MIXED STRUCTURES IN SHIMURA VARIETIESAND AUTOMORPHIC FORMS

ARVIND N. NAIR

Abstract. We study the mixed structures appearing in the cohomology of noncom-pact Shimura varieties and their relation to automorphic forms. The main result isthat a certain spectral sequence converging to the cohomology of the Shimura variety,defined analytically in work of Franke, is in fact a spectral sequence of mixed Hodge-deRham structures. This is proved using Morel’s theory of weight truncations. The E1

term of this spectral sequence is a direct sum of pure structures which are related tointersection complexes of strata of the minimal compactification.

Resume. Nous etudions les structures mixtes qui apparaissent dans la cohomologiedes varietes de Shimura non compactes et leurs liens avec les formes automorphes. Leresultat principal est qu’une certaine suite spectrale aboutissant a la cohomologie dela variete de Shimura, defini analytiquement dans le travail de Franke, est en fait unsuite spectrale des structures de Hodge-de Rham mixtes. Ceci est demontre utilisantla theorie de troncature par poids de Morel. Le terme E1 de cette suite spectrale estune somme directe de structures pures qui sont lies a des complexes d’intersection desstrates de la compactification minimale.

1. Introduction

Let M be a connected component of a Shimura variety at finite level, i.e. a locallysymmetric variety. Thus M = Γ\G(R)/K for a semisimple Q-algebraic group G, amaximal compact subgroup K ⊂ G(R), and an arithmetic subgroup Γ ⊂ G(Q). Let Ebe the local system on M given by a Q-rational representation E of G. We assume thatM is noncompact.

The cohomology H∗(M,E) has a mixed structure – a mixed Hodge structure or amixed Hodge-de Rham structure over the field of definition of M , or a mixed Galoismodule structure (if E is of geometric origin), or a “mixed motive”. The fundamentaltheorem of Franke [Fra98] says that the cohomology of M can be computed using au-tomorphic forms. More precisely, if A (Γ\G(R)) is the space of automorphic forms onΓ\G(R) then the cohomology of M can be computed as Lie algebra cohomology:

H∗(M,E)⊗ C = H∗(LieG(R),K,A (Γ\G(R))⊗ E). (1)

The natural question is to what extent the mixed (Hodge or Hodge-de Rham) structurecan be described in these terms. It is well-known that this question has no simpleanswer. (For example, while the Hodge filtration is easily seen in the description (1),the weight filtration is quite mysterious.)

In [Mor08] S. Morel introduced a general method of weight truncations in categories ofmixed sheaves. The formalism gives a natural spectral sequence converging to H∗(M,E)in which the E1 terms are related to intersection cohomology groups (with coefficients)of strata of the minimal or Baily-Borel compactification j : M ↪→ M∗ of M . In termsof Morel’s weight truncation functors w6a (cf. [Mor08]) the E1 term has the form

Ep,q1 = Hp+q(M∗, w6dimM+w−pw>dimM+w−pRj∗EH) ⇒ Hp+q(M,EH) (2)

MSC 2010 classification: 14G35 (primary), 11G18 (secondary).

1

2 ARVIND NAIR

where p ≤ 0. (This is a second quadrant spectral sequence with edge terms givenby E0,q

1 = IHq(M∗,E).) The superscript H indicates that we are in Saito’s category ofmixed Hodge modules [Sai90], w is the weight of the natural variation of Hodge structuresupported by E, and EH is the associated Hodge module; this is a spectral sequence ofmixed Hodge structures. In fact one can work in any theory of mixed sheaves in whichE has a lift, e.g. mixed Hodge modules with de Rham rational structure [Sai06] or, ifE is of geometric origin, mixed l-adic complexes [Mor11]. The analogue of (2) is thena spectral sequence in the appropriate category, e.g. mixed Hodge-de Rham structuresover the field of definition, or Galois representations. In any setting, the E1 term iscanonically split, i.e. a direct sum of pure structures (possibly of different weights).

The main result of this paper (contained in Theorem 5.2.1) is the following:

Theorem. The spectral sequence (2) is automorphic. More precisely, there is a de-creasing filtration on the space of automorphic forms A (Γ\G(R)) such that the spectralsequence induced by (1) is (2) (after tensoring with C).

The filtration in the theorem is defined by conditions on the Fourier expansion ofthe constant term along the split central tori of Levi subgroups. In [Fra98, §6] Frankestudied a family of similarly-defined filtrations and described their graded pieces interms of square-integrable automorphic forms on Levi subgroups. The filtration in thetheorem is not one of Franke’s, but rather a degenerate limiting case of these. (1)

The fact that the E1 terms of (2) above are split means that for this spectral sequenceall the extension data is contained in the differentials. These differentials are related(via Eisenstein series) to subtle questions about the vanishing of L-functions at specialpoints, as in the work of Harder [Har93]. Thus the spectral sequence (2) provides anatural general setting in which to study the relationship between the mixed motives inShimura varieties and automorphic forms.

Franke’s methods in [Fra98, §6] can be adapted to the filtration in the theorem to givea more precise description of the E1 term (⊗C) (similar to that for spectral sequencesassociated to filtrations in [Fra98, §7]). In particular, the description of the discrete L2

spectrum conjectured by Langlands and Arthur should give a precise description of theE1 term in terms of parameters, in a way that the mixed structures can be read off fromthe description. This will be discussed in a sequel to this paper where we will also lookat examples. For the moment we show, using the automorphic interpretation, that thespectral sequence (2) degenerates at E1 if the highest weight of E is regular (Theorem5.2.1 (iii)).

As an easy application of the spectral sequence (2) (requiring no automorphic input),we show that the pure (Hodge or Hodge-de Rham) structures appearing as the weightgraded pieces of the cohomology of H∗(M,E) already appear as subquotients of theintersection cohomology (with coefficients) of smaller Shimura varieties related to M ,namely the Baily-Borel boundary strata (Theorem 5.4.3). Thus the genuine interestin the cohomology of noncompact Shimura varieties (as opposed to the intersectioncohomology of their Baily-Borel compactifications) comes from the study of the mixedmotives appearing therein. (The proof of Theorem 5.4.3 requires only some propertiesof Morel’s truncation functors, together with a result from [LR91, HZ01, BW04]. Thereader interested only in this application can read §2, §3.1-3.3 (for notation), §5.1-5.2(for the definition of (2)) and then skip to §5.4.)

1It is natural to ask if Franke’s filtrations give spectral sequences of mixed structures. This is indeedtrue, cf. Remark 5.3.1 (ii). (They do not have the property that the E1 terms are split mixed structures.)Since we have no immediate applications in mind for this we have not pursued it here.

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 3

There is another natural spectral sequence in the context of the mixed structures ap-pearing in Shimura varieties. The cohomology H∗(∂MBS

,E) of the Borel-Serre bound-ary carries a natural mixed Hodge-de Rham structure such that the long exact sequence· · · → H i

c(M,E)→ H i(M,E)→ H i(∂MBS,E)→ · · · is a sequence of mixed HdR struc-

tures. In a series of papers [HZ94a, HZ94b, HZ01] Harris and Zucker prove that thenerve spectral sequence (of Borel and Harder) for the covering of ∂MBS by its closedfaces is a spectral sequence of mixed HdR structures. Except in the case of Q-rankone there seems to be no simple relation between this spectral sequence and the one(s)above. The E1 term in the nerve spectral sequence is (in general) genuinely mixed, i.e.nonsplit.

We give a brief outline of the proof that (2) is automorphic. The proof proceeds viafirst reinterpreting (2) in terms of the weighted cohomology groups defined by Goresky,Harder, and MacPherson [GHM94]. This amounts to relating their construction (on thereductive Borel-Serre compactification) with Morel’s truncations (on the minimal com-pactification); this is Theorem 4.2.1. That there is a relation is hardly surprising sinceMorel’s construction is meant to be an algebraic analogue of [GHM94], but it requiressome proof and the result is of interest in itself. (A special case is applied in [Nai10].)The necessary ideas are well-known: The key input is the action of the ubiquitous localHecke operator introduced by Looijenga [Loo88] and the relation between Hecke weightsand weights in the sense of mixed Hodge theory or Frobenius established in Looijenga-Rapoport [LR91]. As a corollary, we see that the weighted cohomology groups of alocally symmetric variety carry natural mixed Hodge-de Rham structures and also Ga-lois representations. (2) Thanks to Morel’s formalism we can prove some properties ofthese mixed structures (e.g. determine the top weight quotient or bottom weight piece)in some cases. Once the link with [GHM94] is established, results of [Nai99] allows usto appeal to [Fra98] to conclude that the spectral sequence is automorphic (Theorem5.2.1). (At this point properties of the Q-root system of groups giving rise to Shimuravarieties come into play.)

We mention some applications which will be taken up elsewhere. In [Fra08] Frankeshowed that the subspace of Hecke-invariant classes in H∗(M,Q) is a direct summand(as Hecke-modules) and has a particularly simple topological model in terms of an opensubset of the compact dual. It should be possible to describe this summand completelyas a mixed structure (it is of mixed Tate type) using the results here. Secondly, inthe regular case we should have a more detailed description of the mixed structureH∗(M,E) thanks to the degeneration at E1 of the spectral sequence (2). Finally, thevanishing theorems of Borel [Bor87] and Saper [Sap05] for the cohomology of linearlocally symmetric spaces can now be interpreted as vanishing theorems for summandsof the E1 term. In some examples these should give constraints on the weight filtrationof H∗(M,E).

The arguments in the paper are written in the context of mixed Hodge modules, butmany of them generalize with minor changes to other settings of mixed sheaves; ratherthan work in a general setting we make remarks about the necessary changes from timeto time. We have also avoided the formalism of Shimura varieties for simplicity, althoughthis would have been more elegant at some places.

The organization of this paper is as follows. In section 2 we recall Morel’s theoryof weight truncations, as transposed to the setting of mixed Hodge modules (or, moregenerally, any theory of mixed sheaves for varieties over a subfield of C). In section3 we recall background material on locally symmetric varieties and the minimal and

2It may well be that some of these mixed structures are easier to understand in automorphic termsthan H∗(M, E).

4 ARVIND NAIR

reductive Borel-Serre compactifications. In section 4 we prove the basic relation betweenthe weighted complexes and the weight truncations. We also prove some basic propertiesof the resulting mixed structures on weighted cohomology. In section 5 we discuss thespectral sequence (2), its relation with automorphic forms, and the pure structuresappearing as the weight-graded pieces of M . The appendix contains a root systemcalculation required in §4, 5.

Acknowledgements. I am grateful to the Department of Science and Technology of theGovernment of India for its support through a Swarnajayanti Fellowship (DST/SF/05/2006).

2. Weight truncation

2.1. We recall some facts from Saito’s theory of mixed Hodge modules (cf. [Sai88,Sai90], especially §4 of the latter). For any complex algebraic variety X, Saito definesthe abelian category MHM(X) of (algebraic) mixed Hodge modules on X. This is theanalogue in Hodge theory of the category of mixed perverse complexes of [BBD82].When X is a point MHM(X) is equivalent to the category of rational mixed Hodgestructures with polarizable pure subquotients. When X is smooth MHM(X) containspolarizable variations of Hodge structure on X and, more generally, admissible variationsof mixed Hodge structure with polarizable pure subquotients. In general, there is afaithful and exact functor rat from MHM(X) to the category of perverse complexes ofQX -sheaves constructible for an algebraic stratification. rat gives a functor on boundedderived categories:

rat : DbMHM(X)→ Dbc(QX).

For algebraic maps f : X → Y there are functors Rf∗, f∗, Rf!, f! and ⊗ and RHom

between the appropriate derived categories of mixed Hodge modules, and these arecompatible under rat with the corresponding functors on derived categories of complexesof Q-sheaves (3). The cohomology functor H0 : DbMHM(X) → MHM(X) goes to theperverse cohomology functor pH0 on Db

c(QX) under rat, i.e. rat ◦H0 = pH0 ◦ rat.Mixed Hodge modules have weight filtrations and for K ∈ MHM(X) and a ∈ Z there

is a canonical short exact sequence in MHM(X)

0 −→ w6aK −→ K −→ w>aK −→ 0

where w6aK is the maximal subobject of K with weights 6 a and w>aK is the maximalquotient object of K with weights > a (equivalently, weights > a + 1). This is, up tounique isomorphism, the unique short exact sequence 0→ K ′ → K → K ′′ → 0 with K ′

having weights 6 a and K ′′ having weights > a. By strictness of morphisms in MHM(X)with respect to weight filtrations, this implies that K 7→ w6aK and K 7→ w>aK =w>a+1K are functors. Represent a complex K of DbMHM(X) by a bounded complex(Ci)i∈Z of objects in MHM(X). Setting (w6aK)i := w6aC

i and (w>aK)i := w>aCi can

be shown to define functors w6a, w>a : DbMHM(X)→ DbMHM(X).S. Morel’s original definition of these functors (rather, the analogues in the `-adic

context) is via a t-structure. Following [Mor08, §3], define wD6a = wD6aMHM(X)(respectively, wD>a) to be the full subcategory of DbMHM(X) with objects

Ob wD6a = {K ∈ DbMHM(X) : H i(K) has weights 6 a for all i}.(respectively, with objects

Ob wD>a = {K ∈ DbMHM(X) : H i(K) has weights > a for all i}).Then (wD6a,wD>a) is a t-structure on DbMHM(X). (The proof of this from [Mor08]carries over verbatim, the main point being that Homi(K,L) = 0 if K has weights 6 a

3We follow Saito’s notation in [Sai88, Sai90], except that we use the old-fashioned notation Rf∗ etc.for functors for mixed Hodge modules and for Q-sheaves.

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 5

and L has weights > a+ i [Sai90, 4.5.3]. Recall that a complex K ∈ DbMHM(X) is saidto have weights 6 a (resp. > a) if H i(K) is of weights 6 i + a (resp. > i + a) for all i([Sai90, 4.5]).) The corresponding truncation functors are

w6a, w>a : DbMHM(X)→ DbMHM(X)

(the right adjoint to wD6a ⊂ DbMHM(X) and the left adjoint to wD>a ⊂ DbMHM(X),respectively). This t-structure has some unusual properties: wD6a and wD>a are fulltriangulated subcategories of DbMHM(X). The functors w6a and w>a commute withshift and take distinguished triangles to distinguished triangles. Any object in the heartwD6a ∩ wD>a of the t-structure is isomorphic to zero. The cohomology sequences (forthe usual cohomology functor H0) associated with the distinguished triangle w6aK −→K −→ w>aK

+1−→ are short exact and H i(w6aK) = w6aHi(K) and H i(w>aK) =

w>aHi(K). (The proofs of these facts in [Mor08] carry over, mutatis mutandis, to our

setting.)A key result is the following:

2.1.1. Theorem. (Morel [Mor08, Thm 3.1.4]) If j : U ↪→ X is a nonempty open subsetand K is a pure polarizable Hodge module of weight a on U then w>aj!K = j!∗K =w6aRj∗K.

Suppose now that we are given a stratification of X, i.e. a partition X =∐ri=0 Si

into locally closed subvarieties with each Sd open in X − t06j<dSj . Let iSd : Sd ↪→ Xbe the inclusion. Let a = (a0, . . . , ar) ∈ (Z ∪ {±∞})r+1. As in [BBD82, 1.4], one canglue the t-structures (wD6ai(Si),wD>ai(Si)) on DbMHM(Si) to get a new t-structure(wD6a,wD>a) on DbMHM(X). Thus wD6a (respectively, wD>a) is the full subcategoryof DbMHM(X) consisting of complexes K such that i∗SdK ∈

wD6ad(Sd) for d ∈ {0, . . . , r}(respectively, i!SdK ∈ wD>ad(Sd) for d ∈ {0, . . . , r}). The corresponding truncationfunctors are denoted

w6a, w>a : DbMHM(X)→ DbMHM(X).

These take distinguished triangles to distinguished triangles and commute with the shift([Mor08, Prop. 3.4.1]). When a = (a, . . . , a) we recover the previous construction, i.e.w6(a,...,a) = w6a ([Mor08, Lemme 3.3.3]).

The following lemma, which we will use later, is stronger than the Hom-vanishingrequired to prove that (wD6a,wD>a) is a t-structure (which follows by taking H0). Theproof in [Mor08] works mutatis mutandis.

2.1.2. Lemma. ([Mor08, Prop. 3.4.1]) If K ∈ wD6a and L ∈ wD>a then RHom(K,L) =0.

For k ∈ {0, . . . , r} and a ∈ Z∪ {±∞} define wk6a := w6(+∞,...,+∞,a,+∞,...,+∞) where aappears in the kth place.

2.1.3. Lemma. ([Mor08, Prop. 3.3.4]) (i) w6(a0,...,ar) = wr6ar ◦ wr−16ar−1

◦ · · · ◦ w06a0

.(ii) For K ∈ DbMHM(X) and k ∈ {0, . . . , r} there is a distinguished triangle wk6aK −→

K −→ Rik∗w>ai∗kK

+1−→.

2.2. We collect some facts for use later. For X =∐ri=0 Si and d = 0, . . . , r let Ud :=∐d

i=0 Sd and let

Ud−1� � jd // Ud Sd? _

idoo (2.2.1)

be the inclusions.

6 ARVIND NAIR

2.2.1. Lemma. There are natural isomorphisms of functors

w6(a0,...,ar)R(jr · · · j1)∗ = wr6arRjr∗ ◦ wr−16ar−1

Rjr−1∗ ◦ · · · ◦ w16a1

Rj1∗ ◦ w6a0 .

= wr6arRjr∗ ◦ w6(a0,...,ar−1)R(jr−1 · · · j1)∗.

Proof. The lemma follows from Lemma 2.1.3 (ii) and the observation that for k < d wehave the identity wk6aRjd∗ = Rjd∗w

k6a. To prove this identity, it suffices to show that

the two obvious natural transformations

wk6aRjd∗ ← wk6aRjd∗wk6a → Rjd∗w

k6a

are isomorphisms. For the second transformation, note thatRjd∗ takes wD6(+∞,...,a,...,+∞)(Ud−1)to wD6(+∞,...,a,...,+∞,+∞)(Ud) (where the a is in the kth place in each case), so that wk6a isthe identity on any object like Rjd∗wk6aK. For the first, note that for K on Ud−1, apply-ing wk6a◦Rjd∗ to the t-structure distinguished triangle for (wD6(+∞,...,a,...,+∞)(Ud−1),wD>(+∞,...,a,...,+∞)(Ud−1))gives a distinguished triangle

wk6aRjd∗wk6aK −→ wk6aRjd∗K −→ wk6aRjd∗w>(+∞,...,a,...,+∞)K

+1−→ .

The third term is zero sinceRjd∗ takes wD>(+∞,...,a,...,+∞)(Ud−1) into wD>(+∞,...,a,...,+∞,+∞)(Ud)(because i!dRjd∗ = 0) and wD6a∩wD>a = {0} for any a, in particular for a = (+∞, . . . , a, . . . ,+∞,+∞).�

2.2.2. Lemma. (i) If a 6 b (i.e. ad 6 bd for all d) then w6aw6b = w6a and w>aw6b =w6bw>a.

(ii) If (a0, · · · , ar) is nonincreasing then w6a is right t-exact, i.e. w6a(D60) ⊂ D60.

Proof. (i) follows from the inclusion wD6a ⊂ wD6b. (ii) If ad > ad+1 for all d, then

w6(a0,...,ar) = w6(a0,...,ar) ◦ · · · ◦ w6(a0,a1,...,a1) ◦ w6(a0,...,a0)

= w6(+∞,...,+∞,ar) ◦ · · · ◦ w6(+∞,a1,...,a1) ◦ w6(a0,...,a0)

Now w6(+∞,...,+∞,ad,...,ad) is the functor w6(+∞,ad) for the coarser stratification of Xwith two strata S′0 = tk6d−1Sk and S′1 = tk≥dSk. In such a situation any w6(+∞,b)preserves D60. Indeed, if K ∈ D60, then i∗S′0

w6(+∞,b)K = i∗S′0K ∈ D60 by definition

and i∗S′1w6(+∞,b)K = w6bi

∗S′1K ∈ D60 since i∗S′1K ∈ D

60 and w6b is t-exact. �

For later use we note some properties of the “diagonal” functors w6a = w6(a,...,a).

2.2.3. Lemma. We have(i) w>aw6b = w6bw>a

(ii) For K ∈ wD6a ∩ wD>a there is a canonical isomorphism K = ⊕kHk(K)[−k].(iii) For K,L ∈ wD6a ∩ wD>a, Hom(K,L) = ⊕kHom(Hk(K), Hk(L)).

Proof. (i) is a special case of Lemma 2.2.2(i).(ii) We prove this by induction on the cardinality of {k ∈ Z : Hk(K) 6= 0}, the case

of cardinality one being obvious. Let K ∈ wD6a ∩ wD>a with card({k ∈ Z : Hk(K) 6=0}) = i. Let k0 be the minimal element of this set, so that τ6k0K = Hk0(K)[−k0]. Inthe distinguished triangle

Hk0(K)[−k0] −→ K −→ τ>k0+1K+1−→

the first two terms belong to wD6a ∩ wD>a and hence τ>k0+1K belongs to wD6a ∩wD>a. By the induction hypothesis there is a canonical isomorphism τ>k0+1K =

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 7

⊕k>k0+1Hk(K)[−k]. Now Hk(K)[−k] is pure of weight a− k, so τ>k0+1K is of weights

6 a − k0 − 1, while Hk0(K)[−k0 + 1] is pure of weight a − k0 + 1. Thus in the dis-tinguished triangle above the +1 morphism must be zero, i.e. the triangle must split.The splitting is unique: Two splittings differ by a morphism τ>k0+1K → Hk0(K)[−k0],which must be zero for weight reasons. So there is a canonical isomorphism K =Hk0(K)[−k0]⊕ τ>k0+1K, completing the inductive step.

(iii) In the canonical isomorphisms given by (ii), an element of Hom(K,L) is a sumof its components in Hom(Hk(K)[−k], Hj(L)[−j]) for various k, j. If k < j thenHom(Hk(K)[−k], Hj(L)[−j]) = 0 by the standard t-structure vanishing. If k > j thenHom(Hk(K)[−k], Hj(L)[−j]) = 0 since Hk(K)[−k] has weight a−k and Hj(L)[−j] hasweight a− j > a− k. �

In (ii), each Hk(K)[−k] is pure (of weight a − k) and so further breaks up canoni-cally as a sum of shifts of simple pure Hodge modules supported on irreducible closedsubvarieties.

There is a pointwise criterion for the weights of a complex K ∈ DbMHM(X): K isof weights 6 a if and only if for any point ix : {x} ↪→ X, the mixed Hodge structureH i(i∗xK) has weights 6 i + a ([Sai90, 4.6.1]). There is a t-structure (cD60, cD>0) onDbMHM(X) related to the classical t-structure on Db

c(QX) (see [Sai90, 4.6]). It isdefined by K ∈ cD60 if and only if rat(K) ∈ Db

c(QX)60. Let cH0 be the correspondingcohomology functor, so that rat ◦ cH0 = H0 ◦ rat.

For a smooth variety X we say that a mixed Hodge module K ∈ MHM(X) is smoothif rat(K)[−dimX] is a local system. (4) A complex K ∈ DbMHM(X) is smooth ifH i(K) is smooth for all i.

2.2.4. Lemma. Suppose that X is smooth and K ∈ DbMHM(X) is smooth. Then(i) H i(K) = cH i−dimX(K)[dimX] for all i(ii) K ∈ wD6a if, and only if, for each x ∈ X, H i(i∗xK) has weights 6 a − dimX

for all i.

Proof. (i) follows from the identity pH i(rat(K)) = H i−dimX(rat(K))[dimX] in Dbc(QX).

(ii) Since K is smooth, H i(i∗xK) = cH i(i∗xK) = i∗xcH i(K) = i∗xH

i+dimX(K)[−dimX].By definition, K ∈ wD6a if H i(K) has weights 6 a for all i, which translates to thecondition that H i(i∗xK) has weights 6 a− dimX for all i. �

2.3. Mixed sheaves. Everything so far works in any theory of A-mixed sheaves (inthe sense of Saito [Sai06], A a field of characteristic zero) for varieties over a fixedfield k ⊂ C. Recall that such a theory gives an A-linear abelian category M (X) forevery variety X/k with an A-linear forgetful functor For to perverse A-sheaves onX(C), satisfying a number of axioms similar to the properties of mixed Hodge modules[Sai88, Sai90]. In particular, objects have weights filtrations, there are four functorsassociated with a morphism of varieties satisfying conditions with respect to weights,and a (Verdier) duality functor. The basic example is the theory of Q-mixed sheavesgiven by M (X) = MHM(X(C)) with For = rat.

A second example is the theory of Q-mixed sheaves defined in [Sai06, 1.8(ii)], which wewill denote MHM(X/k) and refer to as mixed Hodge modules with de Rham structure.In essence, for X/k, MHM(X/k) is the category of mixed Hodge modules on X(C) suchthat the underlying bifiltered D-module on X(C) comes (by extension of scalars) from a

4By Theorem 3.27 of [Sai90] (cf. also the remark after the proof of the theorem regarding the algebraiccase) a smooth Hodge module on a smooth variety X is the same thing as an admissible variation ofmixed Hodge structure with polarizable pure subquotients.

8 ARVIND NAIR

bifiltered D-module on X/k. (For X = Spec(k) this gives the category of mixed Hodge-de Rham structures over k with polarizable graded pieces.) Working in this theoryof mixed sheaves allows us to keep track of the de Rham rational structure in variouscohomology groups, i.e. to work with Hodge-de Rham structures everywhere.

In [Mor11] Morel constructs a theory of mixed l-adic complexes on varieties over anumber field k ⊂ C (l is any fixed prime). ([Mor11] is written with k = Q but works overany number field.) The category of l-adic mixed perverse complexes having a weightfiltration is denoted Ml(X); this has a forgetful functor to the category of perverse Ql-complexes in the classical topology on X(C) which factors through the perverse l-adiccomplexes in the etale topology. Morel [Mor11] shows that given a morphism f : X → Ythe four functors f∗, f !, Rf∗, Rf! on l-adic complexes can be lifted to functors betweenthe derived categories DbMl(X) and DbMl(Y ). Working in this theory will allow us toconclude that some statements hold in the context of Gal(Q/k)-modules. (The theoryMl(·) is not quite a theory of mixed Ql-sheaves in the sense of Saito as one does notknow semisimplicity of pure objects in Ml(·). Nevertheless, the preceding discussiongoes through in this theory, with the exception of the remark after Lemma 2.2.3 aboutthe further decomposition of cohomology objects into simple ones, i.e. intersectioncomplexes.)

2.3.1. Remark (Effectivity). Recall that a Hodge structure H is effective if Hp,q 6= 0 onlyif p, q ≥ 0. A mixed Hodge structure is effective if its weight graded pieces are effective.(In particular, its weights are ≥ 0.) For example, if X is a complex algebraic variety themixed Hodge structures on H∗(X,Q) and H∗c (X,Q) given by Deligne’s Hodge theory[Del74] are effective. More generally, if X is smooth and V is an effective variation ofHodge structure on X then H∗(X,V) and H∗c (X,V) are effective (cf. [Sai90, 3.10]).

For a complex K ∈ DbMHM(X) we will say K is effective if the mixed Hodgestructure Hk(i∗xK) is effective for all ix : {x} ↪→ X and k ∈ Z. Effective complexes forma triangulated subcategory of DbMHM(X). The following can be proved by standardarguments: (1) Effectivity is preserved by the four functors Rf∗, Rf!, f

∗, f ! associatedwith a morphism f of algebraic varieties (2) For a Whitney stratification, the functorsw6a and w>a applied to objects constructible for the stratification preserve effectivity.Thus with the notation of 2.2 objects like w6aRj∗QH

U are effective. (We will not usethese statements below so we do not give complete proofs here.)

A mixed Hodge-de Rham structure (respectively, complex in DbMHM(X/k)) is effec-tive if and only if its underlying Hodge structure (respectively, underlying complex inDbMHM(X(C))) is effective. So the statements (1) and (2) also hold in this context.

3. Locally symmetric varieties and their compactifications

3.1. Locally symmetric varieties. Let G be a simply-connected Q-algebraic groupsuch that the adjoint group Gad is Q-simple. We assume that the symmetric space D ofmaximal compact subgroups of G(R) is a Hermitian symmetric domain. Let Γ ⊂ G(Q)be a neat arithmetic subgroup. The quotient

M = Γ\D

is a smooth quasiprojective complex algebraic variety. We assume that M is noncompact(equivalently, that G is Q-isotropic).

As a notational convention, for subgroups J / H ⊂ G, let ΓH := Γ ∩ H(R) andΓH/J := ΓH/ΓJ . Since Γ is neat, so are ΓH and ΓH/J .

3.2. Rational parabolic subgroups. (cf. [AMRT75, III.4.1–III.4.2] or [LR91, 6.1])Let P be a rational parabolic subgroup of G. The Levi quotient M = P/RuP has an

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 9

almost-direct product decomposition

M = M` ·A ·Mh

where A is the Q-split part of the centre of M and such that(i) Mh contains no nontrivial proper Q-anisotropic connected subgroup(ii) the symmetric space of Mh(R) is of Hermitian type.

Since G is simply-connected, so is Mder, and therefore Mder = Mder` ×Mh. Thus Mh

and Mder` are simply-connected. Note that Mh satisfies the same assumptions as G in

(3.1) (except that it may be Q-anisotropic).If P is a maximal rational parabolic subgroup, then in addition to (i) and (ii),(iii) If U is the centre of W := RuP then the quotient V := W/U is abelian. The

adjoint action of A = Gm on LieU(R) is by the square χ2 of a generator of thecharacter group. If V = W/U is nontrivial then A acts by χ on LieV (R).

(iv) Mh centralizes U(v) M`(R)A(R)0 acts (via the adjoint action) properly and transitively on an open

cone C ⊂ LieU(R). The stabilizer of a point in C is maximal compact, identi-fying C/A(R)0 with the symmetric space of Mder

` (R).The projection from a parabolic P to its Levi quotient will be denoted νP .

Fix a minimal parabolic P0. The standard (with respect to P0) maximal parabolicsubgroups are partially ordered by

Q ≺ Q′ ⇔ U ⊂ U ′.Under our assumption that Gad is Q-simple this is a total order. Define a map P 7→ P+

from standard rational parabolics to standard maximal rational parabolics as follows: IfP = Q1∩· · ·∩Qd with Qi ≺ Qi+1, then P+ = Qd, i.e. P+ is the last maximal parabolic(i.e. the one with largest U) containing P in the ordering.

3.2.1. Lemma. The map P 7→ P+ can be extended to all rational parabolics compatiblywith conjugation, i.e. with (gPg−1)+ = gP+g−1. It has the following properties:

(i) If P ⊂ Q then P+ � Q+.(ii) If P+ = Q and Q/W = MhAM` then νQ(P ) contains Mh.

(iii) If P+ = Q+ then P and Q are Γ-conjugate if and only if they are ΓQ+-conjugate.(iv) If Q is maximal and Q/W = M`AMh, then there is a bijection between Γ-

conjugacy classes in {P : P+ is Γ-conjugate to Q} and ΓM`-conjugacy classes

of parabolics in M`. (The bijection is the following: If γP+γ−1 = Q then P 7→νQ(γPγ−1) ∩M`.)

(v) Two standard parabolics P1 and P2 are associates (i.e. have conjugate Levisubgroups) if and only if P+

1 = P+2 (and hence M1,h = M2,h) and M1,` and M2,`

are conjugate in M+1,` = M+

2,`.

3.3. Minimal (or Satake-Baily-Borel) compactification. This is a normal projec-tive algebraic variety M∗ with an open immersion

j : M ↪→M∗.

As a topological space, M∗ is the quotient M∗ = Γ\D∗ where

D∗ =∐

F rationalF (3.3.1)

is the union of all rational boundary components equipped with the Satake topology.The action of G(Q) on D extends to a continuous action on D∗; the stabilizer of arational boundary component F is a maximal Q-parabolic subgroup or G itself (thecase F = D). Thus the proper rational boundary components (i.e. those other than D)

10 ARVIND NAIR

are indexed bijectively by maximal Q-parabolic subgroups, and hence M∗ has a naturalstratification induced by (3.3.1) in which M is the open and dense stratum and theboundary strata are indexed by Γ-conjugacy classes of such subgroups. If F ⊂ D∗ isa rational boundary component and Q the associated parabolic, then (M`A)(R) actstrivially on F while Mh(R) acts properly transitively, identifying F with the symmetricspace of Mh(R). The stratum S of M∗ covered by F is S = ΓMh

\F . Thus M∗ has astratification in which each stratum is a smooth locally symmetric variety satisfying thesame assumptions as M in 3.1.

The Baily-Borel theory [BB66] puts an analytic structure on M∗ inducing the givencomplex structure on each stratum, and this structure is unique. Moreover, M∗ hasa unique structure of projective algebraic variety compatible with this analytic struc-ture. The stratification induced by (3.3.1) is a Whitney stratification by smooth sub-varieties. The closure relations between the strata are determined by those betweenrational boundary components in D∗, which in turn are determined by the associatedparabolic subgroups: Let F, F ′ be rational boundary components with stabilizers Q,Q′.Then

F ′ ⊂F ⇐⇒ U ′ ⊃ U (3.3.2)The minimal compactification has a hereditary nature: The closure of a boundary stra-tum in M∗ is isomorphic to its minimal compactification. If Γ′ ⊂ Γ is of finite indexand M ′∗ = Γ′\D∗ then the map M ′∗ →M∗ is a finite morphism of varieties.

For Gad 6= PGL(2) the variety M∗ is the closure of M in the projective embeddinggiven by a sufficiently large power of the canonical bundle; this shows that j : M ↪→M∗

is defined over the same field as M . (If Gad = PGL(2) this is clear anyway.)

3.4. Reductive Borel-Serre (RBS) compactification. The RBS compactificationM , first constructed by Zucker in [Zuc83] by modifying the earlier construction of Boreland Serre, should perhaps be thought of as a desingularization of M∗, although it is ingeneral neither algebraic nor smooth. The construction is similar to that of M∗: Thereis an extension D of D which is the union of locally closed strata indexed by Q-parabolicsubgroups of G:

D =∐

PDP . (3.4.1)

If M = P/RuP is the Levi quotient, DP = M(R)/A(R)0KM where A is the Q-splitcentre of M and KM is the image of K ∩ P (R) in M(R). (5) D carries a topology forwhich DP is contained in the closure of DQ if and only if P ⊂ Q. There is a continuousaction of G(Q) on D and the quotient

M = Γ\Dis the RBS compactification of M . (3.4.1) induces a decomposition of M into locallyclosed strata indexed by Γ-conjugacy classes of Q-parabolic subgroups of G:

M =∐

Γ\{P}

MP , where MP = ΓM\DP = ΓM\M(R)/A(R)0KM . (3.4.2)

The RBS construction can be made for spaces like MP and the closure of a stratumin M is naturally identified with its RBS compactification MP . Note that M can havestrata of odd real dimension.

The identity mapping of M extends to a proper continuous map

p : M →M∗.

5In general, M(R)/A(R)0KM is the product of a symmetric space and a Euclidean space; we willrefer to it as a symmetric space, and its quotient by an arithmetic group as a locally symmetric space inwhat follows. RBS compactifications can be constructed for arbitrary locally symmetric spaces in thissense.

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 11

This was proved in [Zuc83]; we describe p following [GHM94, §22]. For a rationalparabolic P let M = P/RuP and M = MhAM` as in (3.2). Then we have

DP = M`(R)/KM`×Mh(R)/KMh

where KMhand KM`

are the respective intersections with Mh(R) and M`(R) of KM .Recall that the rational boundary component associated with P+ is F = Mh(R)/KMh

.Thus projection to the second factor defines a map DP → F . Varying P defines aΓ-equivariant continuous extension of the identity

p : D → D∗

and this gives p : M → M∗ (cf. [GHM94, Cor. 22.5]). Using Lemma 3.2.1 it is easyto see that on the closure MP of the P -stratum in M the map p has the followingdescription: Suppose that M = M`AMh is the decomposition of the Levi quotient of P .Then

MP = ΓM`\M`(R)/KM`

× ΓMh\Mh(R)/KMh

and p|MPis given by projection to the second factor followed by pMh

: ΓMh\Mh(R)/KMh

→(ΓMh

\Mh(R)/KMh)∗. (Recall that the target is identified with the closure of the P+-

stratum in M∗.) In particular, over the P+-stratum of M∗, the fibre is isomorphic toΓM`\M`(R)/KM`

.

3.5. Correspondences. There is a plentiful supply of finite correspondences on M andthe compactifications mentioned above. For g ∈ Γ and Γ′ ⊂ Γ ∩ g−1Γg of finite indexlet M ′ = Γ′\D. The pair of morphisms

(c1, c2) : M ′ ⇒M by c1(Γ′x) = Γxc2(Γ′x) = Γgx (3.5.1)

give a finite correspondence on M . When Γ′ = Γ ∩ g−1Γg this is called the Heckecorrespondence associated with g; in general (3.5.1) is a finite cover of the Hecke corre-spondence associated with g. (3.5.1) extends to a finite correspondence

(c1, c2) : M ′∗ ⇒M∗ (3.5.2)

where M ′∗ = Γ′\D∗. (3.5.1) also extends to RBS compactifications: For g ∈ G(Q) andΓ′ ⊂ Γ∩ g−1Γg of finite index, letM ′ be the RBS compactification of M ′ = Γ′\D. Thenthere is a correspondence

(c1, c2) : M ′ ⇒M (3.5.3)which covers (in the obvious sense) (c1, c2) : M ′∗ ⇒ M∗ and has finite fibres ([GM03,6.3]).

3.6. Local Hecke actions. (cf. [Loo88, LR91, GHM94]) Let S be a stratum of M∗.Fix a rational boundary component F covering it and let Q be the maximal parabolicstabilizing it. Fix a lift A ⊂ Q of the split centre of the Levi quotient. By [Loo88, 3.6]there exist elements a ∈ A(Q) such that:

(i) χ(a) ∈ Z>0

(ii) aΓQa−1 ⊂ ΓQ.(Here χ is the generator of X∗(A) as in 3.2.) Such elements are called divisible. Thefollowing summarizes some well-known facts about Looijenga’s local Hecke operators:

3.6.1. Lemma. Let a ∈ A(Q) be divisible. Let W = RuQ.(i) There is a closed W (R)-invariant neighbourhood BF of F in Star(F ) such that(a) ΓQ\Star(F )→ Γ\D∗ identifies ΓQ\BF with a closed nbhd of S in Star(S)(b) the endomorphism of ΓQ\Star(F ) defined by Φ(ΓQx) = ΓQax restricts to an

endomorphism of ΓQ\BF fixing S pointwise.

12 ARVIND NAIR

(ii) There is a closed W (R)-invariant neighbourhood V Q of DQ in D such that

(a) ΓQ\V Q is identified with a closed nbhd of MQ in M

(b) Φ(ΓQx) = ΓQax defines an endomorphism of ΓQ\V Q fixing MQ pointwise.(c) Φ respects the stratification of ΓQ\V Q induced by M =

∐P MP .

The morphism Φ is a component of the restriction to ΓQ\V Q of the Hecke correspondencegiven by ΓaΓ.

(iii) Let p : D → D∗ be as in 3.4. If V Q is as in (ii), let

BQ :=∐

P+=Q

V Q ∩MP .

(This is preserved by Φ by (ii)(c).) Then BF = p(BQ) satisfies (a) and (b) of (i), andΦ and Φ are related by p ◦ Φ = Φ ◦ p.

Proof. (i)(a) is well-known (cf. [Loo88]): For BF one may take any closed W (R)-invariant nbhd of F in Star(F ) which in ΓQ-invariant and on which Γ-equivalence andΓQ-equivalence coincide. If, in addition, BF ∩D is invariant under the geodesic actionof the semigroup {a ∈ A(R)0 : χ(a) > 1} then (b) also holds.

For (ii) we recall some facts from [GM03] about how Hecke correspondences break upnear the boundary:

(1) For g ∈ G(Q) and a parabolic P , the double coset ΓgΓ meets P (Q) in finitelymany double cosets for ΓP :

ΓgΓ ∩ P (Q) =∐

iΓP giΓP (3.6.1)

For a double coset ΓP giΓP here write g = γ1giγ2. Let Γ′ = Γ ∩ g−1Γg. ThenP 7→ γ1Pγ

−11 = g−1γ−1

2 Pγ2g gives a bijection between the double cosets on theright of (3.6.1) and the Γ′-conjugacy classes of parabolics Q with both Q andgQg−1 Γ-conjugate to P . (See [GM03, 7.3].)

(2) Consider the Hecke correspondence (c1, c2) in (3.5.1) with Γ′ = Γ ∩ g−1Γg.The restriction to the P -boundary stratum MP is the correspondence (c1, c2) :c−1

1 (MP ) ∩ c−12 (MP )⇒MP . This breaks up according to (3.6.1):

c−11 (MP ) ∩ c−1

2 (MP ) =∐

iM ′Qi

where Qi corresponds to gi under the bijection and M ′Q = Γ′\DQ is the Q-stratum of M ′. The restriction of (c1, c2) to M ′Qi is isomorphic to the cor-respondence on MP of the form (3.5.1) given by νP (gi) ∈ (P/RuP )(Q) forΓ′P/RuP = νP (ΓP ∩ g−1

i ΓP gi) (cf. [GM03, 8.1, 8.6]).(3) In fact this happens over a suitable neighbourhood of MP in M ([GM03, Prop.

7.3]). Let VP be a closed ΓP -invariant and (RuP )(R)-invariant neighbour-hood of DP in D on which Γ-equivalence and ΓP -equivalence coincide. As-sume further that VP is invariant under the geodesic action of the semigroup{a ∈ A(R)0 : χ(a) > 1} (Without the (RuP )(R)-invariance condition this iscalled a Γ-parabolic neighbourhood of DP in [GM03].) Then ΓP \VP is a neigh-bourhood of MP in M and c−1

1 (ΓP \VP ) ∩ c−12 (ΓP \VP ) is a disjoint union of

components indexed by (3.6.1). The gi component is a neighbourhood of M ′QiinM ′ with properties similar to VP .

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 13

Now if a ∈ P (Q) there is a distinguished double coset in (3.6.1), namely ΓPaΓP . LetV ′P = VP ∩ a−1VP . The corresponding component of the restriction of the Hecke corre-spondence is

(c1, c2) : Γ′P \V ′P ⇒ ΓP \VPc1(Γ′Px) = ΓP xc2(Γ′Px) = ΓP ax

(3.6.2)

where Γ′P = ΓP ∩ a−1ΓPa. Suppose further that a ∈ (RdP )(Q) (the split radical). Inthis case (3.6.2) is a covering of the trivial correspondence when restricted to MP .

Now let Q, A be as in the lemma. If P ⊂ Q then RdQ ⊂ RdP , so it follows thata ∈ A(Q) ⊂ (RdQ)(Q) induces local correspondences on neighbourhoods ΓP \VP foreach P ⊂ Q. Letting V Q := ∪P⊂QVP , V ′Q := ∪P⊂QV ′P , and Γ′Q = ΓQ ∩ a−1ΓQa we geta local correspondence

(c1, c2) : Γ′Q\V′Q ⇒ ΓQ\V Q (3.6.3)

on the closed neighbourhood ΓQ\V Q of the closure MQ, and which is a covering of thetrivial correspondence on each MP ⊂MQ.

Finally, suppose that a is divisible. Since a−1ΓQa ⊃ ΓQ we have a−1ΓPa ⊃ ΓP for allP ⊂ Q since a ∈ P (Q). Thus Γ′P = ΓP for all P ⊂ Q. Moreover, we can take V ′P = VPfor all P since aVP ⊂ VP . Thus Φ = c2 defines an endomorphism of ΓQ\V Q fixing eachMP for P ⊂ Q pointwise.

(iii) Since p ◦ Φ = Φ ◦ p on (ΓQ\BQ) ∩M they are the same. �

3.7. Coefficient systems. A finite-dimensional algebraic representation of G(C) on acomplex vector space E defines the local system EM = E×ΓD = E×D/(e, x) ∼ (γ e, γ x)on M . Under our assumptions in 3.1, Zucker [Zuc81, §4] shows that EM underlies(nonuniquely) a polarizable variation of Hodge structure. We describe this, following[Zuc81, §4] and [LR91, §4]. (6)

There is a natural choice of metric on EM . Fix a basepoint x0 ∈ D and let K0 be itsisotropy. Choose a Hermitian form 〈 , 〉0 on E which is K0-invariant and in which p0 (theorthogonal complement of LieK0 in the Cartan decomposition LieG(R) = LieK0 ⊕ p0)acts by self-adjoint transformations. This gives a metric on the trivial bundle D × Eby setting 〈e, f〉gx0 := 〈g−1e, g−1f〉0. This metric is G(R)-invariant, hence descends toa metric on EM . Such a metric is called admissible. It is unique up to scalar multiplesif E is irreducible.

For x ∈ D let Kx be the isotropy subgroup. The connected centre of Kx is isomorphicto U(1) and its character group can be identified with a subgroup of Q isomorphic toZ in such a way that the character of the adjoint action on TxD is 1. Let E be anirreducible complex representation of G(C). Considering E as a U(1)-module via uxgives a decomposition

E = E(m)x ⊕ E(m− 1)x ⊕ · · · ⊕ E(m− w)xwhere m ∈ Q, w ∈ Z are such that E(m)x and E(m−w)x are nonzero. The assignmentx 7→ ⊕k6m−pE(k)x defines a filtration of the trivial bundle E ×D by G-invariant holo-morphic subbundles. Dividing by Γ gives a filtration of EM which defines a variation ofcomplex Hodge structure of weight w on D.

If E is a real representation, i.e. E is a real vector space with a homomorphism GR →GL(E), then w = 2m (cf. [Zuc81, 1.7]) and the above gives a variation of real Hodgestructure on D × E: Write E ⊗ C as a sum of irreducibles and take the above complexvariation on each one. (This does not depend on the decomposition into irreducibles.)We can choose the Hermitian form 〈 , 〉 on E ⊗ C to be the Hermitian extension ofan R-bilinear form on E. The variation is polarized by the form (e, f)x := 〈C−1

x e, f〉x6This would be much more elegant in the setting of Shimura varieties.

14 ARVIND NAIR

where Cx is the Weil operator of the Hodge structure on EM,x⊗C = E⊗C. (Note thatthe metric and the polarization determine each other.) This polarization is evidentlyG(R)-invariant, so dividing by Γ gives a polarized variation of real Hodge structure onM , pure of a single weight w if E is irreducible.

Finally, if E is a Q-irreducible rational representation of G there is a nondegeneratesymmetric (if w is even) or antisymmetric (if w is odd) G-invariant bilinear form ( , ) onE such that the associated form 〈e, f〉x := (Cxe, f)x is an admissible Hermitian metricon EM . Thus we have a polarizable irreducible variation of rational Hodge structure ofweight w on EM . By [Sai90, Theorem 3.27] (cf. also the remark after 3.27 regarding thealgebraic case) a polarizable variation of Hodge structure gives a smooth Hodge module

EHM [dimM ] ∈ MHM(M)

which is pure of weight w+dimM , irreducible, and polarizable, and has rat(EHM (dimM ]) =EM [dimM ]. Note that EHM is effective.

3.7.1. Remark. It is well-known that M , being a connected component of a Shimuravariety at finite level, has a model over a number field k contained in C. (The theoryof canonical models shows that k is, up to an abelian extension, independent of thecongruence subgroup Γ.) Let M (·) be a theory of A-mixed sheaves over k (cf. 2.3).If the rational local system EM is of geometric origin (in the sense of [BBD82, Sai06])then EM ⊗A underlies an object EM

M ∈ DbM (M). (If A = Q and the forgetful functorto perverse sheaves on X(C) factors through mixed Hodge modules then EM

M maps toEHM .) The local systems EM are known to be of geometric origin in many cases whereM can be related to a moduli problem involving abelian varieties, cf. [Pin92, 5.6], butnot in general.

For the particular theory of mixed sheaves MHM(·/k) in [Sai06, 1.8(ii)] (mentionedearlier in 2.3), the local system EM coming from a Q-rational representation of G un-derlies an object in the category MHM(M/k) where k is the field of definition of M .

A well-known prescription (of Langlands, cf. [Pin92, 5.1]) makes EM ⊗ Ql into anl-adic sheaf. For Morel’s theory of mixed l-adic complexes (cf. 2.3) EM ⊗ Ql underliesa mixed l-adic sheaf if EM is of geometric origin (in the sense of [BBD82]).

4. Mixed structures

4.1. Weighted complexes. Let P0 be the fixed minimal parabolic and A0 the splitcentre of the Levi quotient M0. For a standard parabolic P ⊃ P0 with Levi quotient MA(i.e. A is the split centre of the Levi and M is its natural complement), the inclusionof split radicals RdP ⊂ RdP0 induces an inclusion A ⊂ A0. Thus ν ∈ X∗(A0)Q givesν|A ∈ X∗(A)Q by restriction. For nonstandard P an element of X∗(A)Q is defined byconjugation. (Since X∗(A) = X∗(P ) one sees that this is well-defined.) Thus ν definesa (quasi)character on every split part of a Levi quotient.

Let EM be the local system given by an irreducible rational representation of G (asin 3.7). For each ν ∈ X∗(A0)Q, the weighted cohomology complex

W>νC(EM )

is an object in the derived category of complexes of Q-sheaves on M with cohomologysheaves constructible for the standard stratification ofM . It is defined in [GHM94] via anexplicit complex of Q-sheaves on M . In fact, two definitions are given, one via a complexof Q-sheaves [GHM94, IV] and another via a complex of C-sheaves [GHM94, II] andtheir agreement (in the derived category) is proved [GHM94, §29]. [Nai99] gives anothercomplex of C-sheaves quasi-isomorphic to weighted cohomology using differential forms

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 15

with square-integrability conditions from [Fra98]. We will not need to recall the actualdefinitions but only some properties of these complexes established in [GHM94].

The natural correspondences of 3.5 lift to weighted complexes: For (c1, c2) : M ′ ⇒Mas (3.5.1) there are natural isomorphisms

c∗2W>νC(EM ) = W>νC(EM ′) = c!

1W>νC(EM )

(cf. e.g. [GM03, §13]). Applying Rp∗ gives a lift of the correspondence (c1, c2) :M ′∗ ⇒ M∗ to Rp∗W>νC(EM ). For a maximal parabolic Q with associated boundarystratum S ⊂M∗ restricting the lifts to the relevant neighbourhoods as in Lemma 3.6.1gives compatible lifts of the local Hecke operator Φ to W>νC(EM )

∣∣ΓQ\V Q

and of Φ to

Rp∗W>νC(EM )

∣∣ΓQ\BF

.The key property of the weighted complexes is a certain splitting property with respect

to this action of local operators, which we now recall. Recall that we fixed (in 3.2) anenumeration Q1, . . . , Qr of the standard maximal parabolics with the property thati < j ⇐⇒ Qi ≺ Qj (⇐⇒ Ui ⊂ Uj ⇐⇒ Fj ⊂ F ∗i ). For d = 0, 1, . . . , r define D∗6d tobe the union of D with the rational boundary components corresponding to parabolicsubgroups conjugate to Qi for some i ∈ {1, . . . , d}. Let M∗6d := Γ\D∗6d. This defines afiltration

M = M∗60 ⊂M∗61 ⊂ . . . ⊂M∗6r = M∗ (4.1.1)

by Zariski-open and dense sets with Md := M∗6d − M∗6d−1 (and M0 = M) a unionof finitely many copies of the Hermitian locally symmetric space associated with theHermitian part Md,h of the standard parabolic Qd. This is preserved by the correspon-dences, i.e. (c1, c2) as in (3.5.2) restricts to (c1, c2) : M ′∗6d ⇒ M∗6d for each d. A localHecke operator Φ preserves this filtration locally (in the obvious sense). The filtration(4.1.1) of M∗ defines one of M by setting M6d := p−1(M∗d ). This is a filtration by opensets

M = M60 ⊂M61 ⊂ . . . ⊂M6r = M (4.1.2)

such that p−1(Md) = M6d −M6d−1 contains, as a dense open subset, finitely manycopies of the RBS boundary stratum MQd attached to Qd. The correspondence (3.5.3)restricts to (c1, c2) : M ′6d ⇒ M6d for each d. A local Hecke operator Φ preserves thisfiltration locally. We fix notation for inclusions as follows:

M6d−1� � jd //

��

M6d

p��

M6d −M6d−1? _idoo

��

M∗6d−1� � jd // M∗6d Md? _

idoo

(4.1.3)

For a C-vector space V with an endomorphism write V6λ (resp. V>λ) for the sum ofeigenspaces with eigenvalue of modulus 6 λ (resp. > λ). The basic splitting propertyof weighted complexes is the following:

4.1.1. Lemma. (cf. [GHM94, Prop. 16.4]) (i) For any ν the triangle on M∗6d

Rp∗W>νC(EM ) −→ Rjd∗j

∗dRp∗W

>νC(EM ) −→ id∗i!dRp∗W

>νC(EM )[1] +1−→ (4.1.4)

is (noncanonically) split when restricted to Md = M∗6d −M∗6d−1, i.e. after applying i∗d.(ii) Let S be a component of M∗6d − M∗6d−1 and Q a maximal parabolic subgroup

stabilizing a boundary component covering S. Let A ⊂ Q be a lift of the split centre and

16 ARVIND NAIR

let Φ be given by a divisible element a ∈ A(Q). For any subset Z ⊂ S the action of Φ inHi(Z,Rjd∗j∗dRp∗W

>νC(EM )) is semisimple and (4.1.4) gives the short-exact sequence

0 −→ Hi(Z,Rjd∗j∗dRp∗W>νC(EM ))6ν(a−1) −→ Hi(Z,Rjd∗j∗dRp∗W

>νC(EM )) −→−→ Hi(Z,Rjd∗j∗dRp∗W

>νC(EM ))>ν(a−1) −→ 0 (4.1.5)

given by decomposing with respect to weights of Φ.

Proof. Proposition 16.4 of [GHM94] says that the distinguished triangle on M6d

W>νC(EM ) −→ Rjd∗j∗dW

>νC(EM ) −→ id∗i!dW

>νC(EM )[1] +1−→

splits when restricted to M6d −M6d−1. Moreover (see also Prop. 15.5 of loc. cit.) forany subset Y ⊂M6d−M6d−1 contained in a single connected component the resultingshort-exact sequence in hypercohomology is the short-exact sequence

0 −→ Hi(Y,Rjd∗j∗dW

>νC(EM ))6ν(a−1) −→ Hi(Y,Rjd∗j∗dW

>νC(EM )) −→−→ Hi(Y,Rjd∗j

∗dW

>νC(EM ))>ν(a−1) −→ 0 (4.1.6)

given by decomposing Hi(Y,Rjd∗j∗dW

>νC(EM )) with respect to weights of the localHecke operator Φ, which acts semisimply over Q. (This is proved for the weightedcomplex in Db

c(CM ); it follows for the version in Dbc(QM ) since the action of local

Hecke operators is diagonalized rationally.) Taking the direct image by the proper mapp : M →M∗ gives the lemma since Φ ◦ p = p ◦ Φ on neighbourhoods as in Lemma 3.6.1.�

4.1.2. Remark. The decomposition of Hi(Z,Rjd∗j∗dRp∗W>νC(EM )) (respectively, of

Hi(Y,Rjd∗j∗dW

>νC(EM ))) into eigenspaces for Φ (respectively, for Φ) depends on thechoice of a lift A ⊂ Q, but once a lift is chosen it is independent of the choice of divisibleelement a ∈ A(Q). The sum of eigenspaces with eigenvalues 6 ν(a−1) and the shortexact sequence (4.1.5) are independent of choices.

4.2. Pushforward of weighted complexes. For d = 1, . . . , r, let Q1, . . . , Qd be themaximal parabolic subgroups containing P0 and let χd denote the distinguished genera-tor of the character group of Ad (cf. 3.2). Recall the inclusion Ad ⊂ A0 induced by theinclusion of split radicals RdQd ⊂ RdP0. If ν ∈ X∗(A0)Q satisfies

ν|Ad = χndd (for d = 1, . . . , r)

we associate with it the sequence (a1, . . . , ar) ∈ Qr defined by

ad = dimMd − nd (for d = 1, . . . , r). (4.2.1)

4.2.1. Theorem. Let EM be the local system of rational vector spaces on M coming froman irreducible representation of G. Let w be the weight of the associated shifted Hodgemodule EHM . For (a1, . . . , ar) associated with ν ∈ X∗(A0)Q (i.e. satisfying (4.2.1)) anda0 = dimM there are natural isomorphisms in Db

c(QM∗):

Rp∗W>νC(EM ) = rat(w6(a0+w,a1+w,...,ar+w)Rj∗EHM )

= rat(w>(a0+w,a1+1+w,...,ar+1+w)j!EHM ).

Proof. The second equality is a special case of [Mor08, Prop. 3.4.2], so it is enough toprove the first. The proof is by induction on the stratification M∗ =

∐d=0,...,rMd, which

is a stratification in the sense used in 2.1. (The objects in question are constructiblefor this stratification.) Over M∗60 = M both sides are isomorphic to the local system

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 17

EM , compatibly with correspondences. The induction hypothesis is that we have anisomorphism

Rp∗W>νC(EM )

∣∣M∗6d−1

= rat(w6(a0+w,...,ar+w)Rj∗EHM )∣∣M∗6d−1

(4.2.2)

extending the one onM and compatible with Hecke correspondences. LetK ∈ DbMHM(M∗6d−1)be the complex appearing here, i.e.:

K := w6(a0+w,...,ar+w)Rj∗EHM∣∣M∗6d−1

= w6(a0+w,...,ad−1+w)R(jd−1 · · · j1)∗EHM .

We have the following diagram

Rp∗W>νC(EHM )

∣∣M∗6d

α

��

α

))TTTTTTTTTTTTTTT

uul l l l l l l

rat(wd6ad+wRjd∗K) // rat(Rjd∗K) // id∗rat(w>ad+wi∗dRjd∗K) +1 //

The bottom triangle comes from Lemma 2.1.3 and the homomorphism α comes fromthe adjunction id→ Rjd∗j

∗d over M∗6d and (4.2.2):

Rp∗W>νC(EHM )

∣∣M∗6d−→ Rjd∗(Rp∗W>νC(EHM )

∣∣M∗6d−1

) = rat(Rjd∗K).

By Lemma 2.2.1,

wd6ad+wRjd∗K = w6(a0+w,...,ad+w)R(jd · · · j1)∗EHM = w6(a0+w,...,ar+w)Rj∗EHM∣∣M∗6d

.

To extend (4.2.2) over M∗6d it is enough to show that α = 0 and that the induced brokenarrow is a quasi-isomorphism. We do this after the next two lemmas.

4.2.2. Lemma. Let A,B be complexes in Dbc(QX) with endomorphisms ΦA,ΦB. If

for all j, k there are no nonzero (ΦA,ΦB)-equivariant homomorphisms pHj(A)[−j] →pHk(B)[−k], then there are no nonzero (ΦA,ΦB)-equivariant homomorphisms A→ B.

Proof. This is proved by the usual devissage: Assume pHk(A) = 0 for k < a andpHk(B) = 0 for k > d. Consider first pτ6aA→ B induced by an equivariant homomor-phism φ : A → B. The composition pτ6aA = pHa(A)[−a] → pτ>dB = pHd(B)[−d] iszero by assumption, so it factors through pτ>aA −→ pτ6d−1B. A decreasing inductionon k using that pτ6aA has no equivariant homomorphism to pHk(B)[−k] gives thatpτ6aA → B is zero, so φ factors through pτ>a+1A → B. Now repeating the argu-ment using the fact that pHa+1(A)[−a − 1] has no equivariant homomorphisms to anypHk(B)[−k] shows that it factors through pτ>a+2(A) → B. An induction then showsthat φ = 0. �

The following lemma (from [LR91], cf. especially Prop. 6.4 of loc. cit.) gives theconnection between weights (in mixed Hodge theory) and Φ-weights:

4.2.3. Lemma. Let S be a component of Md, and Q the maximal parabolic subgroupstabilizing a boundary component covering S. Let A ⊂ Q be a lift of the split centre andlet Φ be given by a divisible element a ∈ A(Q). Let K = w6(a0+w,...,ar+w)Rj∗EHM

∣∣M∗6d−1

.

Applying pH i+dimMd ◦ i∗S to the triangle

rat(wd6ad+wRjd∗K) −→ rat(Rjd∗K) −→ id∗rat(w>ad+wi∗dRjd∗K) +1−→

18 ARVIND NAIR

gives the short exact sequence of local systems on S

0 −→ H i(rat(i∗SRjd∗K))6χ(a)ad−dimMd −→ H i(rat(i∗SRjd∗K)) −→

−→ H i(rat(i∗SRjd∗K))>χ(a)ad−dimMd −→ 0 (4.2.3)

given by decomposing H i(i∗Srat(Rjd∗K)) with respect to weights of Φ.

Proof. Applying i∗S to wd6ad+wRjd∗K → Rjd∗K → id∗w>ad+wi∗dRjd∗K

+1→ gives the trian-gle for the t-structure (wD6ad+w(S),wD>ad+w(S)), i.e. w6ad+wi

∗SRjd∗K → i∗SRjd∗K →

w>ad+wi∗SRjd∗K

+1→. Now if χ(a)2 = q then the results in [LR91, §2] (specifically, (2.14)and (2.13) of loc. cit., noting also that the truncation functors w6ak make sense inthe setting of q-Hodge modules of loc. cit.) show that χ(a)wΦ∗ = qw/2Φ∗ acts as aq-endomorphism on each term in the triangle, making each into a (mixed) q-Hodgemodule in the terminology of [LR91]. Thus for x ∈ S, χ(a)wΦ∗ splits the weight fil-tration on H i(i∗xi

∗SRjd∗K), i.e. it acts by eigenvalues of modulus qk/2 = χ(a)k on

GrWk ( · ) of this space. Now by Lemma 2.2.4, for each i, H i(i∗xwd6ad+wRjd∗K) is the

subspace of H i(i∗xRjd∗K) with Hodge weights 6 ad + w − dimMd. Since χ(a)wΦ∗ is aq-endomorphism, this is precisely the subspace on which Φ∗ has eigenvalues of modulus6 χ(a)ad−dimMd . Applying rat gives the statement of the lemma. �

Let us complete the proof of Theorem 4.2.1. Let L := Rp∗W>νC(EM )|M∗6d . Then

α ∈ Hom(L, id∗rat(w>ad+wi∗dRjd∗K)) = Hom(i∗dL, rat(w>ad+wi

∗dRjd∗K))

= ⊕S Hom(i∗SL, rat(w>ad+wi∗SRjd∗K)).

The S-component of α is equivariant for the action of a local Hecke operator Φ associatedwith S. Lemmas 4.2.2 and 4.2.3 imply that α = 0 if χ(a)ad−dimMd > ν(a−1) = χ(a)−nd .Lemmas 4.1.1 and 4.2.3 imply that the induced broken arrow is a quasi-isomorphismalong S if χ(a)ad−dimMd = ν(a−1) = χ(a)−nd . �

4.2.4. Remarks. (i) The proof of the theorem shows that

Rp∗W>νC(EM ) = rat(w6(a0+w,...,ar+w)Rj∗EHM )

holds for any a0 > dimM and

Rp∗W>νC(EM ) = rat(w>(a0+w,a1+1+w,...,ar+1+w)j!EHM )

holds for any a0 6 dimM .(ii) If ν is very positive (or very negative) the theorem is trivial as we have j!EM (or

Rj∗EM ) on both sides.(iii) If ν = −ρ is minus the half-sum of positive roots then −ρ|Ad = χ−codimMd

d ford = 1, . . . , r and hence ad = dimMd + codimMd = dimM for all d. In this case thereare natural isomorphisms

Rp∗W>−ρC(EM ) = (j!∗EM [dimM ])[−dimM ] = rat(w6dimM+wRj∗EHM ) (4.2.4)

by [GHM94, Theorem 23.2] and [Mor08, Theorem 3.1.4], which gives the theorem. (Infact, either one of these equalities, together with (4.2.1), implies the other.)

(iv) For ν = 0 and E trivial we get Rp∗QM = rat(w6(dimM,dimM1,...,dimMr)Rj∗QHM ).

(v) For general ν, the resulting equality in the Grothendieck group of Dbc(QM∗) is in

[Mor08, Rem. 4.2.4].(vi) The theorem implies that the direct images of weighted complexes give a de

Rham description for w6(a0+w,...,ar+w)Rj∗EHM . In particular, we get an analogue ofProp. 6.3 of [LR91], namely if we fix a stratum S ⊂ M∗ and a rational boundary

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 19

component F covering S with associated Q,Mh, A etc., then the action of (MhA)(Q)on the cohomology sheaves Hk(i∗Sw6(a0+w,...,ar+w)Rj∗EHM ) and on their weight-gradedpieces via local Hecke correspondences is part of an action of the group (MhA)(R) by analgebraic representation. (We only defined the local Hecke action of divisible elementsin 3.6, but all of (MhA)(Q) acts by local correspondences, cf. [LR91, 6.3], [Loo88, p.18f].) (In fact, this can also be deduced from from [BW04] as in 5.4 below.)

4.2.5. Remark. The theorem and its proof work in some other categories of mixedsheaves. If (1) EM underlies an object in the theory and (2) the forgetful functor Forto perverse complexes factors through rat, i.e. through mixed Hodge modules, thenthe proof is exactly the same since Lemma 4.2.3 is available. This is the case with thetheory MHM(·/k) of 2.3 of mixed Hodge modules with de Rham structure.

In the theory Ml(·) of Morel, if EM ⊗ Ql underlies an object in Ml(M) (e.g. if EMis of geometric origin) one can argue as follows: By definition we have a mixed l-adiccomplex on a model of M∗ over an open subset of Spec(Ok). The weights of such anobject are given by reducing modulo a prime and then looking at Frobenius weights.Now the arguments in 6.9–6.11 of [LR91] show how to deduce the analogue of Lemma4.2.3 in this situation. The result is the following version of Theorem 4.2.1: There is anatural isomorphism Rp∗W

>νC(EM ) ⊗ Ql = For(w6(a0+w,a1+w,...,ar+w)Rj∗EMlM ) where

For(EMlM ) = EM ⊗Ql.

4.3. In this subsection we use the notation M, j : M → M∗, but the statements andproofs hold in the following situation: j : M ↪→ M∗ is an open immersion of a smoothvariety M as a Zariski-dense subset of a complete variety M∗, EHM [dimM ] is a pureHodge module of weight w+ dimM on M , and M =

∐iMi is a stratification for which

Rj∗EHM and j!EHM are constructible.

4.3.1. Lemma. (i) Suppose that (a1, . . . , ar) is nonincreasing, ai < dimM for all i > 1,and a0 > dimM . Then

rat(w6(a0+w,...,ar+w)Rj∗EHM )[dimM ] ∈ pD60(QM∗) (4.3.1)

andw>dimM+ww6(a0+w,...,ar+w)Rj∗EHM [dimM ] = j!∗(EHM [dimM ]). (4.3.2)

The mixed structure on Hi(M∗, rat(w6(a0+w,...,ar+w)Rj∗EHM )) has weights 6 i+ w.(ii) Suppose that (a1, . . . , ar) is nondecreasing, ai > dimM for i > 1 and a0 > dimM .

Thenrat(w6(a0+w,...,ar+w)Rj∗EHM )[dimM ] ∈ pD>0(QM∗) (4.3.3)

andw6dimM+ww6(a0+w,...,ar+w)Rj∗EHM [dimM ] = j!∗(EHM [dimM ]). (4.3.4)

The mixed structure on Hi(M∗, rat(w6(a0+w,...,ar+w)Rj∗EHM )) has weights > i+ w.

Proof. Write d0 := dimM to simplify the notation.(i) If (a1, . . . , ar) is nonincreasing, ai 6 d0 for all i and a0 = d0 then by Lemma

2.2.2(i)

w6(a0+w,...,ar+w)Rj∗EHM [d0] = w6(a0+w,...,ar+w)w6d0+wRj∗EHM [d0] (4.3.5)

and this lies in D60 by Lemma 2.2.2 since w6d0+wRj∗EHM [d0] = j!∗EHM [d0] ∈ D60. Thisproves (4.3.1) if (a1, . . . , ar) is nonincreasing and a0 > d0 since we can always replacea0 by d0 without changing w6(a0+w,...,ar+w)Rj∗EHM [d0].

For a0 = d0, we have:

w>d0+ww6(a0+w,...,ar+w)Rj∗EHM [d0] = w>d0+ww>(a0+w,a1+1+w,...,ar+1+w)j!EHM [d0]

= w>d0+wj!EHM [d0] = j!∗EHM [d0]

20 ARVIND NAIR

by Lemma 2.2.2(i) and Theorem 2.1.1. This proves (4.3.2) if a0 = d0; the general case fol-lows since we can replace a0 = d0 by any a0 > d0 without changing w6(a0+w,...,ar+w)Rj∗EHM .

For the assertion about weights of the mixed structure it is enough to prove thatw6(a0+w,...,ar+w)Rj∗EHM [d0] has weights 6 d0 +w. Suppose X = S0 t S1 with S1 closed.If K on X is of weights 6 a then i∗1w6(+∞,b)K = w6bi

∗1K and Hk(i∗1w6(+∞,b)K) =

w6bHk(i∗1K) has weights 6 min(b, k + a) 6 k + a. So i∗1w6(+∞,b)K has weights 6 a.

Since i∗0w(6(+∞,b)K = i∗0K has weights 6 a, we conclude that w6(+∞,b)K has weights6 a. Thus w6(+∞,b) preserves the subcategory of complexes with weights 6 a. Now for(a0, . . . , ar) nonincreasing we have w6(a0,...,ar) = w6(+∞,...,+∞,ar) ◦ · · · ◦ w6(+∞,a1,...,a1) ◦w6(a0,...,a0). Each functor here is of the form w6(+∞,b) for a suitable restratification ofM∗. So we conclude that if K is of weights 6 a and (a0, . . . , ar) is nonincreasing, thenw6(a0,...,ar)K is of weights 6 a. Combining this with (4.3.5) gives the required assertionin the case a0 = d0 and the general case then follows.

(ii) follows by duality from (i). �

In some cases we can say a little more:

4.3.2. Proposition. (i) For a 6 dimM the weights of

Hi(M∗, w>a+wj!EHM [dimM ]) = Hi(M∗, w6(dimM+w,a+w−1,...,a+w−1)Rj∗EHM [dimM ])

are 6 i+ dimM +w. The top weight quotient is the image in IH i+dimM (M∗, E) underthe map induced by (4.3.2).

(ii) For b > dimM the weights of Hi(M∗, w6b+wRj∗EHM [dimM ]) are 6 i+dimM+w.The bottom weight piece is the image of IH i+dimM (M∗, E) under the map induced by(4.3.4).

(iii) The mixed structure on Hi(M∗, w6(dimM,dimM1,...,dimMr)Rj∗EHM ) has weights 6i+w. The top weight quotient is the image in IH i+dimM (M∗, E) under the map inducedby (4.3.2).

Proof. Let d0 := dimM . In each of (i)–(iii) the first assertion follows from the previouslemma, so we concentrate on the second assertions.

(i) From the long exact sequence in cohomology associated with the triangle

w<d0+ww>a+wj!EHM [d0]→ w>a+wj!EHM [d0]→ w>d0+wj!EHM [d0] +1→it suffices to show that for a 6 d0, w<d0+ww>a+wj!EHM [d0] has weights < d0 + w. Since

w<d0+ww>a+wj!EHM [d0] = w>a+ww<d0+wj!EHM [d0]

and w>a+w preserves the complexes of weight < d0 + w, it is enough to show thatw<d0+wj!EHM [d0] has weights < d0 + w. The triangle w<d0+wj!EHM [d0] → j!EHM [d0] →w>d0+wj!EHM [d0] +1→ shows that for any d > 1,

i∗dw<d0+wj!EHM [d0] = i∗dw>d0+wj!EHM [d0] [−1]

has weights 6 d0 + w − 1 since w>d0+wj!EHM [d0] = j!∗EHM [d0] is of weights 6 d0 + w.Since i∗0w<d0+wj!EHM [d0] = 0 we have that w<d0+wj!EHM [d0] has weights 6 d0 + w − 1.

(ii) follows from (i) by duality.(iii) Let di = dimMi. From the triangle

w6(d0+w,...,dr+w)Rj∗EHM → w6d0+wRj∗EHM → w>(d0+w,...,dr+w)w6d0+wRj∗EHM+1→

we see that it suffices to show that K := w>(d0+w,...,dr+w)w6d0+wRj∗EHM has weights6 w. For this we will use the pointwise criterion (cf. 2.2): K has weights 6 w if andonly if for every point ix : {x} ↪→ M∗, H i(i∗xK) has weights 6 i + w. If x ∈ M0 thisobviously holds, so assume x ∈ Mk for k > 1. Then i∗kw6(d0+w,...,dr+w)Rj∗EHM belongs

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 21

to wD6dk+w, i.e. H i(i∗kw6(d0+w,...,dr+w)Rj∗EHM ) has weights 6 dk + w for all i. Sincethese cohomology objects are smooth, Lemma 2.2.4 implies that for any point x ∈Mk,H i(i∗xw6(d0+w,...,dr+w)Rj∗EHM ) has weights 6 w+dk−dk = w for all i. In the long exactsequence

→ H i(i∗xw6d0+wRj∗EHM )→ H i(i∗xK)→ H i+1(i∗xw6(d0+w,...,dr+w)Rj∗EHM )→

the first group has weights 6 i + w (because w6d0+wRj∗EHM = (j!∗EHM [d0])[−d0] hasweight w) and the last group has weights 6 w. So H i(i∗xK) has weights 6 i+ w for alli. �

4.3.3. Remark. Everything in this subsection evidently holds in any theory of A-mixedsheaves in which EM ⊗A underlies an object.

4.4. Mixed structures in cohomology. To translate the above into results aboutweighted cohomology groups we note some facts about roots and weights which comefrom explicit calculations with the Q-root system to be found in the appendix.

First we note that there is a natural condition on ν ensuring that the associated se-quence (a1, . . . , ar) is as in ((i) of) the previous lemma. Our choice of minimal parabolicP0 ⊃ A0 fixes a positive cone in X∗(A0)R and the associated half-sum of positive rootsρ. The dominant cone is the closed cone dual to the positive cone; it is contained in theinterior of the positive cone and contains ρ.

4.4.1. Lemma. If ν + ρ is dominant (respectively, antidominant) then the sequence(a1, . . . , ar) associated with ν is nonincreasing (respectively, nondecreasing).

Proof. It is enough to show that for each fundamental weight$i the sequence m1, . . . ,mr

defined by $i|Ad = χmdd is nondecreasing, because the sequence associated with −ρ+$i

is (dimM, . . . , dimM) − (m1, . . . ,mr) and every dominant weight is a positive linearcombination of the $i. An explicit calculation with the system Φ(A0, G) of rationalroots shows that this is the case (cf. Lemma A.2.2 in the Appendix). �

Next define a weight $1 ∈ X∗(A0) by the conditions

$1|Ad = χd (for d = 1, . . . , r). (4.4.1)

This weight is not always dominant (e.g. for the root system G2). By an explicitcalculation using properties of the Q-root system of a group giving rise to a Hermitiansymmetric domain, we show in the Appendix that:

4.4.2. Lemma. The weight $1 ∈ X∗(A0) is dominant.

(In fact, $1 is a fundamental weight, whence the notation.)The following theorem summarizes the properties of the mixed structures on weighted

cohomology groups. We state it in the context of mixed Hodge structures, but seeRemark 4.4.4 (i) below.

4.4.3. Theorem. (i) The weighted cohomology groups

W>νH i(M,EM ) := Hi(M,W>νC(EM ))

carry canonical mixed Hodge structures with polarized graded quotients such that themorphisms

W>µH i(M,EM )→W>νH i(M,EM ) (for µ 6 ν)

(where µ 6 ν if ν − µ is in the positive cone), the cup products

W>νH i(M,E1,M )⊗W>µHj(M,E2,M )→W>µ+νH i+j(M,E1,M ⊗ E2,M ),

22 ARVIND NAIR

and the duality pairings

W>νH i(M,EM )⊗W>−2ρ−νH2 dimM−i(M, EM )→ Q(−dimM)

are all morphisms of mixed Hodge structures. (7)(ii) If ν + ρ is dominant then the mixed Hodge structure on W>νH i(M,EM ) is like

that of a complete variety, i.e. W>νH i(M,EM ) has weights 6 i+ w.If ν + ρ is antidominant then the mixed Hodge structure on W>νH i(M,EM ) is like

that of a smooth variety, i.e. W>νH i(M,EM ) has weights > i+ w.(iii) If ν = −ρ + n$1 for n > 0 then the top weight quotient is isomorphic to the

image of W>νH i(M,E) in W>−ρH i(M,EM ) = IH i(M∗,EM ), i.e.

GrWi+wW>νH i(M,EM ) = im

(W>νH i(M,EM )→ IH i(M∗,EM )

).

If ν = −ρ+ n$1 for n 6 0 or if ν = 0 then the bottom weight piece is isomorphic tothe image of IH i(M∗,EM ) = W>−ρH i(M,EM ) in W>νH i(M,EM ), i.e.

Wi+w

(W>νH i(M,EM )

)= im

(IH i(M∗,EM )→W>νH i(M),EM

).

Proof. (i) follows from Theorem 4.2.1. (ii) follows from Lemma 4.4.1 and Lemma 4.3.1.(iii) follows from Lemma 4.4.2 and Proposition 4.3.2 after noticing that the sequenceassociated with −ρ− n$1 is (dimM + n, . . . ,dimM + n). �

4.4.4. Remarks. (i) The theorem holds in the theories of mixed sheaves for which The-orem 4.2.1 holds. In particular by Remark 4.2.5 it holds for mixed Hodge-de Rhamstructures and for mixed l-adic complexes. If one were to work in the context of Shimuravarieties all this would obviously work over the reflex field.

(ii) The mixed Hodge-de Rham structures in the theorem are effective (by 2.3.1).(iii) If ν + ρ is antidominant then the groups W>νH∗(M,EM ) are computable us-

ing automorphic forms ([Nai99, Fra98]). So the theorem provides a supply of mixedstructures related to automorphic forms. One might expect (or hope) that the relationbetween extensions and automorphic forms is easier in some of these than in H∗(M,EM ).

(iv) It is an interesting question whether the Hodge structures in weighted cohomologygroups can be seen in more classical terms, e.g. in terms of explicit bifiltered complexesof differential forms (as in Deligne’s Hodge theory for smooth varieties [Del74]). Thiswill be taken up elsewhere.

(iii) If ν = 0 and E is trivial then rat(w6(dimM0,...,dimMr)Rj∗QHM ) = W>0C(QM ) =

QM (cf. [GHM94, §19]). Thus we get a mixed Hodge structure like that of a completevariety on H∗(M). Such a mixed Hodge structure was constructed in [Zuc04] by differentmeans. Theorem 4.4.3(iii) says that the mixed Hodge structure satisfies GrWi H

i(M) =im(H i(M)→ IH i(M∗)

). In fact the particular truncation w6(dimM0,...,dimMr) has many

nice properties for general algebraic varieties, as we show in [NV12]. Also, in [Nai10] weapply this description of H∗(M) and results about automorphic forms to Chern classes ofautomorphic vector bundles and to restriction maps between locally symmetric varieties.

5. Spectral sequences and automorphic forms

5.1. Quasi-filtered objects and spectral sequences. (cf. [Sai88, 5.2.17]) Let Λ ={(p, q) ∈ (Z ∪ {±∞})2 : p 6 q}. For α = (p, q) ∈ Λ we set α1 = p and α2 = q. A partialorder on Λ is given by α > β if αi > βi for i = 1, 2.

A quasi-filtered object in a triangulated category D consists of objects K(α) in D forα ∈ Λ and morphisms uβα : K(α) → K(β) for α > β and νβα : K(α) → K(β)[1] forα2 = β1, satisfying the following conditions:

7The variation on EHM must be twisted by w to get the pairing in Q(−dim M).

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 23

(i) there exists n,m ∈ Z such that K(α) = 0 if α1 > m or α2 6 n.(ii) the following identities hold:

uαα = id for all α ∈ Λuγβuβα = uγα for α > β > γuδβνβα = νδγuγα for β > δ, γ > α, α2 = β1, γ2 = δ1

(iii) for α, β, γ such that α1 = γ2, α2 = β2, β1 = γ1 the triangle

K(α)uβα−→ K(β)

uγβ−→ K(γ)ναγ−→+1

is distinguished in D .Let K = K(−∞,+∞) and GrpK = K(p, p + 1). Given a cohomological functor H :D → A to an abelian category A there is a spectral sequence starting from E1 in Awith

Ep,q1 = Hp+q(GrpK) ⇒ Hp+q(K). (5.1.1)The E∞ term computes the graded of the filtration of H∗(K) defined by

F pH i(K) = im(H i(K(p,∞)→ H i(K)) = ker(H i(K)→ H i(K(−∞, p)))(cf. [Sai90, 5.2.18]).

If D = Db(A ) is the derived category of an abelian category A and K is a complexof objects in A with a finite decreasing filtration F then K(p, q) = F pK/F qK definesa quasi-filtered object in D . If H is the usual cohomology functor then (5.1.1) is theusual spectral sequence of the filtered complex K. This is a special case of the followingmore general situation: If D has a t-structure with truncation functors τ60 and τ>0 thenfor any object K, K(p, q) = τ6−pτ>−qK defines a quasi-filtered object and the spectralsequence Ep,q1 = Hp+q(τ6−pτ>−p(K))⇒ Hp+q(K).

Now consider this example in our context. Suppose D = DbMHM(X) where X is analgebraic variety (or, more generally, D is the derived category of any category of mixedsheaves). For K ∈ DbMHM(X),

K(p, q) := w6−pw>−qK (5.1.2)

defines a quasi-filtered object with K(−∞,+∞) = K and Grp(K) = w6−pw>−pK. Fora cohomological functor H the resulting spectral sequence has

Ep,q1 = Hp+q(w6−pw>−pK) ⇒ Hp+q(K).

More generally, if X comes with a stratification X =∐rk=0 Sk , then for any sequence

a0 6 a1 6 . . . 6 an ∈ (Z ∪ {±∞})r+1 and K ∈ DbMHM(X) with w6anK = K andw>a0

K = K,K(p, q) = w6a−pw>a−qK (5.1.3)

defines a quasi-filtered object with K(−∞,+∞). (It is easy to check using Lemma2.2.2(i) that the necessary morphisms exist. It is understood here that ak = a0 fork < 0 and ak = an for k > n.) For a cohomological functor H the resulting spectralsequence has

Ep,q1 = Hp+q(w6a−pw>a−pK) ⇒ Hp+q(K).

There are two cohomological functors of interest. First, if H = H0 is the cohomologyfunctor of DbMHM(X) we get a spectral sequence of objects in MHM(X). Secondly,if we take H = H0 ◦ Ra∗ where a : X → pt, so that H i is the ith hypercohomologyfunctor, then we get a spectral sequence of mixed Hodge structures converging to themixed Hodge structure on H∗(X,K) with

Ep,q1 = Hp+q(X,w6a−pw>a−pK) ⇒ Hp+q(X,K). (5.1.4)

In the locally symmetric setting of §2 and §3, any increasing sequence ν1 > ν2 > · · · >νr of elements of X∗(A0)Q (for the order that µ 6 ν if ν − µ is in the positive cone)

24 ARVIND NAIR

the filtration W>ν1C(EM ) ⊃ W>ν2C(EM ) ⊃ · · · ⊃ W>νrC(EM ) ⊃ {0} gives a spectralsequence

Ep,q1 = Hp+q(M,W>νpC(EM )/W>νp+1C(EM )) ⇒ W>ν1Hp+q(M,EM ). (5.1.5)

If ν1 is sufficiently negative (i.e. −ν1 is sufficiently positive in the interior of the Weylchamber) then W>ν1C(EM ) = Rj∗EM and we get a spectral sequence converging toH∗(M,EM ). Theorem 4.2.1 identifies these with spectral sequences of the type (5.1.4)so these are spectral sequences of mixed Hodge-de Rham structure.

5.2. A distinguished spectral sequence. Consider M∗ with the canonical stratifi-cation and the quasi-filtration of Rj∗EHM given as in (5.1.3) by the sequence a0 6 a1 6. . . 6 an ∈ (Z ∪ {±∞})r+1 (where n = codimMr if E is trivial) where

ak = (dimM+w+k, . . . , dimM+w+k).

(This is a modification of the quasi-filtration (5.1.2) given by Morel’s t-structure (wD60,wD>0)on M∗.) The nonzero graded pieces are:

K(p, p+ 1) = w6dimM+w−pw>dimM+w−pRj∗EHM (for p ≤ 0). (5.2.1)

The spectral sequence has

Ep,q1 = Hp+q(M∗, w6dimM+w−pw>dimM+w−pRj∗EHM ) ⇒ Hp+q(M,EHM ). (5.2.2)

The following summarizes the properties of this spectral sequence.

5.2.1. Theorem. The spectral sequence (5.2.2) has the following properties:(i) It is a spectral sequence of split mixed Hodge(-de Rham) structures, i.e. the E1 term

is a direct sum of pure structures. The E1 term is (canonically) a sum of intersectioncohomology groups of the minimal compactifications of strata of M∗ with homogeneouscoefficient systems.

(ii) It is automorphic (after ⊗C), i.e. in the isomorphism

H∗(M,EM )C = H∗(g,K; A (Γ\G(R))⊗ E)

of [Fra98] it is the spectral sequence associated with a decreasing filtration on the spaceof automorphic forms.

(iii) If the highest weight of E is regular then it degenerates at the E1 term.

Proof. (i) Lemma 2.2.3(ii) gives a canonical decompositions

w>bw6bRj∗EHM = ⊕iH i(w>bw6bRj∗EHM )[−i] = ⊕iw>bw6bHi(Rj∗EHM )[−i] (5.2.3)

and a further canonical decomposition of each summand on the right (which is a shiftedpure Hodge module) by support, i.e. as a sum of intersection complexes. Applying thisto the E1 term gives (i).

The proofs of (ii) and (iii) will be in the next subsection. �

5.2.2. Remark. The spectral sequence exists and (i) holds in any theory of A-mixedsheaves (in the sense of Saito) if EM ⊗A underlies an object in the theory, in particularif EM is of geometric origin. In Morel’s theory M (·,Ql) of 2.3 the spectral sequenceexists under the geometric origin condition and the E1 term continues to be split, butneed not be a sum of intersection cohomology groups of the minimal compactificationsof strata of M∗ with homogeneous coefficient systems. (Presumably there is a filtrationof the E1 term with subquotients of this form.)

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 25

5.3. Automorphic forms. We recall some results on automorphic forms from [Fra98](cf. also [Wal97]) required to complete the proof of Theorem 5.2.1.

We fix some notation. Recall that we have fixed a minimal Q-parabolic P0 and amaximal Q-split torus A0. This fixes in a0 = X∗(A0)⊗R a root system Φ0 = Φ(A0, G),a system of positive roots Φ+

0 , a positive cone +a0, a positive Weyl chamber a+0 (⊂+a0),

and their closures +a0 and a+0 . The centralizer M0 of A0 is a minimal Levi subgroup.

For any standard Levi subgroup M ⊃M0 the split centre is a torus AM ⊂ A0 and thisgives the dual vector spaces aM = LieAM (R) and aM = X∗(M) ⊗ R = X∗(AM ) ⊗ R.Restriction of characters by M0 ⊂ M gives an embedding aM ⊂ a0. Restriction byAM ⊂ A0 gives a projection a0 � aM inverse to aM ⊂ a0. (8)

Fix a Cartan subalgebra h of gC containing a0 = LieA0(R) (hence h ⊂ (m0)C). Thisgives a root system Φ = Φ(h, gC) and Weyl group W = W (h, gC). Fix a system ofpositive roots Φ+ ⊂ Φ = Φ(h, gC) compatible with Φ+

0 (i.e. if β ∈ Φ+ then β|a0 ∈Φ+

0 ∪ {0}). The half-sum of roots in Φ+ will be denoted ρh; so ρh|a0 = ρ, where ρ isthe half-sum of roots in Φ+

0 . The Harish-Chandra isomorphism Z(g) ∼= S(h)W identifiesinfinitesimal characters with W -orbits in h and ideals of finite codimension in Z(g) withfinite W -invariant sets in h.

Let A (Γ\G(R)) be the space of automorphic forms for Γ; these are the smoothfunctions on Γ\G(R) which are Z(g)-finite and K-finite and satisfy a moderate growthcondition. For a finite W -invariant set Θ ⊂ h, let FinΘA (Γ\G(R)) denote the sub-space of automorphic forms killed by a power of the ideal of Z(g) given by Θ. ThenA (Γ\G(R)) = ⊕ΘFinΘ A (Γ\G(R)) as Θ runs over W -orbits in h.

We need an elementary construction. For λ ∈ a0 denote by λ+ the closest point to λin the closed positive Weyl chamber a+

0 . For a finite set Θ ⊂ h define another finite setΘ+ ⊂ a+

0 as follows:

Θ+ :=⋃

M

{Re(θ|aM )+ : θ ∈ Θ

}.

The union is over standard Levis and we have used the inclusions aM ⊂ a0 ⊂ h. Notethat associated with each λ ∈ Θ+ is a canonical standard parabolic subgroup Pλ andstandard Levi Mλ, maximal with the property that λ ∈ aMλ

⊂ a0.Let f ∈ A (Γ\G(R)). For each standard parabolic P the constant term of f along P

admits a Fourier expansion in terms of characters of aM (cf. [Fra98, §6]); the charactersappearing form a finite set, the P -exponents of f :

ExpP (f) ⊂ (aM )C ⊂ (a0)C.

Let Θ := W ·(ρh +λE) where λE ∈ h is the highest weight of E. (This is the infinitesimalcharacter of E.) The exponents of automorphic forms in FinΘ A (Γ\G(R)) satisfy:

f ∈ FinΘ A (Γ\G(R)) and λ ∈ ExpP (f) =⇒ Re(λ)+ ∈ Θ+. (5.3.1)

Let Alog(Γ\G(R)) ⊂ A (Γ\G(R)) be the subspace of automorphic forms whichhave all their exponents in the negative cone −+a0, i.e. f ∈ Alog(Γ\G(R)) if and onlyif Re(λ)+ = 0 for all λ ∈ ExpP (f). (This is not the definition of Alog(Γ\G(R)) in[Fra98] but is equivalent. The functions in Alog(Γ\G(R)) are square-integrable up tosome logarithmic terms; in particular, square-integrable automorphic forms belong toAlog(Γ\G(R)).)

We will filter FinΘA (Γ\G(R)) by conditions on the exponents. (This is a degeneratelimiting case of the filtrations used by Franke [Fra98], which we will also recall below.)

8The notation here is slightly different from that in §2: M is a Levi subgroup of P (not the Leviquotient) and so AM is a lift of the split centre A of the Levi quotient from 3.2.

26 ARVIND NAIR

Let $1 be the dominant weight defined by equation (4.4.1). Let N be the smallestinteger such that

λ ∈ Θ+ ⇒ λ 6 N$1.

(This depends on the representation E; if E is trivial then N = codimMr.) Define anindexing function T : Θ+ → Z≥0 by

T (λ) = p if (N − p+ 1)$1 < λ 6 (N − p)$1. (5.3.2)

Define a finite decreasing filtration F • of FinΘA (Γ\G(R)) by the condition

f ∈ F p ⇐⇒ for all P and λ ∈ ExpP (f) we have T (Re(λ)+) > p. (5.3.3)

Thus F 0 = FinΘA (Γ\G(R)), FN = FinΘAlog(Γ\G(R)), and FN+1 = {0}. (Notice thatthe number of nontrivial graded pieces is independent of E.)

Now Franke’s fundamental theorem [Fra98] says that the cohomology of M can becomputed using automorphic forms:

H∗(M,EM )C = H∗(g,K; FinΘA (Γ\G(R))⊗ E).

Then F • induces a filtration of the complex computing Lie algebra cohomology andhence a spectral sequence with

Ep,q1 = Hp+q(g,K;F p/F p+1 ⊗ E) ⇒ H∗(M,EM )C. (5.3.4)

We can now complete the proof of the theorem.

Proof of Theorem 5.2.1. (i) was proved above, so we consider (ii). We will show that(5.2.1) is identified with (5.3.4) after tensoring with C. This will follow by combiningTheorem 4.2.1 with [Nai99] and [Fra98]. Let $1 be the weight defined in 4.4.1. ByTheorem 4.2.1 we have

W>−ρ−p$1H∗(M,EM ) = H∗(M∗, w6dimM+w+pRj∗EHM ).

By Lemma 4.4.2 $1 is dominant and so ρ+ p$1 is dominant and hence

H∗(g,K;F pA (Γ\G(R))⊗ E) = W>−ρ−p$1H∗(M,EM )

where we have combined Theorem A of [Nai99] and Theorem 16 of [Fra98] (the latterrequires the dominance condition). This proves (ii).

(iii) It is enough to work ⊗C. Let T ′ : Θ+ → Z≥0 be a function which satisfies thecondition

T ′(λ) < T ′(µ) if µ 6 λ, µ 6= λ. (5.3.5)

This defines a filtration of the space of automorphic forms by the condition (5.3.3); let′Ep,q1 be the E1 term of the spectral sequence it gives. Then∑

p,q

dimEp,q1 =∑p,q

dim ′Ep,q1

so that the spectral sequence associated with T degenerates at E1 if and only if the oneassociated with T ′ does. By [Fra98, Theorem 19], if the highest weight of E is regularthe spectral sequence associated with T ′ degenerates at E1. �

5.3.1. Remarks. (i) For a filtration coming from an indexing function T on Θ+ satisfying(5.3.5), Theorem 14 of [Fra98] gives a detailed decomposition of the graded pieces ofthis filtration. This decomposition has the following coarsening:

F p/F p+1 =⊕

ν∈Θ+,T (ν)=p

IndGPν(CMνν ⊗ FinΘ−ν Alog

(ΓMνAν(R)0\Mν(R)

))(5.3.6)

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 27

(Here IndGP is a suitable induction functor on (g,K)-modules and CMνν is the one-

dimensional space on which Mν acts via the character ν ∈ aMν .) Using standard ma-nipulations in Lie algebra cohomology ([Fra98, p. 256]) this gives a spectral sequencewith

Ep,q1 =⊕

ν∈Θ+,T (ν)=p

Hp+q−r (mν ,KMν ; CMνν ⊗ FinΘ−νAlog

(ΓMνAν(R)0\Mν(R)

)⊗Hr(nν , E)

).

(5.3.7)This admits a further decomposition using Eisenstein series.

There are similar decompositions of the E1 term of the distinguished spectral se-quence. This does not follow immediately from the results of [Fra98, §6] because thefunction T in (5.3.2) does not satisfy (5.3.5), so some adaptation of arguments in [Fra98]is necessary. It is an interesting question to what extent these decompositions respectmixed structures. We plan to take this up elsewhere.

(ii) It seems likely that all of Franke’s spectral sequences (i.e. those associated withany filtration of the space of automorphic forms by (5.3.1) where T : Θ+ → Z≥0 satisfiesT (µ) ≥ T (λ) if µ 6 λ) are spectral sequences of mixed structure. This would require a(straightforward) generalization of Morel’s construction in the category of automorphiccomplexes (on a locally symmetric variety, see the next subsection), a refinement of theweighted cohomology construction of [GHM94] along the lines suggested in [GHM94,35.4], and also a refinement of the main result of [Nai99] relating these to automorphicforms.

5.4. Pure and mixed. We use the spectral sequence (5.2.2) to show that the purestructures appearing in the cohomology of noncompact Shimura varieties (i.e. thingslike GrWi H

j) can already be found in the intersection cohomology (with homogeneouscoefficients) of noncompact Shimura varieties of smaller dimension, at least in the settingof Hodge (or Hodge-de Rham) structures. The proof of this is geometric, i.e. does notuse the automorphic interpretation of the spectral sequence. We will only use someresults from §2, the setting and notation of §3.1–§3.3, the existence of the spectralsequence (5.2.2), and a result of [BW04].

We will need to use the notion of automorphic complexes of mixed Hodge modules.Following [LR91, §6]), let DbMHMaut(M∗) be the (full) triangulated subcategory ofDbMHM(M∗) whose objects are complexes K ∈ DbMHM(M∗) such that for each stra-tum S (of the canonical stratification of M∗) and integer k the cohomology objectHk(i∗SK) ∈ MHM(S) is smooth and satisfies

(1) its underlying local system is the local system associated with a finite-dimensionalrational representation of Mh and

(2) the weight graded pieces GrWj Hk(i∗SK) are pure Hodge modules associated

with the locally homogeneous variation coming from the Mh-representation onGrWj H

k(i∗SK) (as in 3.7).

The definition makes sense for any subvariety X ⊂ M∗ which is a union of strata, forexample X = M∗6d for d ∈ {0, . . . , r} (as in (4.1.1)) or X = S∗ (the closure of a stratum).

5.4.1. Lemma. Let X ⊂ Y be subvarieties of M∗ which are unions of strata of the canon-ical stratification and let i : X ↪→ Y be the inclusion. Then the functors Ri∗, Ri!, i∗, i!

define functors between categories of automorphic complexes of mixed Hodge modules.

Proof. This is stated in [LR91, p. 256], but only the weight fitration is treated there.In [BW04, Theorem 2.6] Burgos and Wildeshaus show that Rj∗ preserves automorphiccomplexes in the case where j is the inclusion of a locally symmetric variety in itsminimal compactification. (Alternately, one could use the results of Harris and Zucker,

28 ARVIND NAIR

cf. [HZ01, III.4.3] here.) The general case follows from this and from the standardtriangles for open-closed pairs. �

The notion of automorphic complex makes sense in (the derived category of) anycategory of A-mixed sheaves M on a union of strata of M∗ as the conditions (1) and (2)of the previous paragraph do so. (If S is smooth an object M ∈M (S) is called smoothif For(M) is a shifted local system of A-vector spaces.) The proof of the lemma showsthat if Rj∗ preserves the automorphy condition, where j is the inclusion of a stratum inits closure (i.e. in its minimal compactification), then the other functors also do so, i.e.the lemma holds for M (·).

5.4.2. Lemma. The functors w6a, w>a preserve categories of automorphic complexes ofmixed Hodge modules.

Proof. If X and K are smooth and K is automorphic then H i(w6aK) ↪→ H i(K) for alli, so that w6aK is smooth and automorphic. It follows that w>aK is also smooth andautomorphic. Now consider a general X. For an automorphic complex K and a stratumSk consider the standard triangle

wk6aK −→ K −→ RiSk∗w>ai∗SkK

+1−→

from Lemma 2.1.3 (ii). The third term is automorphic because i∗SkK is smooth andRiSk∗ is the composite of pushforward by an open immersion (which preserves automor-phy by the previous lemma and pushforward by a closed immersion (which preservesautomorphy trivially). Since K is automorphic, we conclude that wk6aK is automorphic.Now Lemma 2.1.3 (i) and an induction proves the lemma. �

The same proof works in any category of mixed sheaves such that the direct imageby an open immersion preserves automorphy.

5.4.3. Theorem. Assume that we are working in a theory of A-mixed sheaves M (·) suchthat (a) the direct image preserves automorphy and (b) EM ⊗ A underlies an object inthe theory.

Then the weight-graded pieces GrWj Hi(M,EM

M ) of the mixed structure on H i(M,EMM )

appear as subquotients of the intersection cohomology of (minimal compactifications of)boundary strata of M∗ with homogeneous local systems as coefficients.

In particular this holds in the following situations: (i) A = Q and the theory of mixedHodge modules (ii) A = Q and the theory of mixed Hodge modules with de Rham rationalstructure.

Proof. Assume that (a) and (b) hold for M (·). Consider the spectral sequence (5.2.2)with

Ep,q1 = Hp+q(M∗, w62 dimM−pw>2 dimM−pRj∗EMM ) ⇒ Hp+q(M,EM

M ).

By Lemma 2.2.3 (ii) and the remark following it, the E1 term breaks up as a sum ofintersection complexes supported on the closures of boundary strata. These closures arethemselves minimal compactifications of locally symmetric varieties, and by the previouslemma the intersection complexes in question are associated with homogeneous localsystems and their canonical variations.

If A = Q and M (X) = MHM(X) we have established (a) and (b) above. For A = Qand M (X) = MHM(X/k) for k the field of definition of the canonical stratification ofM∗, (b) holds and (a) also holds because MHM(X/k) is a subcategory of MHM(X). �

MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS 29

5.4.4. Remark. As remarked in 2.3, pure objects in Morel’s theory Ml(·) are not knownto be semisimple. Thus the further decomposition of the E1 term in terms of intersectioncomplexes is not available to us. We also do not know if (a) holds in Morel’s theoryMl(·). (After reduction to finite fields it is contained in [Pin92]; perhaps (a) in Ml(·)can be deduced from this.)

Appendix A. Relative root system

For a reductive group G over a field k, by the k-root system of G we mean the systemof roots with respect to a maximal k-split torus. This root system may be reducible andnonreduced.

A.1. Suppose that the group G is as in §2, i.e. semisimple and simply-connected withGad Q-simple. Then there is a number field k and a simply connected group H over kwith Had absolutely simple, such that G = Resk/QH. Under our assumption that thesymmetric space of G(R) is Hermitian the number field k is totally real and the R-rootsystem of H is of type Cs or BCs. It follows ([BB66, 2.9]) that the k-root system ofH is of type Cr or BCr. Now if T is a maximal k-split torus in H then a maximallyQ-split subtorus A0 ⊂ Resk/Q T is maximal Q-split in G and the root systems Φ(A0, G)and Φ(T,H) are identified. Thus the Q-root system Φ(A0, G) of G is of type Cr or BCr.

A.2. Suppose G is as above, i.e. as in §2. Fix a maximal Q-split torus A0 in G. Interms of a maximal orthogonal system of rational roots β1, . . . , βn, the roots of A0 in Gare the nonzero elements of{

±βi ± βj2

}1≤i≤j≤n

in the Cn case{±βi ± βj

2,±βi

2

}1≤i≤j≤n

in the BCn case.

A system of simple roots {α1, . . . , αn} is given by

αi :=βi − βi+1

2(for 1 ≤ i < n) and αn :=

{βn for Cnβn2 for BCn.

The basis dual to the simple coroots {α∨i := 2αi/(αi, αi)}1≤i≤n is

$i =β1 + · · ·+ βi

2(for 1 ≤ i < n) and $n =

{β1+···+βn

2 for Cnβ1+···+βn

4 for BCn.

The lattice {λ ∈ X∗(A0)Q|(λ, α∨) ∈ Z for α∨ ∈ Φ∨} is spanned by $1, . . . , $n in theCn case and $1, . . . , $n−1, 2$n in the BCn case. (In the BCn case 2αn is a root and($n, (2αn)∨) = ($n, βn) = 1/2, so that $n is not a weight.) Under our assumption thatG is simply connected this is the weight lattice X∗(A0).

Now letAk = (∩i 6=k ker(αi))

0 .

On Ak we have β1 = · · · = βk and βk+1 = · · · = βn = 0. For Cn we have:

$i|Ak =

{i$1|Ak if i ≤ kk$1|Ak if i > k.

(A.2.1)

For BCn we have the same identities for i < n plus the identity

2$n|Ak = k$1|Ak (A.2.2)

for each k = 1, . . . , n. Since X∗(A0)� X∗(Ak) we arrive at:

30 ARVIND NAIR

A.2.1. Lemma. The restriction of the weight $1 = β1/2 ∈ X∗(A0) generates the char-acter group of Ak for k = 1, . . . , n.

From (A.2.1) and (A.2.2) we have:

A.2.2. Lemma. For each i = 1, . . . , r, the sequence of integers (m1, . . . ,mr) defined by$i|Ak = mk$1|Ak is nondecreasing.

References

[AMRT75] A. Ash, D. Mumford, M. Rapoport and Y.-S. Tai, Smooth Compactification of Locally Sym-metric Varieties, Math Sci Press, Brookline, MA, 1975.

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E-mail address: arvind@math.tifr.res.in

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Colaba, Mumbai 400005, India