Journal of Lie Theory Volume 10 (2000) 81–91 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag Berlin

Poisson liftings of holomorphic automorphic forms on semisimple Lie groups

Min Ho Lee and Hyo Chul Myung^{∗}

Communicated by E. Zelmanov

Abstract. Let G be a semisimple Lie group of Hermitian type, K ⊂ G a maximal compact subgroup, and P ⊂ G a minimal parabolic subgroup associated to K. If σ is a finite-dimensional representation of K in a complex vector space, it determines the associated homogeneous vector bundles on the homogeneous manifolds G/P and G/K. The Poisson transform associates to each section of the bundle over G/P a section of the bundle over G/K, and it generalizes the classical Poisson integral. Given a discrete subgroup Γ of G, we prove that the image of a Γ-invariant section of the bundle over G/P under the Poisson transform is a holomorphic automorphic form on G/K for Γ. We also discuss the special case of symplectic groups in connection with holomorphic forms on families of abelian varieties.

1. Introduction

In classical harmonic function theory, it is well-known that the Poisson integral of a complex-valued integrable function defined on a unit circle in the complex plane determines a harmonic function on the corresponding unit disk. The purpose of this paper is to discuss the Lie-theoretic analogue of the Poisson integral that transforms sections of a certain vector bundle to holomorphic automorphic forms on a semisimple Lie group of Hermitian type.

Let S be the unit circle in the complex plane ℂ given by S = {e^{iθ} | −π ≤ θ < π}, and let L^{1}(S) denote the space of complex-valued integrable functions on S. Let f ∈ L^{1}(S), and set f̂(θ) = f(e^{iθ}) for −π ≤ θ < π. Then the classical Poisson integral Pf of f is a function defined on the unit disk U = {z ∈ ℂ | |z| < 1} given by

Pf(re^{iϕ}) = \frac{1}{2π} ∫_{−π}^{π} \frac{1 − r^{2}}{1 − 2r cos(ϕ − θ) + r^{2}} f̂(θ)dθ  (1)

for 0 ≤ r < 1 and ϕ ∈ ℝ, and it is known that Pf is a harmonic function (see e.g. [8]). If we use the normalized measure ds for S, then the expression in (1)

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag can now be written in the form

∗ Supported by KIAS Grant M98004

(Pf)(z) = ∫_{S} \frac{1 − |z|^{2}}{|z − s|^{2}} f(s)ds  (2)

for all z ∈ U. We can interpret (2) in terms of Lie groups as follows. Let G_{0} be the Lie group SU(1, 1) which can be written in the form

G_{0} = \{(\begin{matrix}a & b \\ \overline{b} & \overline{a} \end{matrix}) | a, b ∈ ℂ, |a|^{2} − |b|^{2} = 1\}.

Then G_{0} acts on the unit disk U by

(\begin{matrix}a & b \\ \overline{b} & \overline{a} \end{matrix}) · z = \frac{az + b}{\overline{b}z + \overline{a}}

for z ∈ U and ( \begin{matrix}a & b \\ \overline{b} & \overline{a} \end{matrix} ) ∈ G_{0}. We consider the subgroups A_{0}, K_{0}, M_{0} and N_{0} of G_{0} given by

A_{0} = \{(\begin{matrix}cosh t & sinh t \\ sinh t & cosh t \end{matrix}) | t ∈ ℝ\}, K_{0} = \{(\begin{matrix}e^{iθ} & 0 \\ 0 & e^{−iθ} \end{matrix}) | θ ∈ ℝ\},

M_{0} = \{(\begin{matrix}ε & 0 \\ 0 & ε \end{matrix}) | ε^{2} = 1\}, N_{0} = \{(\begin{matrix}1 + ix & −ix \\ ix & 1 − ix \end{matrix}) | x ∈ ℝ\}.

Then the unit circle S and the unit disk U can be identified with the quotient spaces

S = G_{0}/K_{0}, U = G_{0}/M_{0}A_{0}N_{0}

(cf. [2, Chapter I]), and therefore we see that the Poisson integral associates a harmonic function on U to each integrable function on S. Such an interpretation of the Poisson integral suggests the possibility of extending (2) to the case of a more general semisimple Lie group.

Let G be a semisimple Lie group of Hermitian type, and let σ be a finitedimensional representation of a maximal compact subgroup K of G in a complex vector space. Let 𝔤 and 𝔨 be the Lie algebras of G and K, respectively, and let 𝔭 be the orthogonal complement of 𝔨 in 𝔤 relative to the Killing form. Let P be a minimal parabolic subgroup of G, and let 𝔞 be a maximal abelian subgroup of 𝔭. If 𝔞_{ℂ} denotes the complexification of a maximal abelian subspace 𝔞 of 𝔭 and if λ ∈ 𝔞_{ℂ}^{∗}, then σ can be extended to a representation σ_{P,λ} of P. Thus we can consider the homogeneous vector bundle W(σ_{P,λ}) (resp. V(σ)) over G/P (resp. G/K) associated to the representation σ_{P,λ} (resp. σ). Now as the analogue of the classical Poisson integral we can consider the Poisson transform P_{λ,P} which assigns to each section ϕ of W(σ_{P,λ}) a section of V(σ) (see Section 4 for details; see also [2, §II.3.4], [5], [11]).

Let Γ be a discrete subgroup of G that acts on the Riemannian symmetric space G/K properly discontinuously without fixed points. Then we can discuss holomorphic automorphic forms on G/K for Γ associated to an automorphy factor of Γ. In this paper, we show that the Γ-invariant sections of W(σ_{P,λ}) are liftings of such automorphic forms. More precisely, we prove that the Poisson transform P_{λ,P} ϕ of a Γ-invariant section ϕ of W(σ_{P,λ}) is a holomorphic automorphic form on G/K for Γ with respect to the canonical automorphy factor of G associated to σ. As an application we show that for symplectic groups such liftings can be regarded as liftings of some holomorphic forms on a family of abelian varieties.

2. Automorphic forms

In this section we discuss holomorphic automorphic forms on semisimple Lie groups of Hermitian type and their interpretation in terms of sections of certain vector bundles over locally symmetric spaces. We also describe canonical automorphy factors associated to semisimple Lie groups of Hermitian type.

Let G be a connected semisimple Lie group Hermitian type, and let K be a maximal compact subgroup of G. Thus the associated Riemannian symmetric space D = G/K is a Hermitian symmetric domain that has a G-invariant complex structure, and it can be realized as a bounded domain in ℂ^{k} with k dim_{ℂ} D. Let W be a finite-dimensional complex vector space, and let GL(W) denote the group of all invertible endomorphism of W. If G′ is a subgroup of G, a map j : G′ × D → GL(W) is called an automorphy factor of G′ if it satisfies the following conditions:

(i) For fixed g ∈ G′, the map z ↦ j(g, z), D → GL(W) is holomorphic. (ii) For all g_{1}, g_{2} ∈ G′ and z ∈ D, we have

j(g_{1}g_{2}, z) = j(g_{1}, g_{2}z) · j(g_{2}, z)  (3)

Let Γ be a torsion-free discrete subgroup of G. Then the complex structure on D induces the structure of a complex manifold on the locally symmetric space X = Γ\D, and we can define automorphic forms on D as follows (cf. [1]).

Definition 2.1. Let j : Γ × D → GL(W) be an automorphy factor of Γ. A holomorphic automorphic form on D of type j for Γ is a holomorphic map f : D → W that satisfies

f(γz) = j(γ, z) · f(z)  (4)

for all z ∈ D and γ ∈ Γ.

Given an automorphy factor j : Γ × D → GL(W), we can construct an associated vector bundle on the locally symmetric space X = Γ\D as follows. Let the discrete subgroup Γ of G act on D × W by

γ · (z, w) = (γz, j(γ, z)w)

for all γ ∈ Γ and (z, w) ∈ D × W. The fact that this operation indeed defines an action of Γ on D × W follows from the condition (3). We set

A(Γ, j) = Γ\D × W,

where the quotient is taken with respect to the above action of Γ on D×W. Then the natural projection D → Γ\D induces the structure of a holomorphic vector bundle on the induced map

ϖ : A(Γ, j) → X = Γ\D

with fiber W. Let Γ_{0}(X, A(Γ, j)) denote the space of holomorphic sections of A(Γ, j) over X, that is, holomorphic maps s : X → A(Γ, j) such that ϖ ∘s = 1_{X}.

Lemma 2.2. Let j : Γ × D → GL(W) be an automorphy factor. Then each element of Γ_{0}(X, A(Γ, j)) can be identified with an automorphic form on D of type j for Γ.

Proof. See for example [6].

We shall now describe the construction of the canonical automorphy factor of G. Let I be a G-invariant complex structure on D = G/K. Then it determines a complex structure I_{z} on the tangent space T_{z}(D) for each z ∈ D. Let 𝔤 and 𝔨 be the Lie algebras of G and K, respectively, and let 𝔤 = 𝔨 + 𝔭 be the corresponding Cartan decomposition of 𝔤. If z_{0} is the point in D with Kz_{0} = z_{0}, then we can identify 𝔭 with the tangent space T_{z_{0}}(D). Thus we obtain a complex structure I_{z_{0}} on 𝔭. We set

𝔭_{±} = {X ∈ 𝔭_{ℂ} | I_{z_{0}}(X) = ±iX},

and denote by P_{+}, P_{−} the ℂ-subgroups of G_{ℂ} corresponding to 𝔭_{+}, 𝔭_{−}, respectively; here (·)_{ℂ} denotes the complexification. Then we have

P_{+} ∩ K_{ℂ}P_{−} = {1}, G ⊂ P_{+}K_{ℂ}P_{−}, G ∩ K_{ℂ}P_{−} = K

(cf. [9, Lemma II.4.2], [6]). If g ∈ P_{+}K_{ℂ}P_{−} ⊂ G_{ℂ}, we denote by (g)_{+} ∈ P_{+}, (g)_{0} ∈ K_{ℂ} and (g)_{−} ∈ P_{−} the components of g such that

g = (g)_{+} · (g)_{0} · (g)_{−}.

Let (G_{ℂ} × 𝔭_{+})_{∗} denote the subset of G_{ℂ} × 𝔭_{+} consisting of elements (g, z) such that g · exp z ∈ P_{+}K_{ℂ}P_{−}, and set

J(g, z) = (g · exp z)_{0} ∈ K_{ℂ}.

for (g, z) ∈ (G_{ℂ} × 𝔭_{+})_{∗}. If we identify the Hermitian symmetric domain D with a subset of 𝔭_{+} using the Harish-Chandra embedding D ↪ 𝔭_{+} (cf. [9, §II.4]), then we have

G × D ⊂ (G_{ℂ} × 𝔭_{+})_{∗}.

Thus we obtain a map J : G × D → K_{ℂ} which satisfies the condition

J(g_{1}g_{2}, z) = J(g_{1}, g_{2}z) · J(g_{2}, z)  (5)

for g_{1}g_{2} ∈ G and z ∈ D. Let σ : K → GL(W) be a representation of K in W, and extend it to a representation of K_{ℂ}. From (5) we see that σ ∘J : G×D → GL(W) is an automorphy factor.

Definition 2.3. The automorphy factor J_{σ} = σ ∘ J : G × D → GL(W) is called the canonical automorphy factor of G associated to σ.

3. Homogeneous vector bundles

Let G be a Lie group, and let H be a closed subgroup of G. Let G/H denote the set {gH | g ∈ G} of left cosets modulo H, and let p : G → G/H be the natural projection p(g) = gH. Then G/H has a unique manifold structure such that p is smooth and for each gH ∈ G/H there is a neighborhood U of gH and a smooth map μ : U → G such that p ∘ μ = id_{U}. The quotient space G/H with such a manifold structure is called a homogeneous manifold.

Definition 3.1. Let M = G/H be a homogeneous manifold. A vector bundle E over M is called a homogeneous vector bundle if G acts on E on the left and the G-action satisfies the following conditions:

(i) If E_{x} denotes the fiber of E over x ∈ M, then g · E_{x} = E_{gx} for all x ∈ M and g ∈ G.

(ii) The map E_{x} → E_{gx} induced by g is linear for all x ∈ M and g ∈ G. We shall now construct a homogeneous vector bundle associated to a rep-

resentation of H. Let M = G/H be a homogeneous manifold, and let τ be a representation of H in a finite-dimensional complex vector space V . Then H acts on the product G × V on the right by

(g, v) · h = (gh, τ(h)^{−1}v)  (6)

for all g, g′ ∈ G and v ∈ V . We set

V = G × V/H,

where the quotient is taken with respect to the action of H on G × V given by (6). The natural projection G × V → G induces the map π : V → M which has the structure of a vector bundle with fiber V (cf. [10]). It can be shown that V is a homogeneous vector bundle over M. Let Γ_{0}(M, V) be the space of sections of V, that is, smooth maps s : M → V such that π ∘ s = 1_{M}.

Lemma 3.2. A section s ∈ Γ_{0}(M, V) of V can be identified with a smooth function f : G → V on G satisfying

f(gh) = τ(h)^{−1}f(g)

for all (g, v) ∈ G × V and h ∈ H.

Proof. See for example [10].

4. Poisson transforms

Let G, K, 𝔤, 𝔨 and 𝔭 be as in Section 2 with 𝔤 = 𝔨 + 𝔭, and let 𝔞 be a maximal abelian subspace of 𝔭. Then we obtain the Iwasawa decomposition

𝔤 = 𝔨 + 𝔞 + 𝔫

of 𝔤, where 𝔫 is a nilpotent subalgebra of 𝔤 (see e.g. [3, §V.2]). Let A and N be subgroups of G corresponding to the Lie algebras 𝔞 and 𝔫, respectively, so that we obtain the Iwasawa decomposition G = KAN of G. Let M be the centralizer of A in K, and set P = MAN, which is a minimal parabolic subgroup of G. We shall write any element g ∈ G in the form g = κ(g) · e^{H(g)} · n with κ(g) ∈ K, H(g) ∈ 𝔞 and n ∈ N.

Let σ be an irreducible representation of K in a finite-dimensional complex vector space W. Given an element λ ∈ 𝔞_{ℂ}^{∗} we define the representation σ_{λ,P} of P in W by

σ_{λ,P} (man) = e^{(−λ+ρ)H(a)}σ(m)  (7)

for all m ∈ M, a ∈ A and n ∈ N, where ρ is the half-sum of dim(𝔤_{α})α over the positive roots α of (𝔤, 𝔞).

Let V(σ) (resp. W(σ_{λ,P})) be the homogeneous vector bundle on G/K (resp. G/P) associated to the representations σ and σ_{λ,P}, respectively. Using Lemma 3.2 and (7), we see that each section ϕ ∈ Γ_{0}(G/P, W(σ_{λ,P})) of the bundle W(σ_{λ,P}) can be regarded as a function ϕ : G → W satisfying

ϕ(gman) = σ_{λ,P} (man)^{−1}ϕ(g) = e^{(λ−ρ)H(a)}σ(m)^{−1}ϕ(g)  (8)

for g ∈ G, m ∈ M, a ∈ A and n ∈ N. Similarly, a section ψ ∈ Γ_{0}(G/K, V(σ)) of V(σ) can be identified with a function ψ : G → W such that

ψ(gk) = σ(k)^{−1}ψ(g)  (9)

for all k ∈ K and g ∈ G. We now define the Poisson transform which may be regarded as the Lie-theoretic analogue of the classical Poisson integral (cf. [7], [2, §II.3.4], [5]).

Definition 4.1. Let ϕ be an element of Γ_{0}(G/P, W(σ_{λ,P})), that is, a smooth function ϕ : G → W satisfying (8). The Poisson transform P_{λ,P} ϕ of ϕ is a W -valued function on G given by

(P_{λ,P} ϕ)(g) = ∫_{K} σ(k)ϕ(gk)dk  (10)

for all g ∈ G.

Lemma 4.2. Let f is a continuous function on K that is right invariant under K ∩ M. Then we have

∫_{K} f(k)dk = ∫_{K} e^{−2ρH(g^{−1}k)}f(κ(g^{−1}k))dk

for g ∈ G.

Proof. See for example [3, p. 170].

Lemma 4.3. The Poisson transform P_{λ,P} ϕ of an element ϕ ∈ Γ_{0}(G/P, W(σ_{λ,P})) can be written in the form

(P_{λ,P} ϕ)(g) = ∫_{K} e^{−(λ+ρ)H(g^{−1}k)}σ(κ(g^{−1}k))ϕ(k)dk  (11)

for all g ∈ G.

Proof. If m ∈ K ∩ M, then by 8 we have

σ(km)ϕ(gkm) = σ(k)σ(m)σ(m)^{−1}ϕ(g)

for g ∈ G and k ∈ K. Hence, applying Lemma 4.2 to the function k ↦ σ(k)ϕ(gk), we obtain

∫_{K} σ(k)ϕ(gk)dk = ∫_{K} e^{−2ρH(g^{−1}k)}σ(κ(g^{−1}k))ϕ(gκ(g^{−1}k))dk

for g ∈ G. Let g^{−1}k = κ(g^{−1}k) · a_{1} · n_{1} with a_{1} ∈ A and n_{1} ∈ N. Then we have

gκ(g^{−1}k) = kn_{1}^{−1}a_{1}^{−1} = ka_{1}^{−1}n′

for some n′ ∈ N. Hence we see that

ϕ(gκ(g^{−1}k)) = ϕ(ka_{1}^{−1}n′) = e^{(λ−ρ)H(a_{1}^{−1})}ϕ(k)
= e^{−(λ−ρ)H(a_{1})}ϕ(k) = e^{−(λ−ρ)H(g^{−1}k)}ϕ(k).

Thus we have

∫_{K} σ(k)ϕ(gk)dk = ∫_{K} e^{−2ρH(g^{−1}k)}σ(κ(g^{−1}k))e^{−(λ−ρ)H(g^{−1}k)}ϕ(k)dk
= ∫_{K} e^{−(λ+ρ)H(g^{−1}k)}σ(κ(g^{−1}k))ϕ(k)dk,

and the lemma follows.

Let Γ be a torsion-free discrete subgroup of G as in Section 2, and consider the left-action of Γ on G × W defined by

γ · (g, w) = (γg, w)  (12)

for γ ∈ Γ and (g, w) ∈ G × W. Since this action commutes with the action given by (6) for τ = σ and V = W that was used for the construction of a homogeneous vector bundle, the homogeneous vector bundle V(σ) → D = G/K associated to σ induces the vector bundle V(Γ, σ) = Γ\V(σ) over the locally symmetric space X = Γ\D = Γ\G/K with fiber W. Similarly, we obtain the vector bundle W(Γ, σ_{λ,P}) = Γ\W(σ_{λ,P}) over the space Γ\G/P whose fiber is again W. Thus each section ϕ ∈ Γ_{0}(Γ\G/P, W(Γ, σ_{λ,P})) of W(Γ, σ_{λ,P}) is a Γ-invariant section of W(Γ, σ_{λ,P}), and it can be identified with a smooth function ϕ : G → W on G satisfying (8) and

ϕ(γg) = ϕ(g)  (13)

for all γ ∈ Γ and g ∈ G. In the same way, a section of V(Γ, σ) can be regarded as a smooth function ψ : G → W satisfying (9) and ψ(γg) = ψ(g) for γ ∈ Γ and g ∈ G.

Lemma 4.4. Let J_{σ} : G × D → GL(W) be the canonical automorphy factor given in Definition 2.3, and let z_{0} ∈ D = G/K be the point with Kz_{0} = z_{0}. Then the map [(g, w)] ↦ [(gz_{0}, w)] determines a canonical isomorphism of the vector bundles

V(Γ, σ) = Γ\G × W/K ≅ A(Γ, J_{σ}) = Γ\D × W.

Proof. See [6, Theorem II.4.1].

Now we state our main theorem in this paper, which implies that each section of the homogeneous vector bundle W(Γ, σ_{λ,P}) can be regarded as a lifting, via the Poisson transform map P_{λ,P}, of a holomorphic automorphic form on D for Γ of type J_{σ}, the canonical automorphy factor of G associated to σ.

Theorem 4.5. Let ϕ be an element of Γ_{0}(Γ\G/P, W(Γ, σ_{λ,P})) considered as a W -valued smooth function on G satisfying (8) and (13), and let J_{σ} be the canonical automorphy factor associated to σ described in Definition 2.3. Then the Poisson transform P_{λ,P} ϕ of ϕ is a holomorphic automorphic form on D of type J_{σ} for Γ.

Proof. Let ϕ be an element of Γ_{0}(Γ\G/P, W(Γ, σ_{λ,P})). Using Lemma 4.3, for each g ∈ G and k_{1} ∈ K we obtain

(P_{λ,P} ϕ)(gk_{1}) = ∫_{K} e^{−(λ+ρ)H((gk_{1})^{−1}k)}σ(κ((gk_{1})^{−1}k))ϕ(k)dk
= ∫_{K} e^{−(λ+ρ)H(k_{1}^{−1}g^{−1}k)}σ(κ(k_{1}^{−1}g^{−1}k))ϕ(k)dk
= ∫_{K} e^{−(λ+ρ)H(g^{−1}k)}σ(k_{1}^{−1}κ(g^{−1}k))ϕ(k)dk
= σ(k_{1}^{−1}) · ∫_{K} e^{−(λ+ρ)H(g^{−1}k)}σ(κ(g^{−1}k))ϕ(k)dk
= σ(k_{1})^{−1} · (P_{λ,P} ϕ)(g),

which implies that P_{λ,P} ϕ is a smooth section of V(σ). Since the function g ↦ e^{−(λ+ρ)H(g^{−1})}σ(κ(g^{−1})) is analytic and K is compact, using (11), we see that P_{λ,P} ϕ is also analytic. Furthermore, since D is assumed to have a G-invariant complex structure, it follows that P_{λ,P} ϕ is in fact a holomorphic section of the bundle V(σ). On the other hand, using (10), we obtain

(P_{λ,P} ϕ)(γg) = ∫_{K} σ(k)ϕ(γgk)dk = (P_{λ,P} ϕ)(g)

for all γ ∈ Γ and g ∈ G. Therefore P_{λ,P} ϕ is Γ-invariant, and hence it follows that P_{λ,P} ϕ is an element of Γ_{0}(X, V(Γ, σ)). Now the theorem follows by using this and the canonical isomorphism described in Lemma 4.4.

5. Symplectic groups

It is well-known that an arithmetic quotient of a Siegel upper half space can be considered as the parameter space of a family of polarized abelian varieties. In this section we show that sections of a certain vector bundle associated to a symplectic group can be regarded as liftings of some holomorphic forms on such a family of abelian varieties.

Throughout this section, let G_{0} be the symplectic group Sp(n, ℝ), and let K_{0} be a maximal compact subgroup of G_{0}. Then the associated Hermitian symmetric domain G_{0}/K_{0} can be identified with the Siegel upper half space H_{n} of degree n. Let Γ_{0} be a torsion-free subgroup of Sp(n, ℤ) of finite index. Consider the semidirect product Γ_{0} ⋉ (ℤ^{n} × ℤ^{n})^{m} whose multiplication operation is given by

(g, (u_{1}, v_{1}), …, (u_{m}, v_{m})) · (g′, (u′_{1}, v′_{1}), …, (u′_{m}, v′_{m}))
= (gg′, (u_{1}, v_{1})g′ + (u′_{1}, v′_{1}), …, (u_{m}, v_{m})g′ + (u′_{m}, v′_{m}))

for all g, g′ ∈ Γ_{0} ⊂ Sp(n, ℝ) and (u_{j}, v_{j}) ∈ ℤ^{n} × ℤ^{n} for 1 ≤ j ≤ m. Then it acts on the space H_{n} × ℂ^{mn} by

(( \begin{matrix}a & b \\ c & d \end{matrix}), (u_{1}, v_{1}), …, (u_{m}, v_{m})) · (z, ζ_{1}, …, ζ_{m})
= ((az + b)(cz + d)^{−1}, (u_{1}z + v_{1} + ζ_{1})(cz + d)^{−1}, …
…, (u_{m}z + v_{m} + ζ_{m})(cz + d)^{−1})

for all ζ_{1}, …, ζ_{m} ∈ ℂ^{n}, z ∈ H_{n}, ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}, and (u_{j}, v_{j}) ∈ ℤ^{n}×ℤ^{n} for 1 ≤ j ≤ m.

We denote by Y_{0}^{m} the associated quotient space, that is,

Y_{0}^{m} = Γ_{0} ⋉ (ℤ^{n} × ℤ^{n})^{m}\H_{n} × ℂ^{mn}.  (14)

Then the natural projection H_{n} × ℂ^{mn} → H_{n} induces a map π_{0} : Y_{0}^{m} → X_{0} = Γ_{0}\H_{n}, which gives Y_{0}^{m} the structure of a fiber bundle over X_{0} whose fiber is isomorphic to the complex torus

(ℤ^{n} × ℤ^{n})^{m}\ℂ^{mn} ≅ (ℤ × ℤ\ℂ)^{mn}.

By (14) a holomorphic differential form on Y_{0}^{m} can be regarded as a holomorphic form on H_{n} × ℂ^{mn} that is invariant under the action of Γ_{0} ⋉ (ℤ^{n} × ℤ^{n})^{m}.

Given a nonnegative integer k, a holomorphic function f : H_{n} → ℂ is a Siegel modular form of weight k for Γ_{0} if it satisfies

f((az + b)(cz + d)^{−1}) = (cz + d)^{k}f(z)

for all ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0} ⊂ Sp(n, ℝ). We shall denote by M_{k}(Γ_{0}) the space of Siegel modular forms of weight k for Γ_{0}.

Proposition 5.1. Let n ≥ 2 and ⟨n⟩ = n(n + 1)/2. There is a canonical isomorphism between the space M_{m+n+1}(Γ_{0}) of Siegel modular forms for Γ_{0} of weight m + n + 1 and the space H^{0}(Y_{0}^{m}, Ω^{⟨n⟩+mn}) of holomorphic forms on Y_{0}^{m} of degree ⟨n⟩ + mn.

Proof. The proof can be sketched as follows. Let

z = (z_{1}, …, z_{⟨n⟩}), ζ = (ζ_{1}, …, ζ_{m})

with ζ_{j} = (ζ_{j}^{1}, …, ζ_{j}^{n}) for 1 ≤ j ≤ m be the canonical coordinate systems for H_{n}, ℂ^{mn}, respectively. Let f be a Siegel modular form of weight m + n + 1 for Γ_{0}, and define the associated holomorphic form Φ_{f} on H_{n} × ℂ^{mn} by

Φ_{f}(z, ζ) = f(z)dz ∧ dζ_{1} ∧ ⋯ ∧ dζ_{m},

where dz = dz_{1} ∧ ⋯ ∧ dz_{⟨n⟩} and dζ_{j} = dζ_{j}^{1} ∧ ⋯ ∧ ζ_{j}^{n} for 1 ≤ j ≤ m. Then it can be shown that Φ_{f} is invariant under the action of Γ_{0} ⋉ (ℤ^{n} × ℤ^{n})^{m}, and therefore Φ_{f} is a holomorphic form on Y_{0}^{m} of degree ⟨n⟩ + mn. On the other hand, given a (Γ_{0} ⋉ (ℤ^{n} × ℤ^{n})^{m})-invariant holomorphic form

Φ(z, ζ) = h(z, ζ)dz ∧ dζ_{1} ∧ ⋯ ∧ dζ_{m}

on H_{n} × ℂ^{mn} of degree ⟨n⟩ + mn, it can be shown that h(z, ζ) is a function of z only and that it is a Siegel modular form for Γ_{0} of degree m + n + 1 (see e.g. the arguments in the proof of [4, Theorem 4.2]). Thus the map f ↦ Φ_{f} determines the desired isomorphism.

Let G_{0} = K_{0}A_{0}N_{0} be the Iwasawa decomposition of G_{0} described in Section 4, and let P_{0} = M_{0}A_{0}N_{0} be the corresponding minimal parabolic subgroup of G_{0}, where M_{0} is the centralizer of A_{0} in K_{0}. Let σ_{0} : K_{0} → GL_{1}(ℂ) = ℂ^{×} be the complex one-dimensional representation of K_{0} given by

σ_{0}(k) = det(k)^{m+n+1}

for all k ∈ K_{0}. Let λ_{0} be an element of (𝔞_{0})_{ℂ}^{∗} with 𝔞_{0} = Lie A_{0}, and let

P_{λ_{0},P_{0}} : Γ_{0}(Γ_{0}\G_{0}/P_{0}, W(Γ_{0}, (σ_{0})_{λ_{0},P_{0}})) → Γ_{0}(Γ_{0}\H_{n}, V(Γ_{0}, σ_{0}))

be the associated Poisson transform.

Theorem 5.2. Let ϕ be an element of Γ_{0}(Γ_{0}\G_{0}/P_{0}, W(Γ_{0}, (σ_{0})_{λ_{0},P_{0}})). Then the Poisson transform P_{λ_{0},P_{0}}ϕ of ϕ is a Siegel modular form of weight m + n + 1 for Γ_{0}, and therefore by Proposition 5.1 it can be identified with a holomorphic form of degree ⟨n⟩+mn on the family Y_{0}^{m} of abelian varieties parametrized by the Siegel modular variety X_{0} = Γ_{0}\H_{n}.

Proof. By Theorem 4.5 the Poisson transform P_{λ_{0},P_{0}}ϕ of ϕ is a holomorphic automorphic form on H_{n} of type σ_{0} ∘ J_{0} for Γ_{0}, where J_{0} : G_{0} × H_{n} → K_{0} is the canonical automorphy factor of G_{0}. However, for the symplectic group G_{0} = Sp(n, ℝ) the canonical automorphy factor is given by

J_{0}(g, z) = cz + d

for g = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ G_{0} and z ∈ H_{n} (cf. [9]). Thus we have

(P_{λ_{0},P_{0}}ϕ)(γz) = σ_{0}(J_{0}(γ, z))(P_{λ_{0},P_{0}}ϕ)(z)
= det(cz + d)^{m+n+1}(P_{λ_{0},P_{0}}ϕ)(z)

for all γ ∈ Γ_{0} and z ∈ H_{n}, and hence P_{λ_{0},P_{0}}ϕ is a Siegel modular form for Γ_{0} of weight m + n + 1. Thus the theorem follows from this and Proposition 5.1.

References

[1] Borel, A., Introduction to automorphic forms, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence, 1966, 199–210.

[2] Helgason, S., “Geometric analysis on symmetric spaces, ” Amer. Math. Soc., Providence, 1994.

[3] Knapp, A., “Representation theory of semisimple groups: An overview based on examples,” Princeton University Press, Princeton, 1986.

[4] Lee, M. H., Mixed automorphic vector bundles on Shimura varieties, Pacific J. Math. 173 (1996), 105–126.

[5] Minemura, K., Invariant differential operators and spherical sections on a homogeneous vector bundle, Tokyo J. Math. 15 (1992), 231–245.

[6] Murakami, S., “Cohomology of vector-valued forms on symmetric spaces,” Univ. of Chicago, Chicago, 1966.

[7] Okamoto, K., Harmonic analysis on homogeneous vector bundles, Lecture Notes in Math. 266, Springer-Verlag, Heidelberg, 1972, 255–271.

[8] Rudin, W., “Real and complex analysis,” 3^{rd} ed., McGraw-Hill, New York, 1987.

[9] Satake, I., “Algebraic structures of symmetric domains,” Princeton Univ. Press, Princeton, 1980.

[10] Wallach, N., “Harmonic analysis in homogeneous spaces,” Marcel Dekker, New York, 1973.

[11] Wallach, N., “Real reductive groups II,” Academic Press, New York, 1992.

Min Ho Lee Department of Mathematics University of Northern Iowa Cedar Falls, Iowa 50614 U. S. A. lee@math.uni.edu Hyo Chul Myung Korea Institute for Advanced Study and KAIST, 207-43 Chunryanri-dong, Dongdaemoon-ku Seoul 130-012, Korea hm@kias.re.kr

Received August 5, 1998 and in final form April 26, 1999