SHIMURA DATUM 1. Review Notations: Let U1 Be the Complex

SHIMURA DATUM

HAO (BILLY) LEE

Abstract. These are notes I created for a seminar talk, following the paper of “Introduction to Shimura Varities” by Milne. I claim no credit to the originality of the contents of these notes. Nor do I claim that they are without errors, nor readable.

1. Review

Notations: Let U1 be the complex unit circle

Definition 1.1. Let M be a complex manifold. It is homogeneous if its automorphism group (automorphism can be Riemannian, Hermitian, ...) acts transitively (for all p, q there exists an automorphism sending on to the other). It is symmetric if it is homogeneous, and for some p (hence for all) there exists an involution with p as an isolated fixed point (some nbhd st. p is the only fixed point). A connected symmetric Hermitian manifold is called a Hermitian symmetric space. It is a Hermitian symmetric domain if it is a hermitian symmetric space of non-compact type (non-positive, but not identically zero sectional curvature).

Proposition 1.2. Let (M, g) be a Hermitian symmetric domain and h be the Lie algebra of Hol(M)+ (this is a Lie Group, in fact, so is Is (M, g)). There is a unique connected algebraic subgroup G of GL(h) such that

+ + G(R) = Hol(M) inside GL(h).

For such a G, + G(R) = G(R) ∩ Hol (M) inside GL(h) and so G(R)+ is the stabilizer of G(R) of M. This group G is adjoint (in particular semisimple) and G(R) is not compact.

Deﬁnition 1.3. G is adjoint if Ad : G → Aut(g) is faithful (non-trivial center). It’s semi-simple.

Proposition 1.4. Let D be a HSD, for each p ∈ D, there exists a unique homomorphism up : U1 → Hol(D) = G(R) such that up(z) ﬁxes p and acts on TpD as multiplication by z.

Remark 1.5. These homomorphisms are algebraic, because the representations of U1 have the same description when regarded as a Lie group or as an algebraic group

Theorem 1.6. Let D be a Hermitian symmetric domain, G the associated real adjoint algebraic group. The homomorphisms up : U1 → G attached to each p ∈ D satisﬁes: −1 (1) Only the characters z, 1, z occur in the representation of U1 on Lie(G)C deﬁned by Ad ◦ up (2) ad (up(−1)) is a Cartan Involution

(3) up(−1) does not project to 1 in any simple factor of G

Conversely, let G be a real adjoint algebraic group, and let u : U1 → G satisfy the above. Then the set D of conjugates of u by elements of G(R)+ has a natural structure of a hermitian symmetric domain, for which G(R)+ = Hol(D)+ and u(−1) is the symmetry at u (viewed as a point of D).

Lemma 1.7. Let H be an adjoint real Lie group (semisimple with trivial center), and let u : U1 → H be a homomorphism satisfying SU1 and SU2. Then TFAE: (1) u(−1) = 1 (2) u is trivial (u(z) = 1 for all z) 1 SHIMURA DATUM 2

(3) H is compact

2 ±1 Proof. First is because u factors through U1 → U1 so z can’t occur in the representation of U1.

The second part is H compact (H(R)) iﬀ identity map on G is a Cartan involution, so adu(−1) = idG. Hence, u(−1) ∈ Z(H) and since Z(H) = 1, u(−1) = 1. This means that the third condition of the theorem, is really just saying that G has no compact simple factors. (some non-compactness condition).

2. Connected Shimura Datum

Note.

• Since extensions and quotients of solvable algebraic groups are solvable (G ⊇ G1 ⊇ ... ⊇ e normal, where each quotient is commutative), G contains a maximal connected solvable (commutative) normal subgroup called the radical R(G) of G. G defined over an algebraically closed field is semisimple if R(G) = e. Else, say it’s semisimple if it is after base extension. (for us, always connected) • Simple algebraic or Lie group is a semi-simple group whose only proper normal subgroups are finite (almost-simple). • Group G is unipotent if every representation of G has a non-zero fixed vector. Also fixed under extensions and quotients. Therefore, every connected group variety has a maximal connected unipotent normal subgroup variety. Define reductive similarly • Gad is the image of G under adjoitn representation • Gder is [G, G], the intersection of all normal subgroups N of G such that G/N is commutative. It’s the smallest subgroup such that G/Gder is commutative. (s.s.).

Definition 2.1. A connected Shimura Datum is a pair (G, D) of a semi-simple algebraic group G over Q and a Gad(R)+-conjugacy class D of homomorphisms u : U → Gad satisfying: 1 R −1 ad • SU1: for all u ∈ D, only the characters z, 1, z occur in the representation of U1 on Lie G defined by C ad ad Ad ◦ u : U1 → G → Aut Lie G . C • SU2: for all u ∈ D, ad (u(−1)) is a Cartan involution on Gad. R • SU3: Gad has no Q-factor H such that H(R) is compact Remark 2.2. We can also talk about pairs (G, u) and if the u satisfies the conditions SU1, and SU2, so will any conjugate of it. Therefore, it determines a connected Shimura datum, but this is better, to avoid having a distinguished point. × Let S be the algebraic trosu over R obtains from Gm over C by restriction of scalars (Deligne torus). That is, S(R) = C and SC = Gm × Gm. We have an exact sequence of tori ω 0 → Gm → S → U1 → 0 where in the real values gives −1 z7→ z × r7→r × z¯ 0 → R → C → U1 → 0. z When H is a s.s. real alg group with trivial center, u : U1 → H gives a homomorphism h : S → H by the rule h(z) = u( z¯). −1 z z¯ ad Then acting by characters z, 1, z is the same as S acting via z¯, 1, z . Converse also holds. Change, ... and adh(i), G has no Q-factor on which the projection of h is trivial. For G a connected algebraic group, there exists a Cartan involution iff G is reductive (any two are conjugate by G(R)). Proof: Satake. ! 2 a b Example 2.3. u : U1 → P GL2(R) be the homomorphism sending z = (a + bi) to mod ± I. Let D be the −b a ! a b −1 set of conjugates (so A A mod ± I) for A ∈ SL2(R). Hence, (SL2,D) is a Shimura datum. −b a This is really just up from above when p = i ∈ H. SHIMURA DATUM 3

Deﬁnition 2.4. A semi-simple algebraic group G/Q is of compact type if G(R) is compact. It’s of non-compact type if it does not contain a non-trivial normal subgroup. G semi-simple over Q, then there exists isogeny

G1 × ... × Gr → G,

(fact in Milne) where Gi’s are simple. All Gi’s are compact (non-compact) iﬀ all Gi’s are non-compact. If it’s simply connected or adjoint, this is non-compact iﬀ no simple factor is of compact type.

Proposition 2.5. To give a connected Shimura Datum is the same as to give

• a semisimple algebraic group G over Q of non-compact type • a Hermitian symmetric domain D • an action of G(R)+ on D deﬁned by a surjective homomorphism Gad(R)+ → Hol(D)+with compact kernel

Proof. Let (G, D) be a connected Shimura datum, and let u ∈ D. Decompose Gad into a product of its simple factos R H1 × ... × Hs, and write u = (u1, ..., us) each projection onto the i-th factor.

If Hi is compact, then ui = 1 (by the lemma). Otherwise, Hi is not compact, and so ui satisﬁes the conditions of the 0 + 0 + 0 real version, so we get an irreducible HSD Di such that Hi(R) = Hol (Di) and Di is in natural 1-1 correspondence + with the set Di of Hi(R) -conjugates of ui. The 0 Y 0 D = Di + ad + + is an HSD on which Hi(R) acts via a surjective homomorphism G (R) → Hol(D) with compact kernel (the Hi’s 0 Q where ui = 1). There’s a natural identiﬁcation D with D = Di.

3. Connected Shimura Variety

Proposition 3.1. Let D be a hermitian symmetric domain, and Γ a discrete subgroup of Hol(D)+. If Γ is torsion free, then Γ acts on D freely. Additionally, there is a unique complex structure on Γ\D that makes the quotient map a local isomorphism.

Remark 3.2. Want BIG groups Γ so that we have ﬁnite co-volume.

Deﬁnition 3.3. Let G be an algebraic group over Q. A subgroup Γ of G(Q) is arithmetic if it is commensurable (equivalence relation) with G(Q) ∩ GLn(Z) (ﬁnite index in both) for some embedding G,→ GLn (then true for every embedding into GLn0 ). Let H be a connected real Lie group. A subgroup Γ of H is arithmetric, if there is an algebraic group G over Q, a + + surjective homomorphism G(R) → Hol(D) with compact kernel, and an arithmetric subgroup Γ0 of G(Q) such that + Γ0 ∩ G(R) maps onto Γ.

Proposition 3.4. Let H be a s.s. real Lie group which admits a faithful finite-dimensional rep. Every arithmetic subgroup Γ of H is discrete of finite covolume, and contains a torsion-free subgroup of finite index.

Theorem 3.5. (Baily and Borel). Let D(Γ) = Γ\D (called locally symmetric variety) be the quotient of a HSD by a torsion-free arithmetric subgroup Γ of Hol(D)+. Then D(Γ) has a canonical realization as a Zariski-open subset of a projective algebraic variety D(Γ)? (so it has a canonical structure of an algebraic variety). In fact, the strucutre of an algebraic variety on D(Γ) is unique. D(Γ) ,→ D(Γ)? is called minimal compactiﬁcation.

Remark 3.6. Recall that in the modular form case, if we let H? = H ∪ P1(Q) then the quotient Γ\H? is a compact Riemann surface, and we can show that it is actually algebraic. We basically omit a ﬁnite number of points to get to Γ\H, so it’s a Zariski-open subset. Here, D(Γ)? = Γ\D? where D? is the union of D with certain “rational boundary components” endowed with the Satake topology. Then automorphic forms of a suﬃciently high weight map Γ\D? isomorphically onto a closed subvariety SHIMURA DATUM 4 of a projective space, and Γ\D is a Zariski open subvariety. Apparently D(Γ)? is usually very singular, when D(Γ) is compact, this can be constructed using Kodaira embedding theorem. This theorem still holds when Γ has torsion, but then D(Γ) is a normal complex analytic space (not manifold) and it has the structure of a normal algebraic variety (rather than non-singular).

Theorem 3.7. (Borel) Let D(Γ) = Γ\D be as above. Let V be a non-singular quasi-projective variety over C. Then every holomorphic map f : V an → D(Γ)an is regular.

Deﬁnition 3.8. G be a reductive algebraic group over Q. Choose an embedding G,→ GLn and deﬁne

Γ(N) = G(Q) ∩ {g ∈ GLn(Z): g ≡ I mod N} .

A congurence subgroup of G(Q) is any subgroup containing Γ(N) as a subgroup of ﬁnite index. (This deﬁnition does not depend on the choice of embedding).

Let (G, D) be a connected Shimura datum, view D as HSD with G(R)+ acting on it. Since Gad(R)+ → Hol(D)+ has compact kernel, the image Γ¯ of any arithmetic subgroup Γ of Gad(Q)+ in Hol(D)+ is arithmetic; and the kernel of Γ → Γ¯ is ﬁnite. Point is, the two theorems above apply to D(Γ). For every Γ0 ⊆ Γ, we have natural regular map

D(Γ) ← D(Γ0).

The inverse system of all D(Γ) where Γ is an airthmetric subgroup of Gad(Q)+ containing the images of congruence subgroups of G(Q)+ and such that Γ¯ is torsion-free is denoted Sh◦(G, D) is called the connected Shimura variety attached to (G, D). Each D(Γ) is called a connected Shimura variety relative to (G, D). • The varieties D(Γ) for Γ a congruence subgroup are coﬁnal • Why congruence subgroups, because arithmetic properties of quotients of HSD by noncongruence arithmetic subgroups are not well understood. Also, congruence subgroups arise naturally when we work adelically.

Remark 3.9. Γ¯ doesn’t have to be torsion free, but then things may be singular, and the two theorems may not apply.

Let (G, D) be a connected Shimura datum, and τ be the topology on Gad(Q) for which the images of the congruence subgroups of G(Q) form a fundamental system of neighbourhoods of 1. D(Γ) are just those Γs that are open and image is torsion free. Since D was a conjugacy class of u : U → Gad, an element g ∈ Gad( )+ deﬁnes a holomorphic map D → D, 1 R Q and so a map Γ\D → gΓg−1\D which is holomorphic and regular (by the Borel Baily theorems). Conjugation is a homeomorphism under this τ topology, so Gad(Q)+ acts on the inverse system Sh0 (G, D) and extends by continuity to action of G\ad(Q)+.

Example 3.10. Let B be a quaternion algebra over a totally real ﬁeld F . Then Y B ⊗Q R = B ⊗F,v R v: F,→R where each B ⊗F,v R is isomorphic to either the usual quaternions H or to M2(R). Let G be the semi-simple algebraic group over Q such that × × G(Q) = ker Nm : B → F , then ∼ ×1 ×1 G(R) = H × ... × H × SL2(R) × ... × SL2(R) ×1 × × where H = ker (Nm : H → R ) (the ﬁrst comes from the parts where B ⊗F,v R is isomorphic to H, and the latter from SL2(R) because the norm is just the determinant). Let D be a product of copies of the upper half plane, one for each copy of SL2(R) then the above isomorphism determines an action of G(R) on D, which will satisfy all the conditions and so is a Shimura datum. SHIMURA DATUM 5 ! ∼ 1 1 If B = M2(F ), then G(Q) has unipotent elements (like ) and so D(Γ) is not compact (theorem). This D(Γ) 0 1 is then called a Hilbert modular variety. If B is a division algebra, then G(Q) has no unipotent elements, and so D(Γ) is compact.

Theorem 3.11. G reductive group over Q and Γ an arithmetic subgroup of G(Q). Γ\G(R) is compact iﬀ hom (G, Gm) = 0 and G(Q) contains no unipotent element (other than 1).

Remark 3.12. The idea is that the unipotent elements correspond to cusps (think SL2 acting on H1), so to have no rational unipotent elements means that there are no cusps (projective and smooth).

4. Adelic Description Q0 Proposition 4.1. Let Af = ` Q`(ranging through all ﬁnite primes). Let K be a compact open subgroup of G(Af ). Then K ∩ G(Q) is a congruence subgroup of G(Q) and every congruence subgroup is of this form. Q Proof. Let K(N) = ` K` where  G(Z`) ` - N K` = r {g ∈ G(Z`): g ≡ 1 mod ` ` } r` = ord`(N) Q Q Then K(N) is a compact open subgroup of G(Af ) (topology is v∈S Uv × v∈ /S Ov where Uv ⊆ Qv is open), and

K(N) ∩ G(Q) = Γ(N).

It follows that the compact open subgroups of G(Af ) containing K(N) intersect G(Q) exactly in the congruence subgroups of G(Q) containing Γ(N). Since every compact open subgroup of G(Af ) contains K(N) for some N, done.

Deﬁnition 4.2. A semisimple group G over Q is of compact type if G(R) is compact. It’s of non-compact type if it does not contain a non-trivial normal subgroup of compact type.

Theorem 4.3. (Borel Density Theorem) Let G be a semisimple group over Q of non-compact type. Then every arithmetic subgroup Γ of G(Q) is Zariski dense in G.

Theorem 4.4. (Strong Approximation Theorem) Let G be a s.s. group over Q that is also simply connected and of 0 non-compact type, then G(Q) is dense in G(Af ). Here, simply connected means that every isogeny G → G (surjective with ﬁnite kernel) is an isomorphism (technically only over characteristic 0). Every s.s. algebraic group has a unique isogeny G˜ → G connected and simply connected.

Fact 4.5. If G is s.s. and simply connected group over Q, then G(R) is connected, and so G(Q) ⊆ G(R)+ ⊆ G(R). This means that G(Q) acts on D via G(R)+.

Theorem 4.6. Let (G, D) be a connected Shimura datum with G simply connected. Let K be a compact open subgroup of G(Af ), and let Γ = K ∩ G(Q) (congruence subgroup of G(Q)), then the map x 7→ [x, 1] deﬁnes a homeomorphism ∼ Γ\D = G(Q)\D × G(Af )/K and here, G(Q) acts on D and G(Af ) on the left, and K acts on G(Af ) on the right. Giving D the usual topology, and G(Af ) the adelic topology (or discrete), this is a homeomorphism. Additionally,

lim G( )\D × G( f )/K = lim D(Γ) = G( )\D × G( f ). ← Q A ← Q A

Proof. K open, so by the strong approximation theorem, G(Af ) = G(Q) · K. Therefore, any element of

G (Q) \D × G(Af )/K SHIMURA DATUM 6 is represented by some element of the form [x, 1]. We have [x, 1] = [x0, 1] iﬀ there exists q ∈ G(Q) and k ∈ K such that x0 = qx and 1 = qk. The second implies that q = k−1 ∈ Γ. Therefore, x0 = x mod Γ is the same element in Γ\D. Consider

x7→(x,[1]) D D × (G(Af )/K)

[x]7→[x,1] Γ\D G(Q)\D × G(Af )/K

Since K is open, G(Af )/K is discrete. This means that the upper map is a homeomorphism of D onto its image, which is open. It follows that the bottom is a homemorphism. Second part is hard.

Remark 4.7. Why do we care? Well, this description makes it obvious that there’s a G(Af ) on the inverse system. Double cosets are important in Langlands program. The inverse limit is actually a scheme, and it’s locally noetherian and regular, but not of ﬁnite type over C. We can recover the inverse system from the limit, so we can really just work with the limit (as Deligne does). There is yet another description of connected Shimura datum, that I skipped.

5. Shimura Datum

Deﬁnition 5.1. A Shimura datum is a pair (G, X) consisting of a reductive group G over Q and a G(R)-conjugacy class

X of homomorphisms h : S → GR satisfying the following:

(1) For all h ∈ X, the Hodge structure on Lie(GR) deﬁned by Ad ◦ h is of type {(−1, 1), (0, 0), (1, −1)} .

That is, the only characters that appear in Lie(GR)C deﬁned by z/z,¯ 1, z/z¯ occur in the representation. Since, Lie(G ) = Lie (Z ) ⊕ Lie Gad and Ad (h(z)) automatically acts trivially on Lie(Z ), this really is the same as C C C C before. (2) For all h ∈ X, ad (h(i)) is a Cartan involution of Gad R (3) Gad has no Q-factor on which the projection of h is trivial

Example 5.2. Let G = GL2 (over Q) and let X be the GL2(R)-conjugates of the homomorphism h0 : S → GL2, deﬁned ! R a b by h0 (a + ib) = . Then (G, X) is a Shimura datum. −b a −1 × There is a natural bijection X → C\R, by h0 7→ i and gh0g 7→ gi. More intrinsically, h ↔ z iﬀ h(C ) is the stabilizer of z in GL2(R) and h(z) acts on the tangent space at z as multiplication by z/z¯.

Deﬁnition 5.3. Gder or [G, G] is the intersection of all normal subgroups N of G such that G/N is commutative. It’s the smallest subgroup such that G/Gder is commutative and Gder is semisimple. For G reductive, let T be G/Gder the largest commutative quotient of G (torus).

Proposition 5.4. Let G be a reductive group over R. For a homomorphism h : S → G, let h¯ be the composite of h with the projection G → Gad. Let X be the G(R)-conjugacy class of hom S → G and X¯ be the Gad(R)-conjugacy class hom S → Gad containing h¯ for h ∈ X. (1) The map X → X¯ given by h 7→ h¯ is injective, and its image is a union of connected components. (2) Let X+ be a connected component of X, and X¯ + be its image in X.¯ If (G, X) satisﬁes the above 3 conditions, der ¯ + + then G , X satisﬁes the same for semisimple. Moreover, the stabilizer of X in G(R) is G(R)+ (the group ad ad + of elements of G(R) whose image in G (R) lies in its identity component G (R) . For later, G(Q)+ = G(Q) ∩ G(R)+). SHIMURA DATUM 7

(3) Let X+ be a connected component, regarded as a G( )+-conjugacy class of hom → Gad, then Gder,X+ is a R S R connected Shimura datum, and so X is a ﬁnite disjoint union of HSD.

6. Shimura Varieties

Deﬁnition 6.1. For a compact open subgroup K of G(Af ), deﬁne

ShK (G, X) = G(Q)\X × G(Af )/K where G(Q) acts on X and G(Af ) on the left, and K acts on G(Af ) on the right. When K is small, this is called a Shimura variety relative to (G, X).

+ Proposition 6.2. Let C be the set of representatives for the double coset space G(Q)+\G(Af )/K , and let X be a connected component of X, then ∼ + G(Q)\X × G(Af )/K = tg∈C Γg\X , −1 where Γg is the subgroup gKg ∩ G(Q)+ (homeomorphism, with usual topology on X and adelic on G(Af )).

Remark 6.3. G(Q)+\G(Af )/K is a ﬁnite set.

Proof. For g ∈ C, consider the map

+ + Γg\X → G(Q)+\X × G(Af )/K, given by [x] 7→ [x, g].

Want to show that this is injective for each g, and RHS is the disjoint union of the images for diﬀering g. Then by the ∼ + lemma, G(Q)\X × G(Af )/K = tg∈C Γg\X . 0 0 −1 −1 If [x, g] = [x , g] then x = qx and g = qgk for some q ∈ G(Q)+ and k ∈ K. From the second equation, q = gk g ∈ Γg and so [x] = [x0]. This proves injectivity. −1 For the second part, let (x, a) ∈ G(Af ). Then a = qgk for some q ∈ G(Q)+, g ∈ C, k ∈ K. Now [x, a] = q x, g + which lies in the image of Γg\X . Suppose [x, g] = [x0, g0] for g, g0 ∈ C. Then x0 = qx and g0 = qgk. The second equation implies that g0 = g, by deﬁnition.

Theorem 6.4. (Real approximation) Let G be a connected algebraic group over Q. Then G(Q) is desne in G(R). More generally, G(Q) is dense in G(R) if every connected component of G contains Q-point.

Proof. Deligne: "on se ramène aisément au cas des tores" (easily get back to the case of tori).

Lemma 6.5. For every connected component X+ of X, the natural map

+ G(Q)+\X × G(Af ) → G(Q)\X × G(Af ) is a bijection.

Proof. Since G(Q) is dense in G(R) and G(R) acts transitively on X, every x ∈ X is of the form qx+ with q ∈ G(Q) and x+ ∈ X+. This shows that the map above is surjective. 0 0 + Let (x, a) and (x , a ) ∈ X × G(Af ). If they are equal on the RHS, then

0 0 x = qx, a = qa for some q ∈ G(Q).

0 + + 0 0 Since x, x are both in X , q stabilizes X and so lies in G(R)+. Therefore, [x, a] = [x , a ] in G(Q)+\X × G(Af ).

+ Remark 6.6. Γg\X are arithmetic locally symmetric varieties, so ShK (G, X) are ﬁnite disjoint unions of these. Given 0 an inclusion K ⊆ K suﬃciently small compact open subgroups of G(Af ), the map ShK0 (G, X) → ShK (G, X) is regular.

Deﬁnition 6.7. The inverse system Sh (G, X) is called the Shimura variety attached to (G, X). SHIMURA DATUM 8

−1 Remark 6.8. There is a natural action of G(Af ) on the system, by g ∈ G(Af ) we map K 7→ g Kg maps compact open subgroups to compact open subgroups, so we get

T (g): ShK (G, X) → Shg−1Kg(G, X).

On points, it’s [x, a] 7→ [x, ag] (right action). Use “deﬁned by” or associated with”, but not “associated to”, which, strictly speaking, is not English. Careful writers distinguish “attach to” from “associated with” and look with horror on “associate to”.

7. Morphisms

Deﬁnition 7.1. The (G, X) and (G0,X0) be Shimura data. (1) A morphism of Shimura data is just a homomorphism G → G0 of algebraic groups sending X into X0 (2) A morphism of Shimura varieties Sh (G, X) → Sh (G0,X0) is an inverse system of regular maps of algebraic

varieties compatible with the action of G(Af )

Theorem 7.2. Morphism of Shimura data deﬁnes a morphism of Shimura varieties. Closed immersion if injective

8. Structure of a Shimura Variety

General case worked out in Deligne 1979 Variété de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. When Gder is simply connected, the set of connected components is a “zero-dimensional Shimura variety” and each connected component is a connected Shimura variety. × × Since h : S → GR we act through those characters, we see that for all r ∈ R ⊆ C = S(R), h(r) acts trivially on ad ad Lie(GR)C. The action of G on Lie(G), factors through G and Ad : G → GL(Lie(G)) is faithful, h(r) ∈ Z(R) where Z is −1 w × r7→r × z7→z/z¯ the centre. Thus, h |Gm is independent of h (0 → Gm → S → U1 → 0, take R-points, get 0 → R → C → U1 → 0 just depends on X). Let the reciprocal be denoted by ωX , called the weight homomorphism. From Michael’s talk, we know that the weight homomorphism determines a representation of S, which is the same as a Hodge structure. Let

ρ : GR → GL(V ) be a representation, then ρ ◦ ωX deﬁnes a decomposition V = ⊕Vn, which is the weight decomposition of the Hodge structure (V, ρ ◦ h) for every h ∈ X.

Proposition 8.1. Let (G, X) be a Shimura datum, then X has a unique structure of a complex manifold, such that for every representation ρ : GR → GL(V ), (V, ρ ◦ h)h∈X is a holomorphic family of Hodge structures. For this complex structure, each family is a variation of Hodge structures, and X is a ﬁnite disjoint union of HSD.