenstein series plays a fundamental role in the description of the spectral decomposition of L 2 .nG/. This space is the direct sum of the discrete spectrum L 2 dis .nG/, i.e. the span of the irreducible closed G-submodules of L 2 .nG/, and the continuous spectrum L 2 ct .nG/. The former space contains as a G-invariant subspace, the space L 2 cusp .nG/ of cuspidal automorphic forms, i.e. the cuspidal spectrum. The orthogonal complement in L 2 dis .nG/ is the residual spectrum, to be denoted L 2 res .nG/, thus, there is a direct sum decomposition L 2 dis .nG/ D L 2 cusp .nG/˚L 2 res .nG/: The discrete spectrum is a countable Hilbert direct sum of irreducible G-modules with finite multiplicities, say 1) By the work of Langlands each of the constituents of the residual spectrum L 2 res .G 1 = / can be structurally described in terms of residues of Eisenstein series attached to irreducible representations occurring in the discrete spectra of the Levi components of proper parabolic subgroups of G.
Our object of concern is the cohomology of with values in E, to be given in terms of relative Lie algebra cohomology as where C 1 .nG/ denotes the space of C 1 -functions on nG. 2) This cohomology space contains as a natural subspace the square integrable cohomology H .sq/ .; E/, defined as the image of the homomorphism j dis W H .g; KI L 2;1 dis .nG/˝E/ ! H .g; KI C 1 .nG/˝E/ induced in cohomology by the natural inclusion of the space of C 1 -vectors in the discrete spectrum of nG into C 1 .nG/. In general, the homomorphism j dis is not injective whereas the homomorphism induced by the inclusion of the space of C 1 -vectors in the cuspidal spectrum into C 1 .nG/ is injective; its image is called the cuspidal cohomology of , to be denoted H cusp .nX; E/ Given  An analogous results holds in the case of the group SLṅ .R/, and, with a slight modification regarding the number of irreducible unitary representations which are cuspidal and cohomological, for the group SL n (see [28,Proposition 3.5

.]).
Remark 2.2. When m 3, the group SL m has the congruence subgroup property. Therefore, we can infer from the adelic result of Franke and the preceding discussion, that, for m > 3, there is no cuspidal cohomology in degree cd./ D m.m 1/ 2 , thus, the total cohomology in this degree comes from the faces in the Borel-Serre compactification. Which ones actually contribute is determined later on. Example 2.3. We consider the case of the group GL 2 .R/, that is, m D 2. Let V .r/, r 2, denote the irreducible two-dimensional representation of the orthogonal group O.2/ which is fully induced by the character k Â 7 ! e i rÂ of the subgroup SO.2/ of rotations k Â , Â 2 OE0; 2 , in O.2/ of index two. Given an integer` 2, we denote by D`the discrete series representation of GL.2; R/ of lowest O.2/-type V .`/. The representation D`is squareintegrable and characterized by the fact that its restriction to the maximal compact subgroup O.2/ of GL.2; R/ decomposes as an algebraic sum of the form In this labelling of the discrete series representations of GL.2; R/ the Harish-Chandra parameter of D`,` 2, is` 1.
Let . k ; F k /, k 0; be the irreducible finite-dimensional representation of GL.2; R/ which originates from the algebraic representation of GL.2; Q/ of highest weight k D k ! (where ! denotes the fundamental dominant weight of GL.2; Q/), thus, dim F k D k C 1. The cohomology H .gl 2 .R/; O.2/I D`˝F k / vanishes if k ¤` 2 since the infinitesimal character D`d iffers from the one of the contragredient representation of . k ; F k /. In the case k D` 2 one has H q .gl 2 .R/; O.2/I D`˝F` 2 / D C for q D 1; it vanishes otherwise.

2.5.
Cohomology of a face e 0 .P/ in the boundary of nX . Let e 0 .P /, P 2 P , P ¤ G be a single face in the boundary .nX /. The spectral sequence in cohomology associated to the fibration (2.1) of e 0 .P / degenerates at E 2 (see [27,Theorem 2.7]). The de Rham cohomology spaces of the fiber N nN can be identified with the Lie algebra cohomology H .n; E/ of the Lie algebra n of N . This latter cohomology space carries a natural structure as an M -module; its restriction to M coincides with the natural action of M on the cohomology of the fiber. This gives rise to the identification where the sum ranges over w in the set W P of the minimal coset representatives for the left cosets of the Weyl group W D W .g C ; h C / modulo the Weyl group where ƒ 2 L a P 0 ;C is the highest weight of . ; E/. The weights w are all dominant and distinct and, given a fixed degree q, only the weights w with length`.w/ D q occur in the decomposition of H q .n; E/ into irreducibles. As usual, the length of w 2 W is meant with respect to the set of simple reflections w˛2 W;˛2 . The set W P can also be described as the set Since M is the direct product A P 0 M of the split component A P and 0 M , it follows that the module F w , viewed as 0 M -module, is irreducible. With respect to the A P -action, we get the analogue of the decomposition (2.4) where F w is viewed as an A P -module of highest weight w ja P . If the infinitesimal character of does not coincide with the infinitesimal character of the representation contragredient to F w , the cohomology space H . 0 m; K M I V ˝F w / vanishes, that is, there are no classes of type . ; w/. If the module F w is not isomorphic to its complex conjugate contragredient F w , the Lie algebra cohomology H . 0 m; K M I V ˝F w / vanishes, since this condition implies that the complex contragredient of F w and V have distinct infinitesimal character.

Eisenstein cohomology classes
We review how the theory of Eisenstein series can be used to construct certain cohomology classes in H .nX I E/ which are represented by a regular value of a suitable Eisenstein series attached to cuspidal cohomology classes in H cusp .e 0 .P /; E/. In this section and the subsequent ones we have to assume some familiarity with the theory of Eisenstein series as given in Langlands [17] and Harish-Chandra [11] and the general results regarding the construction carried through in [25], [27], and [29]. This Eisenstein series is first defined for all in .
where .a / C D ¹ 2 a j . ;˛/ > 0 for all˛2 .P; A/º and is holomorphic in that region. Via analytic continuation it admits a meromorphic continuation to all of a C . We fix 0 2 a C . If the Eisenstein series E. ; / is holomorphic at this point, then E. ; 0 / is an E-valued, -invariant differential form on X. The following result [27,Theorem 4.11.] is decisive for the construction of Eisenstein cohomology classes.
Theorem 3.1. Let P be a proper parabolic Q-subgroup of G, and let A P be the uniquely determined split component of P D P.R/ which is stable under the Cartan involution ‚ K . Let OE be a non-trivial cohomology class in H cusp .e 0 .P /; E/ D H cusp . M nZ M ; H .n; E// of type . ; w/, with an irreducible unitary representation occurring in the cuspidal spectrum of M n 0 M and w 2 W P , represented by a differential form 2 . M nZ M ; H .n; E//. If the Eisenstein series E. ; / assigned to OE is holomorphic at the point 0 D w.ƒ C / ja P (which is real and uniquely determined by OE ), then E. ; 0 / is a closed harmonic differential form on nX and represents a non-trivial cohomology class OEE. ; 0 / in H .nX; E/. We call such a cohomology class a regular Eisenstein cohomology class. For later use, we have to recall the following case where the notion of associate parabolic subgroups plays a role. Let P; Q be two parabolic Q-subgroups of G, and let A P ; A Q be the corresponding ‚ K -stable split components. We denote by W .A P ; A Q / the set of isomorphisms A P ! A Q which are induced by those inner automorphism of G.Q/ defining a Q-rational isomorphism M P ! M Q . Equivalently, the split components A P and A Q are conjugate in G under an element in G.Q/. We say that P and Q are associate if W .A P ; A Q / ¤ ;. The notion of being associated defines an equivalence relation on the set of parabolic Q-subgroups G. Let C be the set of classes of associate parabolic Q-subgroups of G. Note that the minimal parabolic Q-subgroups form one class ¹P 0 º 2 C, represented by the standard minimal parabolic Q-subgroup P 0 . Now suppose that the parabolic Q-subgroups P and Q are in the same associate class. Then we have where c.s; 0 / s 0 W . P nX; E/ ! . Q nX; E/ is a certain "intertwining" operator defined in [27,Section 4.10]. The cohomology class OEc.s; 0 The latter equation refers to an equality for the infinitesimal character of the representation s of 0 M Q obtained from the representation of 0 M P under the twist induced by the element s 2 W .A P ; A Q /.

The case SL n =Q -The faces which matter
Now we consider the Q-split simple simply connected special linear Q-group G D SL n of Q-rank n 1, where n 2. We fix the maximal compact subgroup K D SO.n/ in the real Lie group G D SL n .R/. The symmetric space X associated to G has dimension d.SL n / D .n 1/.n C 2/ 2 : Given an arithmetic torsion-free subgroup of SL n .Q/, in the following sections, we study the construction of Eisenstein cohomology classes in the cohomology H .nXI E/ in the degree given by the cohomological dimension cd./ D n.n 1/

2
. By the very definition of this notion, we have H r .nX I E/ D ¹0º for r > cd./.
In this section, we determine which classes ¹Pº of associate parabolic Q-subgroups of G D SL n may contribute to the Eisenstein cohomology in the decomposition (1.1) in Section 1.

4.1.
Let P 0 be in SL n =Q the minimal parabolic Q-subgroup of upper triangular matrices, and P 0 D L 0 N 0 its Levi decomposition where N 0 is the unipotent radical. The maximal split torus L 0 is given as the group ¹diag.t 1 ; : : : ; t n / DW t 2 SL n .Q/º of diagonal matrices with determinant one.
LetˆQ,ˆC Q , Q denote the corresponding sets of roots, positive roots, simple roots, respectively. We may (and will) identifyˆQ andˆR resp.ˆ. If e i is the projection of L 0 to its i th component, then Q D ¹˛i D e i e 1 i C1 j i D 1; : : : ; n 1º. The conjugacy classes of parabolic Q-subgroups of SL n are in bijection with the subsets of the set Q of simple roots. Each of the following standard parabolic Q-subgroups represents a SL n .Q/-conjugacy class. Given J Q , let S J D .
T˛2 J Ker.˛// ı be the corresponding subtorus in L 0 . We denote by P J the corresponding parabolic Q-subgroup containing P 0 , defined to be P J D L J N J , where L J is the Levi factor, i.e. the centralizer of S J , and N J is the unipotent radical. The characters of L 0 in the Lie algebra of N j are the positive roots which contain at least one simple root outside of J . The roots of L J are those roots whose simple components are in J . Evidently, P ; D P 0 and P Q D SL n , and, given two subsets I; J , where I J , then P I P J . Moreover, P I \ P J D P I \J .

4.2.
A choice of faces in the adjunction of corners. We are interested to construct non-vanishing classes in the degree given by the cohomological dimension cd./ of , that is, in the highest possible degree in which cohomology may exist at all. Thus, being interested in Eisenstein cohomology classes, we have to determine for which proper parabolic subgroups P, up to -conjugacy, the corresponding cuspidal cohomology groups H cd./ cusp .e 0 .P /; E/ are non-zero in this degree. However, in most cases, these groups are zero. This is a consequence of the vanishing result (2.4) concerning the cuspidal cohomology for arithmetic subgroups in some GL m =Q in degrees outside a certain range centered around the middle dimension.
The following proper parabolic Q-subgroups will be at our disposal for the envisaged construction: First, say case .I/, a minimal parabolic Q-subgroup B (also called Borel subgroup) is G.Q/-conjugate to the standard minimal parabolic Q-subgroup P 0 D P ; . The corresponding set W B of minimal coset representatives coincides with the Weyl group W , and the longest element w B in W has length .w B / D dim N 0 D n.n 1/ 2 D cd./: Thus, we have F w B D H cd./ .n 0 ; E/.
Second, labelled case .II/, denote by J cd the family of non-empty sets J Q subject to the condition that if˛i ;˛i C1 2 J , with i 2 ¹1; : : : ; n 2º, then˛i C2 … J . The corresponding standard parabolic subgroup P J presents itself as the stabilizer within SL n of the flag V i 1¨V i 2¨ ¨V i r where the tuple .i 1 ; i 2 ; : : : ; i r / is given by the set of indices in ascending order of the simple roots in the complement CJ of J in Q . Given 1 Ä s Ä n 1, V s is defined to be Qf 1 C Qf 2 C C Qf s with regard to the standard basis of Q n . This condition assures that only blocks of size one, two or three make up the Levi subgroup of P J . In the latter two cases, the upper bound v o .m/ of the range in which the group GL m has possible nonvanishing cuspidal cohomology classes coincides with m.m 1/ 2 when m D 2; 3. Consequently, adding up this upper bound over all blocks of size two or three, we get

Construction of Eisenstein cohomology classes -Case (II)
In this section we carry through the construction of non-vanishing cohomology classes in the top cohomology H cd./ .nX I E/ which originate from faces e 0 .Q/ where the associate class of the corresponding parabolic Q-subgroup falls into case .II/. The main result is Theorem 5.6.

Block-2 parabolic subgroups of G.
For the sake of clarity we first deal with the following specific family of parabolic Q-subgroups belonging to case .II/. Let J D ¹˛i 1 : : : : ;˛i r º Q be a non-empty set of simple roots of SL n =Q subject to the conditions that if˛i ;˛j 2 J , then ji j j 2, and let P J be the corresponding parabolic Q-subgroup of G=Q D SL n =Q. The parabolic rank of P J equals .n 1/ r. Observe that the blocks in the Levi subgroup of P J can only have size one or two, and at least one block of size two occurs. Therefore we call a parabolic Q-subgroup P J where J satisfies this condition a block-2 parabolic Q-subgroup of G.
We give two examples: First, we suppose that n D 2m is even, and n 4. We take the set J D ¹˛1;˛3; : : : ;˛n 3 ;˛n 1 º: Then the Levi subgroup L J of the corresponding parabolic Q-subgroup P J is given via "diagonal matrices" as L J D ¹diag.A 1 ; A 3 ; : : : ; A n 3 ; A n 1 / 2 SL n .Q/ j A i 2 GL 2 .Q/º: The parabolic rank of P J is n 2 . Second, let n 3, and take J 0 D ¹˛j º for any simple root in Q . The Levi subgroup of the corresponding parabolic Q-subgroup P J 0 is given as L J 0 D ¹diag.a 1 ; : : : ; a j 1 ; A; a j C2 ; : : : ; a n / 2 SL n .Q/ j A 2 GL 2 .Q/; a i 2 GL 1 .Q/º: Theorem 5.1. Given n 3, let P D P J be the parabolic Q-subgroup of SL n =Q defined by the non-empty set J D ¹˛i 1 : : : : ;˛i r º Q subject to the conditions that if˛i ;˛j 2 J , then ji j j 2: Given an arithmetic torsion-free subgroup SL n .Q/, and a rational finitedimensional representation . ; E/ of SL n .R/ of highest weight ƒ, let OE 2 H cd./ cusp .e 0 .P /; E/, OE ¤ 0, be a cuspidal cohomology class of type . ; w P J /, where is an irreducible unitary representation of 0 M J and w P J denotes the longest element in W P J . Then the Eisenstein series E. ; / attached to the differential form is holomorphic in 0 D w P J .ƒ C / ja J , and OEE. ; 0 / is a non-zero class in H cd./ .nX; E/ which is represented by the closed, harmonic differential form E. ; 0 /.
Proof. As stated in Theorem 3.1, the point in question 0 D w P J .ƒ C / ja P J is real valued. Let w G be the longest element in W , and w M the longest element in W P J ; we havè .w G / D jˆCj and`.w M / D jˆC M j: The product w P J D w M w G has length`.w G / `.w M / D dim N P , and w P J is the longest element in the set W P of minimal coset representatives for the cosets W M nW G . Since W M acts trivially on a P J , we obtain 0 D w G .ƒ C / ja P J . The highest weight ƒ of . ; E/ is transferred under w G into the highest weight Q ƒ of the representation which is contragredient to . ; E/. In particular, we have w G . / D . Therefore we obtain 0 D Q ƒ C ja P J : The highest weight Q ƒ is dominant because ƒ is dominant. Thus we have the estimate (5.1) . 0 ;˛/ D . Q ƒ ja P J ;˛/ C . ja P J ;˛/ . P J ;˛/ for all˛2 .P J ; A P J /: For the sake of simplicity, since the parabolic group P J is fixed, we now write a instead of a P J . The region of absolute convergence of the Eisenstein series E. ; / is given as .a C / C D ¹ 2 a C j Re 2 P C .a / C º; where .a / C D ¹ 2 a j . ;˛/ > 0 for all˛2 .P; A/º. The Eisenstein series is holomorphic there and admits a meromorphic continuation to all of a C .
Suppose that the Eisenstein series E. ; / has a pole at 0 . Given the data .P J ; ; 0 /, the corresponding Langlands quotient J P J ; ; 0 would carry as a .sl n .R/; K R /-module a unitary structure. Note that P J ¤ P 0 and that the tempered representation is not the trivial representation. Therefore, using a criterion, due to Wallach (see [3, Chapter IV, Theorem 5.2] resp. [32, Theorem 6.2]), this implies P J .a/ > Re 0 .a/ for all a 2 C`.a C / D ¹a 2 a jˇ.a/ 0;ˇ2 .P J ; A P J /º. However, this contradicts (5.1). Therefore, the Eisenstein series E. ; / is holomorphic at the point 0 D w P J .ƒ C / ja P J . Then Theorem 3.1 implies the claim.
We define H cd./ .nXI E/ e 0 .P / to be the subspace of H cd./ .nX; E/ which is generated by all non-zero Eisenstein cohomology classes OEE. ; 0 / where OE ranges over all non-zero cuspidal classes in H cd./ `.w P / . M nZ M ; F w P /. We are interested in the relation of this space to analogous spaces attached to a different parabolic Q-subgroup which is not -conjugate to P. The theory of Eisenstein series, in particular, the properties of the constant terms of a given Eisenstein series, require to group together the various contributions originating from faces e 0 .P /, where, up to -conjugacy, P ranges over the finitely many elements in a given associate class ¹Qº 2 C of parabolic subgroups.
We retain the notation and assumptions of Theorem 5.1. Suppose that OEE. ; 0 / is a nonzero class in H cd./ .nX; E/ which is represented by the closed, harmonic differential form E. ; 0 / and originates from a non-zero cuspidal cohomology class OE in H cusp .e 0 .P /; E/ of type . ; w P J / where is an irreducible unitary representation of 0 M J and w P J denotes the longest element in W P J . Let Q be a proper parabolic Q-subgroup of G. If prk.Q/ > prk.P J / or if prk.Q/ D prk.P J / and Q and P J are not associate to one another, then the constant Fourier coefficient E. ; / Q along Q vanishes identically (see [11, Corollary 2 of Lemma 33]). Therefore, we obtain OEE. ; 0 / Q je 0 .Q/ D .0/. Now we consider the case that the parabolic Q-subgroup Q is associated to the standard parabolic P J we started with. The following assertion concerning the associate class of a given block-2 parabolic subgroup P J is a simple observation, based on interpreting an element in L J as a diagonal matrix with block entries of size at most two.
Lemma 5.2. Let P J and P J 0 be two block-2 parabolic Q-subgroups of G such that jJ j D jJ 0 j, that is, the number of blocks of size two in the corresponding Levi subgroups is the same, then P J and P J 0 are in the same associate class.
In particular, if there exists J 0 ¤ J , with jJ j D jJ 0 j, we see that the associate class ¹P J º contains parabolic Q-subgroups which are not G.Q/-conjugate to P J . This is already the case if n D 3, and J D ¹˛1º, J 0 D ¹˛2º. Theorem 5.3. Let OEE. ; 0 / be a non-zero class in H cd./ .nX; E/ which is represented by the closed, harmonic differential form E. ; 0 / and which originates with a non-zero cuspidal cohomology class OE in H cusp .e 0 .P J /; E/ of type . ; w P J / where is an irreducible unitary representation of 0 M J and w P J denotes the longest element in W P J . Let Q be a proper parabolic Q-subgroup of G. We suppose that Q is associate to P J but Q and P J are not -conjugate. Then r cd./ Q .OEE. ; 0 // D 0: Proof. Since the parabolic subgroups P J and Q are in the same associate class, we have (cf. Section 3) where c.s; 0 / s 0 W . P J nX; E/ ! . Q nX; E/ is a certain "intertwining" operator defined in [27,Section 4.10]. It arises from the corresponding intertwining operator which occurs in the constant Fourier coefficient of the Eisenstein series in question along the parabolic Q.
First, we consider the case that Q is conjugate under G.Q/ to P J . Thus, there is g 2 G.Q/ with P g J D Q and A g J D A Q is a split component of Q, and W .A P J ; A Q / is non-trivial. Let s 2 W .A P J ; A Q /. Using [11, Lemma 106 and its Corollary] (see also [17,Lemma 4.5 (ii)]), the intertwining operator is identically zero unless Q is conjugate under to P J . However, this case is excluded by our assumption.
Second, we consider the case that Q 2 ¹P J º but Q is not conjugate under G.Q/ to P J . Then there is a subset J 0 Q , J 0 ¤ J with jJ 0 j D jJ j, such that Q is conjugate under G.Q/ to P J 0 . We may assume Q D P J 0 , and, for the sake of simplicity, we write P D P J . Recall that both parabolic Q-subgroups Q and P are block-2 parabolics.
Given an element s 2 W .A P ; A Q /, the cohomology class OEc.s; 0 A straightforward computation shows that the only element in W Q which can satisfy the former condition is v s D 1; see [27, pp. 128-129], for a specific case. Note that the corresponding cohomology class OEc.s; 0 / s 0 . 0 / je 0 .Q/ is a class of degree cd./, that is, it is an there is no non-trivial class of the required weight. Consequently, r cd./ Q .OEE. ; 0 / D 0.
Let J D ¹˛i 1 ; : : : ;˛i r º Q be a non-empty set of simple roots subject to the conditions that if˛i ;˛j 2 J , then ji j j 2, and let P J be the corresponding parabolic Q-subgroup of G=Q D SL n =Q. The parabolic rank of P J is equal to .n 1/ r. Let ¹P J º be the corresponding associate class of parabolic Q-subgroups of G. By Lemma 5.2, the class ¹P J º consists of all parabolic Q-subgroups P J 0 of G and their G.Q/-conjugates, where jJ 0 j D jJ j and P J 0 is a block-2 parabolic.
Given an arithmetic torsion-free subgroup G.Q/, the -conjugacy classes of elements in ¹P J º are in one-to-one correspondence to the faces e 0 .Q/ in .nX /, where Q ranges over a set of representatives for ¹P J º= . Given such a representative, say a parabolic Q-subgroup Q, using the construction in Theorem 5.1, there is a corresponding subspace The decomposition of the latter space, as exhibited in (5.2), can be simplified by subdividing the set ¹P J º= which parametrizes the individual summands. To simplify matters, we now assume that the coefficient system is given by the trivial representation.
If two parabolic Q-subgroups Q and R of G are conjugate under G.Q/, then they lie in the same associate class. Evidently, the converse is not correct. Thus, a given associate class ¹Qº of parabolic Q-subgroups of G falls into a finite number of G.Q/-conjugacy classes. Given an arithmetic subgroup G, each of these G.Q/-conjugacy classes decomposes into a finite set of -conjugacy classes. In the case of the previously considered block-2 parabolic Q-subgroups we have the following result regarding the number of G.Q/-conjugacy classes within an associate class ¹Qº.
Lemma 5.4. Let P be a parabolic Q-subgroup of G=Q D SL n =Q, where n 3, whose G.Q/-conjugacy class is represented by a standard parabolic Q-subgroup P J indexed by the set J D ¹˛i 1 ; : : : ;˛i r º Q , with J ¤ ;, subject to the conditions that if˛i ;˛j 2 J , then ji j j 2: Let conj G OE¹Pº denote the number of G.Q/-conjugacy classes of parabolic Q-subgroups in the associate class ¹Pº. Then we have conj G OE¹Pº D n r r ! : Proof. 3) By Lemma 5.2, this problem amounts to determine the cardinality of the family A of subsets I Q which are subject to the analogous conditions as J and jJ j D jI j. Consider the set A D ¹1; : : : ; n 1º of indices of the elements in Q . Given a subset T A, we enumerate its elements in increasing order 1 Ä t 1 < t 2 < < t r Ä n 1: Since the given conditions are read as t i > t i 1 C 1, where i D 1; : : : ; r, we can transform this sequence into the sequence 1 Ä t 1 < t 2 1 < < t r .r 1/ Ä n 1 .r 1/ D n r: Conversely, given a sequence 1 Ä c 1 < c 2 < < c r Ä n r, we can define an increasing sequence 1 Ä c 1 < c 2 C 1 < < c r C .r 1/ Ä n r C .r 1/ D n 1; which is subject to the conditions as given. Therefore, there is a one-to-one correspondence between the elements in A and the family of subsets C of ¹1; : : : ; n rº with r elements. Thus, the cardinality of A equals n r r .
Proposition 5.5. Let P be a parabolic Q-subgroup of G=Q D SL n =Q, n 3, whose G.Q/-conjugacy class is represented by a standard parabolic Q-subgroup P J determined by the non-empty set J D ¹˛i 1 : : : : ;˛i r º Q , subject to the conditions that if˛i ;˛j 2 J , then Proof. Without loss of generality we may assume that P D P J . First, consider the case where J consists of one simple root, say˛j 2 Q . Then 0 M D ¹diag.˙1; : : : ; A; : : : ;˙1/ j A 2 SL˙.R/º: Consequently, since F w P J Š C and`.w P J / D cd./ 1, we obtain The latter space is described in terms of relative Lie algebra cohomology groups by Second, in the case where jJ j D r > 1, P J is a parabolic Q-subgroup whose Levi component L J is isomorphic, up to finite index, to a direct product of r copies of GL 2 =Q. On the real points GL 2 .R/ of each of these copies we take the discrete series representation D 2 of GL 2 .R/. This gives rise to a representation of 0 M J , and we have Let P be a proper parabolic Q-subgroup of G. If SL n .Z/ is a subgroup of finite index, the G.Q/-conjugacy class of parabolic Q-subgroups of G determined by P falls into a finite set of -conjugacy classes. We denote its cardinality by conj .P/.
Theorem 5.6. Let P be a parabolic Q-subgroup of G=Q D SL n =Q, n 3, whose G.Q/-conjugacy class is represented by a standard parabolic Q-subgroup P J indexed by the set J D ¹˛i 1 : : : : ;˛i r º Q , with J ¤ ;, subject to the conditions that if˛i ;˛j 2 J , then generated by Eisenstein cohomology classes as constructed in Theorem 5.1 for each of the faces e 0 .Q/; Q 2 ¹Pº, is non-trivial, and its dimension is given by ¹Pº .nXI C/ D conj G OE¹Pº conj .P/ dim C H cd./ cusp .e 0 .P/; C/ where conj G OE¹Pº D n r r .
Proof. Let P J 0 be a standard parabolic Q-subgroup which belongs to the associate class ¹Pº. Then jJ 0 j D jJ j, and there is an inner automorphism of G.Q/ which defines a Q-rational isomorphism M J ! M J 0 . This implies that the corresponding cuspidal cohomology spaces are isomorphic, that is, cusp .e 0 .P J /; C/ Š H cd./ cusp .e 0 .P J 0 /; C/: Since these spaces are non-zero by Proposition 5.5, the corresponding Eisenstein cohomology spaces H cd./ .nXI C/ e 0 .P J / and H cd./ .nX I C/ e 0 .P J 0 / are isomorphic and non-trivial. Therefore the space H cd./ ¹Pº .nX I C/ is non-zero. The formula for its dimension follows from the previous argument and arranging the -conjugacy classes of parabolic Q-subgroups in the associate class ¹Pº according to the G.Q/-conjugacy class to which they belong.
If one is interested to determine the size of a specific summand H cd./ ¹Pº .nXI C/, it is necessary to analyze conj .P/ and the cuspidal cohomology H cd./ cusp .e 0 .P/; C/. Let q D p > 2 be a prime power, and let .q/ SL n .Z/ be the principal congruence subgroup of level q.
Lemma 5.7. Let P be a parabolic Q-subgroup of the group G=Q D SL n =Q, where n 3, which is G.Q/-conjugate to a standard parabolic Q-subgroup P J indexed by a set J D ¹˛i 1 ; : : : ;˛i r º Q , J ¤ ;, subject to the conditions that if˛i ;˛j 2 J then ji j j 2: Then the number of .q/-conjugacy classes of P is conj .q/ .P/ D 1 2 n r 1 Q n i D2 .q i 1/ .q 2 1/ r : Proof. By definition, the principal congruence subgroup .q/ is the kernel of the homomorphism SL n .Z/ ! SL n .Z=qZ/ given by taking the entries of a matrix mod q. Since this homomorphism is surjective, we have SL n .Z/= .q/ Š SL n .F q / where F q D Z=qZ is the finite field with q elements.
We may assume that P D P J where J D ¹˛i 1 ; : : : ;˛i r º Q , J ¤ ; subject to the conditions that if˛i ;˛j 2 J then ji j j 2:. Then we have the Levi decomposition P J D L J N J , and, accordingly for the group of real points 0 P J D 0 M J N J . Since D .q/ is fixed, we may write .q/ P J D \ P J ; .q/ M J D \ M J ; .q/ N J D \ N J : We note that P J 0 P J D 0 M J N J . By the strong approximation property the group SL n =Q has, the SL n .Z/-conjugacy class of P J coincides with the SL n .Q/-conjugacy class of P J . Moreover, the normalizer of P J in SL n .Q/ is the group P J itself. Therefore, conj .q/ .P/ D jSL n .Z/= .q/j j.SL n .Z/ \ P J /= .q/ P J j 1 : With regard to the first factor, the cardinality of the special linear group SL n .Z/= .q/ Š SL n .F q / over the finite field F q is .q i 1/: In order to determine the second factor in formula (5.3), we may assume, without loss of generality, that P J has the index set J D ¹˛1;˛2 C1 ; : : : ;˛r C.r 1/ º. Then the Levi component of P J has the form L J D ¹diag.A 1 ; A 3 ; : : : ; A 2r 1 ; a 2rC1 ; : : : ; a n / 2 SL n .Q/ j A i 2 GL 2 .Q/; a j 2 GL 1 .Q/º: Taking the determinant of each of the components of such a "diagonal matrix" induces a homomorphism ! W SL n .Z/ \ M J ! .˙1/ n r 1 ; formally given by the assignment diag.A 1 ; A 3 ; : : : ; A 2r 1 ; a 2rC1 ; : : : ; a n / 7 ! .det A 1 ; : : : ; det A 2r 1 ; det a 2rC1 ; : : : ; det a n 1 /: and .q/ \ P J 0 P J D 0 M J N J . Therefore the kernel of ! is ker ! D ¹diag.A 1 ; A 3 ; : : : ; A 2r 1 ; a 2rC1 ; : : : ; a n / 2 SL n .Z/ j A i 2 SL 2 .Z/; a j D 1º: Since q > 2, an element diag.A 1 ; A 3 ; : : : ; A 2r 1 ; a 2rC1 ; : : : ; a n / 2 .q/ \ P J satisfies that det A i D 1 and a j D 1, thus, it lies in the kernel of !. It follows that j.SL n .Z/ \ P J /= .q/ P J j D 2 n r 1 OEq.q 2 1/ r q n.n 1/ 2 r ; where OEq.q 2 1/ D jSL 2 .F q /j, and the last factor is the contribution from the "unipotent" part .SL n .Z/ \ N J /= .q/ N J .
Ultimately, the non-vanishing Eisenstein cohomology classes in H Example 5.8. Let n 3, and let .q/ SL n .Q/, with q D p > 5, p a prime. Take J D ¹˛j º for any simple root in Q . The Levi subgroup of the corresponding parabolic Q-subgroup P J is given as L J D ¹diag.a 1 ; : : : ; a j 1 ; A; a j C2 ; : : : ; a n / 2 SL n .Q/ j A 2 GL 2 .Q/; a i 2 GL 1 .Q/º: Using Hecke's work [12], the dimension of the latter space is given by where g .q/ denotes the genus of the compact Riemann surface attached to the group .q/ by adding the cusps to the surface at infinity. The genus is given by the formula where q D 1 2 q.q 2 1/ is the index of .q/ in SL 2 .Z/=¹˙1º, and the expression in the square brackets is the number of .q/-conjugacy classes of Borel subgroups in SL 2 .Q/. Observe that conj G OE¹P J º D n 1. Thus, we are reduced to determine the number conj .q/ .P J / of .p/-conjugacy classes of parabolic Q-subgroups within the G.Q/-conjugacy class of P J . By Lemma 5.7 we have Taking the different terms together, we obtain where J D ¹˛j º for any simple root in Q .

Mixed cases.
We now turn our attention to those cases of parabolic Q-subgroups of SL n that are in case (II) but not yet covered in the preceding investigation. Recall that the standard parabolic Q-subgroups P J in case (II) are characterized by the subsets J Q subject to the condition that if˛i ;˛i C1 2 J , with i 2 ¹1; : : : ; n 2º, then˛i C2 … J . Previously, in our treatment, we did not discuss how we handle the occurrence of blocks of size three, i.e. GL 3 , in the Levi component Ł J . Though one can expect a result analogous to Theorem 5.6, exact formulas for the dimension of the cuspidal cohomology H cd./ .cusp/ .e 0 .P J /; C/, for example, in the case of a principal congruence subgroup, are currently not known. However, the existence of cuspidal cohomology classes for congruence subgroups of sufficiently high level of SL 3 =Q or GL 3 =Q is a consequence of the general results in [14,Section 3], [15] or [31,Section 5]. Nevertheless, in principle, the anticipated result is a structural one but less precise as to the size of the space of Eisenstein cohomology classes.
The situation is slightly better in the following case where one can give a lower bound for the dimension in question. Let .p/ SL 3 .Q/ be the principal congruence subgroup of level p, where p Á 3 mod 8 and p Á 1 mod 3. Then, by the main theorem in [19], It is clear that, following the lines of arguments as given in the case of block-2 parabolic Q-subgroups, a result analogous to Theorem 5.6 can be obtained. We leave details to the interested reader.
Example 5.9. Let n 4, and let SL n .Q/ be a torsion-free congruence subgroup. Take J D ¹˛j ;˛j C1 º for any simple root in Q . The Levi subgroup of the corresponding parabolic Q-subgroup P J is given as L J D ¹diag.a 1 ; : : : ; a j 1 ; A; a j C3 ; : : : ; a n / 2 SL n .Q/ j A 2 GL 3 .Q/; a i 2 GL 1 .Q/º: Then there exists a space H cd./ ¹P J º .nX I C/ D L Q2¹P J º= H cd./ .nX I C/ e 0 .Q/ which is generated by regular Eisenstein cohomology classes and non-trivial for congruence subgroups of sufficiently high level. Its dimension is given by cusp .e 0 .P J /; C/; where conj G OE¹Pº D n 1.

Construction of Eisenstein cohomology classes -Case (I)
For the sake of completeness we briefly review the results obtained in [25] and [26] which concern the subspace H cd./ ¹P 0 º .nX; C/ of Eisenstein cohomology classes which originate from the cohomology of the faces e 0 .Q/, Q 2 ¹P 0 º, in .nX /. Their construction relies on the theory of Eisenstein series in the framework of the adele group attached to the underlying algebraic group G=Q; a detailed investigation of certain local and global questions regarding the constant terms of the Eisenstein series is decisive.
6.1. Cohomology at infinity. We identify the set of non-archimedean places V Q;f with the set of primes in Q. The group G D SL n has the strong approximation property (with respect to the set of archimedean places), that is, Given the open compact subgroup K G.A f /, we define WD G.Q/ \ K, that is, Then is an arithmetic subgroup of G.Q/, and we have, using (6.1) and bringing the maximal compact subgroup K R WD SO.n/ of G.R/ into play, nG.R/=K R D G.Q/nG.A/=.K R K/: In particular, the group L.1/ WD Q p2Q SL n .Z p / is a maximal open compact subgroup of G.A f /, and, given a natural number m, the group is an open compact subgroup of G.A f /. Then the arithmetic group defined in (6.2) is the principal congruence subgroup .m/ of level m of SL n .Z/. It is torsion-free for m 3. Note that the group L.m/ is a normal subgroup of L.1/ of finite index. Let P 0 SL n =Q be the minimal parabolic Q-subgroup of upper triangular matrices, and P 0 D L 0 N 0 its Levi decomposition where N 0 is the unipotent radical. The maximal split torus L 0 is given as the group ¹diag.t 1 ; : : : ; t n / DW t 2 SL n .Q/º of diagonal matrices with determinant one. To simplify notation we write T WD L 0 for the maximal Q-split torus L 0 . Note that the associate class ¹P 0 º consists of all minimal parabolic Q-subgroups of G.
It is necessary to deal with all the finitely many .m/-conjugacy classes of minimal parabolic Q-subgroups of G at the same time. The face e 0 .P 0 / will be our point of reference in our adelic description of the cohomology at infinity that depends on the associate class ¹P 0 º. With regard to the T.R/-module structure, the cohomology of the face e 0 .P 0 / decomposes as where the sum ranges over the elements w in the Weyl group W P 0 D W , and F w denotes the irreducible finite-dimensional T.C/-module of highest weight w D w. P 0 / P 0 : We choose a harmonic differential form w 2 `.w/ .e 0 .P 0 /; C/ such that F w D COE w , that is, the form w represents a class of weight w . We denote by Á w W 0 T.R/ ! C the character obtained by restricting w to 0 T.R/. The character Á w ; w 2 W , is a sign character. Given the longest element w 0 2 W , Á w 0 is the trivial character.
Given a fixed natural number m, we consider the space of double cosets In particular, given t 1 2 T.R/, .t 1 / D 1.
Next we have to merge this approach with the description of H .e 0 .P 0 /; C/ as given in the decomposition (6.3). The starting point is the following result (see [25,Satz 5.7]): Therefore, in view of the decomposition H .e 0 .P 0 /; C/ Š L w2W F w , a cohomology class in L Q2D m H .e 0 .Q/; C/ may be characterized by an element w 2 W and a character on T m , trivially extended to a quasi-character on T.Q/nT.A/=.T.A f / \ L.m//. The latter character has to be compatible with a given w 2 W in the following sense: Let O T A be the set of characters Given the sign character Á w W 0 T.R/ ! C , indexed by w 2 W , the character has to be an element of the set O T A .Á w / of characters in O T A whose restriction to 0 T.R/ coincides with Á w .
6.2. Eisenstein cohomology classes. The torus T G can be written as a direct product of tori T i , i D 1; : : : ; n 1, and accordingly we write D .
In particular, if tends to infinity, the dimension is unbounded.
Proof. The Eisenstein cohomology classes OEE. w 0 ; ; 0 /, where D . 1 ; : : : ; n 1 / ranges over all characters occurring in the decomposition with i ¤ 1 for all i D 1; : : : ; n 1, generate a subspace in H cd./ ¹P 0 º ..q/nX; C/. Since the characters in question are uniquely determined by their restriction W T p ! C , a counting of the characters which satisfy the given condition leads to the lower bound as claimed.
With regard to the Eisenstein cohomology, using Proposition 4.1, we have where Ass cd G denotes the finite set of classes of associate proper standard parabolic Q-subgroups P J whose defining set J Q belongs to the family J cd . Note that an associate class ¹Pº falls into G.Q/-conjugacy classes of parabolic Q-subgroups, thus, the defining set J is not uniquely determined by ¹Pº.
For each of the summands H where P J 2 ¹Pº. Thus, by taking together the lower bound for dim C H cd./ ¹P 0 º ..q/nX; C/, given in Corollary 6.3, and the results in Sections 4 and 5, we obtain a lower bound for the cohomology group in the top degree.
We exemplify the procedure in the following example: The set Ass cd G attached to the group G D SL 5 =Q consists of four classes of associate proper parabolic Q-subgroups. First, one consists of the four G.Q/-conjugacy classes represented by the standard parabolic subgroups P J of parabolic rank three, that is, where J D ¹˛1º; ¹˛2º; ¹˛3º; and ¹˛4º. Second, the standard parabolic Q-subgroups of parabolic rank two fall into two associate classes, one, say ¹Q 2 º, consists of the block-2 parabolic subgroups whereas the other one, say ¹Q 3 º, consists of the block-3 parabolic subgroups. Finally, there is the mixed case, denoted ¹Rº; it falls into two different G.Q/-conjugacy classes, namely, the one of P J with J D ¹˛1;˛2;˛4º, and the one of P J 0 with J 0 D ¹˛1;˛3;˛4º. The two other maximal standard parabolic Q-subgroups do not account for a class in Ass cd G . Then, given .q/ SL 5 .Q/, we obtain the estimate dim C H cd..q// ..q/nXI C/ Â 1 2 .q 1/ 1 .q i 1/g .q/ C RS 5 .q/; where RS 5 .q/ denotes the sum of the remaining summands dim C H cd./ ¹Q 2 º ..q/nX; C/; dim C H cd./ ¹Q 3 º ..q/nX; C/; dim C H cd./ ¹Rº ..q/nX; C/: A formula, depending only on the level q, for each of these terms requires explicit knowledge of the cuspidal cohomology groups H cd..q// cusp .e 0 .P /; C/, thus, at least for the latter two cases, a dimension formula for spaces of cohomological cuspidal automorphic forms for congruence subgroups of SL 3 .Q/. This is currently out of reach. However, as mentioned before, one has non-vanishing results for congruence groups of sufficiently high level, see [14], [15], [19]. Remark 7.1. For arbitrary n 3 and .q/ SL n .Q/, q D p > 2, p a prime, the lower bound for the dimension of the cohomology of .q/ in the top degree would read as dim C H cd..q// ..q/nX I C/ (7. .q i 1/ g .q/ C RS n .q/; where RS n .q/ has the analogous meaning. Note that the lower bound given by formula (7.1), taken without the term RS n .q/, exceeds the one given in [23, Section 4].
7.3. Other groups and fields. Let be a torsion-free arithmetic subgroup of a connected semi-simple algebraic Q-group G of positive Q-rank. For the sake simplicity we assume that G is Q-split and belongs to the family of classical groups, meaning, beside general linear groups, symplectic groups, orthogonal groups or unitary groups. Given a finite-dimensional irreducible representation . ; E/ of the real Lie group G D G.R/ on a complex vector space, the cohomology H .nX; E/ of the locally symmetric space nX has a direct sum decomposition, H .nX; E/ D H cusp .nX; E/˚M ¹Pº H ¹Pº .nX; E/ into the subspace of classes represented by cuspidal automorphic forms for G with respect to and the Eisenstein cohomology. The latter space is decomposed according to the classes ¹Pº of associate proper parabolic Q-subgroups of G. Each summand is built up by Eisenstein series (derivatives or residues of such) attached to cuspidal automorphic representations on the Levi components of elements in ¹Pº.
Being interested in the cohomology group H cd./ .nX; E/ in the degree of the cohomological dimension of , it should be possible to determine which associate classes ¹Pº of parabolic Q-subgroups of G finally contribute to this decomposition. Recall that a parabolic Q-subgroup is the semi-direct product of its unipotent radical and any Levi subgroup. These Levi subgroups are products of groups of GL-type and of isometry groups.
For example, in the case of the symplectic group Sp n =Q of rank n, up to G.Q/-conjugacy, a maximal parabolic Q-subgroup has the form P r Š L r N r , r D 1; : : : ; n, with Levi subgroup L r Š GL r Sp n r if r < n, and L n Š GL n if r D n, and N r is the unipotent radical. Note that such a parabolic Q-subgroup is conjugate to its opposite parabolic subgroup, thus, the associate class ¹P r º coincides with the G.Q/-conjugacy class P r . All other standard parabolic Q-subgroups are expressible as intersections of the proper standard maximal parabolic Q-subgroups.
A result, analogous to Proposition 4.1, hinges on an explicit knowledge of the range in which the Levi subgroup of a given parabolic Q-subgroup P of G has possibly non-vanishing cuspidal cohomology, that is, H cusp .e 0 .P /; E/ is non-trivial. In view of the final construction of Eisenstein cohomology classes, this requires a thorough understanding of the cuspidal cohomology of arithmetic groups in classical groups. We refer to [22] and [30].
With regard to the existence of Eisenstein cohomology classes in H ¹P 0 º .nX; E/, where ¹P 0 º denotes the associate class of minimal parabolic Q-subgroups of G one has to expect a result similar to Theorem 6.2 and its Corollary. In the case G D Sp n , see [16], and, in the case of a generic coefficient system, see [29].