Journal of Lie Theory Volume 11 (2001) 545–557 ◯^{C}2001 Heldermann Verlag

Jacobi Forms on Symmetric Domains and Torus Bundles over Abelian Schemes

Min Ho Lee

Communicated by M. Moskowitz

Abstract. We introduce Jacobi forms on Hermitian symmetric domains using automorphy factors associated to torus bundles over ableian schemes. We discuss families of modular forms determined by such Jacobi forms and prove that these Jacobi forms reduce to the usual Jacobi forms of several variables when the Hermitian symmetric domain is a Siegel upper half space.

1. Introduction

Jacobi forms on the Poincaré upper half plane, which were developed systematically by Eichler and Zagier [5], occur naturally in various contexts in number theory, and they have been studied extensively in recent years (see e.g. [3], [6]). Jacobi forms of several variables have been considered mainly on Siegel upper half spaces (cf. [15], [16]) and a few other special types of Hermitian symmetric domains (see [4], [7]). The goal of this paper is to introduce Jacobi forms on Hermitian symmetric domains which allow equivariant holomorphic maps into Siegel upper half spaces.

Torus bundles over abelian schemes occur naturally in the study of compactifications of locally symmetric spaces. Let D be a Hermitian symmetric domain, and let Γ ⊂ Aut(D) be a discrete subgroup of arithmetic type. Then the locally symmetric space Γ\D can be compactified by the method of Baily and Borel (cf. [2]). Thus Γ\D can be canonically embedded as a Zariski open subset in a projective variety \overline{Γ\D}. Unfortunately, the Baily-Borel compactification \overline{Γ\D} may have complicated singularities along the boundary, which prevents us from using it to obtain information on automorphic forms on D, for example. On the other hand the method of toroidal compactification provides smooth compactifications of locally symmetric spaces (see [1], [11], [12]). Toroidal compactifications are defined using torus embeddings, and there are in general different toroidal compactifications associated to different systems of cone decompositions involved. The toroidal compactification of Γ\D dominates the Baily-Borel compactification \overline{Γ\D}, and it possesses a natural stratification compatible with that of the Baily-Borel compactification. The strata of toroidal compactifications are torus bundles over abelian

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag schemes. Abelian schemes here are families of abelian varieties parametrized by the locally symmetric space Γ\D, which play an important role in number theory and algebraic geometry.

In [14] Satake constructed explicitly abelian schemes over a locally symmetric space of the form Γ\D and torus bundles over such abelian schemes, and he obtained results on the Chern classes and the projectivity of those torus bundles. His construction involves certain automorphy factors on the product D × V_{+} of D and a complex vector space V_{+}. In fact, such automorphy factors also play a crucial role for the proof of the algebracity of abelian schemes over which torus bundles reside (see [13, Chapter 4]). In this paper we define Jacobi forms using such automorphy factors and discuss families of modular forms on Hermitian symmetric domains associated to such Jacobi forms. We also prove that these Jacobi forms reduce to the usual Jacobi forms of several variables investigated by Ziegler [16] when the Hermitian symmetric domain is a Siegel upper half space.

2. Torus bundles over abelian schemes

In this section we describe the construction of a torus bundle over an abelian scheme following Satake [14]. Let V be a real vector space of dimension 2m > 0 defined over ℚ, and let Alt(V) denote the space of all alternating bilinear forms on V .

Definition 2.1. A Hermitian structure on V is a pair (α, I) consisting of all elements α ∈ Alt(V) and I ∈ GL(V) with I^{2} = −1_{V} such that the bilinear map

V × V → ℝ, (v, v′) ↦ α(v, Iv′)

is symmetric and positive definite. We shall denote by Herm(V) the space of all Hermitian structures on V .

Given α ∈ Alt(V), we denote by Sp(V, α) the symplectic group associated to α, that is,

Sp(V, α) = {g ∈ GL(V) | α(gv, gv′) = α(v, v′) for all v, v′ ∈ V }.

We fix a Hermitian structure (α_{0}, I_{0}) ∈ Herm(V) on V with α_{0} ∈ Alt(V) defined over ℚ. Then the Hermitian symmetric domain associated to the symplectic group Sp(V, α_{0}) can be identified with the space

H = H(V, α_{0}) = {I ∈ GL(V) | (α_{0}, I) ∈ Herm(V)}  (1)

on which Sp(V, α_{0}) acts by

(g, I) ↦ gIg^{−1}

for all g ∈ Sp(V, α_{0}) and I ∈ H(V, α_{0}).

Let G be a semisimple Lie group of Hermitian type defined over ℚ. Thus G = 𝔾(ℝ) for some semisimple linear algebraic group 𝔾 defined over ℚ, and, if K is a maximal compact subgroup of G, the associated Riemannian symmetric space D = G/K is a Hermitian symmetric domain. We assume that there are a homomorphism ρ : G → Sp(V, α_{0}) of Lie groups defined over ℚ and a holomorphic map τ : D → H such that I_{0} ∈ τ(D) and

τ(gz) = ρ(g)τ(z)

for all g ∈ G and z ∈ D. We set

U^{∗} = {α ∈ Alt(V) | ρ(G) ⊂ Sp(V, α)}.

Then U^{∗} is a subspace of Alt(V) defined over ℚ, and we have α_{0} ∈ U^{∗}. Let U = (U^{∗})^{∗} be the dual space of U^{∗}. Then we obtain an alternating bilinear map A : V × V → U defined over ℚ by

A(v, v′)(α) = α(v, v′)  (2)

for all α ∈ U^{∗} and v, v′ ∈ V .

Following Satake (cf. [13, §III.5], [14]), we consider the generalized Heisenberg group ℍ associated to A consisting of all elements of V × U together with a multiplication operation given by

(v, u) · (v′, u′) = (v + v′, u + u′ − A(v, v′)/2)  (3)

for all (v, u), (v′, u′) ∈ V × U. Then the group G operates on ℍ by

g · (v, u) = (ρ(g)v, u)  (4)

for all g ∈ G and (v, u) ∈ ℍ, and we can form the semidirect product G ⋉ ℍ with respect to this operation. Thus G ⋉ ℍ consists of the elements (g, v, u) of G × V × U whose multiplication operation is given by

(g, v, u) · (g′, v′, u′) = (gg′, (v, u) · (ρ(g)v′, u′))  (5)
= (gg′, v + ρ(g)v′, u + u′ − A(v, ρ(g)v′)/2).

Let I ∈ GL(V) be an element of H = H(V, α_{0}). We extend the complex structure I on V linearly to the complexification V_{ℂ} = V ⊗_{ℝ} ℂ of V , and set

V_{+}(I) = {v ∈ V_{ℂ} | Iv = iv}, V_{−}(I) = {v ∈ V_{ℂ} | Iv = −iv}.

When I is equal to I_{0} considered above, we shall write V_{+} = V_{+}(I_{0}) and V_{−} = V_{−}(I_{0}).

Lemma 2.2. To each complex structure I on V there corresponds a unique complex linear map ξ_{I} : V_{−} → V_{+} satisfying

V_{−}(I) = (1 + ξ_{I})V_{−}.  (6)

Furthermore, the map I ↦ ξ_{I} determines a bijection between H = H(V, α_{0}) and the set of ℂ-linear maps ξ : V_{−} → V_{+} such that 1 − ξ\overline{ξ} is positive definite and ξ^{t} = ξ, where the transpose is taken with respect to the bilinear map α_{0}.

Proof. This follows from [13, Lemma II.7.2].

If ξ is an element of Hom_{ℂ}(V_{−}, V_{+}) in Lemma 2.2 corresponding to an element I ∈ H, then we shall write I = I_{ξ}.

Lemma 2.3. For each ξ ∈ Hom_{ℂ}(V_{−}, V_{+}) with I_{ξ} ∈ H the map

Ξ_{ξ} : (V, I_{ξ}) → V_{+}, v ↦ v_{+} − ξv_{−}

determines an isomorphism of vector spaces over ℂ, where v = v_{+} + v_{−} ∈ V ⊂ V_{ℂ} with v_{±} ∈ V_{±}.

Proof. Given ξ ∈ Hom_{ℂ}(V_{−}, V_{+}) with I_{ξ} ∈ H, the map Ξ_{ξ} is linear. Since dim_{ℂ} V = dim_{ℂ} V_{+}, it suffices to show that Ker Ξ_{ξ} = {0}. Suppose that v ∈ V satisfies

Ξ_{ξ}(v) = v_{+} − ξv_{−} = 0.  (7)

By (6) there exists v′ ∈ V_{−}(I_{ξ}) such that

v′ = v_{−} + ξv_{−}.  (8)

¿From (7) and (8) we see that v′ = v_{+} + v_{−} = v ∈ V . Since V ∩ V_{−}(I_{ξ}) = {0}, we have v′ = v = 0; hence it follows that Ker Ξ_{ξ} = {0}

By Lemma 2.2 we may identify the symmetric domain H in (1) with the set of elements z ∈ Hom_{ℂ}(V_{−}, V_{+}) with z^{t} = z and 1 − z\overline{z} ≫ 0. Then the symplectic group Sp(V, α_{0}) operates on H in (1) by

g(z) = (az + b)(cz + d)^{−1}

for all z ∈ H and

g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp(V, α_{0});

here we wrote g ∈ Sp(V, α_{0}) as a 2 × 2 block matrix with respect to the decomposition V_{ℂ} = V_{+} + V_{−}. The canonical automorphy factor J of Sp(V, α_{0}) is the map on Sp(V, α_{0}) × H with values in GL(V_{ℂ}) given by

J(g, z) = (\begin{matrix}J_{+}(g, z) & 0 \\ 0 & J_{−}(g, z) \end{matrix})  (9)

for all g ∈ Sp(V, α_{0}) and z ∈ H, where

J_{+}(g, z) = a − g(z)c, J_{−}(g, z) = cz + d.  (10)

If τ : D → H is the holomorphic map equivariant with respect to the homomorphism ρ : G → Sp(V, α_{0}) as before and if A is as in (2), then we set

J ((g, r, s), (z, v)) = s − A(ρ(g)r, J_{+}(ρ(g), τ(z))r_{z})/2  (11)
− A(ρ(g)v, J_{+}(ρ(g), τ(z))v)/2
− A(ρ(g)r, J_{+}(ρ(g), τ(z))v)

for g ∈ G, (r, s) ∈ ℍ and (z, v) ∈ D × V_{+}, where

r_{z} = r_{+} − zr_{−} ∈ V_{+}  (12)

for r = (r_{+}, r_{−}) ∈ V ⊂ V_{ℂ} = V_{+} ⊕ V_{−}. Then it is known (see [13, §III.5]) that the group G ⋉ ℍ operates on D × V_{+} × U_{ℂ} by

(g, r, s) · (z, v, u) = (gz, J_{+}(ρ(g), τ(z))(v + r_{z}), u + J ((g, r, s), (z, v))). (13)

By restricting this action to D × V_{+} we obtain the action of G ⋉ ℍ on D × V_{+} given by

(g, r, s) · (z, v) = (gz, J_{+}(ρ(g), τ(z))(v + r_{z}))

for g ∈ G, (r, s) ∈ ℍ and (z, v) ∈ D × V_{+}.

Proposition 2.4. The map J : (G ⋉ ℍ) × (D × V_{+}) → U_{ℂ} given by (11) satisfies

J ((g′, r′, s′) · (g, r, s), (z, v)) = J ((g′, r′, s′), (g, r, s) · (z, v)) + J ((g, r, s), (z, v))

for all (g′, r′, s′), (g, r, s) ∈ G ⋉ ℍ and (z, v) ∈ D × V_{+}.

Proof. Given (g′, r′, s′), (g, r, s) ∈ G ⋉ ℍ and (z, v, u) ∈ D × V_{+} × U_{ℂ}, we set

(z_{1}, v_{1}, u_{1}) = (g, r, s) · (z, v, u)
(z_{2}, v_{2}, u_{2}) = (g′, r′, s′) · (z_{1}, v_{1}, u_{1})
(z_{3}, v_{3}, u_{3}) = ((g′, r′, s′) · (g, r, s)) · (z, v, u).

Then by (13) we see that

u_{1} = u + J ((g, r, s), (z, v))
u_{2} = u_{1} + J ((g′, r′, s′), (g, r, s) · (z, v))
u_{3} = u + J ((g′, r′, s′) · (g, r, s), (z, v)).

Since G⋉ℍ acts on D ×V_{+} ×U_{ℂ}, we have u_{2} = u_{3}; hence the proposition follows.

Let L_{ℍ} be an arithmetic subgroup of ℍ, and set

L = p_{V} (L_{ℍ}), L_{U} = p_{U}(L_{ℍ}),  (14)

where p_{V} : ℍ → V and p_{U} : ℍ → U are the natural projection maps. Then L and L_{U} are lattices in V and U, respectively, and we have L = L_{ℍ}/L_{U}. Given elements l, l′ ∈ L, we have (l, 0), (l′, 0) ∈ L_{ℍ}; hence by (3) we see that

(l, 0) · (l′, 0) = (l + l′, −A(l, l′)/2) ∈ L_{ℍ}.

Since (l + l′, 0)^{−1} = (−l − l′, 0) ∈ L_{ℍ}, we have

(l + l′, −A(l, l′)/2) · (−l − l′, 0) = (0, −A(l, l′)/2) ∈ L_{ℍ}.

Thus it follows that A(L, L) ⊂ L_{U}. Let γ be a torsion-free arithmetic subgroup of G. Using the isomorphism

Γ ⋉ L_{ℍ}/L_{U} ≅ Γ ⋉ L,

we see that the action of G ⋉ ℍ on D × V_{+} × U_{ℂ} induces actions of the discrete groups Γ⋉L_{ℍ}, Γ⋉L and Γ on the spaces D×V_{+}×U_{ℂ}, D×V_{+} and D, respectively. We denote the associated quotient spaces by

W = Γ ⋉ L_{ℍ}\D × V_{+} × U_{ℂ}, Y = Γ ⋉ L\D × V_{+}, X = Γ\D.

Then each of the spaces W, Y and X has a natural structure of a complex manifold, and there are natural projections

W ⟶^{π_{1}} Y ⟶^{π_{2}} X.

The complex manifold Y is an abelian scheme over the arithmetic variety X, called a Kuga fiber variety, whose fiber V_{+}/L has the structure of a polarized abelian variety (see e.g. [8], [9], [10], [13]), and W is a torus bundle over the Kuga fiber variety Y whose fiber is isomorphic to the complex torus U_{ℂ}/L_{U}.

3. Modular forms and Jacobi forms

Let τ : D → H be the holomorphic map that is equivariant with respect to the homomorphism ρ : G → Sp(V, α_{0}) considered in Section 2.. In this section we define Jacobi forms and modular forms on Hermitian symmetric domains using the map J in (11) and construct modular forms on D associated to Jacobi forms.

Let K_{ℂ}^{Sp} be the subgroup of GL(V_{ℂ}) given by

K_{ℂ}^{Sp} = \{(\begin{matrix}p & 0 \\ 0 & q \end{matrix}) ∈ GL(V_{ℂ}) | p = (q^{t})^{−1}\},  (15)

where the matrix is written with respect to the decomposition V_{ℂ} = V_{+}+V_{−}. Then it is known that K_{ℂ}^{Sp} is the complexification of a maximal compact subgroup K^{Sp} of Sp(V, α_{0}). Let J, J_{+} and J_{−} be as in (9), and let g = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp(V, α_{0}) and ζ ∈ H. Since the matrix gζ = (aζ + b)(cζ + d)^{−1} ∈ H is symmetric, by (10) we obtain

J_{+}(g, ζ) = a − (aζ + b)(cζ + d)^{−1}c  (16)
= a − (ζc^{t} + d^{t})^{−1}(ζa^{t} + b^{t})c
= (ζc^{t} + d^{t})^{−1}(ζ(c^{t}a − a^{t}c) + d^{t}a − b^{t}c)
= (ζc^{t} + d^{t})^{−1} = (J_{−}(g, ζ)^{t})^{−1}.

Hence it follows that J_{+}(g, ζ) ∈ K_{ℂ}^{Sp}.

Let σ : K_{ℂ}^{Sp} → GL(Z) be a representation of K_{ℂ}^{Sp} in a finite-dimensional complex vector space Z. Given a holomorphic map f : D → Z, we set

(f |_{σ} γ)(z) = σ(J(ρ(γ), τ(z)))^{−1}f(γz)  (17)

for γ ∈ G and z ∈ D. Then it can be shown that

f |_{σ} γ |_{σ} γ′ = f |_{σ} γγ′,

for γ, γ′ ∈ Γ. Let Γ ⊂ G be a torsion-free arithmetic subgroup as before.

Definition 3.1. A holomorphic map f : D → Z is a modular form for Γ associated to σ if it satisfies

f |_{σ} γ = f

for all γ ∈ Γ.

Let χ : U_{ℂ} → ℂ^{×} be a character of U_{ℂ} with χ(s) = 1 for all s ∈ L_{U}, where L_{U} is as in (14). Then by Proposition 2.4 we see that χ∘J : (G⋉ℍ)×(D×V_{+}) → ℂ is an automorphy factor, that is, it satisfies

(χ ∘ J)(g̃g̃′, z̃) = (χ ∘ J)(g̃, g̃′z̃) · (χ ∘ J)(g̃′, z̃)

for all g̃, g̃′ ∈ G ⋉ ℍ and z̃ ∈ D × V_{+}. Given a holomorphic map F : D × V_{+} → Z, we set

(F |_{σ,χ} (γ, r, s))(z, w) = χ(−J ((γ, r, s), (z, w))) · σ(J(ρ(γ), τ(z)))^{−1}  (18)
× F (γz, J_{+}(ρ(γ), τ(z))(w + r_{z}))

for all (z, w) ∈ D × V_{+} and (γ, r, s) ∈ G ⋉ ℍ, where r_{z} is as in (12). Using the fact that χ ∘ J is an automorphy factor, we see that

F |_{σ,χ} (γ, r, s) |_{σ,χ} (γ′, r′, s′) = F |_{σ,χ} ((γ, r, s) · (γ′, r′, s′))

for γ, γ′ ∈ G and (r, s), (r′, s′) ∈ ℍ.

Definition 3.2. A holomorphic map F : D × V_{+} → Z is a Jacobi form for Γ ⋉ L_{ℍ} associated to σ and χ if it satisfies

F |_{σ,χ} (γ, r, s) = F  (19)

for all (γ, r, s) ∈ Γ ⋉ L_{ℍ}.

We can obtain a family of modular forms on D parametrized by the rational points of ℍ as is described in the next theorem.

Theorem 3.3. Let F : D × V_{+} → Z be a Jacobi form for Γ ⋉ L_{ℍ} associated to σ and χ, and let (r, s) ∈ ℍ_{ℚ} = V_{ℚ} × U_{ℚ} with r = (r_{+}, r_{−}) ∈ V ⊂ V_{ℂ} = V_{+} ⊕ V_{−}. If A : V × V → U is the bilinear map in (2), we set

f(z) = χ(−A(r, zr_{−})/2) · F (z, r_{z})  (20)

for all z ∈ D. Then f is a modular form for an arithmetic subgroup Γ′ ⊂ Γ of G associated to σ.

Proof. Let ε be the identity element of G, and let (r, s) ∈ ℍ_{ℚ}. Then, for each z ∈ D, J(ρ(ε), τ(z)) and J_{+}(ρ(ε), τ(z)) are identity matrices, and in particular σ(J(ρ(ε), τ(z))) is the identity element in GL(Z). Thus, using (11) and (18), we see that

J ((ε, r, s), (z, 0)) = s − A(r, r_{z})/2 − A(v, v)/2 − A(r, 0) = s − A(r, r_{z})/2,
(F |_{σ,χ} (ε, r, s))(z, 0) = χ(−s + A(r, r_{z})/2)F (z, r_{z})

for all z ∈ D. Hence by (20) we obtain

f(z) = χ(s − A(r, r_{+} − zr_{−})/2 − A(r, zr_{−})/2)(F |_{σ,χ} (ε, r, s))(z, 0)
= χ(s − A(r, r_{+})/2)(F |_{σ,χ} (ε, r, s))(z, 0).

Thus it suffices to show that the function F_{(r,s)} : D → Z given by

F_{(r,s)}(z) = (F |_{σ,χ} (ε, r, s))(z, 0)

is a modular form for an arithmetic subgroup Γ′ ⊂ Γ associated to σ. Given an element γ ∈ Γ, by (17) we have

(F_{(r,s)} |_{σ} γ)(z) = σ(J(ρ(γ), τ(z)))^{−1}F_{(r,s)}(γz)
= σ(J(ρ(γ), τ(z)))^{−1}(F |_{σ,χ} (ε, r, s))(γz, 0)
= ((F |_{σ,χ} (ε, r, s)) |_{σ,χ} (γ, 0, 0))(z, 0)

for all z ∈ D. However, by (5) we see that

(ε, r, s) · (γ, 0, 0) = (γ, r, s) = (γ, 0, 0) · (ε, ρ(γ)^{−1}r, s).

Using this and the fact that F is a Jacobi form for Γ, we obtain

(F_{(r,s)} |_{σ} γ)(z) = ((F |_{σ,χ} (γ, 0, 0)) |_{σ,χ} (ε, ρ(γ)^{−1}r, s))(z, 0)
= (F |_{σ,χ} (ε, ρ(γ)^{−1}r, s)(z, 0)
= F_{γ^{−1}·(r,s)}(z),

where γ^{−1} · (r, s) = (ρ(γ)^{−1}r, s) by (4). Let Γ_{(r,s)} be the subgroup of Γ consisting of the elements γ ∈ Γ satisfying

γ^{−1} · (r, s) = (0, s_{1}) · (r_{2}, s_{2}) · (r, s)

with s_{1} ∈ L_{U} and (r_{2}, s_{2}) ∈ L_{ℍ}. Then Γ_{(r,s)} is an arithmetic subgroup, and, if γ ∈ Γ_{(r,s)}, we have

(F_{(r,s)} |_{σ} γ)(z) = F_{γ^{−1}·(r,s)}(z) = F_{(0,s_{1})·(r_{2},s_{2})·(r,s)}(z)
= (F |_{σ,χ} (ε, 0, s_{1}) |_{σ,χ} (ε, r_{2}, s_{2}) |_{σ,χ} (ε, r, s))(z, 0)
= (F |_{σ,χ} (ε, r, s)(z, 0) = F_{(r,s)}(z)

for all z ∈ D. Hence F_{(r,s)} is a modular form for the arithmetic subgroup Γ_{(r,s)} ⊂ Γ associated to σ, and therefore the proof of the theorem is complete.

4. Jacobi forms for symplectic groups

In this section we specialize the equivariant pair (ρ, τ) considered in Section 2. to the case where G is a symplectic group and ρ and τ are identity maps. We then prove that the associated Jacobi forms in the sense of Definition 3.2 coincide with the usual Jacobi forms of several variables (cf. [16]).

We first describe the definition of Jacobi forms of several variables introduced by Ziegler (see [16] for details). If a and b are positive integers and R is a ring, we shall denote by R^{(a,b)} the set of a × b matrices over R. Let ℍ_{ℝ}^{n,j} be the Heisenberg group consisting of the elements

(λ, μ, κ) ∈ ℝ^{(j,n)} × ℝ^{(j,n)} × ℝ^{(j,j)}

such that the matrix κ + μλ^{t} is symmetric. The multiplication operation in ℍ_{ℝ}^{n,j} is given by

(λ, μ, κ) · (λ′, μ′, κ′) = (λ + λ′, μ + μ, κ + κ′ + λμ′^{t} − μλ′^{t}),

and the symplectic group Sp(n, ℝ) acts on ℍ_{ℝ}^{n,j} on the right by

(λ, μ, κ) · M = ((λ, μ) · M, κ)

for M ∈ Sp(n, ℝ) and (λ, μ, κ) ∈ ℍ_{ℝ}^{n,j}. This action enables us to define the Jacobi group

G_{ℝ}^{n,j} = Sp(n, ℝ) ⋉ ℍ_{ℝ}^{n,j}

whose multiplication is given by

(M, (λ, μ, κ)) · (M′, (λ′, μ′, κ′)) = (MM′, ((λ, μ, κ) · M′) · ζ′)

= (MM′, (λ̃ + λ′, μ̃ + μ′, κ + κ′ + λ̃μ′^{t} − μ̃λ′^{t})),

where (λ̃, μ̃) = (λ, μ) · M′. Then G_{ℝ}^{n,j} acts on H_{n} × ℂ^{(j,n)} by

(M, (λ, μ, κ)) · (z, w) = ((az + b)(cz + d)^{−1}, (w + λz + μ)(cz + d)^{−1})

for M = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp(n, ℝ).

Let η : GL(n, ℂ) → GL(Z) be a representation of GL(n, ℂ) in a finitedimensional complex vector space Z, and let 𝔐 = (m_{αβ}) ∈ ℝ^{(j,j)} be a positive symmetric matrix with 2m_{αβ}, m_{αα} ∈ ℤ for 1 ≤ α, β ≤ j. Given M = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp(n, ℝ), ξ = (λ, μ, κ) ∈ ℍ_{ℝ}^{n,j} and a holomorphic map Φ : H_{n} × ℂ^{(j,n)} → Z, we set

(Φ |_{η,𝔐} M)(z, w) = η(cz + d)^{−1} exp{−2πi Tr(𝔐w(cz + d)^{−1}cw^{t})}
× Φ((az + b)(cz + d)^{−1}, (w + λz + μ)(cz + d)^{−1}),
(Φ |_{𝔐} ξ)(z, w) = exp{−2πi Tr(𝔐(λzλ^{t} + 2λw^{t} + μλ^{t} + κ))}
× Φ(z, w + λz + μ),

for all (z, w) ∈ H_{n} × ℂ^{(j,n)}, where Tr denotes the trace. Let ℍ_{ℤ}^{n,j} be a discrete subgroup of ℍ_{ℝ}^{n,j} given by

ℍ_{ℤ}^{n,j} = {(λ, μ, κ) ∈ ℍ_{ℝ}^{n,j} | λ, μ ∈ ℤ^{(j,n)}, κ ∈ ℤ^{(j,j)}}  (21)

and let Γ ⊂ Sp(n, ℤ) be a subgroup of finite index.

Definition 4.1. Let η : GL(n, ℂ) → GL(Z) and 𝔐 ∈ ℝ^{(j,j)} be as above. A holomorphic map Φ : H_{n} × ℂ^{(j,n)} → Z is a Jacobi form of index 𝔐 with respect to η for Γ if it satisfies

Φ |_{η,𝔐} M = Φ, Φ |_{𝔐} ξ = Φ

for all M ∈ Γ and ξ ∈ ℍ_{ℤ}^{n,j}.

Thus, if Φ is a Jacobi form of index 𝔐 with respect to η for Γ and if

M = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ and ξ = (λ, μ, κ) ∈ ℍ_{ℤ}^{n,j}, we have

Φ((az + b)(cz+d)^{−1}, (w + λz + μ)(cz + d)^{−1})  (22)
= exp{2πi Tr(𝔐((w + λz + μ)(cz + d)^{−1}c(w^{t} + zλ^{t} + μ^{t})
− λw^{t} − (w + λz + μ)λ^{t}))} · η(cz + d) · Φ(z, w)

for all (z, w) ∈ H_{n} × ℂ^{(j,n)}.

Now we specialize the Jacobi forms described in Section 2. to the case of real vector spaces V and U given by

V = ℝ^{(j,n)} × ℝ^{(j,n)}, U = ℝ^{(j,j)}  (23)

for some positive integer j. We choose a complex structure I_{0} on V in such a way that, if v = (v_{1}, v_{2}) ∈ ℝ^{(j,n)} × ℝ^{(j,n)} = V , the V_{+} and V_{−} components of v are given by v_{+} = v_{2} and v_{−} = −v_{1}, respectively, where V_{ℂ} = V_{+} ⊕ V_{−} is the decomposition determined by I_{0} (see Section 2.). We define the bilinear maps β, A : V × V → U by

β((v_{+}, v_{−}), (v′_{+}, v′_{−})) = v′_{+}v_{−}^{t} − v′_{−}v_{+}^{t},  (24)
A((v_{+}, v_{−}), (v′_{+}, v′_{−})) = β((v_{+}, v_{−}), (v′_{+}, v′_{−})) + β((v_{+}, v_{−}), (v′_{+}, v′_{−}))^{t}  (25)

for all v_{+}, v′_{+} ∈ V_{+} and v_{−}, v′_{−} ∈ V_{−}. Then we have

β((v′_{+}, v′_{−}), (v_{+}, v_{−})) = −β((v_{+}, v_{−}), (v′_{+}, v′_{−}))^{t};

hence we see that A is an alternating bilinear map on V .

If K_{ℂ}^{Sp} is the compexification of a maximal compact subgroup of Sp(n, ℝ) as in (15) and if σ : K_{ℂ}^{Sp} → GL(Z) is its representation in a finite-dimensional complex vector space Z, then we define a representation η_{σ} : GL(n, ℂ) → GL(Z) of GL(n, ℂ) by

η_{σ}(q) = σ (\begin{matrix}(q^{t})^{−1} & 0 \\ 0 & q \end{matrix})  (26)

for all q ∈ GL(n, ℂ). Given 𝔐 ∈ ℝ^{(j,j)}, let χ_{𝔐} : U_{ℂ} → ℂ^{×} be the character on U_{ℂ} = ℂ^{(j,j)} given by

χ_{𝔐}(u) = exp(−2πi Tr(𝔐 · u))

for all u ∈ ℂ^{(j,j)}.

Theorem 4.2. Let U and V be as in (23), and let 𝔐 = (m_{αβ}) ∈ ℝ^{(j,j)} be a positive symmetric matrix with 2m_{αβ}, m_{αβ} ∈ ℤ for 1 ≤ α, β ≤ j. Let Γ be a subgroup of Sp(n, ℤ) of finite index, and let ℍ_{ℤ}^{n,j} be as in (21). Then a Jacobi form for Γ⋉ℍ_{ℤ}^{n,j} associated to a representation σ : K_{ℂ}^{Sp} → GL(Z) and a character χ_{𝔐} in the sense of Definition 3.2 is a Jacobi form of index 𝔐 with respect to the representation η_{σ} : GL(n, ℂ) → GL(Z) for Γ in the sense of Definition 4.1.

Proof. Let r = (λ, μ) ∈ ℝ^{(j,n)} × ℝ^{(j,n)} = V . Then as an element of V_{+} ⊕ V_{−} it can be written as

r = (μ, −λ) ∈ V_{+} ⊕ V_{−} = V_{ℂ}.

Let F be a Jacobi form for Γ ⋉ L_{ℍ} associated to σ and χ_{𝔐} in the sense of Definition 3.2, Since we are writing vectors as block row matrices instead of block column matrices, the transformation formula in (19) can be written in the form

F (γz, (w + r_{z})J_{+}(ρ(γ), τ(z))^{t}) = χ_{𝔐}(J ((γ, r, s), (z, w)))σ(J(γ, z))F (z, w)

for all (z, w) ∈ D × V_{+} and (γ, r, s) ∈ Γ ⋉ L_{ℍ} with γ = ( \begin{matrix}a & b \\ c & d \end{matrix}), where

J ((g, r, s), (z, w)) = s − A(rg^{t}, r_{z}J_{+}(g, z)^{t})/2 − A(wg^{t}, wJ_{+}(g, z)^{t})/2 (27)
− A(rg^{t}, wJ_{+}(g, z)^{t})
= s − [A(rg^{t}, (w + r_{z})J_{+}(g, z)^{t})
+ A((w + r)g^{t}, wJ_{+}(g, z)^{t})]/2.

by (11). Given an element g = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Sp(n, ℝ), we have

rg^{t} = (μ, −λ) (\begin{matrix}a^{t} & c^{t} \\ b^{t} & d^{t} \end{matrix}) = (μa^{t} − λb^{t}, μc^{t} − λd^{t}).

For z ∈ H_{n} we have

r_{z} = r_{+} − r_{−}z = μ + λz ∈ V_{+}.

By (16) we have J_{+}(g, z)^{t} = J_{−}(g, z)^{−1} = (cz + d)^{−1}. We also see that

(w + r)g^{t} = (μ + w, −λ) (\begin{matrix}a^{t} & c^{t} \\ b^{t} & d^{t} \end{matrix}) = (μa^{t} + wa^{t} − λb^{t}, μc^{t} + wc^{t} − λd^{t}).

Thus by (24) we have

β(rg^{t}, (w + r_{z})J_{+}(g, z)^{t}) = (w + λz + μ)(cz + d)^{−1}(cμ^{t} − dλ^{t}),
β((w + r)g^{t}, wJ_{+}(g, z)^{t})^{t} = (w + μ)c^{t}(zc^{t} + d^{t})^{−1}w^{t} − λd^{t}(zc^{t} + d^{t})^{−1}w^{t}

= (w + λz + μ)c^{t}(zc^{t} + d^{t})^{−1}w^{t} − λ(zc^{t} + d^{t})(zc^{t} + d^{t})^{−1}w^{t}

= (w + λz + μ)(cz + d)^{−1}cw^{t} − λw^{t},

where we used the fact that (cz + d)^{−1}c is symmetric. Hence we obtain

β(rg^{t}, (w + r_{z})J_{+}(g, z)^{t}) + β((w + r)g^{t}, wJ_{+}(g, z)^{t})^{t}

= (w + λz + μ)(cz + d)^{−1}(c(w^{t} + zλ^{t} + μ^{t}) − (cz + d)λ^{t}) − λw^{t}

= (w + λz + μ)(cz + d)^{−1}c(w^{t} + zλ^{t} + μ^{t}) − λw^{t} − (w + λz + μ)λ^{t}.

Using this, (27) and (25), we have

−2J ((g, r, s), (z, w)) = −2s + β(rg^{t}, (w + r_{z})J_{+}(g, z)^{t}) + β((w + r)g^{t}, wJ_{+}(g, z)^{t})^{t}

+ β(rg^{t}, (w + r_{z})J_{+}(g, z)^{t})^{t} + β((w + r)g^{t}, wJ_{+}(g, z)^{t})

= −2s + 2(w + λz + μ)(cz + d)^{−1}c(w^{t} + zλ^{t} + μ^{t})
− 2λw^{t} − 2wλ^{t} − 2λzλ^{t} − μλ^{t} − λμ^{t}.

Since (λ, μ, s) ∈ ℍ_{ℤ}^{n,j} ⊂ ℍ_{ℝ}^{n,j} and

ℍ_{ℝ}^{n,j} = {(λ, μ, s) ∈ ℝ^{(j,n)} × ℝ^{(j,n)} × ℝ^{(j,j)} | (s + μλ^{t})^{t} = s + μλ^{t}},

we see that μλ^{t} + λμ^{t} = 2μλ^{t} + s − s^{t}. Since Tr(𝔐(−2s − s + s^{t})) ∈ ℤ, we obtain

χ_{𝔐}(J ((γ, r, s), (z, w))) = exp(2πi Tr(𝔐 · ((w + λz + μ)(cz + d)^{−1}c(w^{t} + zλ^{t} + μ^{t})
− λw^{t} − (w + λz + μ)λ^{t}))).

On the other hand, by (9), (16) and (26) we have

σ(J(γ, z)) = η_{σ}(J_{−}(γ, z)) = η_{σ}(cz + d).

Hence the transformation formula (19) reduces to (22) with η = η_{σ}, and therefore the proof of the theorem is complete.

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Min Ho Lee Department of Mathematics University of Northern Iowa Cedar Falls, Iowa 50614 U. S. A. lee@math.uni.edu

Received August 4, 2000 and in final form October 18, 2000