Density of members with extra Hodge cycles in a family of Hodge structures

Ching-Li Chai^{∗} University of Pennsylvania ^{†}

Abstract

Let M_{K}(G, X) be a Shimura variety over ℂ; let L be a ℚ-subgroup of G whose G(ℚ)-conjugacy class corresponds to a prescribed type of Hodge cycles. For every subvariety V of M_{K}(G, X), denote by S(V, L) the subset of points of V whose Mumford-Tate group is contained in a G(ℚ)-conjugate of L. We define an invariant c(G, X, L) depending only on the G(ℝ)-conjugacy of L. The main result says that for every subvariety V of codimension at most c(G, X, L), the subset S(V, L) is dense in V in the metric topology. The value of c(G, X, L) is tabulated in many examples.

Contents

0 Introduction 1

1 Formulation 3

2 Further linearization 6

3 Examples 9 3.1 (Sp_{2n}, Sp_{2n_{1}} × ⋯ × Sp_{2n_{r}}), n_{1} + ⋯ n_{r} = n. . . . . . . . . . . 10 3.2 (Sp_{2n}, U(p, q)), p + q = n. . . . . . . . . . . . . . . . . . . . . 12 3.3 (SO(2, 2n − 1), SO(2, 2m − 1) × SO(2n − 2m)), m < n. … 13 3.4 (Sp_{2^{n}}, SO(2, 2n − 1)) . . . . . . . . . . . . . . . . . . . . . . . 14 3.5 (U(p, q), U(p_{1}, q_{1}) × ⋯ U(p_{r}, q_{r})), ∑ p_{i} = p, ∑ q_{i} = q. . . . . . 15 3.6 (SO(2, 2n − 2), SO(2, 2m − 2) × SO(2n − 2m)), m < n. . . . . 16 3.7 (D_{n}^{ℍ}, D_{n_{1}}^{ℍ} × ⋯ D_{n_{r}}^{ℍ}), n_{1} + ⋯ + n_{r} = n. . . . . . . . . . . . . . 16 3.8 (E_{6}, D_{5}^{ℝ}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.9 (E_{6}, A_{5}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.10 (E_{7}, D_{6}^{ℝ}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

∗This research is partially supported by grant DM95-02186 from the National Science Foundation

†email: chai@math.upenn.edu

0 Introduction

The question we address in this paper is part of a more general question: Given a variation of Hodge ℚ-structures over a complex analytic variety V, do the points of V corresponding to members in this family having extra Hodge cycles of a given type form a dense subset of V?

A special case of this question was raised in [1]. Let V ⊂ 𝒜_{g} be a subvariety of the moduli space 𝒜_{g} of principally polarized abelian varieties of dimension g over ℂ, and for any integer k between 1 and g−1 denote by S_{k}(V) the subset of V consisting of all points x ∈ V such that the corresponding abelian variety A_{x} has an abelian subvariety of dimension k. In [1], Colombo and Pirola gave a sufficient condition for S_{k}(V) to be dense in V with respect to the metric topology. They also proved that if V is a subvariety of the Jacobian locus T (ℳ_{g}) of codimension at most g − 1, then S_{1}(V) is dense in V. Here T (ℳ_{g}) denotes the locus of jacobians of complete smooth curves of genus g in 𝒜_{g}. Using the same criterion, Izadi showed in [5] that if V is either a subvariety of 𝒜_{g} of dimension at most g, or V is a subvariety of T (ℳ_{g}) of dimension at most g, then S_{1}(V) is dense in V.

This paper was started by the attempt to understand the meaning of the results of [1, 5] in a more general context. We replace the parameter space 𝒜_{g} by a Shimura variety M_{K}(G, X), and allow Hodge cycles more general than idempotent endomorphisms of a polarized abelian variety to arrive at the more general question stated at the beginning; see §1 for the precise formulation. In our setting, the subset S_{k}(V) of V ⊂ 𝒜_{g} is replaced by a subset S(V, L) of V ⊂ M_{K}(G, X), where L is a reductive subgroup of G defined over ℚ. We define an invariant c(G, X, L) ∈ ℕ which has the property that S(V, L) is dense in V if V has codimension at most c(G, X, L) in M_{K}(G, X). This generalizes [1, 5]. The invariant c(G, X, L) depends only on the G(ℝ)-conjugacy class of L in G; its value in the case considered in [1, 5] is g.

Our method differs from those use in [1, 5] only at the end, where we further linearized the problem to compute c(G, X, L). This is the content of §2; it allowes us to replace the geometric arguments used in [1, 5] by a simpler linear algebra argument, which makes the invariant c(G, X, L) easy to compute. We illustrate this in §3 and tabulate the value of c(G, X, L) in many cases. For instance in the situation considered in [1, 5], our result says that if V ⊂ 𝒜_{g} is a subvariety of codimension at most g, then S_{k}(V) is dense in V for any k between 1 and g − 1. Another example of our result is that for every subvariety V in 𝒜_{2^{9}} of codimension at most 10, the subset of points of V corresponding to abelian varieties attached to a polarized K3 surface form a dense subset in V. The examples in §3 occupy more than half of this paper. The reader is advised to go over just one or two of them to see the results in action.

Although in most irreducible cases the invariant c(G, X, L) is positive when L is reasonably large, it is not always so; we give two such examples. In one example G is of type E_{6} and L^{ad} is of type D_{5}. In another example G is of type E_{7}, and L^{ad} is of type D_{6}.

1 Formulation

We shall follow Deligne’s point of view on Shimura varieties and think of a Shimura variety M_{K}(G, X) as a parameter space of Hodge structures whose Hodge group (or, special Mumford-Tate group) is contained in G; see [2, 3, 4]. The notation and convention in those papers will be generally followed. More specifically, let G be a connected reductive linear algebraic group over ℚ, let X be a G(ℝ)-conjugacy class of ℝ-homomorphisms from 𝕊 = Res_{ℂ/ℝ}𝔾_{m} to G such that the following conditions are satisfied:

1. For one (hence for every) h ∈ X, under the adjoint representation Ad ∘ h of 𝕊 on 𝔤 = Lie(G), 𝔤 is a Hodge ℚ-structure of type ⊆ {(1, −1), (0, 0), (−1, 1)}.

2. For one (hence for every) h ∈ X, Ad ∘ h(\sqrt{−1}) induces a Cartan involution on G^{ad} ×_{Spec ℚ} Spec ℝ.

3. Every nontrivial simple ℚ-factor of G is non-compact.

For every compact open subgroup K ⊂ G(𝔸_{f}) = ∏′_{p} G(ℚ_{p}), the complex analytic variety G(ℚ)\X × G(𝔸_{f})/K has a canonical structure of a complex quasi-projective variety; we denote it by M_{K}(G, X). When K is small enough (for instance if it is neat) M_{K}(G, X) is smooth. Since the properties of M_{K}(G, X) we consider do not depend on the choice of K, we shall assume for simplicity that M_{K}(G, X) is smooth. Because the argument we use is entirely differential geometric, the canonical model and other arithmetic structures of M_{K}(G, X) will not be used in this paper.

To every finite dimensional ℚ-rational representation ρ : G → GL(V_{ρ}) of G on a finite dimensional ℚ-vector space V_{ρ} , there is a variation of ℚ-rational Hodge structure over X naturally attached to ρ; it descends to M_{K}(G, X) if K is small enough. For G = GSp_{2g}, X = ℍ_{g}^{±} = the Siegel upper-and-lower half space of genus g, K = GSp_{2g}(ℤ̂) and ρ the standard representation, we get the family of Hodge structures \underline{H}_{1}(A(ℂ), ℚ) over 𝒜_{g,n}, where f : A → 𝒜_{g,n} is the universal principally polarized abelian variety with level-n-structure over 𝒜_{g,n}, n ≥ 3.

The following construction is used in [1, 5]. To every ℂ-vector subspace W ⊂ Lie(A(ℂ)) of dimension k, one associate to it its underlying ℝ-vector space, thought of as an ℝ-vector subspace of the ℝ-vector space \underline{H}_{1}(A(ℂ), ℝ). This gives a map from the complex Grassmannian of k-dimensional subspaces of Lie(A(ℂ)) to the real Grassmannian of 2k-dimensional subspaces of \underline{H}_{1}(A(ℂ), ℝ). To adapt this construction to our setting, first observe that the only 2k-dimensional ℝ-subspaces of \underline{H}_{1}(A(ℂ), ℝ) which appears are those such that the restriction of the polarization form to them are nondegenerate. So we can replace the target real Grassmanian space by the quotient Sp_{2g}(ℝ)/Sp_{2k}(ℝ) × Sp_{2g−2k}(ℝ). Notice that the source complex Grassmannian can be written as a quotient U_{g}(ℝ)/U_{k}(ℝ) × U_{g−k}(ℝ), and that U_{k}(ℝ) × U_{g−k}(ℝ) is the intersection of Sp_{2g}(ℝ)/Sp_{2k}(ℝ) with the maximal compact subgroup U_{g}(ℝ) of Sp_{2g}(ℝ). Hopefully the above discussion either motivates the definition and the construction below or illustrates them.

Let L be a ℚ-rational reductive subgroups of G such that there exists a homomorphism h : 𝕊 → G in X which factorizes through L.

Definition 1.1. For an analytic subvariety V ⊆ M_{K}(G, X), S(V, L) denotes the subset of V consisting of all points x ∈ V of the form x = [(h, g)] with h ∈ X, g ∈ G(𝔸_{f}) such that h factorizes through γ·L·γ^{−1} for some γ ∈ G(ℚ).

Remark. (i) Clearly S(V, L) = S(V, L′) if L and L′ are conjugate under G(ℚ). The subset S(V, L) is equal to the intersection of S(M_{K}(G, X), L) with V.

(ii) The condition that h factorizes through Ad(γ) · L means that the special Mumford-Tate group for h is contained in Ad(γ) · L. This justifies our assumption that L is reductive, since the special Mumford-Tate group for any Hodge structure is reductive.

(iii) In the case G = GSp_{2g}, X = ℍ_{g}^{±}, and L the intersection of G with the product GSp_{2k} × GSp_{2g−2k} diagonally embedded in GL_{2g}, S(V, L) is equal to the subset S_{k}(V) of V defined in the introduction.

We denote by ℭ the set of all ℝ-rational subgroups L′ of G conjugate to L under G(ℝ); it has a natural structure as a real-analytic manifold. Let Π denote the set consisting of all pairs (h, L′) with h ∈ X, L′ ∈ ℭ such that h factorizes through L′. Again Π is a real-analytic manifold. The two natural projections pr_{1} : Π → X and pr_{2} : Π → ℭ are both realanalytic maps. The group G(ℝ) operates on the left of Π, X and ℭ, and the two projections are both equivariant with respect to G(ℝ). For the given subvariety V ⊂ M_{K}(G, X), let Ṽ be the inverse image of V in Π, and let ϕ : Ṽ → ℭ be the restriction of the projection Π → ℭ to Ṽ.

Proposition 1. Assume that ϕ : Ṽ → ℭ is generically a submersion, i.e. ϕ is a submersion outside a nowhere-dense real-analytic subset Z of Ṽ. Then S(V, L) is dense in V for the metric topology.

Proof. We only need to recall the well-known fact that G(ℚ) is dense in G(ℝ) for the metric topology, and to observe that the projection map pr_{1} : Ṽ → V is surjective. An elementary topological argument concludes the proof: For non-empty open subset U of V, the image of pr_{1}^{−1}(U) − Z under ϕ is open in ℭ, hence contains a G(ℚ)-conjugate of L.

Next we compute the image of the differential of the map ϕ : Ṽ → ℭ. Let y = (h, L_{1}) be a point of Ṽ which lies over a smooth point of V. Then y is a smooth point of Ṽ as well. Denote by N_{G}(L_{1}) the normalizer of L_{1} in G, and let N_{𝔤}(L_{1}) = Lie(N_{G}(L_{1})) be its Lie algebra, which is also equal to the normalizer N_{𝔤}(𝔩_{1}) of 𝔩_{1} = Lie(L_{1}) in 𝔤 = Lie(G). We identify the tangent space of ℭ at the point [L_{1}] ∈ ℭ with 𝔤/N_{𝔤}(𝔩_{1}). Let V̄ be the inverse image of V in X. The fiber at h ∈ V̄ of Ṽ → V̄ maps under ϕ onto the set of all K_{h}-conjugates of L_{1}, where K_{h} denotes the centralizer of h in G. Therefore the differential of ϕ at the point y = (h, L_{1}) sends the pr_{1}-vertical tangent space onto 𝔨_{h} + N_{𝔤}(𝔩_{1})/N_{𝔤}(𝔩_{1}), where 𝔨_{h} = Lie(K_{h}). Following the literature we denote by 𝔭 the (−1)-eigenspace of Ad(h(\sqrt{−1})) in 𝔤, identified with the tangents space of X as usual. The tangent space of V̄ at the point h ∈ V̄ then become an ℝ-vector subspace of 𝔭, which we denote by U. The image of dϕ at (h, L_{1}) is equal to 𝔨_{h} + U + N_{𝔤}(𝔩_{1})/N_{𝔤}(𝔩_{1}) under our notation. The Lie algebra 𝔤 acquires a ℝ-Hodge structure under Ad ∘ h, and 𝔭 is a ℝ-Hodge substructure with Hodge decomposition 𝔭 ⊗_{ℝ} ℂ = 𝔤^{1,−1} ⊕ 𝔤^{−1,1}. Here 𝔤^{1,−1} and 𝔤^{−1,1} are the (1, −1) and (−1, 1)-components of 𝔤 ⊗_{ℝ} ℂ respectively, and 𝔤^{−1,1} is identified with the complex tangent space of X at h. Then U ⊗_{ℝ} ℂ = U^{1,−1} ⊕ U^{−1,1}, with U^{1,−1} ⊆ 𝔤^{1,−1} and U^{−1,1} ⊆ 𝔤^{−1,1}. Moreover U^{−1,1} is the complex tangent space of V̄ at the point h. Since h factorizes through L_{1} ⊆ N_{G}(L_{1}), N_{𝔤}(𝔩_{1}) is an ℝ-Hodge substructure of 𝔤; we write its Hodge decomposition as

N_{𝔤}(𝔩_{1}) ⊗_{ℝ} ℂ = N_{𝔤}(𝔩_{1})^{1,−1} ⊕ N_{𝔤}(𝔩_{1})^{0,0} ⊕ N_{𝔤}(𝔩_{1})^{−1,1}.

Clearly N_{𝔤}(𝔩_{1})^{0,0} = N_{𝔤}(𝔩_{1}) ∩ 𝔨_{h}. Moreover 𝔨_{h} + U + N_{𝔤}(𝔩_{1}) = 𝔤 if and only if U^{−1,1} + N_{𝔤}(𝔩_{1})^{−1,1} = 𝔤^{−1,1}. We record the above in the following:

Proposition 2. Let (h, L_{1}) be a point of Ṽ lying over a smooth point of V. Identify the tangent space of V̄ at h with an ℝ-vector subspace U ⊆ 𝔭_{h} = the (−1)-eigenspace of Ad(h(\sqrt{−1})) in 𝔤. We identify the tangent space of ℭ at L_{1} with 𝔤/N_{𝔤}(𝔩_{1}). Then the image of the differential of ϕ : Ṽ → ℭ at (h, L_{1}) is equal to 𝔨_{h} + U + N_{𝔤}(𝔩_{1}), where 𝔨_{h} denotes the (+1)-eigenspace of Ad(h(\sqrt{−1})) in 𝔤. Consequently ϕ is a submersion at (h, L_{1}) if and only if U^{−1,1} + N_{𝔤}(𝔩_{1})^{−1,1} = 𝔤^{−1,1}.

Corollary 1.1. Let V, Ṽ, V̄ be as above. Suppose that there exists a point h ∈ X lying over a smooth point of V and an element L_{1} ∈ ℭ such that h factorizes through L_{1}, and such that the tangent space U^{−1,1} of the inverse image V̄ of V in X at h is transversal with N_{𝔤}(𝔩_{1})^{−1,1} in 𝔤^{−1,1}. Then S(V, L) is dense in V.

Remark. (i) We shall see that when the codimension of V in M_{K}(G, X) is smaller than or equal to a constant c(G, X, L), then for every smooth point [(h, g)] of V with h ∈ X, g ∈ G(𝔸_{f}), there exists an L_{1} ∈ ℭ satisfying the condition in the corollary.

(ii) Suppose that the condition of the corollary is not satisfied by any point [(h, g)] in V. This gives us a system of equations satisfied by all points of V̄. We can differentiate these equations to get linearized equations. In this paper we shall only use the vertical directions, although it is conceivable that differentiation in the horizontal directions may produce further information.

2 Further linearization

Let G, X, L be as before. For an element h : 𝕊 → G in X, denote by K_{h} the centralizer of h in G. We abbreviate to N the (−1, 1)-component N_{𝔤}(𝔩)^{−1,1} of the normalizer N_{𝔤}(𝔩) of 𝔩 = Lie(L) in 𝔤. Our discussion in the last section led us to make the following definition.

Definition 2.1. Let L_{1} be a G(ℝ)-conjugate of L and h : 𝕊 → L_{1} ⊆ G be an element of X. When N_{𝔤}(𝔩)^{−1,1} ⫋ 𝔤^{−1,1}, define c(G, h, L_{1}) to be the largest non-negative integer such that for every ℂ-vector subspace U ⊆ 𝔤^{−1,1} of codimension ≤ c(G, h, L_{1}), there exists an element k ∈ K_{h}(ℝ) such that U + k · N_{𝔤}(𝔩_{1})^{−1,1} · k^{−1} = 𝔤^{−1,1}. When N_{𝔤}(𝔩)^{−1,1} = 𝔤^{−1,1}, c(G, h, L_{1}) is defined to be ∞. Clearly c(G, h, L_{1}) = c(G, Ad(g) · h, Ad(g) · L_{1}) for every g ∈ G(G(ℝ). It is easy to check that in fact c(G, h, L_{1}) = c(G, h′, L_{2}) for any G(ℝ)-conjugate L_{2} of L and any element h′ : 𝕊 → L_{2} ⊆ G in X. So we write c(G, X, ℭ) instead of c(G, h, L_{1}).

Remark. (i) It is immediate from the Corollary of Proposition 2 that S(V, L) is dense in V for every analytic subvariety V ⊆ M_{K}(G, X) of codimension at most c(G, X, ℭ).

(ii) Clearly c(G, h, L) can be computed after base change to ℝ, i.e. it is insensitive to the ℚ-structure. It is also clear that the invariant c(G, h, L_{1}) can be computed from the K_{h}-representation 𝔤^{−1,1} and the ℂ-vector subspace N_{𝔤}(𝔩_{1})^{−1,1} in 𝔤^{−1,1}. This prompts us to make the following definition.

Definition 2.2. Let ρ be a real-analytic linear representation of a realanalytic Lie group K on a finite dimensional ℂ-vector space V_{ρ}. Let N be a ℂ-vector subspace of V_{ρ}. When N ≠ V_{ρ}, define d(K, ρ, N) to be the largest non-negative integer such that for every ℂ-vector subspace W of V_{ρ} of codimension at most d(K, ρ, N), there exists an element k ∈ K with V_{ρ} = W + ρ(k) · N. When N = V_{ρ}, d(K, ρ, N) is defined to be ∞.

Remark. Clearly c(G, h, L) = d(K_{h}, 𝔤^{−1,1}, N_{𝔤}(𝔩)^{−1,1}) in the context of the two definitions above. Hence the results we have proved so far shows that the subset S(V, L) of V is dense in V for every analytic subvariety V ⊆ M_{K}(G, X) of codimension at most d(K_{h}, 𝔤^{−1,1}, N_{𝔤}(𝔩)^{−1,1}).

We list some trivial properties of the invariant d(K, ρ, N) in the following lemma. The proofs are omitted.

Lemma 2.1. (1) d(K, ρ, N) = d(K, ρ, ρ(k) · N) for every k ∈ K. (2) d(K, ρ, N_{1}) ≥ d(K, ρ, N_{2}) if N_{1} ⊆ N_{2}.

(3) d(K, ρ_{1} ⊕ ρ_{2}, N_{1} ⊕ N_{2}) ≤ min(d(K, ρ_{1}, N_{1}), d(K, ρ_{2}, N_{2})) if N_{1} ⊆ V_{ρ_{1}}, N_{2} ⊆ V_{ρ_{2}}.

(4) d(K, ρ_{1} ⊕ ρ_{2}, N_{1} ⊕ V_{ρ_{2}}) = d(K, ρ_{1}, N_{1}).

(5) d(K, ρ ⊗ λ, N) = d(K, ρ, N) for every 1-dimensional representation λ : K → ℂ^{×} of K. Here the representation space of ρ ⊗ λ is taken to be V_{ρ}, with ρ ⊗ λ(k)v = λ(k)ρ(k)v for all v ∈ V_{ρ}.

Lemma 2.2. Let ρ_{1}, ρ_{2} be two complex linear representations of K as above. Assume either that the real analytic Lie group K is connected, or that K is the group of real points of a connected real linear algebraic group K̃ and the two representations ρ_{1}, ρ_{2} are both algebraic. Then

d(K, ρ_{1} ⊕ ρ_{2}, N_{1} ⊕ N_{2}) = min(d(K, ρ_{1}, N_{1}), d(K, ρ_{2}, N_{2}))

for any ℂ-vector subspace N_{i} ⊆ V_{ρ_{i}}, i = 1, 2.

Proof. We may assume that N_{1} ≠ V_{ρ_{1}}, N_{2} ≠ V_{ρ_{2}}. Let W be a vector subspace of V_{ρ_{1}} ⊕V_{ρ_{2}} of codimension at most min(d(K, ρ_{1}, N_{1}), d(K, ρ_{2}, N_{2})). Consider the non-empty subsets U_{i} ⊂ K consisting of all elements k ∈ K such that V_{ρ_{i}} = ρ_{i}(k) · N_{i} + (V_{ρ_{i}} ∩ W), i = 1, 2. Each U_{i} is the complement of a closed real-analytic subset of K. So U_{1} ∩ U_{2} ≠ ∅ when K a connect real analytic group. When K is the real points of a connected real linear algebraic group K̃ and ρ_{1}, ρ_{2} are algebraic, each U_{i} is the set of real points of a non-empty Zariski open subset Ũ_{i} of K̃. So Ũ_{1} ∩ Ũ_{2} ≠ ∅, so U_{1} ∩ U_{2} = (Ũ_{1} ∩ Ũ_{2})(ℝ) ≠ ∅ since Ũ_{1} ∩Ũ_{2} is unirational over ℝ. Clearly W +(ρ_{1} ⊕ρ_{2})(k)·(N_{1} ⊕N_{2}) = V_{ρ_{1}} ⊕V_{ρ_{2}} for any k ∈ U_{1} ∩ U_{2} and the lemma is proved.

Remark. In the situation we are interested in, K is the centralizer of an element h ∈ X, therefore K is a connected real linear algebraic group and Lemma 2.2 applies. This often reduces the computation of the invariant c(G, X, ℭ) to the cases when X is irreducible. Then Lemma 2.1 allows one to reduce further to the case when G is simple over ℝ.

Proposition 3. Let ρ be a finite dimension complex linear representation of a real Lie group K. Assume either that K is a connected real-analytic group, or that K is the group of real points of a connected real algebraic group K̃ and ρ is an algebraic representation of K. Let 𝔨 = Lie(K). Let N, W be ℂ-vector subspaces of V_{ρ}. Then dim(W + N) ≥ dim(W + ρ(k) · N) for all k ∈ K if and only if ρ(𝔨) · (N ∩ W) ⊆ W + N.

Proof. The proof is an exercise, but we give it here since it does not seem to be well-known. We first choose a basis v_{1}, …, v_{a} of W ∩ N. Then extend it on the one hand to a basis v_{1}, …, v_{a}, w_{1}, …, w_{b} of W, and on the other hand to a basis v_{1}, …, v_{a}, v_{a+1}, …, v_{a+c} of N. For 1 ≤ i ≤ a, consider the function f_{i} : K → ⋀^{a+b+c+1} on K, defined by

f_{i}(x) = ρ(x)v_{i} ∧ ρ(x)v_{a+1} ∧ ⋯ ∧ ρ(x)v_{a+c} ∧ v_{1} ∧ ⋯ ∧ v_{a} ∧ w_{1} ∧ ⋯ ∧ w_{b} .

Clearly each f_{i} is a real-analytic function on K, and it is a regular rational function on K in the algebraic case. First suppose that dim(W + N) ≥ dim(W + ρ(k) · N) for all k ∈ K. Then each f_{i}(x) is identically 0 on K. Differentiating f_{i}, we see that ρ(𝔨) · v_{i} ∈ W + N for each i = 1, …, a. Conversely, if ρ(𝔨) · (N ∩ W) ⊆ W + N, then the derivatives of each f_{i} are zero, hence f_{i} is identically zero on K. The same argument shows that the function

x ↦ ρ(x)u_{1} ∧ ⋯ ∧ ρ(x)u_{c+1} ∧ v_{1} ∧ ⋯ ∧ v_{a} ∧ w_{1} ∧ ⋯ ∧ w_{b}

is identically zero on K for any choice u_{1}, …, u_{c+1} of c + 1 elements among the basis v_{1}, …, v_{a}, v_{a+1}, …, v_{a+c} of N. This just means that dim(W +ρ(k)· N) ≤ a + b + c = dim(W + N) for all k ∈ K.

Corollary 2.1. Let K, ρ be as in Proposition 3. Let N be a vector subspace of V_{ρ}. Then

d(K, ρ, N) = min{codim(N_{1}, N)|N_{1} ⊆ N, N + ρ(𝔨) · N_{1} ⫋ V_{ρ}} .

Remark. (i) This corollary turns the computation of d(K, ρ, N) into a question in finite dimensional representations of Lie algebras, and is quite useful. Also notice that we can complexify the representation ρ of the Lie algebra 𝔨 in computing d(K, ρ, N).

(ii) Although it is not feasible to give the value of d(K, ρ, N) in every case of interest, we will give many examples to illustrate our result. The irreducible representations of interest to us include the second symmetric product of the standard representation for A_{n}, the second exterior product of the standard representation of A_{n}, the tensor product of the two standard representations for A_{m} ×A_{n}, the standard representation of B_{n}, the standard representation of D_{n}, (one of the two of) the half-spin representation of D_{5} and (one of the two of) the miniscule representation of E_{6}.

3 Examples

We asserted that the invariant

c = c(G, X, ℭ) = c(G, h, L) = d(K_{h}, N_{𝔤}(𝔩)^{−1,1})

is easy to compute in any given situation. In this section we illustrate this point in many examples. As we remarked before, this invariant depends only on the real groups underlying G, the G(ℝ)-conjugacy class of the real group underlying L, and of course the space X. So in frow now on we assume that G is a connected reductive linear algebraic group over ℝ, and that L is a reductive subgroup of G over ℝ. In all the examples below the connected components of X are irreducible hermitian symmetric spaces, so G is simple modulo center. Because of Lemma 2.1 (5), the value of c does not change if we replace G by G^{ad}, so we may freely change G to G′ with G^{ad} = G′^{ad}. It is well-known that G is the inner twist by Ad(h(\sqrt{−1})) of its compact form, and μ_{h} = h∘μ is the fundamental coweight corresponding to a special vertex of the Dynkin diagram of the compact form of G, cf. [3] §1. In describing our examples, we shall give just a representative G among those with the same derived group, indicate the conjugacy class of μ_{h}, and give a description of (a representative of)L. We also give the Dynkin diagram for (G, X); the vertex labeled with μ is the special vertex corresponding to the fundamental coweight μ_{h}.

3.1 (Sp_{2n}, Sp_{2n_{1}} × ⋯ × Sp_{2n_{r}}), n_{1} + ⋯ n_{r} = n.

First we bring up the case discussed in the introduction. Here • G = GSp_{2n}^{ad}, X is the Siegel upper-and-lower half space of genus n. The

homomorphism h which sends an element a + b\sqrt{−1} ∈ ℂ^{×} to

( \begin{matrix}a & −b \\ b & a \\ & & . & . & . \\ & & & & & a & −b \\ & & & & & b & a \end{matrix} )

gives an element of X.

• The group K_{h} is centrally isogenous to the unitary group U(n).

• The Dynkin diagram is:

[figure: α_{1}; μ; α_{2}; α_{3}; <; α_{n−1}; α_{n}]

• The roots of G are ±x_{i} ± x_{j}, 1 ≤ i ≤ j ≤ n. The simple roots are

α_{1} = x_{1} − x_{2}, …, α_{n−1} = x_{n−1} − x_{n}, α_{n} = 2x_{n} .

• The subgroup L in this example is the image of the diagonally embedded Sp_{2n_{1}} × ⋯ × Sp_{2n_{r}}, where each n_{i} ≥ 1, and n_{1} + ⋯ + n_{r} = n, r ≥ 2.

• The Lie algebra of the normalizer of L in G is equal to 𝔩 itself.

• After replacing K_{h} by U(n), the representation 𝔤^{−1,1} becomes the second symmetric product of the standard representation of U(n); the subspace N ⊂ 𝔤^{−1,1} is identified with

S^{2}(ℂ^{n_{1}}) ⊕ ⋯ ⊕ S^{2}(ℂ^{n_{r}}) ⊂ S^{2}(ℂ^{n}) .

• The value of the invariant c in this case is

c = min {n_{i} + n_{j}|1 ≤ i ≤ j ≤ r} .

Our result says that for every analytic subvariety V ⊆ 𝒜_{n} of codimension at most this value of c, the set of all points in V such that the corresponding principally polarized abelian variety is isogenous to the product of r abelian varieties of dimension n_{1}, ⋯, n_{r} is dense in V for the metric topology.

Proof. We give a proof for the case r = 2; the general case is similar. Moreover the same method apply to other examples, so we omit the proofs in most cases later. After complexifying the representation, we see that we are dealing with the second symmetric product of the standard representation GL(V_{1} ⊕ V_{2}), with dim(V_{1}) = n_{1}, dim(V_{2}) = n_{2}, n = n_{1} + n_{2}. The subspace N is S^{2}(V_{1}) ⊕ S^{2}(V_{2}). We choose a basis e_{1}, …, e_{n_{1}} for V_{1} and a basis e_{n_{1}+1}, …, e_{n_{1}+n_{2}} for V_{2}. This gives us a diagonal maximal torus in GL(V_{1} ⊕ V_{2}). The representation space S^{2}(V_{1} ⊕ V_{2}) has weights x_{a} + x_{b}, 1 ≤ a ≤ b ≤ n_{1} + n_{2}, each with multiplicity one. The subspace N is a direct sum of weight spaces, with weights x_{a} + x_{b} with 1 ≤ a ≤ b ≤ n_{1} and x_{α} + x_{β} with n_{1} + 1 ≤ α ≤ β ≤ n_{1} + n_{2}.

First we show that c ≤ n_{1} + n_{2}. Let N_{1} be the direct sum of weight spaces with weights x_{a} + x_{b} with 2 ≤ a ≤ b ≤ n_{1} and x_{α} + x_{β} with n_{1} + 2 ≤ α ≤ β ≤ n_{1} + n_{2}. Clearly x_{1} + x_{n_{1}+1} is not a weight of 𝔤𝔩_{n} · N_{1} + N, so 𝔤𝔩_{n} · N_{1} + N ⫋ S^{2}(V_{1} ⊕ V_{2}). Hence c ≤ codim(N_{1}, N) = n_{1} + n_{2}.

Conversely, let W ⊆ N be a subspace of codimension at most n_{1} +n_{2} −1. The intersection of W with the direct sum of weight spaces with weights x_{1} + x_{b} with 1 ≤ b ≤ n_{1} and x_{n_{1}} + x_{β} with n_{1} + 1 ≤ β ≤ n_{2} contains a non-zero vector q. One quickly sees that 𝔤𝔩_{n} · q + N contains the weight space with weight x_{1} + x_{n_{1}+1}. Similarly one sees that 𝔤𝔩_{n} · W + N contains every weight space x_{a} + x_{α} with 1 ≤ a ≤ n_{1}, n_{1} + 1 ≤ α ≤ n_{1} + n_{2}. Hence 𝔤𝔩_{n} · W + N = S^{2}(V_{1} ⊕ V_{2}).

3.2 (Sp_{2n}, U(p, q)), p + q = n.

• In this example, the pair (G, X) is essentially the same as in Example 3.1 above, except that we take G to be GSp_{2n}, rather than the adjoint group of it. So K is isogenous to U(n) × U(1).

• The group L is isogenous to U(p, q)×U(1) via a central isogeny, p+q = n. Its derived group L^{der} has roots ±(x_{a} − x_{b}) with 1 ≤ a < b ≤ p and ±(x_{α} − x_{β}) with p + 1 ≤ α < β ≤ p + q. We assume that p ≤ q.

The symplectic embedding L ↪ G arises in a PEL-type Shimura variety situation as follows. Start with an imaginary quadratic field K, a 2n-dimensional ℚ-vector space H with a K-action, a non-degenerate alternating pairing ⟨, ⟩ on H, and a ℂ-action on H ⊗_{ℚ} ℝ. Assume that the involution on End(H) defined by the alternating form ⟨, ⟩ leaves the image of K and ℂ stable, and induces the complex conjugation on them. Moreover, assume that the bilinear form

(v_{1}, v_{2}) ↦ ⟨v_{1}, −i · v_{2}⟩

on H ⊗_{ℚ} ℝ is symmetric and positive definite. The algebra K ⊗_{ℚ} ℂ ≅ ℂ × ℂ operates on H ⊗_{ℚ}ℝ, and has ℝ-dimension 2p, 2q when localized at the two maximal ideals of K ⊗_{ℚ} ℂ ≅ ℂ × ℂ. Such a PEL-data gives a simple algebra with involution (End_{K}(H), ∗), which gives rise to the symplectic embedding L ⊂ G. With the ℂaction coming from the ℂ^{×}-action as in Example 3.1, we may assume that the K ⊗_{ℚ} ℝ ≅ ℂ-action on H ⊗_{ℚ} ℝ is such that a + b\sqrt{−1} ∈ ℂ operates via the matrix

( \overbrace{\begin{aligned}a & −b \\ b & a \\ & & . & . & . \\ & & & & & a & −b \\ & & & & & b & a \end{aligned}}^{2p×2p} ; \underbrace{\begin{aligned}a & b \\ −b & a \\ & & . & . & . \\ & & & & & a & b \\ & & & & & −b & a \end{aligned}}_{2q×2q} )

• The roots of L are ±(x_{a} − x_{b}) with 1 ≤ a < b ≤ p, ±(x_{α} − x_{β}) with p + 1 ≤ α < β ≤ p + q, and ±(x_{a} + x_{β}) with 1 ≤ a ≤ p and p + 1 ≤ β ≤ p + q.

• The Lie algebra of the normalizer of 𝔩 = Lie(L) is equal to 𝔩 itself. The subspace N = 𝔩^{−1,1} has dimension pq, with roots (x_{a} + x_{β}) with 1 ≤ a ≤ p and p + 1 ≤ β ≤ p + q.

• c = p = min(p, n − p).

The direct sum of the weight spaces in N with weights (x_{a} +x_{β}), where 2 ≤ a ≤ p and p + 1 ≤ β ≤ p + q, is a subspace N_{1} of codimension p in N. Moreover the weight space for 2x_{1} is not contained in N_{1} + 𝔤𝔩_{n} · N_{1}. This shows that c ≤ p. The other half of the proof is omitted.

3.3 (SO(2, 2n − 1), SO(2, 2m − 1) × SO(2n − 2m)), m < n.

• In this example, G is equal to SO(2, 2n − 1), and L is the standardly embedded SO(2, 2m − 1) × SO(2n − 2m).

• The Dynkin diagram is

[figure: α_{1}; μ; α_{2}; α_{3}; >; α_{n−1}; α_{n}]

• The roots of G are ±x_{i} ±x_{j} with 1 ≤ i < j ≤ n, and x_{i} with 1 ≤ i ≤ n. The simple roots are

α_{1} = x_{1} − x_{2}, … α_{n−1} = x_{n−1} − x_{n}, α_{n} = x_{n} .

The coweight μ_{h} corresponds to the only special vertex α_{1}.

• The maximal compact subgroup K is isomorphic to SO(2)×SO(2n−1). The roots of K^{der} are ±x_{i} ± x_{j} with 2 ≤ i < j ≤ n, and x_{i} with 2 ≤ i ≤ n. The representation 𝔤^{−1,1} of K has weights x_{1} and x_{1} ± x_{a} with 2 ≤ a ≤ n. As a representation of K^{der} ≅ SO(2n − 1) it is isomorphic to the standard representation.

• The roots of L are ±x_{i} ± x_{j} with 1 ≤ i < j ≤ m, ±x_{i} ± x_{j} with m + 1 ≤ i < j ≤ n, x_{i} with 1 ≤ i ≤ m. The normalizer of 𝔩 in 𝔨 is equal to 𝔩.

• The subspace 𝔩^{−1,1} of 𝔤^{−1,1} has weights x_{1} and x_{1} ± x_{a} with 2 ≤ a ≤ m.

• The invariant c = 2m − 2.

3.4 (Sp_{2^{n}}, SO(2, 2n − 1))

The roots and Dynkin diagram for B_{n} are already recalled in Example 3.3 above. Here we start with a group L of type B_{n} and embed it in a symplectic group. An instance of this embedding is the Kuga-Satake construction of abelian varieties associated to a polarized K3 surface using the Clifford algebra.

• The group L is the Spin double cover of SO(2, 2n − 1). The spin representation is the fundamental representation of L with highest weight \frac{1}{2}(x_{1} + ⋯ + x_{n}). It has dimension 2^{n}. Its weights are

x_{ν} = \frac{1}{2} ∑_{1≤ϵ≤n} ϵ(i)x_{i} ,

where ϵ = (ϵ(1), …, ϵ(n)), with each ϵ(i) = ±1. Let Λ be the abelian group defined by 2^{n} generators λ_{ϵ}, where the ϵ’s are as above, and 2^{n−1} relations λ_{ϵ} + λ_{−ϵ} = 0. Clearly Λ is a free abelian group of rank 2^{n−1}. This spin representation gives an embedding of L into Sp_{2^{n}}. Let the weights of the standard representation of Sp_{2^{n}} be the 2^{n} generators λ_{ϵ}, and the spin representation is such that the weight λ_{ϵ} of G goes to the weight x_{λ} of L. The roots of G are of course the λ_{ϵ_{1}} − λ_{ϵ_{2}}’s.

• The coweight μ_{h} of L corresponds to the homomorphism which sends x_{ϵ} to \frac{1}{2}ϵ(1), hence also the the homomorphism which sends λ_{ϵ} to \frac{1}{2}ϵ(1). From this one see that 𝔤^{−1,1} has weights λ_{ϵ_{1}} − λ_{ϵ_{2}} with ϵ_{1}(1) = 1 and ϵ_{2}(1) = −1.

• We have Lie(N_{G}(L)) = Lie(L) since the spin representation is irreducible. The embedding N = 𝔩^{−1,1} ↪ 𝔤^{−1,1} is determined from the map between the weights by duality.

• We have c = n in the present situation.

Let N_{1} ⊂ N be the sum of weight spaces with weights x_{1}+x_{2}, … x_{1}+x_{n}. Then N_{1} has codimension n in N, and one checks that N +𝔨·N_{1} ⫋ 𝔤^{−1,1}. Hence c ≤ n. The other part of the proof is omitted.

3.5 (U(p, q), U(p_{1}, q_{1}) × ⋯ U(p_{r}, q_{r})), ∑ p_{i} = p, ∑ q_{i} = q.

• In this example the real group G is equal to U(p, q), defined by the diagonal hermitian form with signature (p, q). The subgroup L is equal to U(p_{1}, q_{1}) × ⋯ U(p_{r}, q_{r})), with p_{1} + ⋯ + p_{r} = p, q_{1} + ⋯ + q_{r} = q, diagonally embedded in G.

• The Dynkin diagram for (G, X) is

[figure: α_{1}; α_{2}; α_{p}; μ; α_{n−1}; α_{p+q−1}]

• The simple roots of G are α_{1} = x_{1} − x_{2}, …, α_{p+q−1} = x_{p+q−1} − x_{p+q}. Similarly for each U_{(}p_{i}, q_{i}). The maximal compact subgroup K of G is U(p) × U(q). The normalizer of the 𝔩 in 𝔤 is equal to 𝔩.

• The weights of 𝔤^{−1,1} are x_{a} −x_{β}, with 1 ≤ a ≤ p and p_{1} ≤ β ≤ p+q. As a representation of K it is isomorphic to the standard representation of U(p) tensored with the dual of the standard representation of U(q). The weights of 𝔩^{−1,1} consists of those x_{a} − x_{β} such that there exists an i between 1 and r, with p_{1} + ⋯ + p_{i−1} + 1 ≤ a ≤ p_{1} + ⋯ + p_{i} and q_{1} + ⋯ + q_{i−1} + 1 ≤ β ≤ q_{1} + ⋯ + q_{i}.

• The invariant c = min {min(p_{i}, q_{i})|p_{i} · q_{i} ≠ 0, 1 ≤ i ≤ r}.

3.6 (SO(2, 2n − 2), SO(2, 2m − 2) × SO(2n − 2m)), m < n.

• The Dynkin diagram for (G, X) is

[figure: α_{1}; μ; α_{2}; α_{3}; α_{n−3}; α_{n−2}; α_{n−1}; α_{n}]

• The real group G is equal to SO(2, 2n − 2), defined by the diagonal quadratic form with signature (2, 2n − 2). The reductive subgroup L is equal to SO(2, 2m − 2) × SO(2n − 2m), m < n, diagonally embedded in G.

• The roots of G are ±x_{i} ± x_{j}, 1 ≤ i < j ≤ n. The simple roots are α_{1} = x_{1} − x_{2}, …, α_{n−1} = x_{n−1} − x_{n}, α_{n} = x_{n−1} + x_{n}.

• The maximal compact subgroup K of G is SO(2) × SO(2n − 2). The compact roots are ±x_{i} ± x_{j}, 2 ≤ i < j ≤ n. The normalizer of 𝔩 in 𝔤 is equal to 𝔩.

• The weights of 𝔤^{−1,1} are x_{1} ± x_{2}, …, x_{1} ± x_{n}. As a representation of SO(2n − 2) = K^{der} it is isomorphic to the standard representation of SO(2n − 2). The subspace N = 𝔩^{−1,1} has weights x_{1} ± x_{2}, …, x_{1} ± x_{m}.

• The invariant c = 2m − 3

3.7 (D_{n}^{ℍ}, D_{n_{1}}^{ℍ} × ⋯ D_{n_{r}}^{ℍ}), n_{1} + ⋯ + n_{r} = n.

• The Dynkin diagram for (G, X) is

[figure: α_{1}; α_{2}; α_{3}; α_{n−3}; α_{n−2}; α_{n−1}; α_{n}; μ]

• This hermitian symmetric space is denoted D_{n}^{ℍ} in [3]. It occurs for PEL-type-D Shimura varieties. The subgroup L considered here is a diagonally embedded reductive subgroup, which is a product of groups of type D_{n_{1}}^{ℍ}, … D_{n_{r}}^{ℍ}, where r ≥ 2 and each n_{i} ≥ 1.

• The roots and simple roots of G are the same as in Example 3.6 above. The maximal compact subgroup K of G is isogenous to U(n) via a central isogeny. The compact roots are ±(x_{i} − x_{j}), 1 ≤ i < j ≤ n. The roots of L consists of those ±x_{a} ± x_{b} such that there exists an i, 1 ≤ i ≤ r, with n_{1} + ⋯ + n_{i−1} + 1 ≤ a < b ≤ n_{1} + ⋯ + n_{i}.

• The weights for 𝔤^{−1,1} are x_{i} + x_{j}, 1 ≤ i < j ≤ n. Each weight space has dimension one. As a 𝔨-representation it is isomorphic to the second exterior product of the standard representation of 𝔲(n).

• The normalizer of 𝔩 in 𝔤 is equal to 𝔩. The subspace 𝔩^{−1,1} has weights x_{a} + x_{b}, where there exists an i, 1 ≤ i ≤ r, with n_{1} + ⋯ + n_{i−1} + 1 ≤ a < b ≤ n_{1} + ⋯ + n_{i}.

• The invariant c = min{n_{i} + n_{j} − 2|i ≠ j}.

3.8 (E_{6}, D_{5}^{ℝ})

Something will be glaringly missing if we do not include exceptional groups in our examples. For hermitian symmetric domains, the only exceptional groups that appears are E_{6} and E_{7}. We first discuss the E_{6} case.

• The Dynkin diagram is

[figure: α_{1}; μ; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}]

• G is adjoint of type E_{6}. The root space is

{ ∑_{1≤i≤8} ξ_{i} · x_{i}|ξ_{6} = ξ_{7} = −ξ_{8}}

with inner product induced by the standard Euclidean inner product. The roots are

±x_{i} ± x_{j} 1 ≤ i < j ≤ 5

and

±\frac{1}{2}(x_{8} − x_{7} − x_{6} + ∑_{1≤i≤5}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤5} ν(i) ≡ 0 (mod 2). The simple roots are

α_{1} = \frac{1}{2}(x_{1} − x_{2} − x_{3} − x_{4} − x_{5} − x_{6} − x_{7} + x_{8})
α_{2} = x_{1} + x_{2}
α_{3} = x_{2} − x_{1}
α_{4} = x_{3} − x_{2}
α_{5} = x_{4} − x_{3}
α_{6} = x_{5} − x_{4}

• There are two special vertices, α_{1} and α_{6}, exchanged under an outer automorphism. Here the coweight μ_{h} corresponds to the special vertex α_{1}.

• The maximal compact subgroup K is isogenous to SO(10) × SO(2) via a central isogeny. The compact roots are ±x_{i} ± x_{j} with 1 ≤ i < j ≤ 5.

• 𝔤^{−1,1} has weights

\frac{1}{2}(x_{8} − x_{7} − x_{6} + ∑_{1≤i≤5}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤5} ν(i) ≡ 0 (mod 2). This representation of Lie(K^{ad})) ≅ 𝔰𝔬(10) is the half-spin representation of 𝔰𝔬(10) of dimension 16. Each weight space is one-dimensional.

• In this example, the subgroup L ⊂ G is of type D_{5}, with simple roots α_{1}, α_{2}, α_{3}, α_{4}, α_{5}. The roots of L are ±x_{i} ± x_{j}, 1 ≤ i < j ≤ 4 and

±\frac{1}{2}(x_{8} − x_{7} − x_{6} − x_{5} + ∑_{1≤i≤4}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤4} ν(i) ≡ 1 (mod 2). The image of the homomorphism h factorizes through L, giving rise to a hermitian symmetric space of type D_{5}^{ℝ}.

• The normalizer N_{𝔤}(𝔩) of 𝔩 = Lie(L) in 𝔤 is equal to 𝔩 itself.

• The subspace N_{𝔤}(𝔩)^{−1,1} of 𝔤^{−1,1} has dimension 8; its weights are

\frac{1}{2}(x_{8} − x_{7} − x_{6} − x_{5} + ∑_{1≤i≤4}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤4} ν(i) ≡ 1 (mod 2).

• Regarded as a representation of the Lie algebra of the derived group of K, the weights of 𝔤^{−1,1} are \frac{1}{2}(∑_{1≤i≤5}(−1)^{ν(i)}x_{i}), with ∑_{1≤i≤5} ν(i) ≡ 0 (mod 2). The weights of the subspace N = N_{𝔤}(𝔩)^{−1,1} are those for which ν(5) is odd.

• c = 4 in this example.

Proof. The direct sum N_{1} of the weight spaces with weights \frac{1}{2}(x_{1} +x_{2} + x_{3} − x_{4} − x_{5}), \frac{1}{2}(x_{1} + x_{2} − x_{3} + x_{4} − x_{5}), \frac{1}{2}(x_{1} − x_{2} + x_{3} + x_{4} − x_{5}), \frac{1}{2}(−x_{1} + x_{2} + x_{3} + x_{4} − x_{5}) has codimension 4 in 𝔤^{−1,1}, and N + 𝔨 · N_{1} does not contain the weight space of \frac{1}{2}(−x_{1} − x_{2} − x_{3} − x_{4} + x_{5}). So c ≤ 4. Conversely let W be a subspace of N of codimension at most 4. Then the intersection of W with the direct sum of weight spaces with weights \frac{1}{2}(−x_{1} − x_{2} − x_{3} + x_{4} − x_{5}), \frac{1}{2}(−x_{1} − x_{2} + x_{3} − x_{4} − x_{5}), \frac{1}{2}(−x_{1} + x_{2} − x_{3} − x_{4} − x_{5}), \frac{1}{2}(x_{1} − x_{2} − x_{3} − x_{4} − x_{5}) is not zero. So 𝔨 · W contains the weight space of \frac{1}{2}(−x_{1} − x_{2} − x_{3} − x_{4} + x_{5}). Similarly 𝔨 · W contains the weight space of the other three weights with ν(5) even.

3.9 (E_{6}, A_{5})

In this example, the pair (G, X) is the same as in example 3.8 above; the subgroup L is different.

• The subgroup L ⊂ G is isogenous to U(1, 5), so L^{ad} is of type A_{5}. The simple roots of L are α_{1}, α_{3}, α_{4}, α_{5}, α_{6}. There are 20 compact roots among them; they are ±(x_{i} − x_{j}) with 1 ≤ i < j ≤ 5. The normalizer of 𝔩 in 𝔤 is equal to 𝔩 itself.

• The subspace N = N_{𝔤}(𝔩)^{−1,1} = 𝔩^{−1,1} of 𝔤^{−1,1} is 5-dimensional, with weights \frac{1}{2}(x_{1} − x_{2} − x_{3} − x_{4} − x_{5}), \frac{1}{2}(−x_{1} + x_{2} − x_{3} − x_{4} − x_{5}), \frac{1}{2}(−x_{1} − x_{2}+x_{3}−x_{4}−x_{5}), \frac{1}{2}(−x_{1}−x_{2}−x_{3}+x_{4}−x_{5}), \frac{1}{2}(−x_{1}−x_{2}−x_{3}−x_{4}+x_{5}). One sees immediately that the the weight space for \frac{1}{2}(x_{1} +x_{2} +x_{3} +x_{4} +x_{5}) is not contained in 𝔨 · N. Hence

• c = 0.

3.10 (E_{7}, D_{6}^{ℝ})

The other exceptional group to deal with is E_{7}.

• The Dynkin diagram is

[figure: α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{7}; μ; α_{2}]

• G is adjoint of type E_{7}. The root space is

{ ∑_{1≤i≤8} ξ_{i} · x_{i}|ξ_{7} + ξ_{8} = 0}

with inner product induced by the standard Euclidean inner product. The roots are

±x_{i} ± x_{j} 1 ≤ i < j ≤ 6; ±(x_{7} − x_{8})

and

±\frac{1}{2}(x_{8} − x_{7} − x_{6} + ∑_{1≤i≤6}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤6} ν(i) ≡ 2 (mod 2). The simple roots are

α_{1} = \frac{1}{2}(x_{1} − x_{2} − x_{3} − x_{4} − x_{5} − x_{6} − x_{7} + x_{8})
α_{2} = x_{1} + x_{2}
α_{3} = x_{2} − x_{1}
α_{4} = x_{3} − x_{2}
α_{5} = x_{4} − x_{3}
α_{6} = x_{5} − x_{4}

The highest root is

α̃ = x_{8} − x_{7} = 2α_{1} + 2α_{2} + 3α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7} .

• The coweight μ_{h} corresponds to the only special vertex α_{7}. The compact subgroup K^{ad} is of type E_{6}. It has 72 roots. They are

\frac{1}{2}(x_{8} − x_{7} − x_{6} + ∑_{1≤i≤5}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤5} ν(i) ≡ 0 (mod 2), and ±x_{i} ± x_{j} with 1 ≤ i < j ≤ 5.

• The subgroup L ⊂ G is a reductive subgroup of the same rank as G. Its adjoint group L^{ad} is of type D_{6}, with simple roots α_{2}, α_{3}, …, α7. The image of h is contained in L, giving rise to a hermitian symmetric space of type D_{6}^{ℝ}. We have N_{𝔤}(𝔩) = 𝔩. The roots of 𝔩 are ±x_{i} ± x_{j}, 1 ≤ i < j ≤ 6; there are 60 such roots.

• The representation 𝔤^{−1,1} of K is irreducible of dimension 27. Its weights are x_{8} − x_{7}, x_{6} ± x_{i} with 1 ≤ i ≤ 5 and

\frac{1}{2}(x_{8} − x_{7} + x_{6} + ∑_{1≤i≤5}(−1)^{ν(i)}x_{i})

with ∑_{1≤i≤5} ν(i) ≡ 1 (mod 2); each weight space dimension one. As a representation of the Lie algebra of the derived group of K, it is the fundamental representation of Lie(K^{der}) corresponding to the vertex α_{1}, a miniscule representation for E_{6}.

• The subspace N = N_{𝔤}(𝔩)^{−1,1} of 𝔤^{−1,1} has dimension 10. Its weights are x_{6} ± x_{i}, 1 ≤ i ≤ 5. We have

𝔨 · N + N ⫋ 𝔤^{−1,1}

since the sum does not contain the weight space for x_{8} − x_{7}. Hence

• c = 0.

References

[1] E. Colombo and G. P. Pirola. Some density results for curves with nonsimple jacobians. Math. Ann., 288:161–178, 1990.

[2] P. Deligne. Travaux de Shimura. In Séminaire Bourbaki 1970/71, volume 244 of Lecture Notes in Math., 1971. Exposé 389.

[3] P. Deligne. Variétés de Shimura: Interprétation modulair, et techniques de construction de modèles canoniques. In A. Borel and W. Casselman, editors, Automorphic Forms, Representations, and L-functions, Part 2, volume 33 of Proc. Symp. Pure Math., pages 247–290, Providence, R.I., 1979. Amer. Math. Soc.

[4] P. Deligne. Hodge cycles on abelian varieties. In Hodge Cycles, Motives, and Shimura Varieties, volume 900 of Lecture Notes in Math., pages 9– 100. Springer-Verlag, 1982. Notes by J. S. Milne.

[5] E. Izadi. Density and completeness of subvarieties of moduli spaces of curves or abelian varieties. to appear in Math. Ann.