A VANISHING THEOREM FOR MODULAR SYMBOLS ON LOCALLY SYMMETRIC SPACES

Toshiyuki KOBAYASHI and Takayuki ODA

Abstract. A modular symbol is the fundamental class of a totally geodesic submanifold Γ′\G′/K′ embedded in a locally Riemannian symmetric space Γ\G/K, which is defined by a subsymmetric space G′/K′ ↪ G/K. In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G, G′), and prove a vanishing theorem with respect to the π-component (π ∈ Ĝ) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction π|_{G′}. In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.

§1 Introduction

Our concern in this paper is to describe the middle Hodge components of the Poincaré duals of certain 2n-cycles on the arithmetic quotients V = Γ\D of the bounded symmetric domain of type IV

D ≅ SO_{0}(2n, 2)/(SO(2n) × SO(2)),

where the 2n-cycles are defined via the embedding SO_{0}(2n, 1) ↪ SO_{0}(2n, 2). These cycles are special cases of generalized modular symbols. To explain our motivation of this investigation, let us recall the origin of modular symbols briefly.

Historically speaking, the idea of modular symbols was introduced and extensively studied for elliptic modular curves C = Γ\ℌ (ℌ is the complex upper half plane) in the work of Manin-Drinfeld and Mazur-Swinnerton-Dyer. Modular symbols are defined to be chains or cycles on the compactification C^{∗} of C, obtained as the images of arcs on ℌ joining two cusps with respect to Γ ⊂ SL_{2}(ℤ). In terms of them, the first homology group H_{1}(C^{∗}, ℤ) is quite explicitly computed. The theory is “elementary” and has applications to the special values of L-functions and to the construction of p-adic L-functions of modular forms. The same cycles appear very naturally in the theory of half-integral weight modular forms (cf. Shintani [21]).

There have been attempts to push forward the investigation since the late ’70’s for higher dimensional arithmetic quotients, i.e. to introduce the object called generalized modular symbols given as follows.

Typeset by AMS-TEX

Let G be a semisimple Lie group and Γ a discrete subgroup such that the volume vol(Γ\G) < ∞. Let G′ be a closed subgroup of G such that Γ′ := G′ ∩ Γ is also discrete and co-volume finite in G′ (this paper will be restricted to the co-compact cases). Then, the inclusion map induces a map of real analytic varieties of double cosets:

ι : Γ′\G′/K′ → Γ\G/K,

where K is a maximal compact subgroup of G and K′ := K ∩ G′. The image of ι defines a chain C_{(G′,ι)} on Γ\G/K, and when Γ\G is compact, C_{(G′,ι)} is a cycle on Γ\G/K. When Γ\G is non-compact, C_{(G′,ι)} is sometimes compactified to give a cycle on a natural compactification of Γ\G/K.

There are a number of results on generalized modular symbols. In some special cases, the periods of automorphic forms along generalized modular symbols are represented as the special values of automorphic L-functions (cf. [4], [18]). Though it is desirable to generalize this type of results, our current knowledge on generalized modular symbols is still very poor. Presently, many of literatures on generalized modular symbols are concerned only in establishing the non-vanishing of these cycles, using the intersection numbers of these cycles (cf. [1]), or using the Weil representations (cf. [11], [13]).

Our main results and the method of the proof are quite different from these previous works. First, we shall show the vanishing of the “non-trivial part” of the (n, n)-components of certain modular symbols. Secondly, the method of our proof is based on a recent theory of unitary representations on the discrete branching laws by T. Kobayashi ([6], [7], [8], [9]). An explicit vanishing theorem for the modular symbols corresponding to G′ ↪ G is then obtained from the criterion on the (non- )existence of continuous spectrum of an irreducible unitary representation of G with non-zero continuous cohomologies when restricted to a non-compact reductive subgroup G′. Then, this method leads us to some heuristic argument that the usual method of the Rankin-Selberg convolution for the construction of automorphic Lfunctions of non-trivial (n, n)-type harmonic (automorphic) forms ω on V gives only zero, hence it gives no information for the periods of non-trivial (n, n)-type harmonic forms on V (cf. Remark 6.1).

Our main results (see Theorem 2.8 and Theorem 4.3) will be formulated for a real reductive Lie group in general. But, we illustrate them by the most interesting case: Let G/K = SO_{0}(2n, 2)/SO(2n) × SO(2) be the Hermite symmetric space of type IV. Its dual symmetric space is G_{U}/K := SO(2n + 2)/SO(2n) × SO(2), that is, the complex quadric. The 2n-th de Rham cohomology group H^{2n}(G_{U}/K; ℂ) is a two dimensional vector space, with generators [ξ] := [∧^{n}κ] and [η]. Here κ is the Kähler form on G_{U}/K and η is a G_{U}-invariant primitive differential form. We note that both [ξ] and [η] lie in the (n, n)-type Hodge components of H^{2n}(G_{U}/K; ℂ) and that (p, q)-type Hodge components of H^{2n}(G_{U}/K; ℂ) (p + q = 2n) are non-zero if and only if p = q = n.

Suppose Γ is a discrete, torsion free co-compact subgroup of G. Then there is a natural injective map of cohomology groups:

Φ: H^{∗}(G_{U}/K; ℂ) → H^{∗}(Γ\G/K; ℂ).

Elements in the image of Φ are said to be universal cohomology classes. We remark that Φ([η]) ∈ H^{n,n}(Γ\G/K; ℂ).

Let G′ := SO_{0}(2n, 1) (⊂ G), and K′ := K ∩ G′ ≃ SO(2n). We assume that Γ′ := Γ ∩ G′ is also a co-compact subgroup of G′. Then the inclusion G′ ↪ G gives rise to a natural embedding of a totally real, totally geodesic submanifold of dimension 2n into the Kähler manifold Γ\G/K of complex dimension 2n:

ι: Y := Γ′\G′/K′ → X := Γ\G/K.

We write [Y] for the fundamental class which is a generator of the homology group H_{2n}(Y; ℤ) ≃ ℤ. We consider the composition of ι_{∗} and the Poincaré duality map:

H_{2n}(Y; ℂ) →_{ι_{∗}} H_{2n}(X; ℂ) →^{∼}_{P} H^{2n}(X; ℂ) ≃ ⨁_{p+q=2n} H^{p,q}(X; ℂ).

We write

M(Y) := P ∘ ι_{∗}[Y] = ∑_{p+q=2n} M^{p,q}(Y) ∈ ⨁_{p+q=2n} H^{p,q}(X; ℂ)

corresponding to the Hodge decomposition. Here is our main result in this setting:

Theorem. The (n, n)-type Hodge component of the Poincaré dual of the modular symbol ι_{∗}[Y] is given by

M^{n,n}(Y) = \frac{vol(Y)}{vol(X)}Φ[η]

with a natural normalization of η.

The above theorem is very easy to prove when n = 1. In fact, in this case SO_{0}(2, 2) is a product SL_{2}(ℝ) × SL_{2}(ℝ) up to central subgroups, and the cycle derived from SO(2, 1) ↪ SO_{0}(2, 2) is (although itself is not algebraic, but) a “partial complex conjugation” of an algebraic cycle with respect to the complex conjugate on the second factor of ℌ × ℌ. Therefore the main theorem in this case is equivalent to the well-known fact that the Poincaré duals of algebraic 2-cycles on a compact algebraic surface over ℂ have no (2, 0) + (0, 2) type Hodge components. The interesting fact is that the similar fact persists in higher dimensional cases !

Here is a brief sketch of the main idea of the proof: We consider the following interactions:

the topology of the embedding of a submanifold i: N ↪ M,
↓↑
the pullback of functions (or forms)  i^{∗}: E^{∗}(N) ← E^{∗}(M).

We could expect to extract some knowledge of the topology of the embedding i: N ↪ M from the understanding of the pullback i^{∗}: E^{∗}(N) ← E^{∗}(M). In particular, if i is equivariant with respect to Lie groups G′ ↪ G, then the pullback i^{∗} factors through the restriction of (infinite dimensional) representations of G with respect to G′. In this way, representation theory gives a clue to the understanding of the topology of the equivariant embedding of a submanifold, although the branching law of a unitary representation is in general a very hard problem for non-compact Lie groups. The idea of our methods in this paper is along this line, to “translate” the information about the restrictions of unitary representations (branching laws) into the topology of modular symbols. The discrete decomposition theorem of L^{2}(Γ\G) due to Gel’fand–Piateski-Shapiro [5] made a bridge between the topology of the locally symmetric space Γ\G/K and representation theory by the Matsushima-Murakami formula ([3], [14]). Similarly, the recent result on the discrete decomposable restriction of the Zuckerman-Vogan derived functor module A_{𝔮} bridges between the topology of the embedding of locally symmetric spaces and representation theory. (We note that what is an equivariant map here is not ι: Y = Γ′\G′/K′ ↪ X = Γ\G/K but the induced map between these universal coverings Ỹ ↪ X̃.)

The paper is organized as follows: In §2 we give an abstract Theorem (Theorem 2.8) which gives a sufficient condition that the modular symbol ι_{∗}[Γ′\G′/K′] is

annihilated by the π-component H^{d(G′)}(π : Γ) (⊂ H_{de Rham}^{d(G′)}(Γ\G/K; ℂ)) for π ∈ Ĝ, where d(G′) = dim G′/K′. In §4 we restate Theorem 2.8 by a very explicit condition in the case where (G, G′) is a semisimple symmetric pair, based on the discrete decomposition theorem (Fact 4.2). This is stated in Theorem 4.3. In §3, we investigate the invariant part of the modular symbols (Theorem 3.1). In §5, we apply the results in §2, §3 and §4 to the case (G, G′) = (SO_{0}(2n, 2), SO_{0}(2n, 1)) and prove the Theorem in this Introduction.

§2 Vanishing theorem of modular symbols in a primitive form

In this section, we give a general framework in representation theory that the modular symbol ι_{∗}[Γ′\G′/K′] is annihilated by the π-component H^{d(G′)}(π : Γ) in the de Rham cohomology group H_{de Rham}^{∗}(Γ\G/K; ℂ) for π ∈ Ĝ.

Suppose that G is a real reductive linear Lie group. Let K be a maximal compact subgroup of G and Γ a torsion free, co-compact discrete subgroup of G. We assume that Γ\G/K is orientable. In particular, this is the case if G is connected.

We say a unitary representation (π, V) of G is G-admissible if (π, V) is decomposed into a discrete Hilbert direct sum of irreducible unitary representations of G with finite multiplicities. The point of the definition is that there is no continuous spectrum in the irreducible decomposition of G.E A result due to Gel’fand–Piateski-Shapiro [5] asserts that the right regular representation on L^{2}(Γ\G) is G-admissible under the assumption that Γ is a co-compact discrete subgroup of G. Namely, we have a discrete Hilbert direct sum decomposition:

L^{2}(Γ\G) ≃ ∑^{⊕} Hom_{G}(π, L^{2}(Γ\G)) ⊗ π = ∑^{⊕}n_{Γ}(π)π.

Here π runs over the unitary dual of G and the multiplicity

n_{Γ}(π) := dim_{ℂ} Hom_{G}(π, L^{2}(Γ\G)) < ∞.

Let 𝔤_{0} = 𝔨_{0}+𝔭_{0} be a Cartan decomposition of the Lie algebra 𝔤_{0} of G and 𝔤 = 𝔨+𝔭 the complexifications. Let E^{p}(Γ\G/K) be the space of p-forms on Γ\G/K. Because the cotangent bundle T^{∗}(Γ\G/K) is isomorphic to the bundle Γ\G×_{K}𝔭^{∗} → Γ\G/K,

which is associated to the principal K-bundle K → Γ\G → Γ\G/K and the Krepresentation K → GL_{ℝ}(𝔭^{∗}), we have

(2.1) E^{p}(Γ\G/K) ≃ (C^{∞}(Γ\G) ⊗ ∧^{p}𝔭^{∗})^{K} ≃ Hom_{K}(∧^{p}𝔭, C^{∞}(Γ\G)) (p ∈ ℕ).

We fix a non-degenerate symmetric bilinear form B on 𝔤_{0} such that B|_{𝔨_{0}×𝔨_{0}} is negative definite, B|_{𝔭_{0}×𝔭_{0}} is positive definite and that 𝔨_{0} and 𝔭_{0} are orthogonal to each other. Then the positive definite bilinear form B|_{𝔭_{0}×𝔭_{0}} induces a G-invariant Riemannian metric on G/K, which makes G/K into a Riemannian symmetric space. Through the covering

ϖ: G/K → Γ\G/K,

the Clifford-Klein form Γ\G/K is equipped with a Riemannian metric so that ϖ is isometric. We write Δ = dd^{∗}+d^{∗}d for the corresponding Laplace-Beltrami operator acting on the space of forms on Γ\G/K. The space of harmonic p-forms is given by

H_{Δ}^{p}(Γ\G/K) := {ω ∈ E^{p}(Γ\G/K) : Δω = 0}.

Then the Hodge theory asserts a natural isomorphism

H_{de Rham}^{∗}(Γ\G/K; ℂ) ≃ H_{Δ}^{∗}(Γ\G/K).

Let (π, V_{π}) be an irreducible (𝔤, K)-module. Using the identification

E^{∗}(Γ\G/K) ≃ (C^{∞}(Γ\G) ⊗ ∧^{∗}𝔭^{∗})^{K} ≃ Hom_{K}(∧^{∗}𝔭, C^{∞}(Γ\G)),

we define a linear map

(2.2) T_{π}: Hom_{K}(∧^{∗}𝔭, π)⊗Hom_{𝔤,K}(π, C^{∞}(Γ\G)) → E^{∗}(Γ\G/K), ψ⊗φ ↦ φ∘ψ.

Let Ĝ be the set of the equivalence classes of irreducible (𝔤, K)-modules that are unitarizable. We note that Ĝ is naturally identified with the unitary dual of G (Harish-Chandra). Let U(𝔤) be the enveloping algebra of the complexified Lie algebra 𝔤 = 𝔤_{0} ⊗_{ℝ} ℂ, Z(𝔤) the center of U(𝔤), and c_{G} ∈ Z(𝔤) the Casimir element defined by the non-degenerate symmetric bilinear form B. We define a subset Ĝ_{0}(⊂ Ĝ) by

Ĝ_{0} := {π ∈ Ĝ : π(c_{G}) = 0}.

Here, we have used the same notation π for the representation of U(𝔤).

The left action of G on G/K induces the representation of U(𝔤) on the space of forms on G/K. In particular, the Casimir element c_{G} (∈ Z(𝔤) ⊂ U(𝔤)) acts on E^{∗}(G/K) as the Laplace-Beltrami operator, since the Riemannian metric on G/K is induced from B|_{𝔭_{0}×𝔭_{0}}. Because the quotient map ϖ: G/K → Γ\G/K is a Riemannian covering, c_{G} induces the Laplace-Beltrami operator Δ on Γ\G/K. Therefore, we have

(2.3.1)  Image T_{π} ⊂ H_{Δ}^{∗}(Γ\G/K) ≃ H_{de Rham}^{∗}(Γ\G/K; ℂ)

if and only if π ∈ Ĝ_{0}. We say that the subspace of H_{de Rham}^{∗}(Γ\G/K; ℂ) corresponding to Image T_{π} is \underline{the π-component}, and write H^{∗}(π : Γ). That is,

(2.3.2)  H^{p}(π : Γ) := Image T_{π} ∩ H_{de Rham}^{p}(Γ\G/K; ℂ)  (p ∈ ℕ),

via the isomorphism (2.3.1). Then the Matsushima-Murakami formula and a result of Borel-Wallach are summarized as:

Lemma 2.4 ([3], [14]). Retain the above setting. We have

(1)  H^{∗}(π : Γ) ≃ n_{Γ}(π)H^{∗}(𝔤, K : π).
(2)  H_{de Rham}^{∗}(Γ\G/K; ℂ) = ⨁_{π∈Ĝ_{0}} H^{∗}(π : Γ).

(3)  H^{dim G/K}(π : Γ) ≠ 0 if and only if π = 1.

Here is an easy application of the Hodge theory.

Lemma 2.5. Suppose ω ∈ E^{∗}(Γ\G/K) is a closed form such that

ω = ∑_{i} T_{π_{i}}(δ_{i}) (finite sum),

where π_{i}’s are irreducible (𝔤, K)-modules and

δ_{i} ∈ Hom_{K}(∧^{∗}𝔭, π_{i}) ⊗ Hom_{𝔤,K}(π_{i}, C^{∞}(Γ\G)).

We define a harmonic form by

ω_{0} := ∑_{π_{i}∈Ĝ_{0}} T_{π_{i}}(δ_{i}).

Then, passing to the cohomology group H_{de Rham}^{∗}(Γ\G/K), we have

[ω] = [ω_{0}].

Proof. Because π_{i} is an irreducible (𝔤, K)-module, the Casimir element c_{G} acts by a scalar, say λ_{i} ∈ ℂ, on the representation space of each π_{i} by Schur’s lemma, so that

Image T_{π_{i}} ⊂ {ω ∈ E^{∗}(Γ\G/K) : Δω = λ_{i}ω}.

We set

ω_{1} := ∑_{π_{i}∈Ĝ\Ĝ_{0}} T_{π_{i}}(δ_{i}).

Then we have ω = ω_{0}+ω_{1}. Since dω = 0 and Δω_{0} = 0, we have dω_{1} = dω−dω_{0} = 0.

Since λ_{i} ≠ 0 for each π_{i} ∈ Ĝ \ Ĝ_{0}, ω_{1} ∈ Image(Δ: E^{∗}(Γ\G/K) → E^{∗}(Γ\G/K)). Denote by G the Green’s operator. Then we have

ω_{1} = ΔGω_{1} = dd^{∗}Gω_{1} + d^{∗}dGω_{1}.

Since G commutes with d, we have

ω_{1} = dd^{∗}Gω_{1} + d^{∗}Gdω_{1}.

Because ω_{1} is a closed form, we have

ω_{1} = dd^{∗}Gω_{1}.

Hence ω_{1} is an exact form. Therefore, [ω] = [ω_{0} + ω_{1}] = [ω_{0}]. □

Let G′ be a closed subgroup that is reductive in G. Assume that Γ′ := Γ ∩ G′ is also a co-compact subgroup of G′. We take a maximal compact subgroup K of G such that K′ := K ∩ G′ is a maximal compact subgroup of G′. For simplicity, we put Y = Γ′\G′/K′ and X = Γ\G/K. Then we have natural embedding maps i and ι:

(2.6)  \begin{aligned}i: & Γ′\G′ & ⟶ & Γ\G \\ & ↓ & & ↓ \\ ι: & Y = Γ′\G′/K′ & ⟶ & X = Γ\G/K. \end{aligned}

Assume moreover that Y is orientable (this is always satisfied if G′ is connected). We write [Y] for the fundamental class which is a generator of the homology group H_{d(G′)}(Y; ℤ) ≃ ℤ. Here, d(G′) := dim Y = dim G′/K′. The homology class

ι_{∗}([Y]) ∈ H_{d(G′)}(X; ℤ)

is said to be the (generalized) modular symbol.

The investigation of the induced homomorphism ι_{∗}: H_{∗}(Y : ℂ) → H_{∗}(X : ℂ) is equivalent to that of the pull-back homomorphism

ι^{∗}: H_{de Rham}^{∗}(X; ℂ) → H_{de Rham}^{∗}(Y; ℂ).

To study the latter object, we recall the notion of the discrete decomposable restriction introduced in [6], [7], [8].

Definition 2.7 ([8], Definition 1.1, 1.2). We say that an irreducible (𝔤, K)-module (π, V_{π}) is \underline{discretely decomposable} , if π is isomorphic as (𝔤′, K′)-modules to an algebraic discrete sum of irreducible (𝔤′, K′)-modules:

π ≃ ⨁_{τ∈Ĝ′} m_{π}(τ)τ,

where m_{π}(τ) ∈ ℕ ∪ {∞}.

Let \overline{π} be the unitary representation defined on the Hilbert completion \overline{V_{π}}. We allow us to use the terminology that the restriction π|_{K′} is K′-admissible for a compact subgroup K′ (⊂ K), if the restriction \overline{π}|_{K′} is K′-admissible, equivalently if dim Hom_{K′}(μ, π) = dim Hom_{K′}(μ, \overline{π}) < ∞ for any μ ∈ K̂′ (we recall that K′ is compact). Then if the restriction π|_{K′} is K′-admissible, then π is discretely decomposable as (𝔤′, K′)-module and m_{π}(τ) < ∞ for any τ ∈ Ĝ′ (see [8], Proposition 1.6(2)).

Theorem 2.8. Suppose we are in the above setting. We assume the following three conditions.

i) π ∈ Ĝ is infinite dimensional. ii) The restriction π|_{K′} is K′-admissible. iii) G′ is semisimple without compact factors. Consider the natural perfect pairing ⟨ , ⟩ given by

⟨ , ⟩: H_{de Rham}^{j}(X; ℂ) × H_{j}(X; ℂ) → ℂ.

Then the modular symbol ι_{∗}[Y] ∈ H_{d(G′)}(X; ℂ) is annihilated by H^{d(G′)}(π : Γ) (⊂

H_{de Rham}^{d(G′)}(X; ℂ)), where d(G′) = dim Y = dim 𝔭′_{0}.

Proof. In view of the isomorphisms (2.1)

E^{∗}(X) ≃ Hom_{K}(∧^{∗}𝔭, C^{∞}(Γ\G)),
E^{∗}(Y) ≃ Hom_{K′}(∧^{∗}𝔭′, C^{∞}(Γ′\G′)),

the pull-back ι^{∗}: E^{∗}(X) → E^{∗}(Y) is identified with

Hom_{K}(∧^{∗}𝔭, C^{∞}(Γ\G)) → Hom_{K′}(∧^{∗}𝔭′, C^{∞}(Γ′\G′)), δ ↦ i^{∗} ∘ δ ∘ j.

Here,

j: ∧^{∗} 𝔭′ → ∧^{∗}𝔭

is an injective K′-homomorphism induced from 𝔭′ ↪ 𝔭, and

i^{∗}: C^{∞}(Γ\G) → C^{∞}(Γ′\G′)

is the pull-back C^{∞}-functions with respect to the embedding i: Γ′\G′ ↪ Γ\G. Suppose an irreducible (𝔤, K)-module (π, V_{π}) is discretely decomposable as (𝔤′, K′)- modules and ω ∈ Image(π). Namely, π is isomorphic to an algebraic direct sum

(2.8.1)  π ≃ ⨁_{τ∈Ĝ′} \underbrace{τ ⊕ ⋯ ⊕ τ}_{m_{π}(τ)}

as a (𝔤′, K′)-module. For each τ ∈ Ĝ′ and each k (1 ≤ k ≤ m_{π}(τ)), let denote by

pr_{τ}^{(k)} : π → τ

the projection onto the k-th component of τ in the direct sum (2.8.1), and by

e_{τ}^{(k)} : τ → π

the injection into the k-th component of τ. Then both pr_{τ}^{(k)} and e_{τ}^{(k)} are (𝔤′, K′)- homomorphisms. We note that if U ⊂ V_{π} is a K-stable finite dimensional subspace, then pr_{τ}^{(k)}(U) = 0 except for finitely many τ ∈ Ĝ′ ([8], Proposition 1.6(1)). In particular, if ψ ∈ Hom_{K}(∧^{∗}𝔭, V_{π}), then the right side of

ψ = ⨁_{τ∈Ĝ′} ⨁_{k=1}^{m_{π}(τ)} pr_{τ}^{(k)} ∘ψ

is actually a finite sum because dim ∧^{∗}𝔭 < ∞. If φ ∈ Hom_{𝔤,K}(V_{π}, C^{∞}(Γ\G)), then we have

(2.8.2)  i^{∗} ∘ φ ∘ ψ ∘ j = ∑_{τ∈Ĝ′} ∑_{k=1}^{m_{π}(τ)}(i^{∗} ∘ φ ∘ e_{τ}^{(k)}) ∘ (pr_{τ}^{(k)} ∘ψ ∘ j).

We note pr_{τ}^{(k)} ∘ψ ∘ j ∈ Hom_{K′}(∧^{∗}𝔭′, τ) and i^{∗} ∘ φ ∘ e_{τ}^{(k)} ∈ Hom_{𝔤′,K′}(τ, C^{∞}(Γ′\G′)). Thus,

(i^{∗} ∘ φ ∘ e_{τ}^{(k)}) ∘ (pr_{τ}^{(k)} ∘ψ ∘ j) ∈ Image T′_{τ} (⊂ E^{∗}(Γ′\G′/K′)).

Here, we have used a notation similar to (2.2):

T′_{τ}: Hom_{K′}(∧^{∗}𝔭′, τ) ⊗ Hom_{𝔤′,K′}(τ, C^{∞}(Γ′\G′)) → E^{∗}(Γ′\G′/K′) = E^{∗}(Y),

for each irreducible (𝔤′, K′)-module τ.

Because the sum (2.8.2) is a finite sum, we have

[i^{∗} ∘ φ ∘ ψ ∘ j] = ∑_{τ∈Ĝ′_{0}} ∑_{k=1}^{m_{π}(τ)}[(i^{∗} ∘ φ ∘ e_{τ}^{(k)}) ∘ (pr_{τ}^{(k)} ∘ψ ∘ j)] ∈ H_{de Rham}^{∗}(Y; ℂ),

from Lemma 2.5. Since the cohomology group of the top degree d(G′) = dim G′/K′, namely,

H^{d(G′)}(τ : Γ′) = Image T_{τ′} ∩ H_{de Rham}^{d(G′)}(Y; ℂ),

is non-zero if only if τ = 1 (Lemma 2.4(3)), we have

(2.8.3)  ⟨[i^{∗} ∘ φ ∘ ψ ∘ j], [Y]⟩ = ∑_{k=1}^{m_{π}(1)}⟨[(i^{∗} ∘ φ ∘ e_{1}^{(k)}) ∘ (pr_{1}^{(k)} ∘ψ ∘ j)], [Y]⟩

with respect to the natural pairing H_{de Rham}^{d(G′)}(Y; ℂ) × H_{d(G′)}(Y : ℂ) → ℂ.

In order to prove the right side of (2.8.3) to be zero, it suffices to show m_{π}(1) = 0. Namely, π|_{G′} does not contain the trivial representation as a direct summand. If it were not the case, then any irreducible unitary representation of G′ arising in the decomposition of the restriction π|_{G′} would be finite dimensional, because any irreducible summands in the restriction \overline{π}|_{G′} has the same associated variety (see [9], Part II, Theorem 3.5; see also [8], Corollary 3.9). On the other hand, if G′ is semisimple without compact factors then the number of equivalence class of irreducible unitary finite dimensional representations of G′ is at most that of connected components of G′, which is finite. This means that the restriction π|_{G′} is decomposed into finitely many, irreducible, finite dimensional representations of G′ with finite multiplicities, which is absurd because π is infinite dimensional. Therefore we conclude that the restriction π|_{G′} does not contain the trivial representation of G′ as a direct summand, namely, m_{π}(1) = 0. Hence we have proved Theorem 2.8. □

Similarly to Theorem 2.8, we formulate a vanishing theorem about the cup products in the de Rham cohomology ring H^{∗}(Γ\G/K; ℂ).

Theorem 2.9. Suppose that G is a connected semisimple linear Lie group without compact factors and that Γ is a torsion free co-compact discrete subgroup of G. Let π, π′ ∈ Ĝ. Assume the following two conditions.

i) At least one of π or π′ is infinite dimensional. ii) The tensor product π ⊗ π′ is K-admissible. Then the cup product α ∪ α′ is zero in H^{d(G)}(Γ\G/K; ℂ) for any α ∈ H^{i}(π : Γ) and α′ ∈ H^{d(G)−i}(π′ : Γ).

Proof. Let α ∈ H^{i}(Γ\G/K; ℂ) and α′ ∈ H^{d(G)−i}(Γ\G/K; ℂ). The cup product α∪α′ is given by the pull back of the cohomology class with respect to the diagonally embedding map

ι: G → G × G, g ↦ (g, g),

namely, we have ι^{∗}(α ∧ α′) = α ∪ α′, where

ι^{∗} : H^{i+(d(G)−i)}((Γ × Γ)\(G × G)/(K × K); ℂ) → H^{d(G)}(Γ\G/K; ℂ).

We note that α ∧ α′ belongs to the π ⊠ π′ components of the cohomology group H^{d(G)}(Γ × Γ\G × G/K × K; ℂ) and that the tensor product π ⊗ π′ is nothing but the restriction of π ⊠ π′ with respect to diag(𝔤, K), which is diagonally embedded into (𝔤 + 𝔤, K × K). Applying Theorem 2.8 with the pair (G × G, G), we see that ι_{∗}[Γ\G/K] is annihilated by the π ⊠ π′ components of the de Rham cohomology group H^{d(G)}(Γ × Γ\G × G/K × K; ℂ). Hence, we have

∫_{Γ\G/K} α ∪ α′ = ∫_{Γ\G/K} ι^{∗}(α ∧ α′) = ∫_{ι_{∗}(Γ\G/K)} α ∧ α′ = 0.

Since G is connected, Γ\G/K is orientable. Therefore α ∪ α′ = 0 □

Applying Theorem 2.9 to π′ = 1, the trivial representation of G, we have:

Corollary 2.10. Retain the setting of Theorem 2.9. Then

H^{i}(1 : Γ)  (⊂ H^{i}(Γ\G/K; ℂ))
⨁_{\begin{matrix}π∈Ĝ_{0} \\ π≠1 \end{matrix}} H^{d(G)−i}(π : Γ) (⊂ H^{d(G)−i}(Γ\G/K; ℂ))

are the annihilators to each other in the perfect pairing

(2.10.1) H^{i}(Γ\G/K; ℂ) × H^{d(G)−i}(Γ\G/K; ℂ) → ℂ, α, β ↦ ∫_{Γ\G/K} α ∪ β.

We note that the restriction of (2.10.1) to the pairing

H^{i}(1 : Γ) × H^{d(G)−i}(1 : Γ) → ℂ

is non-degenerate (cf. Lemma 3.2). Corollary 2.10 will be used in the proof of Theorem 3.1 that gives a comparison of the modular symbols with the corresponding objects for compact homogeneous manifolds.

§3. The invariant part of modular symbols

In this section we examine the invariant part (the 1-component) of modular symbols.

We recall the setting in §2. That is, G is a connected real reductive linear Lie group, θ is a Cartan involution, 𝔤_{0} = 𝔨_{0} + 𝔭_{0} is a Cartan decomposition, K = G^{θ} is a maximal compact subgroup of G, and Γ is a torsion free co-compact subgroup. Suppose G′ is a connected closed subgroup which is θ-stable such that Γ′ = G′ ∩ Γ is co-compact in G′.

We assume that G is contained in a complex Lie group G_{ℂ} with Lie algebra 𝔤 ≃ 𝔤_{0} ⊗ ℂ. Let G_{U} and G′_{U} be the analytic subgroups of G_{ℂ} with Lie algebras 𝔨_{0} + \sqrt{−1}𝔭_{0} and 𝔨′_{0} + \sqrt{−1}𝔭′_{0}, respectively. Then both G_{U} and G are compact. We write Y_{U} := G′_{U}/K′, Y := Γ′\G′/K′, X_{U} := G_{U}/K and X := Γ\G/K. Let ι_{U}: Y_{U} ↪ X_{U} and ι: Y ↪ X be the natural embeddings. The Poincaré dual of the modular symbol, denoted by M(Y), is by definition the image of the fundamental class [Y] of the composition map

H_{dim Y} (Y; ℂ) →_{ι_{∗}} H_{dim Y} (X; ℂ) ⟶^{∼}_{P} H_{de Rham}^{codim Y}(X; ℂ).

Similarly, we define M(Y_{U}) ∈ H^{codim Y} (X; ℂ) to be the image of the fundamental class [Y_{U}] of the composition map

H_{dim Y} (Y_{U}; ℂ) →_{ι_{U∗}} H_{dim Y} (X_{U}; ℂ) ⟶^{∼}_{P_{U}} H_{de Rham}^{codim Y}(X_{U}; ℂ),

where P and P_{U} are the Poincaré duality maps. Here we note that dim Y_{U} = dim Y and dim X_{U} = dim X.

We denote by

pr_{π}: H_{de Rham}^{∗}(X; ℂ) → H^{∗}(π : Γ)

for the projection onto the π-component, corresponding to the direct sum decomposition (see Lemma 2.4)

H_{de Rham}^{∗}(X; ℂ) = ⨁_{π∈Ĝ_{0}} H^{∗}(π : Γ).

The π-component corresponding to π = 1 (the trivial representation), namely,

H^{∗}(𝔤, K; ℂ) = H^{∗}(1 : Γ) ⊂ H_{de Rham}^{∗}(Γ\G/K; ℂ)

is interpreted geometrically by using the diagram:

X_{U} = G_{U}/K ↪ G_{ℂ}/K_{ℂ} ↩ G/K → Γ\G/K = X,

as follows; start with a G_{U}-invariant representative of the de Rham cohomology group on G_{U}/K; extend it to a G_{ℂ}-invariant holomorphic form on G_{ℂ}/K_{ℂ}; restrict it to G/K; and push down to a form on Γ\G/K (this is possible because of the invariance by G, especially by Γ). Then we have defined a linear map

Ψ: H_{de Rham}^{∗}(X_{U}; ℂ) → H_{de Rham}^{∗}(X; ℂ).

Similarly, we define

Ψ′: H_{de Rham}^{∗}(Y_{U}; ℂ) → H_{de Rham}^{∗}(Y; ℂ).

Theorem 3.1. With notation as above, we have

pr_{1}(M(Y)) = \frac{vol(Y)}{vol(X)}Ψ (M(Y_{U})) .

Before proving Theorem 3.1, we recall:

Lemma 3.2.

1) Ψ is a well-defined, injective ring homomorphism with image H^{∗}(1 : Γ).

2) Ψ sends characteristic classes on X_{U} to the corresponding characteristic classes on X.

3) ∫_{X} Ψ(ω) = vol(X) ∫_{X_{U}} ω for any ω ∈ H_{de Rham}^{dim X}(X_{U}; ℂ).

4) The following diagram commutes:

H^{∗}(𝔤, K; ℂ) ≃H_{de Rham}^{∗}(X_{U}; ℂ) ⟶^{∼}_{Ψ} H^{∗}(1 : Γ) ⊂H_{de Rham}^{∗}(X; ℂ)
↓^{i_{U}^{∗}}  ↓_{i^{∗}}
H^{∗}(𝔤′, K′; ℂ)≃H_{de Rham}^{∗}(Y_{U}; ℂ) ⟶^{∼}_{Ψ′} H^{∗}(1 : Γ′)⊂H_{de Rham}^{∗}(Y; ℂ).

Proof of Lemma 3.2. See the main theorem in [10] (treated in a more general setting than here). □

Proof of Theorem 3.1. We fix an arbitrary α ∈ H_{de Rham}^{dim Y}(X_{U}; ℂ). Then

∫_{X} M(Y) ∧ Ψ(α) = ∫_{X} P ∘ ι_{∗}[Y] ∧ Ψ(α)
= ∫_{ι_{∗}[Y]} Ψ(α)  (Poincaré duality)

= ∫_{[Y]} ι^{∗} ∘ Ψ(α)

= ∫_{[Y]} Ψ′ ∘ ι_{U}^{∗}α  (Lemma 3.2 (4))

= vol(Y) ∫_{Y_{U}} ι_{U}^{∗}α  (Lemma 3.2 (3)).

Similarly, we have

∫_{X} Ψ(M(Y_{U})) ∧ Ψ(α) = ∫_{X} Ψ(M(Y_{U}) ∧ α)  (Lemma 3.2 (1))
= vol(X) ∫_{ι_{∗}[Y_{U}]} α  (Lemma 3.2 (3))

= vol(X) ∫_{Y_{U}} ι_{U}^{∗}α.

These two formulae mean that

M(Y) − Ψ(\frac{vol(Y)}{vol(X)}M(Y_{U})) ∈ H_{de Rham}^{codim Y}(X; ℂ)

is annihilated by

Image (Ψ: H_{de Rham}^{dim Y}(X_{U}; ℂ) → H_{de Rham}^{dim Y}(X; ℂ)) = H^{dim Y} (1 : Γ).

It follows from Corollary 2.10 that

M(Y) − \frac{vol(Y)}{vol(X)}Ψ(M(Y_{U})) ∈ ⨁_{\begin{matrix}π∈Ĝ_{0} \\ π≠1 \end{matrix}} H^{codim Y} (π : Γ).

Hence pr_{1}(M(Y)) = \frac{vol(Y)}{vol(X)}Ψ (M(Y_{U})) . □

§4 An explicit vanishing theorem for modular symbols

Any irreducible, infinitesimally unitarizable (𝔤, K)-module with non-zero (𝔤, K)- cohomology groups is represented as Vogan-Zuckerman’s derived functor module A_{𝔮} ([26]). Therefore, in order to apply Theorem 2.8 for actual computation of modular symbols, it is important to tell an explicit condition such that \overline{A_{𝔮}}|_{K′} is K′-admissible. Such a criterion was studied in [6] and [7]. Thus we can reformulate the vanishing theorem of modular symbols defined by a symmetric pair by an explicit condition by means of the root system. This is the main ingredients of this section (see Theorem 4.3).

Let G be a real reductive linear Lie group. We recall the notation in §2. In particular, 𝔤_{0} = 𝔨_{0} + 𝔭_{0} is a Cartan decomposition and 𝔤 = 𝔨 + 𝔭 denotes the complexifications. Take a Cartan subalgebra 𝔱_{0} of 𝔨_{0}. Let

𝔤(𝔱; α) := {X ∈ 𝔤 : [H, X] = α(H)X for H ∈ 𝔱}.

Given an element λ ∈ \sqrt{−1}𝔱_{0}^{∗}. Let

L ≡ L(λ) := {g ∈ G : Ad^{∗}(g)λ = λ},
𝔩 ≡ 𝔩(λ) := Lie (L(λ))⊗_{ℝ}ℂ = ∑_{⟨λ,α⟩=0} 𝔤(𝔱; α),

𝔲 ≡ 𝔲(λ) := ∑_{⟨λ,α⟩>0} 𝔤(𝔱; α),

𝔲̄ ≡ 𝔲̄(λ) := ∑_{⟨λ,α⟩<0} 𝔤(𝔱; α),
𝔮 ≡ 𝔮(λ) := 𝔩(λ) + 𝔲(λ).

The subalgebra 𝔮 ≡ 𝔮(λ) is said to be a θ-stable subalgebra of 𝔤 ([23]). The homogeneous manifold G/L(λ) carries a G-invariant complex structure with the holomorphic tangent bundle being G ×_{L(λ)} 𝔲̄(λ). We define A_{𝔮} to be the underlying (𝔤, K)-module of the S := dim_{ℂ} K/L ∩ K-th Dolbeault cohomology group HS ; ¯∂(G/L(λ), Ω) with the coefficient in the canonical line bundle Ω ≡ Ω_{G/L(λ)} ([20], [28]). Then A_{𝔮} is an irreducible Harish-Chandra module which is unitarizable ([24], [27]). The corresponding unitary representation is denoted by \overline{A_{𝔮}} ∈ Ĝ_{0} (⊂ Ĝ). Then it is a result due to Vogan-Zuckerman ([26]) that

Ĝ_{VZ} := {A_{𝔮} : 𝔮 is a θ-stable parabolic subalgebra of 𝔤} (⊂ Ĝ_{0} ⊂ Ĝ)

is the totality of irreducible unitary representations of G whose underlying (𝔤, K)- modules have non-zero (𝔤, K)-cohomology groups. We note that if 𝔮 is a Borel subalgebra and if rank G = rank K then \overline{A_{𝔮}} is a discrete series representation of G.

Suppose 𝔮̃ = 𝔩̃ + 𝔲̃ and 𝔮 = 𝔩 + 𝔲 are θ-stable parabolic subalgebras such that 𝔩̃ ⊃ 𝔩 and 𝔲̃ ⊂ 𝔲. If L̃/L is compact, then there is a compact normal subgroup L′ of L̃ such that L̃ = L′L. Then we have A𝔮̃ ≃ A_{𝔮} because the spectral sequence of induction by stages ([23], Proposition 6.3.6) collapses thanks to the Borel-Weil-Bott theorem for the compact group L′. So, we consider the following condition on a θ-stable parabolic subalgebra 𝔮:

(4.1) There exists no θ-stable parabolic subalgebra 𝔮̃ = 𝔩̃+ 𝔲̃ of 𝔤 that satisfies both (4.1.1) and (4.1.2):

𝔩̃ ⫌ 𝔩, 𝔲̃ ⫋ 𝔲, (4.1.1)

(4.1.2)  L̃/L is compact.

Then, the set of θ-stable parabolic subalgebras 𝔮 satisfying (4.1) is sufficient to describe Ĝ_{VZ}. That is,

Ĝ_{VZ} = {A_{𝔮} : 𝔮 is a θ-stable parabolic subalgebra of 𝔤 satisfying (4.1)}.

Next, we suppose that σ is an involutive automorphism of G. If G′ is an open subgroup of G^{σ} := {g ∈ G : σg = g}, then the pair (G, G′) is said to be a reductive symmetric pair (or a semisimple symmetric pair if G is semisimple). We fix a Cartan involution θ of G such that σθ = θσ. Then θG′ = G′, K′ := K ∩ G′ is a maximal compact subgroup of G′ and the pair (K, K′) forms a compact symmetric pair. We write 𝔨_{0±} := {X ∈ 𝔨_{0} : σ(X) = ±X}. We take a σ-stable Cartan subalgebra 𝔱_{0} of 𝔨_{0} so that 𝔱_{0−} := 𝔱_{0} ∩ 𝔨_{0−} is a maximal abelian subspace in 𝔨_{0−}. Let Δ(𝔨, 𝔱) be the root system and Σ(𝔨, 𝔱_{−}) the restricted root system of 𝔨 with respect to 𝔱 and 𝔱_{−}, respectively. We fix a positive system Σ^{+}(𝔨, 𝔱_{−}) and take a positive system Δ^{+}(𝔨, 𝔱) which is compatible with Σ^{+}(𝔨, 𝔱_{−}). We shall regard as \sqrt{−1}(𝔱_{0−})^{∗} ⊂ \sqrt{−1}𝔱_{0}^{∗} according to the direct sum decomposition 𝔱_{0} = 𝔱_{0−} + (𝔱_{0} ∩ 𝔨_{0}).

Suppose 𝔮 = 𝔮(λ) is a θ-stable parabolic subalgebra of 𝔤 defined by λ ∈ \sqrt{−1}𝔱_{0}^{∗} as before. After conjugation by an element of K, we may and do assume that λ ∈ \sqrt{−1}𝔱_{0}^{∗} is dominant with respect to Δ^{+}(𝔨, 𝔱). Then a closed cone \sqrt{−1}𝔱_{0}^{∗} is defined by

ℝ_{+}⟨𝔲 ∩ 𝔭⟩ := \{ ∑_{β∈Δ(𝔲∩𝔭;𝔱)} n_{β}β : n_{β} ≥ 0\} ⊂ \sqrt{−1}𝔱_{0}^{∗}.

Fact 4.2. Retain the notation as above. Then the following three conditions on (𝔮, G, G′) are equivalent:

1) ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ \sqrt{−1}(𝔱_{0−})^{∗} = \{0\}.

2) \overline{A_{𝔮}}|_{K′} is K′-admissible.

3) A_{𝔮} is discretely decomposable as (𝔤′, K′)-modules.

Proof. See [6], Theorem 3.2 for (1) ⇒ (2); and [8], Theorem 4.2 for other implications. □

Now, we have

Theorem 4.3. Let (G, G′) be a reductive symmetric pair, and Γ a torsion free, cocompact discrete subgroup of G such that Γ′ := Γ∩G′ is co-compact in G′. Suppose that 𝔮 = 𝔩 + 𝔲 is a θ-stable parabolic subalgebra of 𝔤. Assume the following three conditions are satisfied:

i) 𝔩 ⊅ 𝔭, ii) ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ \sqrt{−1}(𝔱_{0−})^{∗} = \{0\}, iii) G′ is semisimple without compact factors.

Then we have

⟨H^{d(G′)}(\overline{A_{𝔮}} : Γ), ι_{∗}([Γ′\G′/K′])⟩ = 0.

Proof. The assumption (i) implies that \overline{A_{𝔮}} is infinite dimensional. By Fact 4.2, the restriction \overline{A_{𝔮}}_{|_{K′}} is K′-admissible because of the assumption (ii). Now Theorem 4.3 is an immediate consequence of Theorem 2.3. □

Remark 4.4. If 𝔮 satisfies (4.1) then the assumption (i) in Theorem 4.3 is equivalent to the condition that 𝔩 ≠ 𝔤.

Remark 4.5. We can generalize Theorem 4.3 to the non-symmetric case by using [7], Theorem 2.9.

§5 Modular symbols on Hermitian symmetric spaces of type IV

In this section, we apply the results in §2, §3 and §4 to the symmetric pair

(G, G′) = (SO_{0}(2n, 2), SO_{0}(2n, 1)).

Here, SO_{0}(p, q) denotes the identity component of the orthogonal group O(p, q) of signature (p, q). We shall study the middle Hodge component of the modular symbol defined by G′. The key step is the following:

Lemma 5.1. If the π_{K} is a unitarizable irreducible (𝔤, K)-module such that

H^{n,n}(𝔤, K; π_{K}) ≠ 0,

then π_{K}|_{K′} is K′-admissible. In particular, π_{K} is discretely decomposable as a (𝔤′, K′)-module.

This lemma will be proved after Lemma 5.3, 5.4 and 5.5. First we fix notation. A maximal compact subgroup K of G is isomorphic to SO(2n) × SO(2). Take a Cartan subalgebra 𝔱_{0} of 𝔨_{0} = Lie K, which is also a Cartan subalgebra 𝔱_{0} of 𝔤_{0} because of rank K = rank G = n + 1. We take a basis \{f_{1}, ⋯, f_{n+1}\} of \sqrt{−1}𝔱_{0}^{∗} so that the root systems of 𝔨 and 𝔤 are represented as

Δ(𝔨, 𝔱) = \{±(f_{i} ± f_{j}) : 1 ≤ i < j ≤ n\} ⊂ \sqrt{−1}𝔱_{0}^{∗},
Δ(𝔤, 𝔱) = \{±(f_{i} ± f_{j}) : 1 ≤ i < j ≤ n + 1\} ⊂ \sqrt{−1}𝔱_{0}^{∗}.

Let 𝔤 = 𝔨 + 𝔭 = 𝔨 ⊕ 𝔭_{+} ⊕ 𝔭_{−} be the irreducible decomposition as Ad(K)-modules where the subspaces 𝔭_{±} are defined by the weights with respect to 𝔱 as follows:

Δ(𝔭_{+}, 𝔱) = {±f_{i} + f_{n+1} : 1 ≤ i ≤ n},
Δ(𝔭_{−}, 𝔱) = {±f_{i} − f_{n+1} : 1 ≤ i ≤ n}.

Here we note that 𝔭_{±} are Ad(K)-stable, and consequently ad(𝔱)-stable.

We first define some special θ-stable parabolic subalgebras of 𝔤, denoted by 𝔮_{0}, ⋯, 𝔮_{n−1}, 𝔮_{n}^{±}, which contribute (𝔤, K)-cohomology groups of degree (n, n) (see Lemma 5.3 and Lemma 5.4). We fix a positive system of Δ(𝔨, 𝔱) once and for all as follows:

Δ^{+}(𝔨, 𝔱) := {f_{i} ± f_{j} : 1 ≤ i < j ≤ n}.

For k = 0, 1, ⋯, n − 1, we take λ = ∑_{i=1}^{n+1} λ_{i}f_{i} ∈ \sqrt{−1}𝔱_{0}^{∗} so that

(5.2.1)  λ_{1} > ⋯ > λ_{k} > λ_{k+1} = ⋯ = λ_{n} = λ_{n+1} = 0.

According to §4, we define a θ-stable parabolic subalgebra of 𝔤 by

𝔮_{k} := 𝔮(λ).

This definition is independent of the choice of λ as far as λ satisfies (5.2.1). Note that 𝔮_{0} = 𝔤.

Similarly, we take λ = ∑_{i=1}^{n+1} λ_{i}f_{i} ∈ \sqrt{−1}𝔱_{0}^{∗} so that

(5.2.2)  λ_{1} > ⋯ > λ_{n−1} > |λ_{n}| > λ_{n+1} = 0

and define a θ-stable subalgebra 𝔮(λ). This definition depends only on the signature of λ_{n}. So, we write

𝔮_{n}^{ϵ} := 𝔮(λ), ϵ := sgn λ_{n} ∈ {±}.

Then Levi subgroups corresponding to 𝔮_{k}, 𝔮_{n}^{±} are given respectively as follows:

L_{k} ≃ 𝕋^{k} × SO_{0}(2n − 2k, 2) (0 ≤ k ≤ n − 1),
L_{n}^{±} ≃ 𝕋^{n}.

The unitary representations \overline{A_{𝔮_{n}^{±}}} are discrete series representations for G, while

\overline{A_{𝔮_{k}}} are non-tempered representations of G. We recall that if G/K is a Hermitian symmetric space, then the Hodge component of (𝔤, K)-cohomology groups is given by

H^{p,q}(𝔤, K; A_{𝔮}) ≃ H^{p+R_{+},q+R_{−}}(𝔩, L ∩ K; ℂ)

where R_{±} := dim 𝔲 ∩ 𝔭_{±} (see [3]). For the reader’s convenience, we present the computation of (𝔤, K)-cohomology groups of A_{𝔮_{k}} (0 ≤ k ≤ n − 1) and A_{𝔮_{n}^{±}}.

Lemma 5.3 (see [12],[26]). Let 0 ≤ k ≤ n − 1.

H^{i,j}(𝔤, K; A_{𝔮_{k}}) = \{ \begin{matrix}ℂ + ℂ & if i = j = n, \\ ℂ & if k ≤ i = j ≤ 2n − k, i ≠ n, \\ 0 & otherwise, \end{matrix}

H^{i,j}(𝔤, K; A_{𝔮_{n}^{±}}) ≃ \{ \begin{matrix}ℂ & if i = j = n, \\ 0 & otherwise. \end{matrix}

Thus, A_{𝔮_{k}} (0 ≤ k ≤ n − 1) and A_{𝔮_{n}^{±}} contribute to the (𝔤, K)-cohomology groups of degree (n, n). Conversely, no other irreducible infinitesimally unitarizable (𝔤, K)- modules contribute (𝔤, K)-cohomology groups of degree (n, n). That is, we have the following:

Lemma 5.4. Let G = SO_{0}(2n, 2), and π_{K} an irreducible unitarizable (𝔤, K)- module such that

(5.4.1)  H^{n,n}(𝔤, K; π_{K}) ≠ {0}

Then π_{K} ≃ A_{𝔮} as (𝔤, K)-modules, where 𝔮 is one of 𝔮_{0}, 𝔮_{1}, ⋯, 𝔮_{n−1}, 𝔮_{n}^{+} or 𝔮_{n}^{−}.

Proof. By a theorem of Vogan-Zuckerman [26], π_{K} is isomorphic to A_{𝔮} where 𝔮 = 𝔩 + 𝔲 some θ-stable parabolic subalgebra of 𝔤 = Lie SO_{0}(2n, 2) ⊗_{ℝ} ℂ. We may assume that 𝔮 satisfies (4.1). The condition (5.4.1) implies that

(5.4.1′)  dim(𝔲 ∩ 𝔭_{+}) = dim(𝔲 ∩ 𝔭_{−}).

We shall prove that 𝔮 is conjugate to one of 𝔮_{0}, 𝔮_{1}, ⋯, 𝔮_{n−1}, 𝔮_{n}^{+} or 𝔮_{n}^{−} by the adjoint action of K if 𝔮 satisfies (4.1) and (5.4.1′). After a conjugation by an element of K, we may assume that 𝔮 ≡ 𝔮(λ) = 𝔩 + 𝔲 ≡ 𝔩(λ) + 𝔲(λ) is defined by λ that is dominant with respect to Δ^{+}(𝔨, 𝔱). That is, if we write

λ = ∑_{i=1}^{n+1} λ_{i}f_{i} ∈ \sqrt{−1}𝔱_{0}^{∗},

then we have λ_{1} ≥ ⋯ ≥ λ_{n−1} ≥ |λ_{n}|, since the root system Δ(𝔨, 𝔱) is of type D_{n}. Then we have

Δ(𝔲(λ) ∩ 𝔭_{+}, 𝔱) = {f_{i} + f_{n+1} : 1 ≤ i ≤ n, λ_{i} + λ_{n+1} > 0}
∪ {−f_{i} + f_{n+1} : 1 ≤ i ≤ n, −λ_{i} + λ_{n+1} > 0},
Δ(𝔲(λ) ∩ 𝔭_{−}, 𝔱) = {f_{i} − f_{n+1} : 1 ≤ i ≤ n, λ_{i} − λ_{n+1} > 0}
∪ {−f_{i} − f_{n+1} : 1 ≤ i ≤ n, −λ_{i} − λ_{n+1} > 0}.

Therefore, the assumption dim(𝔲(λ) ∩ 𝔭_{+}) = dim(𝔲(λ) ∩ 𝔭_{−}) amounts to

(5.4.2) #{i : 1 ≤ i ≤ n, λ_{i} + λ_{n+1} > 0} + #{i : 1 ≤ i ≤ n, −λ_{i} + λ_{n+1} > 0} = #{i : 1 ≤ i ≤ n, λ_{i} − λ_{n+1} > 0} + #{i : 1 ≤ i ≤ n, −λ_{i} − λ_{n+1} > 0}.

We set

k := #{i : 1 ≤ i ≤ n, |λ_{i}| > |λ_{n+1}|} ∈ {0, 1, ⋯, n},
m := #{i : 1 ≤ i ≤ n, |λ_{i}| = |λ_{n+1}|} ∈ {0, 1, ⋯, n}.

Obviously k + m ≤ n. Without loss of generality, we may assume λ_{n+1} ≥ 0 hereafter. Then the definition of k and m leads to

\{ \begin{matrix}λ_{1} ≥ ⋯ ≥ λ_{k} > λ_{k+1} = ⋯ = λ_{k+m} = λ_{n+1} \\ > λ_{k+m+1} ≥ ⋯ ≥ λ_{n−1} ≥ |λ_{n}| & (k + m < n), \\ λ_{1} ≥ ⋯ ≥ λ_{k} > λ_{k+1} = ⋯ = λ_{n−1} = |λ_{n}| = λ_{n+1} & (k + m = n, k ≠ n), \\ λ_{1} ≥ ⋯ ≥ λ_{n−1} ≥ |λ_{n}| > λ_{n+1} & (k + m = n, k = n). \end{matrix}

Then it is an elementary computation to see

the left side of (5.4.2) = \{ \begin{matrix}n + (n − k − m) & if k + m < n, \\ n & if k + m = n, λ_{n+1} > 0, \\ k & if k + m = n, λ_{n+1} = 0, \end{matrix}

the right side of (5.4.2) = k.

In view of k + m ≤ n, the equation (5.4.2) holds if and only if either (5.4.3) or (5.4.4) is satisfied:

(5.4.3)  k + m = n, λ_{n+1} > 0, k = n,
(5.4.4)  k + m = n, λ_{n+1} = 0.

Then (5.4.3) and (5.4.4) imply

(5.4.3′)  λ_{1} ≥ ⋯ ≥ λ_{n−1} ≥ |λ_{n}| ≥ λ_{n+1} > 0,
(5.4.4′)  λ_{1} ≥ ⋯ ≥ λ_{k} > λ_{k+1} = ⋯ = λ_{n} = λ_{n+1} = 0,

respectively. Let us compare 𝔮 = 𝔮(λ) with 𝔮_{0}, 𝔮_{1}, ⋯, 𝔮_{n−1}, 𝔮_{n}^{±}. If λ satisfies (5.4.3′), then 𝔮 ⊃ 𝔮_{n}^{+} and L/L_{n}^{+} is compact. If λ satisfies (5.4.4′), then 𝔮 ⊃ 𝔮_{k} and L/L_{k} is compact. Because 𝔮 satisfies (4.1), λ = (λ_{1}, ⋯, λ_{n+1}) must satisfy

(5.4.3′′)  λ_{1} > ⋯ > λ_{n−1} > |λ_{n}| > λ_{n+1} > 0,
(5.4.4′′)  λ_{1} > ⋯ > λ_{k} > λ_{k+1} = ⋯ = λ_{n} > λ_{n+1} = 0,

respectively, which means 𝔮 = 𝔮_{n}^{+} or 𝔮 = 𝔮_{k} (1 ≤ k ≤ n − 1). This is what we wanted to prove. The case λ_{n+1} ≤ 0 is similar. □

Lemma 5.5. Suppose that a θ-stable parabolic subalgebra 𝔮 = 𝔩 + 𝔲 of 𝔤 is one of 𝔮_{0}, 𝔮_{1}, ⋯, 𝔮_{n−1}, 𝔮_{n}^{±}. Then we have

ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ ℝf_{n+1} = {0}.

Proof. The set of weights for 𝔲 ∩ 𝔭 with respect to a Cartan subalgebra 𝔱 of 𝔨 is given by

Δ(𝔲 ∩ 𝔭, 𝔱) = \{ \begin{matrix}\{f_{i} ± f_{n+1} : 1 ≤ i ≤ k\} & if 𝔮 = 𝔮_{k}, \\ \{f_{i} ± f_{n+1} : 1 ≤ i ≤ n − 1\} ∪ \{ϵf_{n} ± f_{n+1}\} & if 𝔮 = 𝔮_{n}^{ϵ}. \end{matrix}

where 0 ≤ k ≤ n − 1 and ϵ = ±. In either case, we have

ℝ_{+}⟨𝔲 ∩ 𝔭⟩ = { ∑_{α∈Δ(𝔲∩𝔭,𝔱)} a_{α}α : a_{α} ≥ 0}

⊂ {∑_{i=1}^{n−1}(a_{i} + b_{i})f_{i} + ϵ(a_{n} + b_{n})f_{n} + ∑_{i=1}^{n}(a_{i} − b_{i})f_{n+1} : a_{i}, b_{i} ≥ 0 1 ≤ i ≤ n + 1}.

Therefore

ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ ℝf_{n+1}
⊂{∑_{i=1}^{n−1}(a_{i} + b_{i})f_{i} + ϵ(a_{n} + b_{n})f_{n} + (∑_{i=1}^{n}(a_{i} − b_{i})) f_{n+1} : a_{i}, b_{i} ≥ 0} ∩ ℝf_{n+1}.

If ∑_{i=1}^{n−1}(a_{i} + b_{i})f_{i} + ϵ(a_{n} + b_{n})f_{n} + (∑_{i=1}^{n}(a_{i} − b_{i})) f_{n+1} ∈ ℝf_{n+1}, then a_{i} + b_{i} = 0 (1 ≤ i ≤ n − 1) and ϵ(a_{n} + b_{n}) = 0. This holds if and only if a_{i} = b_{i} = 0 for all i (1 ≤ i ≤ n) because a_{i}, b_{i} ≥ 0. Hence ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ ℝf_{n+1} = {0}. □

Now, we are ready to prove Lemma 5.1:

Proof of Lemma 5.1. Suppose that π_{K} is an irreducible (𝔤, K)-module satisfying H^{n,n}(𝔤, K; π_{K}) ≠ 0. It follows from Lemma 5.4 that π_{K} ≃ A_{𝔮} as (𝔤, K)-modules, where 𝔮 is one of 𝔮_{0}, ⋯, 𝔮_{n−1} or 𝔮_{n}^{±}.

On the other hand, in view of (K, K′) = (SO(2n) × SO(2), SO(2n)), we have

\sqrt{−1}(𝔱_{0−})^{∗} = ℝf_{n+1}.

Here, we recall that \{f_{1}, ⋯, f_{n+1}\} is a basis of \sqrt{−1}𝔱_{0}^{∗} (see §4 for the definition of 𝔱_{0−}). Hence

ℝ_{+}⟨𝔲 ∩ 𝔭⟩ ∩ \sqrt{−1}(𝔱_{0−})^{∗} = \{0\}

by Lemma 5.5. Therefore, it follows from Fact 4.2 that A_{𝔮}|_{K′} is K′-admissible. □

Theorem 5.6. Let Γ be a torsion free discrete subgroup of G = SO_{0}(2n, 2), G′ := SO_{0}(2n, 1) ⊂ G and Γ′ := Γ ∩ G′. Assume that G/Γ and G′/Γ′ are compact. Let ι: Y = Γ′\G′/K′ → X = Γ\G/K be the natural embedding map as before. Let π(≠ 1) be an irreducible unitary representation of G. We write

pr_{1} : H_{de Rham}^{∗}(X; ℂ) → H^{∗}(1 : Γ) = H^{∗}(𝔤, K; ℂ)

for the projection onto 1-component in the Matsushima-Murakami formula (see Lemma 2.4) and

M(Y) = ∑_{p+q=2n} M^{p,q}(Y) ∈ ⨁_{p+q=2n} H^{p,q}(X; ℂ)

for the Hodge decomposition of the Poincaré dual of the modular symbol ι_{∗}([Y]) ∈ H_{2n}(X; ℂ). Then we have

M^{n,n}(Y) = pr_{1}(M(Y)).

Proof. By Corollary 2.10 for the subspace H^{n,n}(X; ℂ) of H_{de Rham}^{2n}(X; ℂ), it suffices to show

⟨H^{2n}(π : Γ), ι_{∗}[Γ′\G′/K′]⟩ = 0

for any π ∈ Ĝ such that π ≠ 1 and H^{n,n}(𝔤, K; π_{K}) ≠ 0. Because π is not the trivial representation, we have 𝔮 ≠ 𝔤 and so 𝔩 ⊅ 𝔭. It follows from Lemma 5.1 and Theorem 4.3 that

⟨H^{2n}(π : Γ), ι_{∗}([Γ′\G′/K′])⟩ = 0,

which we wanted to prove. □

Now Theorem in Introduction follows from Theorem 3.1 and Theorem 5.6.

§6 Concluding remarks

Remark 6.1. Before finishing this paper, let us note some heuristic argument on the relation between our result and the Rankin convolution method which is used to obtain an integral expression of certain automorphic L-functions and to represent the special values of such L in terms of some trajectory integrals. The typical situation is the following. Let Φ be an automorphic form on the adelization G_{𝔸} of a reductive algebraic group G defined over a global field. Let H be an algebraic subgroup of G. Then we consider some Eisenstein series E(s, h) on ℂ × H_{𝔸} and another cusp form ψ on H_{𝔸}. Then we consider the integral

∫_{H_{k}\H_{𝔸}} Φ(h)ψ(h)E(s, h)dh.

The basic assumption behind, but which rarely referred explicitly, is that the tensor product automorphic representation

(Π_{Φ}|_{H_{𝔸}}) ⊗ π_{ψ}

has a continuous spectrum. Here we denote by Π_{Φ} and π_{ψ} the automorphic representations of G_{𝔸} and H_{𝔸}, respectively, generated by Φ and ψ. Then G_{ℝ} is a real reductive linear group, corresponding to the notation “G” in this paper. Suppose that ψ is the constant. Then the very assumption on the existence of continuous spectrum for the restriction Π_{Φ}|_{H_{𝔸}} fails if the real component Π_{ℝ} of Π_{Φ} satisfies the criterion Fact 4.2, because of the discrete decomposability of the restriction Π_{ℝ}|_{H_{ℝ}}. So the Rankin convolution gives just zero in this case !

Remark 6.2. If G/K is Hermitian, then the arithmetic quotient V = Γ\G/K is an algebraic variety by a classical result of Baily. When Γ is not co-compact in G, modular embeddings H ⊂ G do not give cycles on V, they are just chains in general. Consider the Baily-Borel-Satake compactification V^{∗} of V. Then we can form the closure of modular symbols in V^{∗} and obtain closed real analytic subsets in V^{∗}. One can investigate the perversity of these compactified modular symbols (at least) in some special cases, in the sense of Goretzky-MacPherson. When G = SO_{0}(m, 2) and H = SO_{0}(m, 1), they never have the middle perversity. We refer to a related result in the case m = 3, by Nygaard [15], Theorem 2.1, which was brought to our attention by Dr. Miyazaki. It is an easy exercise to check that the closure of modular symbols has “one-step worse” perversity than the middle perversity. This fact is interesting because it may give an account of the “failure of the Petersson conjecture” for the lifted modular forms and the mixed cohomology associated to such modular forms (cf. [17], Conjecture 5.4)

Remark 6.3. Suppose we are in the setting of §5. In contrast to Lemma 5.1, not all unitarizable irreducible (𝔤, K)-modules π_{K} that contribute to the (p, q)-type Hodge component H^{p,q}(𝔤, K; π_{K}) are discretely decomposable as (𝔤′, K′)-modules. This means that Theorem 4.3 gives only a partial information about the (p, q)-type Hodge component of the modular symbol, M^{p,q}(Y) for (p, q) ≠ (n, n).

Remark 6.4. As Theorem 4.3 suggests, the more there are representations π ∈ Ĝ satisfying (4.3)(ii), the more we obtain information about the modular symbol defined by (G, G′). It might be interesting to investigate in other cases the counterpart in differential geometry (e.g. the middle Hodge component in the case of §5 where G/K is Hermitian), corresponding to the representation theoretic condition (4.3)(ii), which assures the vanishing of the π-component in Theorem 4.3.

Remark 6.5. Suppose that (G, G′) is a reductive symmetric pair and defines a modular symbol Γ′\G′/K′ ↪ Γ\G/K, as in (2.6). Y. Tong and S. Wang proved the non-vanishing theorem of the modular symbol (defined on a certain locally constant sheaf) with respect to the π-component if π ∈ Ĝ is a discrete series representation for G/G′ ([22]). On the other hand, we have proved the vanishing theorem of the modular symbol with respect to the π-component, if the restriction π|_{G′} is discretely decomposable (Theorem 4.3). This observation suggests that the assumption on π ∈ Ĝ in [22] (i.e. π is a discrete series representation for G/G′) and the one in this paper (i.e. π is discretely decomposable with respect to G′) should be exclusive. This is true if G′ is compact because there is no discrete series representation for G/G′, and is proved in general in [8], Theorem 6.2. For example, if (G, G′) = (SO_{0}(2n, 2), SO_{0}(2n, 1)), then any of discrete series representations for G/G′ have highest weight vectors (or of lowest weight vectors), and therefore, the contribution of the de Rham cohomology group H^{2n}(Γ\G/K; ℂ) is the (2n, 0)-type Hodge component (or the (0, 2n)-type Hodge component), which is the opposite extremal case of the (n, n)-type that we have treated in §5.

⟨Acknowledgement⟩

The main part of the manuscript was written while the first author was a guest of the Institut Mittag-Leffler supported by the Royal Swedish Academy of Sciences. He expresses his sincere gratitude to the staff and Professors M. Flensted-Jensen and G. Ólafsson for giving me a warm and stimulating atmosphere of research during the special year 1995-1996 on “Analysis on Lie Groups”.

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Graduate School of Mathematical Sciences, University of Tokyo, Meguro, Komaba, 153, Tokyo, Japan