# Hermitian Jacobi Forms Having Modules as their Index and Vector-Valued
Jacobi Forms

Shaul Zemel

## Introduction

The theory of Jacobi forms is detailed, in the classical setting, in [EZ],
with the Skew-holomorphic analogue described in [Sk]. They serve as a bridge
between (elliptic) modular forms and Siegel modular forms of degree 2, by
showing up as the Fourier–Jacobi coefficients of the latter. This notion was
expanded to have a lattice index initially in [BK], but was developed in many
following works, like [CWR], [BRi], and [BRR], the theses [A] and [Mo], and
recently [BS], as well as [Ze3] in the indefinite lattice index case.

We note that the lattice index can be interpreted, in the appropriate
coordinates, as a matrix index for Jacobi forms. This is important for
considering (scalar-valued) Siegel modular forms of degree $r+1$, as the
coefficients of their Fourier–Jacobi expansions of co-genus $r$ (in the
terminology of [BRa] and [Xi]) are Jacobi forms which have $r\times r$
matrices as their indices.

A crucial relation to (vector-valued) elliptic modular forms is the fact that
Jacobi forms admit a theta decomposition, which is described classically in
[EZ], but has its precursors in the relation with the plus-spaces from [K1]
and [K2], generalized later for some congruence subgroups in [LZ]. This
amounts to the identification of (scalar-valued) Jacobi forms with vector-
valued modular forms with the appropriate Weil representation, and have neat
descriptions for indices which are either scalars, lattices, or matrices.

When the even lattice $L$ is positive definite of degree $b_{+}$, so that the
constructions behave well in terms of holomorphicity, the theta decomposition
produces isomorphisms

$J_{k,L}(\Gamma)\longleftrightarrow
M_{k-b_{+}/2}(\Gamma,\rho_{\overline{L}})\quad\mathrm{and}\quad
J_{k,L}^{!}(\Gamma)\longleftrightarrow
M_{k-b_{+}/2}^{!}(\Gamma,\rho_{\overline{L}})$

for every integral weight $k$ and finite index subgroup $\Gamma$ of the
metaplectic double cover of $\operatorname{SL}_{2}(\mathbb{Z})$ between the
corresponding spaces of holomorphic and weakly holomorphic Jacobi and modular
forms, where $\rho_{\overline{L}}$ is the Weil representation dual to that of
$L$.

Siegel modular forms have Hermitian analogues, now known as Hermitian modular
forms, and whose theory was initiated in [Bra1], [Bra2], and [Bra3]. In degree
2 they also admit Fourier–Jacobi expansions, whose members are known as
Hermitian Jacobi forms. Their basic systematic theory was initiated in [Hav1]
(or the shorter version [Hav2]), and a deeper treatise in the thesis [De] also
involving the relation to the Hermitian modular forms. The thesis [He] was
concerned with some arithmetic calculations involving these objects. Virtually
all of these manuscripts consider Hermitian Jacobi forms with a rational (or
integral) index.

To make the distinction between the notions more visible, note that in the
rank 1 case, the classical (also called _orthogonal_) Jacobi forms have one
variable from the upper half-plane $\mathcal{H}$ and one from $\mathbb{C}$.
However, the Hermitian Jacobi forms considered in the references mentioned
above have the variable from $\mathcal{H}$ and _two_ variables from
$\mathbb{C}$, with the complex conjugation on the field showing up in the
parts of the defining formulae involving one of them. In symbols,

$\Phi(\tau,\zeta)\ -\ \mathrm{orthogonal\ Jacobi\
form},\qquad\Phi(\tau,\zeta,\omega)\ -\ \mathrm{Hermitian\ Jacobi\ form}$

(with an orthogonal lattice $L$, the variable $\zeta$ is from
$L_{\mathbb{C}}$, and when $M$ is a Hermitian lattice, the variables $\zeta$
and $\omega$ come from the appropriate parts of its complexification—see
Definition 7 below).

The interplay between the natural setting of a lattice index and of a matrix
index shows up in many applications, in particular for Siegel modularity
results of cycles, as in [Zh] and [BRa]. Recently, [Xi] established the
Hermitian modularity of cycles in some cases of the unitary setting (namely
when the imaginary quadratic field is norm-Euclidean), a result which involves
Hermitian theta series.

Now, in all these references, the Siegel or Hermitian modular forms are
vector-valued, and their Fourier–Jacobi coefficients can thus have non-
integral matrices as their indices. This point was mentioned briefly in these
references by considering the Jacobi group as living inside the Siegel or
Hermitian modular group, which allowed these references to use these Jacobi
forms without needing to deal with some intricate details of their behavior.

This paper thus has 3 goals, which are interwoven with one another.

  1. 1.

The construction in detail the basic parts of a systematic theory of Hermitian
Jacobi forms having lattice index of arbitrary rank over some order in an
imaginary quadratic field.

  2. 2.

The investigation of vector-valued Jacobi forms (both for orthogonal and
Hermitian lattices).

  3. 3.

The establishment of good definitions and theories for Jacobi forms (again in
both settings) having non-integral matrices as their indices.

Regarding the first goal, it seems that such a theory does not exist in the
literature. The second one allows for, and requires, more restrictive and
delicate periodicity properties, as we shall soon explain. The third goal was
the initial motivation for writing this paper, and it serves as the motivation
for the second one (despite the latter being interesting in its own right),
since the desired Jacobi forms must indeed be vector-valued in general, and
present this more delicate periodicity.

We will give more details after a few remarks. First, orthogonal and Hermitian
Jacobi forms of non-integral matrix index, as in our third goal, are expected
to have many applications, like in the Kudla program, concerning the
modularity of special cycles of any codimension on orthogonal and unitary
Shimura varieties. Indeed, the modularity results from [BRa] and [Xi] involve
vector-valued Siegel and Hermitian modular forms, which are established to be
such using their Fourier–Jacobi expansions, the members of which are precisely
Jacobi forms on the sort investigated in this paper. This is also the case in
our joint work [BRZ], the work on which has led to me investigate the
questions answered in this paper.

Note that many papers proving modularity results, like, e.g., [KM], work in
the Adélic setting, and thus consider more general but less explicit Schwartz
functions on the finite Adélic part in their results and calculations. As the
cycles in [BRa], [Xi], [BRZ], and others come with well-defined indices and
associated finite-dimensional Weil representations, it is useful and important
to have the tools to establish such results in this level of explicitness. The
tools developed in this paper serve this purpose, in addition to the
interesting relations that they satisfy among themselves and with vector-
valued elliptic modular forms.

We remark that while for most applications only the holomorphic, positive
definite setting is required, in [Ze3] we could show how to define Jacobi
forms with indefinite lattice index in a way that all the important properties
(except for holomorphicity in the variable from the upper half-plane) carry
out to this indefinite setting, with the same proofs. In this paper we thus
work with the same generality as in [Ze3]. However, we do give at the end, for
future use, the statements of the results for positive definite lattices, as
they are a but simpler to present, and show up, due to their holomorphicity,
in the applications.

We can now describe the content of this paper with more details. The vector-
valued Jacobi theta function $\Theta_{L}$ of an even lattice $L$ (holomorphic
when $L$ is positive definite, or the more general one defined in [Ze3]) is a
function of $\tau$ from the upper half-plane $\mathcal{H}$ and $\zeta$ from
the complex vector space $L_{\mathbb{C}}$. For fixed $\tau$ it is “periodic”
in the variable $\zeta\in L_{\mathbb{C}}$ with respect to $L$ and $\tau L$,
with some explicit exponential factors. In symbols, when $L$ is positive
definite, this property reads

$\Theta_{L}(\tau,\zeta+\tau\sigma+\nu)=\mathbf{e}\big{(}-\tau\sigma^{2}/2-(\sigma,\zeta)\big{)}\Theta_{L}(\tau,\zeta)\quad\mathrm{for\
}\sigma\mathrm{\ and\ }\nu\mathrm{\ in\ }L.$

Thus any combination of its scalar-valued components, with coefficients that
are functions only of the variable $\tau$ from the upper half-plane
$\mathcal{H}$, also shares this periodicity property.

Conversely, any function on $\mathcal{H}\times L_{\mathbb{C}}$ exhibiting this
behavior, and is smooth so that periodicity in $L$ implies the existence of a
Fourier expansion, must be obtained in this way. This is proved in Lemma 2.1
of [Ze3] in the indefinite orthogonal setting (though for definite lattices it
was clearly known before), and see our Lemma 3.1 for the Hermitian one, to be
explained below. Relating the modularity property on both sides produces the
isomorphisms mentioned above, which explicitly becomes

$F\in
M_{k-b_{+}/2}^{!}(\Gamma,\rho_{L}^{*})\quad\mapsto\quad\textstyle{\Phi:=\sum_{\gamma\in
L^{*}/L}f_{\gamma}\theta_{\gamma}\in J_{k,L}^{!}(\Gamma)}$

(with the image of elements of $M_{k-b_{+}/2}(\Gamma,\rho_{\overline{L}})$
covering precisely $J_{k,L}(\Gamma)$).

However, when one considers the full vector-valued theta function
$\Theta_{L}$, then one obtains periodicity with respect to the larger lattice
$L^{*}$ and $\tau L^{*}$, with the former multiplying the elements of the
natural basis of $\mathbb{C}[L^{*}/L]$ by appropriate roots of unity, while
the latter translates the basis elements of $\mathbb{C}[L^{*}/L]$ in which
$\Theta_{L}$ is valued. Using the notation from Definition 2.5, the
periodicity property from above extends to

$\Theta_{L}(\tau,\zeta+\tau\sigma+\nu)=\mathbf{e}\big{(}-\tau\sigma^{2}/2-(\sigma,\zeta)\big{)}\chi_{\nu}t_{\sigma}\Theta_{L}(\tau,\zeta)\quad\mathrm{for\
}\sigma\mathrm{\ and\ }\nu\mathrm{\ in\ }L^{*}.$

The idea is that by constructing vector-valued combinations of the components
of $\Theta_{L}$, if those actions of $\chi_{\nu}$ and $t_{\sigma}$ remain
well-defined for some $\nu$ and $\sigma$ from $L^{*}$ that are not necessarily
in $L^{*}$, then the periodicity property for such $\nu$ and $\sigma$ carries
over, with these definitions, to the combination in question.

To describe that in a clearer and more explicit manner, take $D$ to be any
discriminant form, let $H$ be an isotropic subgroup of $D_{L}\oplus D$ such
that its projections $H_{L}$ onto $D_{L}$ and $H_{D}$ onto $D$ are isomorphic
to $H$ (such subgroups are called _horizontal_ in Definition 4.3 below), and
set $\Delta_{H}:=H^{\perp}/H$. This is a very natural situation, occurring
when $L$ is a primitive sublattice of some lattice $K$, the discriminant form
$D$ is the one arising from the orthogonal complement $L^{\perp}$ of $L$ in
$K$, and $\Delta_{H}$ produces $D_{K}$ from $D_{L}\oplus D_{L^{\perp}}$. Using
the arrow operator $\downarrow_{H}$ we then obtain a map

$F:\mathcal{H}\to\mathbb{C}[D]\quad\mapsto\quad\Phi:=\downarrow_{H}(\Theta_{L}\otimes
F):\mathcal{H}\times L_{\mathbb{C}}\to\mathbb{C}[\Delta_{H}].$

A crucial point is that the operators $t_{\sigma}$ and $\chi_{\nu}$ remain
well-defined on $\mathbb{C}[\Delta_{H}]$, as long as $\sigma$ and $\nu$ from
$L^{*}$ are such that their $D_{L}$-images are in the subgroup
$H_{L}^{\perp}\subseteq D_{L}$ that is perpendicular to $H_{L}$. Moreover, the
aforementioned extended periodicity property of $\Theta_{L}$ implies that
$\Phi$ exhibits the vector-valued periodicity with respect to elements of
$L^{*}$ and of $\tau L^{*}$ using these operators, provided that the
$D_{L}$-images of the elements from $L^{*}$ are in $H_{L}^{\perp}$ (see
Proposition 4.9 below for the Hermitian setting).

In fact, we prove a much stronger result, in Theorems 4.12 and 4.15 for the
Hermitian and orthogonal settings respectively.

######  Theorem.

The construction from the previous paragraph produces an isomorphism between
the space of modular forms having representation $\rho_{D}$ and vector-valued
Jacobi forms having representation $\rho_{\Delta_{H}}$ and this extended
periodicity property, which in the holomorphic setting can be written as

$J_{k,L}^{H}(\Gamma,\rho_{\Delta_{H}})\longleftrightarrow
M_{k-b_{+}/2}(\Gamma,\rho_{D})\quad\mathrm{and}\quad
J_{k,L}^{H,!}(\Gamma,\rho_{\Delta_{H}})\longleftrightarrow
M_{k-b_{+}/2}^{!}(\Gamma,\rho_{D}).$

Our theorem generalizes the scalar-valued case, which is obtained when we take
$D$ to be $D_{\overline{L}}$ and an appropriate $H$, with trivial
$H_{L}^{\perp}$ and $\Delta_{H}$. In order to understand the subspaces of
Jacobi forms carrying higher periodicity properties (either scalar-valued or
vector-valued), we consider the case where the function $F$ is $\uparrow_{I}G$
for an isotropic subgroup $I$ of $D$ in Proposition 5.8 below, and show in
what sense these higher periodicity properties characterize the images of such
functions $F$. This requires a lot more notations for introducing explicitly,
and the results are given in Theorems 5.13 and 5.15 below. This establishes
the second goal of this paper (in both settings).

Before turning to the third one, we describe the construction of Hermitian
Jacobi forms of lattice index. For this we consider an order $\mathcal{O}$ in
an imaginary quadratic field $\mathbb{K}$, which we view as embedded in
$\mathbb{C}$ by some fixed embedding, and a projective module $M$ over
$\mathcal{O}$. Denote by $V$ the vector space generated by $M$ over
$\mathbb{K}$, and let $\mathcal{D}$ be the different of $\mathcal{O}$, with
inverse $\mathcal{D}^{-1}$. We take a non-degenerate Hermitian pairing

$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{K},\mathrm{\ with\ }\langle
M,M\rangle\subseteq\mathcal{D}^{-1},\mathrm{\ and\ set\
}(\cdot,\cdot):=\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}\circ\langle\cdot,\cdot\rangle:V\times
V\to\mathbb{Q}$

be the associated orthogonal pairing. The dual $M^{*}$ of $M$ defined by
pairing with $M$ to $\mathcal{D}^{-1}$ is the same as its dual as an
orthogonal lattice, and we assume an appropriate even condition. Thus $M$
admits a Hermitian Jacobi theta function $\Theta_{M}$ (which in our extended,
possibly indefinite setting, is a Hermitian Siegel–Jacobi theta function), as
well as a theta function $\widetilde{\Theta}_{M}$ obtained from $M$ as an
orthogonal lattice.

Now, the $\mathcal{O}$-lattice $M$ (or the space $V$) has three
complexifications, one via the embedding of $\mathbb{K}$ into $\mathbb{C}$ (we
denote this one by $V_{\mathbb{C}}$ in Definition 1.1), one through letting
$\mathbb{K}$ be embedded into $\mathbb{C}$ in the complex conjugate way
(producing the space $V_{\overline{\mathbb{C}}}$), and the tensor with
$\mathbb{C}$ over $\mathbb{Q}$, which is the direct sum of these spaces. These
subspaces are the eigenspaces of the action of a traceless element of
$\mathbb{K}$, and this element can be used to reproduce the Hermitian pairing
from the $\mathbb{Q}$-valued one, as it behaves like a generalized complex
structure (see Lemma 1.5 below). The Hermitian Jacobi theta function of $M$
has variables $\tau$ from $\mathcal{H}$, $\zeta$ from $V_{\mathbb{C}}$, and
also $\omega$ from $V_{\overline{\mathbb{C}}}$.

With Proposition 1.6 explaining the relations of the two pairings with the
spaces $V_{\mathbb{C}}$ and $V_{\overline{\mathbb{C}}}$, we have the equality

$\Theta_{M}(\tau,\zeta,\omega)=\widetilde{\Theta}_{M}(\tau,\zeta+\omega)\quad\mathrm{for}\quad\tau\in\mathcal{H},\
\zeta\in V_{\mathbb{C}},\mathrm{\ and\ }\omega\in V_{\overline{\mathbb{C}}},$

with $\zeta+\omega$ considered as a variable from the complexification
$V\otimes_{\mathbb{Q}}\mathbb{C}$. Using this equality (generalized in Lemma
2.4 to the indefinite setting) and the relations from Proposition 1.6, we can
transfer all the important properties of $\widetilde{\Theta}_{M}$ already
established in [Ze3] for $M$ as an even lattice to the Hermitian setting and
the theta function $\Theta_{M}$. Then Hermitian Jacobi forms with lattice
index $M$ are defined in such a way that their properties transfer to those of
orthogonal Jacobi forms of lattice index $M$ over $\mathbb{Z}$, via
$\Theta_{M}$ playing the role of $\widetilde{\Theta}_{M}$.

Note that the signature of $M$ over $\mathbb{Z}$ is twice the one of $M$ over
$\mathcal{O}$, affecting the change of weights between Jacobi forms and the
associated elliptic modular forms. In particular, in the positive definite
setting and holomorphic forms, the isomorphisms will be

$J_{k,M/\mathcal{O}}(\Gamma)\longleftrightarrow
M_{k-b_{+}}(\Gamma,\rho_{\overline{M}})\quad\mathrm{and}\quad
J_{k,M/\mathcal{O}}^{!}(\Gamma)\longleftrightarrow
M_{k-b_{+}}^{!}(\Gamma,\rho_{\overline{M}}),$

with the index $M/\mathcal{O}$ implying Hermitian Jacobi forms, but the Weil
representations are the same ones. As the weights are integral, $\Gamma$ is
just a finite index subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$ (without a
metaplectic cover), a group which for all but finitely many orders
$\mathcal{O}$ coincides with the unitary group
$\operatorname{U}(1,1)(\mathbb{Z})$. For the remaining orders, the latter
group can be a bit larger, and the Weil representation extends in Proposition
2.3 below. This achieves the first goal, and allows us to complete the other
two in the orthogonal and Hermitian settings simultaneously.

We remark that the theory of Jacobi froms over totally real fields as
developed in [Boy] can probably combine with our observations to produce the
basis for a theory of Hermitian Jacobi forms over CM fields of larger degree
over $\mathbb{Q}$.

For the third goal we restrict attention to positive definite lattices, but
consider an extended notion in which the matrix can be positive semidefinite
(this is required in [BRa], [Xi], and [BRZ]). After presenting the form of our
results in the holomorphic setting, we define a Jacobi form of non-integral
matrix index $S$ to take values in $\mathbb{C}[\Delta]$ for some discriminant
form $\Delta$, using a map $\psi$ from integral vectors to $\Delta$ that must
preserve the quadratic values in the appropriate sense. This translates to

$\psi:\mathbb{Z}^{c_{+}}\to\Delta\quad\mathrm{with}\quad\tfrac{(\psi a,\psi
a)}{2}=a^{t}Sa+\mathbb{Z}\quad\mathrm{and}\quad\psi(a)=0\mathrm{\ when\
}Sa=0,$

in the orthogonal setting (where $S$ is of size $c_{+}\times c_{+}$), and
similarly with $\mathcal{O}^{c_{+}}$ and the equality $\langle\psi a,\psi
a\rangle=a^{*}Sa+\mathbb{Z}$ in the Hermitian one. This property of $\psi$ is
the reason why for non-integral $S$ one cannot define scalar-valued Jacobi
forms, and a good notion only exists for vector-valued functions, using the
results established in our second goal.

Indeed, using $\psi$ we determine the lattice which can produce a Jacobi form
with coordinates whose pairings are represented by our matrix $S$ as expected,
but the image of the basis used in the definition must be contained in the
dual lattice, and not just the lattice itself if $S$ if not integral. The
operators analogous to $t_{\sigma}$ and $\chi_{\nu}$ from above are defined in
terms of the map $\psi$, as is the group corresponding to $H_{L}^{\perp}$ (or
$H_{M}^{\perp}$ in the Hermitian setting), and then we can complete the
definition of Jacobi forms with index $S$ as vector-valued Jacobi forms with
the appropriate lattice and enhanced periodicity properties. This is done in
Definition 26 and 24 in the orthogonal and Hermitian settings respectively,
with their respective properties given in Theorems 6.9 and 6.11 below. We
conclude by presenting the form of the Fourier expansions of these Jacobi
forms in Remark 6.12 at the end.

This paper is divided into six sections. Section 1 introduces the fine details
of the notions involving Hermitian linear spaces over imaginary quadratic
fields. Section 2 presents the theory of Hermitian Siegel–Jacobi theta
functions, while Section 3 defines the Hermitian Jacobi forms in question and
establishes their properties. Section 4 then gives the details of the
constructions involving orthogonal and Hermitian Jacobi forms in the vector-
valued setting, and Section 5 investigates the relations with the arrow
operators. Finally, Section 6 contains the statements of the results for
positive definite lattices, and shows how to transform the formulations into
Hermitian matrix indices as well.

I would like to thank E. Rosu, B. Williams, and J. Xia for useful comments
which helped to improve the presentation of this paper. Many thanks also to J.
Bruinier and A. Krieg for interesting discussions around this topic, as well
as the introduction of some references, including the new pre-print [BS].

##  1 Hermitian Spaces and Modules

Let $\mathbb{K}$ be an imaginary quadratic field, so that
$\mathbb{K}=\mathbb{Q}(\sqrt{-d})$ for some integer $d>0$, which we consider
as embedded in $\mathbb{C}$, with $\sqrt{-d}$ standing for $\sqrt{d}\cdot i$.
Then a finite-dimensional vector space $V$ over $\mathbb{K}$ can be viewed as
a vector space over $\mathbb{Q}$, endowed with an endomorphism
$J\in\operatorname{End}_{\mathbb{Q}}(V)$, representing the action of
$\sqrt{-d}\in\mathbb{K}$ on $V$, which satisfies
$J^{2}=-d\operatorname{Id}_{V}$ in $\operatorname{End}_{\mathbb{Q}}(V)$. One
can think of $J$ as some kind of generalized complex structure on $V$.

Using this structure, we define the following tensor products.

######  Definition 1.1.

The shorthand $V_{\mathbb{C}}$ stands for $V\otimes_{\mathbb{K}}\mathbb{C}$,
through the fixed embedding of $\mathbb{K}$ into $\mathbb{C}$. When
$\mathbb{K}$ is embedded via the complex conjugate embedding, the
corresponding tensor product of $V$ over $\mathbb{K}$ with $\mathbb{C}$ is
denoted by $V_{\overline{\mathbb{C}}}$. The notation $V_{\mathbb{R}}$ stands
for $V\otimes_{\mathbb{Q}}\mathbb{R}$.

Indeed, for the last notation in Definition 1.1, the field $\mathbb{K}$ cannot
be embedded into $\mathbb{R}$, so that $V_{\mathbb{R}}$ has the only possible
meaning.

We shall be using the following decomposition of the complexification
$V\otimes_{\mathbb{Q}}\mathbb{C}$, and its projections.

######  Lemma 1.2.

The following assertions hold:

  1. $(i)$

The space $V\otimes_{\mathbb{Q}}\mathbb{C}$ is the direct sum of
$V_{\mathbb{C}}$ and $V_{\overline{\mathbb{C}}}$ from Definition 1.1.

  2. $(ii)$

Given $\lambda\in V\otimes_{\mathbb{Q}}\mathbb{C}$, write it as
$\lambda_{\mathbb{C}}+\lambda_{\overline{\mathbb{C}}}$ according to the
decomposition from part $(i)$. Then we have
$\lambda_{\mathbb{C}}=\frac{\lambda}{2}-\sqrt{-d}\frac{J\lambda}{2d}$ and
$\lambda_{\overline{\mathbb{C}}}=\frac{\lambda}{2}+\sqrt{-d}\frac{J\lambda}{2d}$.

  3. $(iii)$

The complex conjugation on $\mathbb{C}$ interchanges the two summands from
part $(i)$. The space $V_{\mathbb{R}}$ from Definition 1.1 embeds into
$V\otimes_{\mathbb{Q}}\mathbb{C}$ as the set of pairs
$\\{(\lambda,\overline{\lambda})|\lambda\in V_{\mathbb{C}}\\}$, and the
restrictions to it of the projections onto $V_{\mathbb{C}}$ and onto
$V_{\overline{\mathbb{C}}}$ are isomorphisms of real vector spaces.

###### Proof.

The fact that $J^{2}=-d\operatorname{Id}_{V}$ in
$\operatorname{End}_{\mathbb{Q}}(V)$ implies that over $\mathbb{C}$, the
endomorphism $J$ decomposes $V$ into the direct sum of its
$\pm\sqrt{-d}$-eigenspaces, yielding part $(i)$ via Definition 1.1. For part
$(ii)$ we just note that the two asserted vectors $v_{\mathbb{C}}$ and
$v_{\overline{\mathbb{C}}}$ are eigenvectors of $J$ with eigenvalues
$\sqrt{-d}$ and $-\sqrt{-d}$ respectively, and they sum to $V$. As complex
conjugation interchanges the aforementioned eigenvalues of $J$, we deduce the
first assertion in part $(iii)$, from which the second one follows and
immediately implies the third one as well. This proves the lemma. ∎

The following simple consequence of our definitions will turn out crucial in
what follows.

######  Corollary 1.3.

Consider $V$ with its structure as a vector space over $\mathbb{K}$, with
$\mathbb{K}$ embedded in $\mathbb{C}$. Then the two projections from part
$(ii)$ of Lemma 1.2 are $\mathbb{Q}$-linear, with
$\lambda\mapsto\lambda_{\mathbb{C}}$ being $\mathbb{K}$-linear, while
$\lambda\mapsto\lambda_{\overline{\mathbb{C}}}$ is $\mathbb{K}$-conjugate-
linear.

###### Proof.

The fact that both maps are $\mathbb{Q}$-linear is immediate, and for the
statements about $\mathbb{K}$, we only need to consider the action of
$\sqrt{-d}\in\mathbb{K}$. But on $V$ this is precisely the endomorphism $J$,
and using the fact that $J^{2}\lambda=-d\lambda$ one easily verifies from the
formulae that $(J\lambda)_{\mathbb{C}}=\sqrt{-d}\cdot\lambda_{\mathbb{C}}$ in
$V_{\mathbb{C}}$, while in $V_{\overline{\mathbb{C}}}$ we have
$(J\lambda)_{\overline{\mathbb{C}}}=-\sqrt{-d}\cdot\lambda_{\overline{\mathbb{C}}}$.
This proves the corollary. ∎

We now make the following definition, in order to set our conventions for the
rest of the paper.

######  Definition 1.4.

A _Hermitian pairing_ on $V$ is a map $\langle\cdot,\cdot\rangle:V\times
V\to\mathbb{K}$, which is $\mathbb{K}$-linear in the first variable and
$\mathbb{K}$-conjugate-linear in the second one. In this case we say that
$\big{(}V,\langle\cdot,\cdot\rangle\big{)}$ is a _Hermitian space over
$\mathbb{K}$_. Such a pairing also endows $V$ with the $\mathbb{Q}$-bilinear
pairing
$(\cdot,\cdot):=\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}\circ\langle\cdot,\cdot\rangle:V\times
V\to\mathbb{Q}$.

If $\big{(}V,\langle\cdot,\cdot\rangle\big{)}$ is a Hermitian space over
$\mathbb{K}$, as in Definition 1.4, then we shorthand the pairing
$\langle\lambda,\lambda\rangle\in\mathbb{Q}$ of $\lambda\in V$ with itself to
$|\lambda|^{2}$. The resulting pairing $(\cdot,\cdot)$ makes every unitary
space over $\mathbb{K}$ also an orthogonal space over $\mathbb{Q}$. If we
write $(\lambda,\lambda)$ for $\lambda\in V$ as simply $\lambda^{2}$, then the
associated quadratic form on $V$ over $\mathbb{Q}$ takes $\lambda$ to
$\frac{\lambda^{2}}{2}$, which also coincides with $|\lambda|^{2}$.

Given a bilinear pairing on $V$, the fact that it arises from a Hermitian
pairing as in Definition 1.4 yields several consequences, which in turn
characterize the pairings that are obtained in this way.

######  Lemma 1.5.

For a bilinear pairing $(\cdot,\cdot):V\times V\to\mathbb{Q}$, with associated
quadratic form $\lambda\mapsto\frac{\lambda^{2}}{2}$, the following are
equivalent:

  1. $(i)$

The pairing $(\cdot,\cdot)$ is
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}\circ\langle\cdot,\cdot\rangle$
for a Hermitian pairing $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{K}$.

  2. $(ii)$

We have $(J\lambda,J\mu)=d(\lambda,\mu)$ for every $\lambda$ and $\mu$ in $V$.

  3. $(iii)$

The quadratic form satisfies $\frac{(J\lambda)^{2}}{2}=d\frac{\lambda^{2}}{2}$
for each $\lambda\in V$.

  4. $(iv)$

The equality $(J\lambda,\mu)=-(\lambda,J\mu)$ holds for any $\lambda$ and
$\mu$ in $V$.

  5. $(v)$

The pairing $(J\lambda,\lambda)$ vanishes for every element $\lambda\in V$.

  6. $(vi)$

The expression
$\langle\lambda,\mu\rangle:=\frac{(\lambda,\mu)}{2}-\sqrt{-d}\frac{(J\lambda,\mu)}{2d}$
defines a Hermitian pairing $\langle\cdot,\cdot\rangle:V\times
V\to\mathbb{K}$.

  7. $(vii)$

By defining
$\langle\lambda,\mu\rangle:=\frac{(\lambda,\mu)}{2}+\sqrt{-d}\frac{(\lambda,J\mu)}{2d}$
one obtains a Hermitian pairing $\langle\cdot,\cdot\rangle:V\times
V\to\mathbb{K}$.

###### Proof.

By assuming $(i)$ and noting that $\langle
J\lambda,J\mu\rangle=d\langle\lambda,\mu\rangle$ by the action of $J$ on both
sides, we obtain $(ii)$. The equivalence of $(ii)$ and $(iii)$ is an immediate
consequence of the relations between a quadratic form and the associated
bilinear form in characteristic different from 2. If we assume $(ii)$, then
$-d(\lambda,J\mu)=(J^{2}\lambda,J\mu)$ equals $d(J\lambda,\mu)$, and $(iv)$
follows. Conversely, if $(iv)$ holds then $(J\lambda,J\mu)$ equals
$-(J^{2}\lambda,\mu)=d(\lambda,\mu)$, completing the equivalence of $(ii)$ and
$(iv)$. Moreover, setting $\mu=\lambda$ in $(iii)$ immediately implies $(v)$.
As if $(v)$ holds then applying it to $\lambda$, $\mu$, and $\lambda+\mu$ and
using the bilinearity and the linearity of $J$ yields $(iv)$, the equivalence
of $(iv)$ and $(v)$ also follows.

Next, note that both $(vi)$ and $(vii)$ involve $\mathbb{Q}$-bilinear maps
$V\times V\to\mathbb{K}$, and as composing any of them with
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}$ reproduces $(\cdot,\cdot)$, we
deduce that either $(vi)$ or $(vii)$ implies $(i)$. Moreover, note that by
replacing $\lambda$ by $J\mu$ in the pairing from $(v)$ we obtain
$\frac{(J\lambda,\mu)}{2}+\sqrt{-d}\frac{(\lambda,\mu)}{2}=\sqrt{-d}\langle\lambda,\mu\rangle$
(yielding the $\mathbb{K}$-linearity), while writing $J\mu$ instead of $\mu$
in the pairing from $(vi)$ produces
$\frac{(\lambda,J\mu)}{2}-\sqrt{-d}\frac{(\lambda,\mu)}{2}=-\sqrt{-d}\langle\lambda,\mu\rangle$
(showing that this pairing is $\mathbb{K}$-conjugate-linear in the second
variable). But as condition $(iv)$ implies that the pairings from $(vi)$ and
$(vii)$ coincide, this joint pairing is Hermitian, yielding both $(vi)$ and
$(vii)$ as desired. This proves the lemma. ∎

It is also clear from the proof of Lemma 1.5 that when the equivalent
conditions are satisfied, the pairing yielding Condition $(i)$ is unique, and
is given by the formulae from both conditions $(vi)$ and $(vii)$ there (which
were seen to be equal).

When extending scalars, all the operations become linear in the extended
scalars. Therefore we shall distinguish the action of $\sqrt{-d}\in\mathbb{K}$
from that of $\sqrt{-d}$ in some field of extended scalars (which can be
$\mathbb{C}$, or $\mathbb{K}$ again), we shall write $\sqrt{-d}$ for the
latter, and use the endomorphism $J$ for the former. We shall thus obtain the
following relations between the spaces from Definition 1.1 and Lemma 1.2 and
any pairings, related as in Lemma 1.5.

######  Proposition 1.6.

Let $\big{(}V,\langle\cdot,\cdot\rangle\big{)}$ be a Hermitian space over
$\mathbb{K}$, with the associated pairing $(\cdot,\cdot):V\times
V\to\mathbb{Q}$, and extend the scalars from $\mathbb{Q}$ to $\mathbb{C}$.

  1. $(i)$

The spaces $V_{\mathbb{C}}$ and $V_{\overline{\mathbb{C}}}$ are both isotropic
with respect to $(\cdot,\cdot)$.

  2. $(ii)$

If $\zeta\in V_{\mathbb{C}}$ then $\langle\lambda,\zeta\rangle=0$ for every
$\lambda\in V\otimes_{\mathbb{Q}}\mathbb{C}$, and given $\mu\in
V\otimes_{\mathbb{Q}}\mathbb{C}$ we get $\langle\zeta,\mu\rangle=(\zeta,\mu)$.
This also equals $\langle\alpha,\mu\rangle$, when $\alpha\in V_{\mathbb{R}}$
corresponds to $\zeta$ via the isomorphism from part $(iii)$ of Lemma 1.2.

  3. $(iii)$

For $\omega\in V_{\overline{\mathbb{C}}}$ and two elements $\lambda$ and $\mu$
from $V\otimes_{\mathbb{Q}}\mathbb{C}$ we have $\langle\omega,\mu\rangle=0$
and $\langle\lambda,\omega\rangle=(\lambda,\omega)$. If $\omega$ is associated
with $\beta\in V_{\mathbb{R}}$ through part $(iii)$ of Lemma 1.2, then the
latter pairing coincides with $\langle\lambda,\beta\rangle$.

  4. $(iv)$

Given $\zeta\in V_{\mathbb{C}}$ and $\omega\in V_{\overline{\mathbb{C}}}$,
with $\alpha$ and $\beta$ as in parts $(ii)$ and $(iii)$, we have
$\langle\omega,\zeta\rangle=0$ as well as
$\langle\zeta,\omega\rangle=(\zeta,\omega)=\langle\alpha,\beta\rangle$.

###### Proof.

Let $\zeta$ and $\omega$ elements that both belong either to $V_{\mathbb{C}}$
or to $V_{\overline{\mathbb{C}}}$, and consider the pairing
$(J\zeta,J\omega)$. On one hand, the action of $J$ multiplies both vectors by
$\pm\sqrt{-d}$ (with the same sign), meaning, by bilinearity, that this
pairing equals $-d(\zeta,\omega)$. As condition $(ii)$ of Lemma 1.5 implies
(after extending scalars) that this pairing also equals $d(\zeta,\omega)$, we
must have $(\zeta,\omega)=0$, and part $(i)$ follows. In part $(ii)$ we have
$J\zeta=\sqrt{-d}\zeta$, and we express $\langle\lambda,\zeta\rangle$ via part
$(vii)$ of Lemma 1.5 and $\langle\zeta,\mu\rangle$ using part $(vi)$ there.
The asserted expressions then follow, and since
$\zeta=\frac{\alpha}{2}-\sqrt{-d}\frac{J\alpha}{2d}$ via part $(ii)$ of Lemma
1.2, the relation with $\langle\alpha,\mu\rangle$ is now a consequence of the
$\mathbb{K}$-linearity in the first variable. For part $(iii)$ we again use
the same parts for writing $\langle\lambda,\zeta\rangle$ and
$\langle\omega,\mu\rangle$, and use the fact that $J\omega=-\sqrt{-d}\omega$
and $\omega=\frac{\beta}{2}+\sqrt{-d}\frac{J\beta}{2d}$ (with the conjugate-
linearity) to deduce the desired results. Part $(iv)$, together with the
additional values $\langle\omega,\omega\rangle$ and
$\langle\zeta,\beta\rangle$, is now an immediate consequences of parts $(ii)$
and $(iii)$. This proves the proposition. ∎

######  Remark 1.7.

In Definition 1.1 it suffices to consider the $J$-eigenspaces $V_{\mathbb{K}}$
and $V_{\overline{\mathbb{K}}}$ inside $V\otimes_{\mathbb{Q}}\mathbb{K}$ (and
$V$ as the rational subspace inside). Lemmas 1.2 and 1.5, Corollary 1.3, and
Proposition 1.6 hold as stated, with the same formulae and proofs, also in
this algebraic setting. However, as the relevant variables of Hermitian Jacobi
forms below will be taken from $V\otimes_{\mathbb{Q}}\mathbb{C}$ (or its
subspaces that we considered above), we worked in the setting of real and
complex vector spaces throughout.

######  Remark 1.8.

The pairing $\langle\cdot,\cdot\rangle$ determines a $\mathbb{K}$-conjugate-
linear map from $V$ to the dual space
$V^{*}=\operatorname{Hom}_{\mathbb{K}}(V,\mathbb{K})$, by sending $\mu\in V$
to $\lambda\mapsto\langle\lambda,\mu\rangle$. The pairing is _non-degenerate_
when this map is injective (hence bijective, since $\dim V<\infty$), and this
happens if and only if the orthogonal pairing $(\cdot,\cdot)$, on $V$ as a
vector space over $\mathbb{Q}$, is non-degenerate. We then write the signature
of $\big{(}V,\langle\cdot,\cdot\rangle\big{)}$ as a Hermitian space as
$(b_{+},b_{-})$, and the signature of the orthogonal space
$\big{(}V,(\cdot,\cdot)\big{)}$ over $\mathbb{Q}$ is thus $(2b_{+},2b_{-})$
(indeed, elements of an orthogonal basis for $V$ over $\mathbb{K}$ remain
orthogonal over $\mathbb{Q}$, they are also orthogonal to all their
$J$-images, and these $J$-images also form an orthogonal set, with the same
signature via part $(ii)$ of Lemma 1.5, with $d>0$).

Let now $\mathcal{O}$ be an order in $\mathbb{K}$, normalize $-d$ to the
discriminant of $\mathcal{O}$ (so that $\mathcal{O}$ is spanned over
$\mathbb{Z}$ by 1 and $\frac{d+\sqrt{d}}{2}$), and we denote the module
$\frac{1}{\sqrt{-d}}\mathcal{O}$ by $\mathcal{D}^{-1}$ (extending the inverse
different from the case where $\mathcal{O}$ is the maximal order
$\mathcal{O}_{\mathbb{K}}$ in $\mathbb{K}$. The objects we will work with are
the following subgroups of $V$.

######  Definition 1.9.

A subgroup $M$ of $V$ is called an _$\mathcal{O}$ -lattice_ in $V$ if
$\mathcal{O}M\subseteq M$; $M$ spans $V$ over $\mathbb{Q}$ (or equivalently
over $\mathbb{K}$); $\langle M,M\rangle\subseteq\mathcal{D}^{-1}$; And the
multiplication from $\mathcal{O}$ makes $M$ a projective $\mathcal{O}$-module.
Noting that
$|\lambda|^{2}\in\mathcal{D}^{-1}\cap\mathbb{Q}=\frac{1}{2}\mathbb{Z}$ for
every $\lambda$ in an $\mathcal{O}$-lattice $M$, we say that $M$ is _even_
when $|\lambda|^{2}\in\mathbb{Z}$ for any $\lambda\in M$.

######  Remark 1.10.

It is clear from Definition 1.9 that any $\mathcal{O}$-lattice is a torsion-
free, finitely generated Abelian group with an action of $\mathcal{O}$. When
$\mathcal{O}=\mathcal{O}_{\mathbb{K}}$ this already implies the projective
property, but when $\mathcal{O}\subsetneq\mathcal{O}_{\mathbb{K}}$ this need
not be the case, as the example where $V$ is spanned over $\mathbb{K}$ by two
elements, say $u$ and $v$, and
$M=\mathcal{O}x\oplus\mathcal{O}_{\mathbb{K}}y$, shows. Such an
$\mathcal{O}$-module is not considered an $\mathcal{O}$-lattice in Definition
1.9 (regardless of the assumption on the pairing)—see Remark 1.13 below.

For duals of lattices, we make the following observation.

######  Lemma 1.11.

If $M$ is an $\mathcal{O}$-lattice, then an element $y$ satisfies $\langle
y,M\rangle\subseteq\mathcal{D}^{-1}$ if and only if $\langle
M,y\rangle\subseteq\mathcal{D}^{-1}$ if and only if
$(y,M)\subseteq\mathbb{Z}$.

###### Proof.

The fact that $\frac{1}{\sqrt{-d}}$ is inverted by conjugation implies that
$\mathcal{D}^{-1}$ is stable under conjugation, yielding the equivalence of
the first two conditions. Next, we note that $\mathcal{D}^{-1}$ is also the
set of elements $r\in\mathbb{K}$ such that
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(r\mathcal{O})\subseteq\mathbb{Z}$
(indeed, the $\mathbb{Z}$-basis $\frac{1}{\sqrt{-d}}$ and
$\frac{1+\sqrt{-d}}{2}$ of $\mathcal{D}^{-1}$ is the $\mathbb{Q}$-basis of
$\mathbb{K}$ that is dual to our $\mathbb{Z}$-basis for $\mathcal{O}$ via this
bilinear form taking $r$ and $s$ to
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(rs)$), which means that the third
condition follows from any of the two others.

For the converse implication, we use the fact that $M$ is a projective
$\mathcal{O}$-module. Thus there is a module $N$ such that $M\oplus N$ is free
over $\mathcal{O}$, so in particular $N$ is also torsion-free, and we view $N$
as embedded in $W=N\otimes_{\mathcal{O}}\mathbb{K}$ and $M\oplus N$ is
contained in $V\oplus W$. We extend the Hermitian pairing
$\langle\cdot,\cdot\rangle$ from $V$ to $V\oplus W$ by letting $W$ be
isotropic and perpendicular to $V$, hence so is $N$ and the pairing (either
Hermitian or orthogonal) with $M\oplus N$ yields the same values as the one
with $M$.

But if $M\oplus N$ is the free module
$\bigoplus_{i=1}^{r}\mathcal{O}\lambda_{i}$ then it is freely generated over
$\mathbb{Z}$ by the $\lambda_{i}$’s and the elements
$\frac{d+J}{2}\lambda_{i}$, $1\leq i\leq r$. Thus
$(\mu,M)=(M,\mu)\subseteq\mathbb{Z}$ if and only if $(M\oplus
N,\mu)\subseteq\mathbb{Z}$ if and only if $(\lambda_{i},\mu)$ and
$\big{(}\frac{d+J}{2}\lambda_{i},\mu\big{)}$ are in $\mathbb{Z}$ for every
$1\leq i\leq r$, and the fact that $\mathcal{D}^{-1}$ is an
$\mathcal{O}$-module implies that $\langle
M,\mu\rangle\subseteq\mathcal{D}^{-1}$ if and only if $\langle M\oplus
N,\mu\rangle\subseteq\mathcal{D}^{-1}$ if and only if
$\langle\lambda_{i},\mu\rangle\in\mathcal{D}^{-1}$ for any $1\leq i\leq r$.
But condition $(vi)$ of Lemma 1.5 expresses $\langle\lambda_{i},\mu\rangle$ as
$\frac{(\lambda_{i},\mu)}{2}-\sqrt{-d}\frac{(J\lambda_{i},\mu)}{2d}$, which we
can write as
$-\frac{\sqrt{-d}}{d}\big{(}\frac{d+J}{2}\lambda_{i},\mu\big{)}+\frac{1+\sqrt{-d}}{2}\frac{(\lambda_{i},\mu)}{2}$.
As this is a presentation of an element of $\mathbb{K}$ in the basis for
$\mathcal{D}^{-1}$ over $\mathbb{Z}$ from above, our assumption is that the
coefficients are integral, we indeed obtain that desired condition for
establishing the remaining implication. This completes the proof of the lemma.
∎

We shall denote the set of elements satisfying the equivalent conditions from
Lemma 1.11 by $M^{*}$, and call it the _dual lattice_ of $M$. This equivalence
can thus be phrased as the following consequence.

######  Corollary 1.12.

The dual of $M$ as an $\mathcal{O}$-lattice coincides with its dual as an
orthogonal lattice over $\mathbb{Z}$. This dual, as a subgroup of $V^{*}$, is
also an $\mathcal{O}$-module, which contains the image of $M\subseteq V$ under
the map from Remark 1.8, with the quotient $M^{*}/M$ being finite and $M^{*}$
being projective over $\mathcal{O}$ in case the pairings are non-degenerate.

###### Proof.

The coincidence of duals is just the content of Lemma 1.11, and as the action
of $\mathcal{O}$ on $V$ preserves $M$ and $\mathcal{D}^{-1}$ is an
$\mathcal{O}$-module in $\mathbb{K}$, it follows that $M^{*}$ is also
preserved under this action, making it an $\mathcal{O}$-module. The fact that
it contains the image of $M$ follows from the assumption that $\langle
M,M\rangle\subseteq\mathcal{D}^{-1}$ in Definition 1.9. Finally, as when the
pairing is non-degenerate, $M^{*}$ coincides with the subgroup
$\operatorname{Hom}_{\mathcal{O}}(M,\mathcal{D}^{-1})$ (or just
$\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$) of $V^{*}$, which is finitely
generated of the same rank as $M$ and its injective image, the finiteness of
$M^{*}/M$ in this case follows. This also combines with the fact that $M$ is
projective of finite rank over $\mathcal{O}$ and
$\mathcal{D}^{-1}=\frac{1}{\sqrt{-d}}\mathcal{O}$ is free of rank 1 over
$\mathcal{O}$ to show that $M^{*}$ is also projective. This proves the
corollary. ∎

######  Remark 1.13.

The proof of Lemma 1.11 used the projectivity assumption for showing that all
the conditions there are equivalent. We shall use the module from Remark 1.10
to exemplify what happens without it. Assume that $\lambda$ and $\mu$ there
are orthogonal, with $|\lambda|^{2}$ and $|\mu|^{2}$ non-zero inside
$\mathcal{D}^{-1}$ (or more precisely, its intersection with $\mathbb{Q}$,
namely $\frac{1}{2}\mathbb{Z}$), then the $\mathbb{Z}$-dual of $M$ consists of
vectors $a\lambda+b\mu$ with $a$ and $b$ from $\mathbb{K}$ such that
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(a)|\lambda|^{2}$ and
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(b)|\mu|^{2}$ are in $\mathbb{Z}$.
This amounts to $a|\lambda|^{2}$ being in $\mathcal{D}^{-1}$, while
$b|\mu|^{2}$ has to be in the usual inverse different
$\mathcal{D}_{\mathbb{K}}^{-1}$ of $\mathcal{O}_{\mathbb{K}}$. Thus $M$ only
has an $\mathcal{O}$-module structure (and not an
$\mathcal{O}_{\mathbb{K}}$-module structure), but $M^{*}$ cannot be described
in terms of the values of the Hermitian pairing with $M$ in the non-projective
case, showing why the equivalences in Lemma 1.11 need not hold as stated
without the projective property from Definition 1.9.

##  2 Hermitian Jacobi–Siegel Theta Functions

As we will be working in general, indefinite setting, we shall be needing the
symmetric space of the corresponding unitary group. Let
$\big{(}V,\langle\cdot,\cdot\rangle\big{)}$ be a Hermitian space over
$\mathbb{K}$ as in Definition 1.4, which we henceforth assume to be non-
degenerate of some signature $(b_{+},b_{-})$. Then the associated bilinear
pairing $(\cdot,\cdot)$ is also non-degenerate, of $V_{\mathbb{R}}$ is
$(2b_{+},2b_{-})$, and by considering $V_{\mathbb{R}}$ as a real orthogonal
space, we obtain the associated Grassmannian

$\operatorname{Gr}(V_{\mathbb{R}}):=\big{\\{}V_{\mathbb{R}}=v_{+}\oplus
v_{-}\big{|}v_{+}\gg 0,\ v_{-}\ll 0,\ v_{+}\perp v_{-}\big{\\}},$ (1)

which is a real manifold of dimension $4b_{+}b_{-}$.

By viewing $V_{\mathbb{R}}$ as a unitary space (with
$\sqrt{-d}\in\mathbb{K}\subseteq\mathbb{C}$ acting as the endomorphism $J$),
the symmetric space consists of similar decompositions, but of spaces over
$\mathbb{C}$ (namely $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{K}$), thus
producing a complex manifold of dimension $b_{+}b_{-}$ (thus with real
dimension $2b_{+}b_{-}$), which embeds into the manifold from Equation (1) (as
in, e.g., [Ho2]). In fact, being closed under this action of $\mathbb{C}$
amounts to being preserved under $J$, and conditions $(ii)$ and $(iv)$ of
Lemma 1.5 immediately imply that the orthogonal complement of a $J$-invariant
subspace is also $J$-invariant. Thus our complex manifold can be identified
with the submanifold

$\operatorname{Gr}^{J}(V_{\mathbb{R}}):=\big{\\{}V_{\mathbb{R}}=v_{+}\oplus
v_{-}\big{|}v_{+}\gg 0,\ v_{-}\ll 0,\ v_{+}\perp v_{-},\ Jv_{+}\subseteq
v_{+},\ Jv_{-}\subseteq v_{-}\big{\\}},$ (2)

of the one from Equation (1), whose elements are characterized by
$J$-invariance.

Given an element of $\operatorname{Gr}^{J}(V_{\mathbb{R}})$ from Equation (2),
any element $\lambda$ in $V$ (or in $V_{\mathbb{R}}$, or of the two summands
$V_{\mathbb{C}}$ and $V_{\overline{\mathbb{C}}}$ from Definition 1.1, or of
their direct sum $V\otimes_{\mathbb{Q}}\mathbb{C}$—see part $(i)$ of Lemma
1.2) decomposes as the sum of its orthogonal projections $\lambda_{v_{+}}$ and
$\lambda_{v_{-}}$ onto the spaces $v_{+}$ and $v_{-}$ respectively. This
decomposition is the same regardless of whether we consider $v$ as a unitary
decomposition, or as an orthogonal one as in Equations (1) and (2). We thus
have that $|\lambda|^{2}=\frac{\lambda^{2}}{2}$ is the sum of
$|\lambda_{v_{+}}|^{2}=\lambda_{v_{+}}^{2}/2\geq 0$ and of
$|\lambda_{v_{-}}|^{2}=\lambda_{v_{-}}^{2}/2\leq 0$, with
$\langle\lambda_{v_{+}},\lambda_{v_{-}}\rangle=(\lambda_{v_{+}},\lambda_{v_{-}})=0$.

Let now $M$ be an even $\mathcal{O}$-lattice in $V$ as in Definition 1.9, with
dual $M^{*}$ as in Lemma 1.11 and Corollary 1.12, and we shall henceforth
denote the quotient $M^{*}/M$ by $D_{M}$. We recall that a matrix $A=\binom{a\
\ b}{c\ \ d}\in\operatorname{SL}_{2}(\mathbb{R})$ acts on the upper half-plane
$\mathcal{H}:=\\{\tau=x+iy\in\mathbb{C}|y>0\\}$ by the formula
$A\tau=\frac{a\tau+b}{c\tau+d}$, with the factor of automorphy
$j(A,\tau):=c\tau+d$, and we use the notation $T:=\binom{1\ \ 1}{0\ \ 1}$ and
$S:=\binom{0\ \ -1}{1\ \ \ 0\ }$ for the generators of the group
$\operatorname{SL}_{2}(\mathbb{Z})$, as usual. Now, on $M$ as an even lattice
over $\mathbb{Z}$, and on $M^{*}$, the pairing $(\cdot,\cdot)$ and the
quadratic form $\lambda\mapsto\frac{\lambda^{2}}{2}$ yield well-defined
$\mathbb{Q}/\mathbb{Z}$-valued functions on the quotient $D_{M}$, for which we
shall use the same notation. To $M$ one then attaches the _Weil
representation_ of $\operatorname{SL}_{2}(\mathbb{Z})$ on $\mathbb{C}[D_{M}]$,
which is defined on the generators by

$\rho_{M}(T)\mathfrak{e}_{\gamma}=\mathbf{e}\big{(}\tfrac{\gamma^{2}}{2}\big{)}\mathfrak{e}_{\gamma},\qquad\rho_{M}(S)\mathfrak{e}_{\gamma}=\frac{i^{b_{-}-b_{+}}}{\sqrt{|D_{M}|}}\sum_{\delta\in
D_{M}}\mathbf{e}\big{(}-(\gamma,\delta)\big{)}\mathfrak{e}_{\delta},$ (3)

where we have introduced the shorthand $\mathbf{e}(z):=e^{2\pi iz}$ for any
$z$ from $\mathbb{C}$ or from $\mathbb{C}/\mathbb{Z}$ (for the root of unity
in $\rho_{M}(S)$, recall that the signature of $M$ over $\mathbb{Z}$ is
$(2b_{+},2b_{-})$, and the fact that $M$ has even rank over $\mathbb{Z}$
implies that this is indeed a representation of
$\operatorname{SL}_{2}(\mathbb{Z})$, and no metaplectic double covers have to
be considered).

Hermitian theta functions and Jacobi forms play a role in the theory of
Hermitian modular forms, where the natural groups acting are the unitary
groups of the following type.

######  Definition 2.1.

For any $g\geq 1$, the unitary group $\operatorname{U}(g,g)$ is the group of
$2g\times 2g$ matrices $A$ in the $\sqrt{-d}$-extension of the ring of
definition which satisfy $A\binom{0\ \ -I_{g}}{I_{g}\ \ \ 0\ }A^{*}=\binom{0\
\ -I_{g}}{I_{g}\ \ \ 0\ }$, where $A^{*}$ is the conjugate transpose
$\overline{A}^{t}$ of $A$. In particular, for
$\operatorname{U}(g,g)(\mathbb{R})$ we consider matrices in
$\operatorname{GL}_{2g}(\mathbb{C})$ satisfying this equality, the group
$\operatorname{U}(g,g)(\mathbb{Q})$ contains matrices from
$\operatorname{GL}_{2g}(\mathbb{K})$ with this property, and for
$\operatorname{U}(g,g)(\mathbb{Z})$, the entries of the matrices have to come
from $\mathcal{O}_{\mathbb{K}}$. The group $\operatorname{SU}(g,g)$ is
intersection with the corresponding $\operatorname{SL}_{2g}$ group.

The Hermitian modular forms containing Hermitian Jacobi forms in their
Fourier–Jacobi expansion involve the groups $\operatorname{U}(g,g)$ and
$\operatorname{SU}(g,g)$ from Definition 2.1 with $g\geq 2$. However, we will
require the groups $\operatorname{U}(1,1)$ and $\operatorname{SU}(1,1)$,
especially over $\mathbb{Z}$. We now make the following identifications of
groups, already alluded to in [Ho1] and others.

######  Lemma 2.2.

We have
$\operatorname{SU}(1,1)(\mathbb{Z})=\operatorname{SL}_{2}(\mathbb{Z})$, and
$\operatorname{U}(1,1)(\mathbb{Z})=\mu_{\mathbb{K}}\cdot\operatorname{SL}_{2}(\mathbb{Z})$,
where $\mu_{\mathbb{K}}$ is the group of roots of unity in $\mathbb{K}$,
viewed as scalar $2\times 2$ matrices.

###### Proof.

If $A=\binom{a\ \ b}{c\ \ d}\in\operatorname{U}(1,1)(\mathbb{R})$ then the
complex numbers $a\overline{b}$ and $c\overline{d}$ are real. Since $A^{*}$
also lies in that group, we deduce that $\overline{a}c$ and $\overline{b}d$
are real as well. It follows that $a$, $b$, $c$, and $d$ lie in a single real
line inside $\mathbb{C}$, and we can thus write $A$ as the product of a scalar
from $S^{1}$ times a real matrix. Moreover, the
$\operatorname{U}(1,1)$-condition also implies that
$a\overline{d}-b\overline{c}=1$, and as the scalar multiplier from $S^{1}$
does not affect the latter equality, we deduce that the real matrix is from
$\operatorname{SL}_{2}(\mathbb{R})$. As one easily verifies that matrices from
$\operatorname{SL}_{2}(\mathbb{R})$ and scalar matrices from $S^{1}$ both lie
in $\operatorname{U}(1,1)$ (via Definition 2.1), it follows that
$\operatorname{U}(1,1)(\mathbb{R})=S^{1}\cdot\operatorname{SL}_{2}(\mathbb{R})$
(this is not a direct product, as $-I$ lies in the intersection of these
groups).

Now, the determinant of an element from
$S^{1}\cdot\operatorname{SL}_{2}(\mathbb{R})$ is just the square of the scalar
from $S^{1}$, yielding that
$\operatorname{SU}(1,1)(\mathbb{R})=\operatorname{SL}_{2}(\mathbb{R})$. In
particular, by intersecting with
$\operatorname{SL}_{2}(\mathcal{O}_{\mathbb{K}})$, the first assertion
follows. For the second one, we saw that if $A=\binom{a\ \ b}{c\ \
d}\in\operatorname{U}(1,1)(\mathbb{Z})$ then $a$, $b$, $c$, and $d$ from
$\mathcal{O}_{\mathbb{K}}$ all lie in one real line, hence they are all in the
same additive subgroup of $\mathcal{O}_{\mathbb{K}}$, and if $\alpha$
generates this subgroup then they all lie in the ideal generated by $\alpha$.
But then the determinant $ad-bc$, which is invertible in
$\mathcal{O}_{\mathbb{K}}$ for a matrix from
$\operatorname{GL}_{2}(\mathcal{O}_{\mathbb{K}})$, must be a multiple of
$\alpha^{2}$, meaning that $\alpha$ has to be a unit in
$\mathcal{O}_{\mathbb{K}}$ hence a root of unity. But then this $\alpha$ lies
in $S^{1}$, and we obtain that our matrix is in
$\mu_{\mathbb{K}}\cdot\operatorname{SL}_{2}(\mathbb{Z})$, as desired. This
proves the lemma. ∎

Note that over $\mathbb{Q}$ we get
$\operatorname{SU}(1,1)(\mathbb{Q})=\operatorname{SL}_{2}(\mathbb{Q})$ as well
(in fact, $\operatorname{SU}(1,1)$ is the same as $\operatorname{SL}_{2}$ as
algebraic groups over $\mathbb{Z}$), but $\operatorname{U}(1,1)(\mathbb{Q})$
does not decompose as the product of $\operatorname{SL}_{2}(\mathbb{Q})$ and
the group of norm 1 elements of $\mathbb{K}$.

Via Definition 2.1 and Lemma 2.2, we may view the Weil representation from
Equation (3) as a representation of $\operatorname{SU}(1,1)(\mathbb{Z})$.
Moreover, when the discriminant of $\mathbb{K}$ is not $-3$ or $-4$ (or more
precisely, when $\mathcal{O}$ is not
$\mathbb{Z}\big{[}\frac{1+\sqrt{-3}}{2}\big{]}$ or $\mathbb{Z}[i]$), the group
of roots of unity in $\mathcal{O}_{\mathbb{K}}$ (or in $\mathcal{O}$) is just
$\pm 1$, so that this representation is of
$\operatorname{U}(1,1)(\mathbb{Z})$, when the order in question is
$\mathcal{O}$, over which $M$ is projective. However, it will be useful for us
to extend the representation to $\operatorname{U}(1,1)(\mathbb{Z})$ also for
the two orders containing more roots of unity, as follows.

######  Proposition 2.3.

When $M$ is an $\mathcal{O}$-lattice, any $\xi$ in the group
$\mu(\mathcal{O})$ of roots of unity in $\mathcal{O}$ acts on $D_{M}$, and by
setting
$\rho_{M}(\xi)\mathfrak{e}_{\gamma}=\xi^{b_{-}-b_{+}}\mathfrak{e}_{\overline{\xi}\gamma}$
we obtain an extension of the representation from Equation (3) to
$\operatorname{U}(1,1)(\mathbb{Z})$.

We remark again that the signature of $M$ over $\mathbb{Z}$ is even, meaning
that no double covers are required, and the extension from Proposition 2.3 is
indeed to the group $\operatorname{U}(1,1)(\mathbb{Z})$.

###### Proof.

As $\xi\in\mathcal{O}$ acts both on $M$ and on $M^{*}$ (by Corollary 1.12), we
obtain its action also on $D_{M}$. The fact that our formula for $\rho_{M}$
yields a representation of $\mu(\mathcal{O})$ is evident, and we first need to
make sure that on the intersection $\pm I$, the formula resulting from
Equation (3) coincides with the one we defined. But with the even signature
$2b_{+}-2b_{-}$ of $M$, the we have
$\rho_{M}(-I)\mathfrak{e}_{\gamma}=(-1)^{b_{-}-b_{+}}\mathfrak{e}_{-\gamma}$,
which coincides with $\rho_{M}(-1)$ for $-1\in\mu(\mathcal{O})$, and the
action of $I$ (or 1) is trivial in both.

As the multipliers $\operatorname{SU}(1,1)(\mathbb{Z})=\operatorname{SL}_{2}$
and $\mu(\mathcal{O})$ commute inside $\operatorname{U}(1,1)(\mathbb{Z})$, we
will obtain a representation of $\operatorname{U}(1,1)(\mathbb{Z})$ with these
formulae once we show that $\rho_{M}(\xi)$ commutes with $\rho_{M}(A)$ for
every $A\in\operatorname{SL}_{2}$, and it suffices to verify this for $A=T$
and $A=S$. We first note that $\frac{\gamma^{2}}{2}$ is the
$\mathbb{Q}/\mathbb{Z}$-image of $\frac{\lambda^{2}}{2}=|\lambda|^{2}$ for
$\lambda\in M+\gamma$, and that
$|\overline{\xi}\lambda|^{2}=\langle\overline{\xi}\lambda,\overline{\xi}\lambda\rangle=|\overline{\xi}|^{2}\langle\lambda,\lambda\rangle=|\lambda|^{2}$
(with $|\overline{\xi}|^{2}=1$ as a root of unity in $\mathbb{K}$). This means
that
$\frac{(\overline{\xi}\gamma)^{2}}{2}=\frac{\gamma^{2}}{2}\in\mathbb{Q}/\mathbb{Z}$
for every $\gamma\in D_{M}$, and therefore $\rho_{M}(T)$ multiplies both
$\mathfrak{e}_{\gamma}$ and
$\xi^{b_{-}-b_{+}}\mathfrak{e}_{\overline{\xi}\gamma}=\rho_{M}(\xi)\mathfrak{e}_{\gamma}$
by the same scalar $\mathbf{e}\big{(}\frac{\gamma^{2}}{2}\big{)}$, whence it
commutes with $\rho_{M}(\xi)$ on all the generators of $\mathbb{C}[D_{M}]$.

For $S$, applying Equation (3) to
$\rho_{M}(\xi)\mathfrak{e}_{\gamma}=\xi^{b_{-}-b_{+}}\mathfrak{e}_{\overline{\xi}\gamma}$
produces

$\frac{(i\xi)^{b_{-}-b_{+}}}{\sqrt{|D_{M}|}}\sum_{\delta\in
D_{M}}\mathbf{e}\big{(}-(\overline{\xi}\gamma,\delta)\big{)}\mathfrak{e}_{\delta}=\frac{(i\xi)^{b_{-}-b_{+}}}{\sqrt{|D_{M}|}}\sum_{\delta\in
D_{M}}\mathbf{e}\big{(}-(\overline{\xi}\gamma,\overline{\xi}\delta)\big{)}\mathfrak{e}_{\overline{\xi}\delta}$

for any $\gamma\in D_{M}$, where we replaced the summation index $\delta$ by
$\overline{\xi}\delta$. But as the same argument from the previous paragraph
shows that
$\langle\overline{\xi}\lambda,\overline{\xi}\mu\rangle=\langle\lambda,\mu\rangle\in\mathbb{K}$
for $\lambda\in M+\gamma$ and $\mu\in M+\delta$, we obtain that
$(\overline{\xi}\lambda,\overline{\xi}\mu)=(\lambda,\mu)\in\mathbb{Q}$ (via
Definition 1.4) and hence
$(\overline{\xi}\gamma,\overline{\xi}\delta)=(\gamma,\delta)\in\mathbb{Q}/\mathbb{Z}$.
But this compares the latter expression with

$\frac{i^{b_{-}-b_{+}}}{\sqrt{|D_{M}|}}\sum_{\delta\in
D_{M}}\mathbf{e}\big{(}-(\gamma,\delta)\big{)}\xi^{b_{-}-b_{+}}\mathfrak{e}_{\overline{\xi}\delta}=\rho_{M}(\xi)\rho_{M}(S)\mathfrak{e}_{\gamma},$

meaning that $\rho_{M}(\xi)$ commutes with $\rho_{M}(S)$ as well. This
completes the proof of the proposition. ∎

We shall thus henceforth consider the Weil representation $\rho_{M}$ to be the
extension of that from Equation (3) described in Proposition 2.3, namely a
representation of $\operatorname{U}(1,1)(\mathbb{Z})$ (recalling that except
for two possible choices of the order $\mathcal{O}$, this is just a change of
notation for the original group and representation).

We shall write $V_{\mathbb{C}}$, $V_{\overline{\mathbb{C}}}$, and
$V_{\mathbb{R}}$ from Definition 1.1 as $M_{\mathbb{C}}$,
$M_{\overline{\mathbb{C}}}$, and $M_{\mathbb{R}}$ respectively (indeed, they
are $M\otimes_{\mathcal{O}}\mathbb{C}$, the same tensor product with the
complex conjugate embedding, and $M\otimes_{\mathbb{Z}}\mathbb{R}$
respectively). For an element $\tau\in\mathcal{H}$, vectors $\zeta\in
M_{\mathbb{C}}$ and $\omega\in M_{\overline{\mathbb{C}}}$, and an element $v$
in $\operatorname{Gr}^{J}(M_{\mathbb{R}})$ from Equation (2), the (vector-
valued) _Hermitian Jacobi–Siegel theta function_ associated with $M$ is
defined to be

$\Theta_{M}(\tau,\zeta,\omega;v):=\sum_{\gamma\in
D_{M}}\textstyle{\big{[}\sum_{\lambda\in
M+\gamma}\mathbf{e}\big{(}\tau|\lambda_{v_{+}}|^{2}+\overline{\tau}|\lambda_{v_{-}}|^{2}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}\big{]}\mathfrak{e}_{\gamma}},$
(4)

where the coefficient multiplying $\mathfrak{e}_{\gamma}$ in Equation (4) is
the (scalar-valued) Hermitian Jacobi–Siegel theta function
$\theta_{M+\gamma}(\tau,\zeta,\omega;v)$.

When $M$ is considered as an orthogonal lattice over $\mathbb{Z}$, we can
identify the element $v\in\operatorname{Gr}^{J}(M_{\mathbb{R}})$ with its
image in $\operatorname{Gr}(M_{\mathbb{R}})$ from Equation (1), and the pair
$\zeta$ and $\omega$ with the vector $\zeta+\omega\in
M\otimes_{\mathbb{Z}}\mathbb{C}$ as in part $(i)$ of Lemma 1.2. Then Equation
(7) of [Ze3], with these variables, defines the (orthogonal) Siegel–Jacobi
theta function

$\widetilde{\Theta}_{M}(\tau,\zeta+\omega;v):=\sum_{\gamma\in
D_{M}}\Bigg{[}\sum_{\lambda\in
M+\gamma}\mathbf{e}\bigg{(}\tau\frac{\lambda_{v_{+}}^{2}}{2}+\overline{\tau}\frac{\lambda_{v_{-}}^{2}}{2}+(\lambda,\zeta+\omega)\bigg{)}\Bigg{]}\mathfrak{e}_{\gamma}$
(5)

(we will henceforth use $\Theta$ and $\theta$ for Hermitian theta functions,
and $\widetilde{\Theta}$, and possibly $\widetilde{\theta}$, for their
orthogonal counterparts, to emphasize our point of view at every step).

All the properties of our $\Theta_{M}$ essentially follow from the following
comparison, using the results of [Ze3]. However, the form that these
properties take can be a bit more delicate, so we will state most of them
explicitly in the Hermitian setting.

######  Lemma 2.4.

The function from Equation (4) coincides with the orthogonal Siegel–Jacobi
theta function from Equation (5).

###### Proof.

The projections $\lambda_{v_{\pm}}$ were seen to be the same in both
interpretations of $v$, and the Hermitian norms $|\lambda_{v_{\pm}}|^{2}$
become $\lambda_{v_{\pm}}^{2}/2$ using the quadratic form. Moreover, parts
$(ii)$ and $(iii)$ of Proposition 1.6 compare the pairings
$\langle\zeta,\lambda\rangle$ and $\langle\lambda,\omega\rangle$ with
$(\zeta,\lambda)=(\lambda,\zeta)$ and $(\lambda,\omega)$ respectively. As
$M^{*}$ is the same dual for $M$ as an $\mathcal{O}$-lattice and as a
$\mathbb{Z}$-lattice (by Corollary 1.12), so are the two meanings of the
quotient $D_{M}$ and of the cosets $M+\gamma$. Thus both the set of elements
$\lambda$ that show up in the two equations, and the summands associated with
each $\lambda$ in them, coincide, yielding the asserted equality. This proves
the lemma. ∎

Given any element $\sigma\in M\otimes_{\mathbb{Z}}\mathbb{C}$, recall that we
have the projections onto the spaces $v_{\pm}$ associated with
$v\in\operatorname{Gr}^{J}(M_{\mathbb{R}})$ (or their extensions of scalars to
$\mathbb{C}$), as well as the projections onto the components $M_{\mathbb{C}}$
and $M_{\overline{\mathbb{C}}}$, as in Lemma 1.2. These projections commute,
i.e., we have
$(\sigma_{v_{\pm}})_{\mathbb{C}}=(\sigma_{\mathbb{C}})_{v_{\pm}}$ and
$(\sigma_{v_{\pm}})_{\overline{\mathbb{C}}}=(\sigma_{\overline{\mathbb{C}}})_{v_{\pm}}$,
and we will write these vectors as $\sigma_{v_{\pm},\mathbb{C}}$ and
$\sigma_{v_{\pm},\overline{\mathbb{C}}}$ for short. The first property of
$\Theta_{M}$ is the “periodicity” in $\zeta$ and $\omega$, as in Proposition
1.1 of [Ze3], but we shall need (especially for the vector-valued case later)
to extend to translations in elements from $M^{*}$ as well. In order to do
this, we make the following definition.

######  Definition 2.5.

Let $\alpha$ and $\beta$ be elements of $D_{M}$. We then define the
_translation operator_ $t_{\alpha}$ to be the linear operator on
$\mathbb{C}[D_{M}]$ taking the basis vector $\mathfrak{e}_{\gamma}$ to
$\mathfrak{e}_{\gamma-\alpha}$, and the _character operator_ $\chi_{\beta}$ to
be the one multiplying each $\mathfrak{e}_{\gamma}$ by
$\mathbf{e}\big{(}\langle\beta,\gamma\rangle+\langle\gamma,\beta\rangle\big{)}$.
Note that the latter is just $\mathbf{e}\big{(}(\beta,\gamma)\big{)}$ when $M$
is viewed as an orthogonal lattice, and both operators are defined on $D_{M}$
for any orthogonal lattice $M$ (or more precisely, on any discriminant module
in the language of [Sch] and [Ze1], or finite quadratic module in the
terminology of [Str]). For $\sigma$ and $\nu$ in $M^{*}$, we shall write just
$t_{\sigma}$ and $\chi_{\nu}$ for the operators $t_{\sigma+M}$ and
$\chi_{\nu+M}$ respectively. They are trivial if and only if $\sigma$ and
$\nu$ are in $M$.

######  Remark 2.6.

While $\alpha\mapsto t_{\alpha}$ and $\beta\mapsto\chi_{\beta}$ from
Definition 2.5 are group homomorphisms from $D_{M}$ into
$\operatorname{GL}_{2}\big{(}\mathbb{C}[D_{M}]\big{)}$ (and even into the
subgroup of unitary operators), they do not form a homomorphism from the
product, but rather from an appropriate Heisenberg group. This is related to
the fact that the periodicity of theta functions and Jacobi forms is based,
inside the larger, Siegel or Hermitian modular group, on the action of a
Heisenberg group, rather that the Abelian self-product of the lattice (this is
more clearly visible in the associated Lie group). However, we will only be
using the combination $\chi_{\beta}t_{\alpha}$, sending
$\mathfrak{e}_{\gamma}$ to
$\mathbf{e}\big{(}\langle\beta,\gamma-\alpha\rangle+\langle\gamma-\alpha,\beta\rangle\big{)}\mathfrak{e}_{\gamma-\alpha}$
(or just
$\mathbf{e}\big{(}(\beta,\gamma-\alpha)\big{)}\mathfrak{e}_{\gamma-\alpha}$).
In particular, if an element $X$ of $\mathbb{C}[D_{M}]$ is defined by the
absolutely convergent series $\sum_{\lambda\in
M^{*}}a_{\lambda}\mathfrak{e}_{\lambda+M}$ (like our theta functions), then
the series $\sum_{\lambda\in
M^{*}}a_{\lambda+\sigma}\mathbf{e}\big{(}\langle\nu,\lambda\rangle+\langle\lambda,\nu\rangle\big{)}\mathfrak{e}_{\lambda+M}$
(or just $\sum_{\lambda\in
M^{*}}a_{\lambda+\sigma}\mathbf{e}\big{(}(\lambda,\nu)\big{)}\mathfrak{e}_{\lambda+M}$
in the orthogonal notation) produces $\chi_{\nu}t_{\sigma}X$.

We can now state and prove the periodicity formula for our theta functions.

######  Proposition 2.7.

Take two elements $\sigma$ and $\nu$ from $M^{*}$, and let
$\tau\in\mathcal{H}$ and $v\in\operatorname{Gr}^{J}(M_{\mathbb{R}})$ be fixed.
Then, as a function of $\zeta\in M_{\mathbb{C}}$ and $\omega\in
M_{\overline{\mathbb{C}}}$, the Hermitian Jacobi–Siegel theta function from
Equation (4) satisfies the equality

$\Theta_{M}(\tau,\zeta+\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}},\omega+\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}};v)=$
$=\mathbf{e}\big{(}-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle\big{)}\chi_{\nu}t_{\sigma}\Theta_{M}(\tau,\zeta,\omega;v).$

When $\sigma$ and $\nu$ are in $M$, the latter multiplier is just
$\Theta_{M}(\tau,\zeta,\omega;v)$ itself.

###### Proof.

We apply Lemma 2.4 to work with $\widetilde{\Theta}_{M}$ from Equation (5),
and, as in the proof of Proposition 1.1 of [Ze3], we consider the summand
associated with some $\lambda\in M^{*}$ but in which we add
$\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}}$
to $\zeta$ as well as
$\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}}$
to $\omega$. Since every vector is the sum of its projections onto
$M_{\mathbb{C}}$ and $M_{\overline{\mathbb{C}}}$, this amounts to adding just
$\tau\sigma_{v_{+}}+\overline{\tau}\sigma_{v_{-}}+\nu$ to the sum
$\zeta+\omega$, the pairing with which shows up in Equation (5). The fact that
the projections onto $v_{\pm}$ are orthogonal implies that this summand is
multiplied by
$\mathbf{e}\big{(}\tau(\lambda_{v_{+}},\sigma_{v_{+}})+\overline{\tau}(\lambda_{v_{-}},\sigma_{v_{-}})\big{)}$
and also by
$\mathbf{e}\big{(}\langle\nu,\lambda\rangle+\langle\lambda,\nu\rangle\big{)}=\mathbf{e}\big{(}(\lambda,\nu)\big{)}$.

Then this summand involves, aside from the exponent of
$(\lambda,\zeta+\omega)$, also that of $\tau$ or $\overline{\tau}$ multiplied
by
$\frac{\lambda_{v_{\pm}}^{2}}{2}+(\lambda_{v_{\pm}},\sigma_{v_{\pm}})=\frac{(\lambda+\sigma)_{v_{\pm}}^{2}}{2}-\frac{\sigma_{v_{\pm}}^{2}}{2}$.
We thus make the change of variable $\lambda\mapsto\lambda+\sigma$, with the
pairing with $\lambda$ becomes that with $\lambda+\sigma$ minus that with
$\sigma$, and then the multipliers that are independent of $\lambda$ reduce to
the exponent of
$-\tau\sigma_{v_{+}}^{2}/2-\overline{\tau}\sigma_{v_{+}}^{2}/2-(\sigma,\zeta+\omega)$.
For the remaining sum over $\lambda$ we obtain via Remark 2.6, due the
translation by $\sigma$ and the multiplier from the previous paragraph, the
image under $\chi_{\nu}t_{\sigma}$ of the theta function
$\widetilde{\Theta}_{M}(\tau,\zeta+\omega;v)$, which becomes the desired one
on the right hand side via Lemma 2.4, whose proof also shows that the exponent
is the asserted one as well. This establishes the desired general formula,
from which the special case with $\sigma$ and $\nu$ in $M$ follows directly
because $\chi_{\nu}$ and $t_{\sigma}$ from Definition 2.5 are then trivial.
This proves the proposition. ∎

Of course, the Proposition 1.1 of [Ze3] can be extended, as in Proposition
2.7, to show that the theta function from Equation (7) there satisfies such a
periodicity for $\sigma$ and $\nu$ in the dual lattice, using the combination
$\chi_{\nu}t_{\sigma}$ from Definition 2.5 and Remark 2.6.

The second property of our Hermitian theta function $\Theta_{M}$ is the
following modularity transformation.

######  Theorem 2.8.

For every matrix $A\in\operatorname{U}(1,1)(\mathbb{Z})$, and every elements
$\tau\in\mathcal{H}$, $\zeta\in V_{\mathbb{C}}$, and $\omega\in
V_{\overline{\mathbb{C}}}$, we have the equality

$\Theta_{M}\Big{(}A\tau,\tfrac{\zeta_{v_{+}}}{j(A,\tau)}+\tfrac{\zeta_{v_{-}}}{\overline{j^{c}(A,\tau)}},\tfrac{\omega_{v_{+}}}{j^{c}(A,\tau)}+\tfrac{\omega_{v_{-}}}{\overline{j(A,\tau)}};v\Big{)}=$
$=j(A,\tau)^{b_{+}}\overline{j(A,\tau)}^{b_{-}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega;v),$

where we recall that if $A=\binom{a\ \ b}{c\ \ d}$ then $j(A,\tau)=c\tau+d$,
so that $j_{A}^{\prime}$ is the derivative $c$ with respect to $\tau$ of
$j_{A}(\tau)=j(A,\tau)$, and $j^{c}(A,\tau)$ stands for
$\overline{c}\tau+\overline{d}$.

Recall that unless the discriminant of $\mathcal{O}$ is in $\\{-3,-4\\}$, the
group $\operatorname{U}(1,1)(\mathbb{Z})$ is
$\operatorname{SU}(1,1)(\mathbb{Z})=\operatorname{SL}_{2}(\mathbb{Z})$, for
elements $A$ of which we have $j^{c}(A,\tau)=j(A,\tau)$ for any
$\tau\in\mathcal{H}$. However, the form given in Theorem 2.8 is the one that
holds also for these exceptional cases, and is the one whose extension to
Jacobi forms that are not necessarily theta functions generalizes better to
Hermitian Jacobi forms that are modular with respect to
$\operatorname{U}(g,g)$ for larger $g$.

###### Proof.

Consider again the theta function $\widetilde{\Theta}_{M}$ from Equation (5),
take an element $A\in\operatorname{SL}_{2}(\mathbb{Z})$, and then for such
$\tau$, $\zeta$, and $\omega$, we obtain from Theorem 1.2 of [Ze3] (with the
signature $(2b_{+},2b_{-})$, rather than $(2b_{+},b_{-})$ as in that
reference) the fact that
$\widetilde{\Theta}_{M}\big{(}A\tau,(\zeta+\omega)_{v_{+}}\big{/}j(A,\tau)+(\zeta+\omega)_{v_{-}}\big{/}\overline{j(A,\tau)};v\big{)}$
equals

$j(A,\tau)^{b_{+}}\overline{j(A,\tau)}^{b_{-}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}(\zeta+\omega)_{v_{+}}^{2}}{2j(A,\tau)}+\tfrac{j_{A}^{\prime}(\zeta+\omega)_{v_{-}}^{2}}{2\overline{j(A,\tau)}}\Big{)}\rho_{M}(A)\widetilde{\Theta}_{M}(\tau,\zeta+\omega;v).$

Lemma 2.4 then compares this left hand side to the desired one and the theta
function on the right hand side to that of the required one, we have the
equalities $j^{c}(A,\tau)=j(A,\tau)$ and
$\overline{j_{A}^{\prime}}=j_{A}^{\prime}$, and since the
$M_{\mathbb{C}}$-part of $(\zeta+\omega)_{v_{\pm}}$ is $\zeta_{v_{\pm}}$ and
its $M_{\overline{\mathbb{C}}}$-part is $\omega_{v_{\pm}}$, parts $(i)$ and
$(ii)$ of Proposition 1.6 shows that the argument of the exponent is also the
asserted one. This establishes the case
$A\in\operatorname{SL}_{2}(\mathbb{Z})$ (in fact, by using the appropriate co-
cycle relations the proof for such an $A$ one could have reduced a direct
proof to checking the very straightforward case $A=T$ and the case $A=S$, the
latter of which amounts, as usual, to the classical calculations using Fourier
transforms and the Poisson summation formula).

Assume now that $A$ is the product of the scalar $\xi\in\mu(\mathcal{O})$ and
the matrix $B\in\operatorname{SL}_{2}(\mathbb{Z})$. Then $A\tau=B\tau$ (since
$\xi$ is a scalar matrix), $j(A,\tau)=\xi j(B,\tau)$, and
$j^{c}(A,\tau)=\overline{\xi}j(B,\tau)$, and Corollary 1.3 shows that the
action of $\xi\in\mathcal{O}$ on $\zeta$ multiplies it by $\xi$, but it
multiplies $\omega$ by $\overline{\xi}=1/\xi$. This compares the left hand
side with
$\Theta_{M}\Big{(}B\tau,\tfrac{\overline{\xi}\zeta_{v_{+}}}{j(B,\tau)}+\tfrac{\overline{\xi}\zeta_{v_{-}}}{\overline{j(B,\tau)}},\tfrac{\overline{\xi}\omega_{v_{+}}}{j(B,\tau)}+\tfrac{\overline{\xi}\omega_{v_{-}}}{\overline{j(B,\tau)}};v\Big{)}$,
and by what we saw for $B\in\operatorname{SL}_{2}(\mathbb{Z})$ the latter
equals

$j(B,\tau)^{b_{+}}\overline{j(B,\tau)}^{b_{-}}\mathbf{e}\Big{(}\tfrac{j_{B}^{\prime}\langle\overline{\xi}\zeta_{v_{+}},\overline{\xi}\omega_{v_{+}}\rangle}{j(B,\tau)}+\tfrac{j_{B}^{\prime}\langle\overline{\xi}\zeta_{v_{-}},\overline{\xi}\omega_{v_{-}}\rangle}{\overline{j(B,\tau)}}\Big{)}\rho_{M}(B)\Theta_{M}(\tau,\overline{\xi}\zeta,\overline{\xi}\omega;v).$

We have $j_{A}^{\prime}/j(A,\tau)=j_{B}^{\prime}/j(B,\tau)$ (by canceling
$\xi$) and
$\overline{j_{A}^{\prime}}/\overline{j(A,\tau)}=j_{B}^{\prime}/\overline{j(B,\tau)}$
(after canceling $\overline{\xi}$), and as
$\langle\overline{\xi}\zeta_{v_{\pm}},\overline{\xi}\omega_{v_{\pm}}\rangle=\langle\zeta_{v_{\pm}},\omega_{v_{\pm}}\rangle$
as in the proof of Proposition 2.3, the exponent is the desired one. Moreover,
the two external $j$-factors are the asserted ones times $\xi^{b_{-}-b_{+}}$,
and the remaining multiplier is the $\rho_{M}(B)$-image of this root of unity
times $\sum_{\gamma\in
D_{M}}\theta_{M+\overline{\xi}\gamma}(\tau,\overline{\xi}\zeta,\overline{\xi}\omega;v)\mathfrak{e}_{\overline{\xi}\gamma}$.
But in the summand associated with $\overline{\xi}\lambda$ for $\lambda\in
M+\gamma$, the multiplication by $\overline{\xi}$ does not affect the
Hermitian norms $|\lambda_{v_{\pm}}|^{2}$, and the pairings
$\langle\overline{\xi}\zeta,\overline{\xi}\lambda\rangle$ and
$\langle\overline{\xi}\lambda,\overline{\xi}\omega\rangle$ were seen to
reproduce $\langle\zeta,\lambda\rangle$ and $\langle\lambda,\omega\rangle$ yet
again. Our remaining multiplier thus equals $\rho_{M}(B)\sum_{\gamma\in
D_{M}}\theta_{M+\gamma}(\tau,\zeta,\omega;v)\xi^{b_{-}-b_{+}}\mathfrak{e}_{\overline{\xi}\gamma}$,
which via the formula from Proposition 2.3 equals
$\rho_{M}(B)\rho_{M}(\xi)\Theta_{M}(\tau,\zeta,\omega;v)=\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega;v)$,
as desired. This completes the proof of the theorem. ∎

We now turn to the differential properties of Hermitian theta functions. In
the setting of Jacobi forms associated with orthogonal lattices, many
differential operators were considered in [BRR], [CWR], [WR], and [RR]. The
operators that we will work with are similar to those from [Ze3], which were
based on some constructions from [Ze2]. We shall shorthand
$\frac{\partial}{\partial z}$ with respect to any variable $z$ to
$\partial_{z}$, and recall that if $W$ is a non-degenerate $r$-dimensional
real quadratic space, and $\\{\eta_{i}\\}_{i=1}^{r}$ are the coordinates with
respect to a basis for $W$, then by letting $\\{\eta_{i}^{*}\\}_{i=1}^{r}$ be
the coordinates with respect to the dual basis for $W$ through the bilinear
form, the Laplacian operator for $W$ (which is defined, up to a scalar
multiple perhaps, by the action of the Casimir operator of the Lie algebra of
$\operatorname{O}(W)$) is
$\sum_{j=1}^{r}\partial_{\eta_{i}}\partial_{\eta_{i}^{*}}$, acting on smooth
functions on $W$ (and this operator is independent of the basis chosen and is
invariant under the action of $\operatorname{O}(W)$). The only extension from
[Ze2] that we shall need for this operator is by considering functions on
$W\otimes_{\mathbb{R}}\mathbb{C}$, and replacing the derivative with respect
to any real variable $\eta_{i}$ or $\eta_{i}^{*}$ by one with respect to an
associated complex holomorphic one (this is an operator of Hodge weight
$(2,0)$, and the other ones from [Ze2], with the other Hodge weights, will not
show up here). One can show, in fact, that when $W$ comes with an endomorphism
$J$ with $J^{2}=-d\operatorname{Id}_{W}$, and
$W\otimes_{\mathbb{R}}\mathbb{C}$ decomposes as in Definition 1.4 and Lemma
1.2, then each summand in the Laplacian $\Delta_{W}$ will involve a derivative
with respect to a variable from $W_{\mathbb{C}}$ times one associated with a
variable from $W_{\overline{\mathbb{C}}}$.

The spaces for which we will consider these Laplacian operators are the spaces
$v_{\pm}$ associated with an element of
$\operatorname{Gr}^{J}(M_{\mathbb{R}})$ from Equation (2), for which we will
adopt the notation $\Delta_{v_{\pm}}^{h}$ from [Ze1], [Ze2], and others (see
[Ze3] for the relations between these operators, their real counterparts, the
Laplacian of $V$ itself, and the Laplacian that was used in the theta
functions defined in [Bor], or in [Ze5]). Proposition 1.3 of [Ze3] then
combines with Lemma 2.4 to produce the following result.

######  Proposition 2.9.

The theta function $\Theta_{M}$ from Equation (4) is sent to 0 by the
operators $4\pi i\partial_{\tau}-\Delta_{v_{+}}^{h}$, $4\pi
i\partial_{\overline{\tau}}-\Delta_{v_{-}}^{h}$, and any anti-holomorphic
derivative $\partial_{\overline{\zeta}_{i}}$ or
$\partial_{\overline{\omega}_{i}}$ that arises from some basis for
$V_{\mathbb{C}}$ or $V_{\overline{\mathbb{C}}}$.

The annihilation of $\Theta_{M}$ by all the differential operators except for
the first two is just the holomorphicity of this function in the variables
$\zeta$ and $\omega$, as observed above.

##  3 Hermitian Jacobi Forms

Hermitian Jacobi forms with lattice index have theta expansions, similar to
the Jacobi forms in other settings. We will first make this connection, since
it will give us some notions that are needed for defining them formally. In
this section we will only consider scalar-valued Jacobi forms, in an analysis
that is parallel to [Ze3], and the next section, after discussing discriminant
forms in more detail, we will show how to modify these constructions for
vector-valued modular forms.

The first step is the following analysis, involving Fourier expansions, that
follows Lemma 2.1 of [Ze3].

######  Lemma 3.1.

Let $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}$ be a smooth function, that for fixed
$\tau\in\mathcal{H}$ is holomorphic in the variables $\zeta\in M_{\mathbb{C}}$
and $\omega\in M_{\overline{\mathbb{C}}}$, and is invariant under taking
$\zeta$ and $\omega$ to $\zeta+\nu_{\mathbb{C}}$ and
$\omega+\nu_{\overline{\mathbb{C}}}$ with $\nu\in M$. Then there exist smooth
functions $f_{\lambda}$ for $\lambda\in M^{*}$ such that

$\Phi(\tau,\zeta,\omega):=\textstyle{\sum_{\lambda\in
M^{*}}f_{\lambda}(\tau)\mathbf{e}\big{(}\tau|\lambda_{v_{+}}|^{2}+\overline{\tau}|\lambda_{v_{-}}|^{2}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}}.$

Assuming that $\Phi$ satisfies the full scalar-valued relation from
Proposition 2.7 (or more precisely, from Equation (6) in Definition 7 below),
for $\sigma$ and $\nu$ from $M$, the function $f_{\lambda}$ remains invariant
under changing $\lambda$ by an element of $M$.

###### Proof.

When we replace the variables $\zeta$ and $\omega$ by $\zeta+\omega\in
M\otimes_{\mathbb{Z}}\mathbb{C}$, our assumption is that $\Phi$ is holomorphic
and $M$-periodic in this variable. It follows that we can write
$\Phi(\tau,\zeta,\omega)$ as a sum $\sum_{\lambda\in
L^{*}}\tilde{f}_{\lambda}(\tau)\mathbf{e}\big{(}(\lambda,\zeta+\omega)\big{)}$
for smooth functions $\tilde{f}_{\lambda}:\mathcal{H}\to\mathbb{C}$, and the
proof of Lemma 2.4 shows that the argument of the exponent also equals
$\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle$. By writing
$\tilde{f}_{\lambda}$ as
$\tau\mapsto\mathbf{e}\big{(}\tau|\lambda_{v_{+}}|^{2}+\overline{\tau}|\lambda_{v_{-}}|^{2}\big{)}$
times another function $f_{\lambda}$, we obtain the first assertion.

For the second one, we express both sides of the equality from Proposition 2.7
(with $\Phi$ rather than $\Theta_{M}$) using the expansion that we already
have, and then on the left hand side $f_{\lambda}(\tau)$ is multiplied by the
exponent of

$\tau\big{(}|\lambda_{v_{+}}|^{2}+\langle\sigma_{+},\lambda\rangle+\langle\lambda,\sigma_{+}\rangle\big{)}+\overline{\tau}\big{(}|\lambda_{v_{+}}|^{2}+\langle\sigma_{-},\lambda\rangle+\langle\lambda,\sigma_{-}\rangle\big{)}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle$

(note that multiplication by $\tau$ and $\overline{\tau}$ is via the extended
scalars to $\mathbb{C}$, not through the endomorphism $J$, which is why we
have $\mathbb{C}$-linearity in these expressions also in the second variable
of the pairing $\langle\cdot,\cdot\rangle$). On the right hand side, it is
multiplied by the exponent of

$\tau|\lambda_{v_{+}}|^{2}+\overline{\tau}|\lambda_{v_{-}}|^{2}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle,$

and as the pairings of $\lambda$ with $\sigma_{\pm}$ coincide with those of
$\lambda_{\pm}$, it follows that replacing $\lambda$ by $\lambda+\sigma$ in
the latter expression yields the former one. We therefore have an equality
between two Fourier expansions, in which comparing the coefficients yields the
equality $f_{\lambda+\sigma}(\tau)=f_{\lambda}(\tau)$ for every
$\tau\in\mathcal{H}$ and $\lambda\in M^{*}$, as desired. This proves the
lemma. ∎

We saw that $\Theta_{M}$ is a $\mathbb{C}[D_{M}]$-valued function, and we
denote by $\overline{M}$ the lattice $M$ in which all the pairings are
inverted. Then $\mathbb{C}[D_{\overline{M}}]$ is the natural dual of
$\mathbb{C}[D_{M}]$, with its natural basis, which we denote by
$\\{\mathfrak{e}_{\gamma}^{*}\\}_{\gamma\in D_{M}}$, being dual to the natural
one $\\{\mathfrak{e}_{\gamma}\\}_{\gamma\in D_{M}}$ of $\mathbb{C}[D_{M}]$, as
representation spaces of $\operatorname{SL}_{2}(\mathbb{Z})$. Moreover, since
the signature of $\overline{M}$ is the opposite to that of $M$, and the action
of $\mu(\mathcal{O})$ is the same on both, this duality extends to a duality
of representations of $\operatorname{U}_{1,1}(\mathbb{Z})$ via Proposition
2.3. We shall denote the (bilinear) pairing between $\mathbb{C}[D_{M}]$ and
$\mathbb{C}[D_{\overline{M}}]$ by $[\cdot,\cdot]_{M}$, and then from Lemma 3.1
we deduce the following consequence.

######  Corollary 3.2.

A smooth function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}$ satisfying the transformation law from
Proposition 2.7, or Equation (6), can be presented for $\tau\in\mathcal{H}$,
$\zeta\in M_{\mathbb{C}}$ and $\omega\in M_{\overline{\mathbb{C}}}$, as
$\Phi(\tau,\zeta,\omega)=[\Theta_{M}(\tau,\zeta,\omega;v),F(\tau)]_{M}$ for a
unique smooth function $F:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$.

###### Proof.

Lemma 3.1 implies that there are functions $f_{\gamma}$ for $\gamma\in D_{M}$
such that $\Phi(\tau,\zeta,\omega)$ expands as $\sum_{\lambda\in
M^{*}}f_{\lambda+M}(\tau)\mathbf{e}\big{(}\tau|\lambda_{v_{+}}|^{2}+\overline{\tau}|\lambda_{v_{-}}|^{2}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}$.
As $f_{\gamma}(\tau)$ multiplies the sum over $\lambda\in M+\gamma$ of these
exponents, and the latter sum is $\theta_{M+\gamma}(\tau,\zeta,\omega;v)$ by
Equation (4), the desired formula follows from the definition of the pairing
$[\cdot,\cdot]_{M}$. This proves the corollary. ∎

The main object of this paper can now be defined, as follows.

######  Definition 3.3.

Take two integral weights $k$ and $l$, as well as a finite index subgroup
$\Gamma\subseteq\operatorname{U}(1,1)(\mathbb{Z})$, and fix an
$\mathcal{O}$-lattice $M$ as in Definition 1.9 and an element $v$ in
$\operatorname{Gr}^{J}(M_{\mathbb{R}})$ from Equation (2). Then a _Hermitian
Jacobi form of weight $(k,l)$ and index $(M,v)$ with respect to $\Gamma$_ is
defined to be a smooth function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}$, with the following properties. First,
$\Phi$ is holomorphic in the variables from the two vector spaces. Second,
given elements $\sigma$ and $\nu$ from $M$ we have, for any
$\tau\in\mathcal{H}$, $\zeta\in M_{\mathbb{C}}$, and $\omega\in
M_{\overline{\mathbb{C}}}$, the equality

$\Phi(\tau,\zeta+\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}},\omega+\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}})=$
$=\mathbf{e}\big{(}-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle\big{)}\Phi(\tau,\zeta,\omega).$
(6)

Third, given such $\tau$, $\zeta$, and $\omega$, as well as a matrix
$A\in\Gamma$, we have

$\Phi\Big{(}A\tau,\tfrac{\zeta_{v_{+}}}{j(A,\tau)}+\tfrac{\zeta_{v_{-}}}{\overline{j^{c}(A,\tau)}},\tfrac{\omega_{v_{+}}}{j^{c}(A,\tau)}+\tfrac{\omega_{v_{-}}}{\overline{j(A,\tau)}}\Big{)}=$
$=j(A,\tau)^{k}\overline{j(A,\tau)}^{l}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}\Phi(\tau,\zeta,\omega),$
(7)

with $j(A,\tau)$ as usual and $j^{c}(A,\tau)$ as defined in Theorem 2.8. And
fourth, we recall that Equation (6) produces, via Lemma 3.1, the smooth
functions $f_{\lambda}:\mathcal{H}\to\mathbb{C}$, and we demand that their
growth towards each cusp of $\Gamma$ is linear exponential. Given a Hermitian
Jacobi form $\Phi$, we call it _pseudo-holomorphic_ if the operator $4\pi
i\partial_{\overline{\tau}}-\Delta_{v_{-}}^{h}$ from Proposition 2.9 takes
$\Phi$ to 0.

We remark that by working with the other operator $4\pi
i\partial_{\tau}-\Delta_{v_{+}}^{h}$ from Proposition 2.9, we can consider
skew-holomorphic Hermitian Jacobi forms, which can be described analogously to
[Ze3], thus extending the notion from [Sk] and [Hay] (among others). In
addition, Equations (6) and (7) can be seen as invariance under the slash
operators arising from a Hermitian version of a Heisenberg group as in Remark
2.4 of [Ze3], or from a semi-direct product involving this Heisenberg group
and the group $\Gamma$ (or, in the real case, all of
$\operatorname{U}(1,1)(\mathbb{R})$ or of
$\operatorname{SU}(1,1)(\mathbb{R})$). One way of considering groups that
contain non-integral entries involves characteristics as in Section 4 of [Ze3]
or in Theorem 3.8 below, but we shall mainly be using the operators from
Definition 2.5 and their consequences.

We recall that for $k$ and $l$ in $\mathbb{Z}$, a _modular form of weight
$(k,l)$ and representation $\rho_{\overline{M}}$ with respect to $\Gamma$_ is
a function $F:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$ that satisfies, for
every $\tau\in\mathcal{H}$ and $A\in\Gamma$ the functional equation

$F(A\tau)=j(A,\tau)^{k}\overline{j(A,\tau)}^{l}\rho_{\overline{M}}(A)F(\tau),$
(8)

and that is smooth and grows at most linearly exponentially towards any cusp
of $\Gamma$. A weight $(k,0)$ will be written simply as weight $k$, as usual,
with $F$ being _weakly holomorphic_ in case it is holomorphic on $\mathcal{H}$
with this growth condition, _holomorphic_ if in addition it is bounded at the
cusps of $\Gamma$, and _cuspidal_ when in even decreases towards them. Using
this notion we can now state the main relation involving Hermitian Jacobi
forms.

######  Theorem 3.4.

Let $\Phi$ be a Hermitian Jacobi form of weight $(k,l)$ and index $(M,v)$ with
respect to $\Gamma$, as in Definition 7, and let $F$ be the function
associated with $\Phi$ via Corollary 3.2. Then $F$ is a modular form of weight
$(k-b_{+},l-b_{-})$ and representation $\rho_{\overline{M}}$ with respect to
$\Gamma$. If $F$ is such a modular form, then the function defined by
$\Phi_{M,v}^{F}(\tau,\zeta,\omega):=[\Theta_{M}(\tau,\zeta,\omega;v),F(\tau)]_{M}$
is a Hermitian Jacobi form with these parameters.

###### Proof.

It is clear from the construction that the pairing with $\Theta_{M}$ in one
direction, and the expansion from Lemma 3.1 and Corollary 3.2 in the other
one, are inverse mappings between spaces of functions. We hence need to show
that pairing a modular form with $\Theta_{M}$ yields a Hermitian Jacobi form,
and that the function arising from the expansion of a Hermitian Jacobi form is
a vector-valued modular form.

So let $\Phi$ be as in Definition 7, and let
$F:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$ be the associated function from
Corollary 3.2. The assumption on the growth is part of Definition 7, so we
have to check Equation (8). Take $\tau\in\mathcal{H}$, $\zeta\in
M_{\mathbb{C}}$, $\omega\in M_{\overline{\mathbb{C}}}$, and $A\in\Gamma$, and
apply Corollary 3.2 to the two sides in Equation (7). On the right hand side
we get

$j(A,\tau)^{k}\overline{j(A,\tau)}^{l}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}[\Theta_{M}(\tau,\zeta,\omega;v),F(\tau)]_{M},$

and for the left hand side we apply Theorem 2.8 to the theta function in the
pairing and obtain, by bilinearity, the expression

$j(A,\tau)^{b_{+}}\overline{j(A,\tau)}^{b_{-}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}[\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega;v),F(\tau)]_{M}.$

As the pairing is $\operatorname{U}(1,1)(\mathbb{Z})$-equivariant, we can
replace $\rho_{M}(A)$ on one side by $\rho_{\overline{M}}(A)^{-1}$, so by
comparing the two expressions we obtain that the $[\cdot,\cdot]_{M}$-pairing
of $\Theta_{M}(\tau,\zeta,\omega;v)$ with
$j(A,\tau)^{k-b_{+}}\overline{j(A,\tau)}^{l-b_{-}}F(\tau)$ and with
$\rho_{\overline{M}}(A)^{-1}F(A\tau)$, both functions of $\tau$ alone,
coincide as functions $\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}$. But since the proof of Corollary 3.2 shows that
the pairing with $\Theta_{M}$ determines the function on $\mathcal{H}$ (this
is just the uniqueness of Fourier expansions), we obtain that $F$ satisfies
the modularity condition from Equation (8), with the desired weights.

Let now $F$ be a modular form of weight $(k-b_{+},l-b_{-})$ and representation
$\rho_{\overline{M}}$ with respect to $\Gamma$, and set $\Phi_{M,v}^{F}$ by
the formula above. The fact that $F$ does not depend on $\zeta$ and $\omega$
implies, via Proposition 2.7 and the bilinearity of the pairing
$[\cdot,\cdot]_{M}$, that Equation (6) holds for $\Phi=\Phi_{L,v}^{F}$, and
that $\Phi_{L,v}^{F}$ is holomorphic in $\zeta$ and $\omega$. Take now
$A\in\Gamma$, and apply Theorem 2.8 and Equation (8) to transform the left
hand side of Equation (7), with $\Phi=\Phi_{M,v}^{F}$, into

$\mathbf{e}\Big{(}\\!\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\\!\Big{)}j(A,\tau)^{k}\overline{j(A,\tau)}^{l}[\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega;v),\\!\rho_{\overline{M}}(A)F(\tau)]_{M}\\!.$

But the equivariance of the pairing allows us to omit the Weil representations
in the latter formula, yielding the desired right hand side by the definition
of $\Phi_{M,v}^{F}$. The fact that the function whose pairing with
$\Theta_{M}$ yields $\Phi_{M,v}^{F}$ is $F$ by definition, for which the
growth condition holds by assumption, complete the verification that
$\Phi_{M,v}^{F}$ has all the properties required in Definition 7. This proves
the theorem. ∎

######  Remark 3.5.

Consider, in Equation (7), the case where $A\in\Gamma$ is a power $T^{N}$ of
$T$ (for an appropriate $N$, this power will indeed by in $\Gamma$), where $j$
and $j^{c}$ equal 1 and $j^{\prime}$ and its complex conjugate vanish. This
implies, by Definition 7, that if $\Phi$ is a Hermitian Jacobi form, then it
is $N$-periodic in $\tau$ (for fixed $\zeta$ and $\omega$), which turns the
Fourier expansion from Lemma 3.1 into the form

$\textstyle{\sum_{m\in\frac{1}{N}\mathbb{Z}}\sum_{\lambda\in
M^{*}}c_{m,\lambda}(y)\mathbf{e}\big{(}mx+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}e^{-2\pi(|\lambda_{v_{+}}|^{2}-|\lambda_{v_{-}}|^{2})y}}.$

Equation (6) would then imply the equality
$c_{m,\lambda}=c_{m+\langle\sigma,\lambda\rangle+\langle\lambda,\sigma\rangle+|\sigma|^{2},\lambda+\sigma}$
of functions of $y$ for any $\sigma\in M$ and $\lambda\in M^{*}$, resembling
more the relations showing up in [Hav1], [Hav2], and others (the change in $m$
can also be written as $(\lambda,\sigma)+\frac{\sigma^{2}}{2}$ using the
associated orthogonal pairing, as in the orthogonal setting). By setting
$a_{n,\lambda}$ to be $c_{n+|\lambda|^{2},\lambda}$ (as functions of $y$
again) for any $n\in\mathbb{Q}$ for which the first index lies in
$\frac{1}{N}\mathbb{Z}$, we deduce that $a_{n,\lambda}$ depends on $\lambda$
only modulo $M$, allowing us to write this function as $a_{n,\gamma}$ where
$\gamma$ is now in $D_{M}$. Since $c_{m,\lambda}$ is now given by
$a_{m-|\lambda|^{2},\lambda+M}$, we can replace $m$ by $n=m-|\lambda|^{2}$ in
the previous expansion, which will take it to the form

$\sum_{n\in\mathbb{Q}}\sum_{\gamma\in
D_{M}}a_{n,\lambda+L}(y)\mathbf{e}(nx)\sum_{\lambda\in
M+\gamma}\mathbf{e}\big{(}|\lambda|^{2}x+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}e^{-2\pi(|\lambda_{v_{+}}|^{2}-|\lambda_{v_{-}}|^{2})y}.$

But the internal sum over $\lambda$ is just
$\theta_{M+\gamma}(\tau,\zeta,\omega;v)$ via Equation (4), and multiplying
each element of the first sum by $\mathfrak{e}_{\gamma}^{*}$ produces the
Fourier expansion of the function $F$ defined in Corollary 3.2 (namely the
modular form $F$ for which we have $\Phi=\Phi_{M,v}^{F}$ in Theorem 3.4). But
working with $F(\tau)$ itself directly makes the argument easier, as we do not
have to consider convergence issues for such series.

In general, Theorem 3.4 involve non-holomorphic expressions in $\tau$, and
indeed, Jacobi forms of indefinite lattice index will never be holomorphic
(though see Theorem 6.2 below for the positive definite case). The
holomorphicity question for vector-valued modular forms shows up on the
Hermitian Jacobi forms side via the following analogue of Proposition 2.7 of
[Ze3].

######  Proposition 3.6.

Let $F$ be a modular form of weight $(k-b_{+},l-b_{-})$ and representation
$\rho_{\overline{M}}$ with respect to $\Gamma$, with the associated Hermitian
Jacobi form $\Phi_{M,v}^{F}$ from Theorem 3.4. The modular form $F$ is then
weakly holomorphic if and only if $\Phi_{M,v}^{F}$ is pseudo-holomorphic, a
situation which can occur only when the weight of $\Phi_{M,v}^{F}$ is
$(k,b_{-})$.

###### Proof.

Apply the operator $4\pi i\partial_{\overline{\tau}}-\Delta_{v_{-}}^{h}$ from
Definition 7 to $\Phi_{M,v}^{F}(\tau,\zeta,\omega)$, considered as the pairing
of $\Theta_{M}(\tau,\zeta,\omega;v)$ and $F(\tau)$, with the latter
independent of $\zeta$ and $\omega$, and using Leibnitz’ rule for
$\partial_{\overline{\tau}}$, produces the expression

$\big{[}(4\pi
i\partial_{\tau}-\Delta_{v_{-}}^{h})\Theta_{M}(\tau,\zeta,\omega;v),F(\tau)\big{]}_{M}+4\pi
i\big{[}\Theta_{M}(\tau,\zeta,\omega;v),\partial_{\overline{\tau}}F(\tau)\big{]}_{M}.$

Proposition 2.9 yields the vanishing of the first term, and as a
$\mathbb{C}[D_{\overline{M}}]$-valued function of $\tau$ is determined by its
pairing with $\Theta_{M}$, we deduce from Definition 7 that $\Phi_{M,v}^{F}$
is pseudo-holomorphic precisely when $\partial_{\overline{\tau}}F=0$. As the
modular form $F$ is smooth by assumption and grows at most linearly
exponentially at the cusps, the latter condition holds if and only if $F$ is
weakly holomorphic. This proves the first assertion, from which the second one
immediately follows as weakly holomorphic modular forms have weights $(k,0)$,
with $l=0$. This proves the proposition. ∎

######  Remark 3.7.

We replace $a_{n,\gamma}(y)$ from Remark 3.5 by
$\tilde{a}_{n,\gamma}(y)e^{-2\pi ny}$, to present the Fourier expansion of $F$
there using the holomorphic exponentials $\mathbf{e}(n\tau)$. Then
$c_{m,\lambda}(y)=a_{m-|\lambda|^{2},\lambda+M}(y)$ becomes
$\tilde{c}_{m,\lambda}(y)e^{-2\pi my+2\pi|\lambda|^{2}y}$, with the former
multiplier being $\tilde{a}_{m-|\lambda|^{2},\lambda+M}(y)$. The full
expansion $\Phi_{L,v}^{F}(\tau,\zeta)$ then takes the form
$\sum_{m\in\frac{1}{N}\mathbb{Z}}\sum_{\lambda\in
M^{*}}\tilde{c}_{m,\lambda}(y)\mathbf{e}\big{(}m\tau+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}e^{4\pi|\lambda_{v_{-}}|^{2}y}$,
using the fact that
$|\lambda|^{2}=|\lambda_{v_{+}}|^{2}+|\lambda_{v_{-}}|^{2}$. Noting that $F$
is weakly holomorphic precisely when $\tilde{c}_{m,\lambda}$ is a constant for
every $m$ and $\lambda$, which happens if and only if $4\pi
i\partial_{\overline{\tau}}$ acts only on the exponentials in $y$. As
$\Delta_{v_{-}}^{h}$ operates only on the terms with $\zeta$ and $\omega$, the
idea behind Proposition 3.6 is clearer using these conventions.

Remark 3.7 is particularly valuable in the case where $M$ is positive
definite, as seen in Equation (20) below.

Section 3 of [Ze3] contains a lot of relations that are satisfied between
constructions on lattices, modular forms, and Jacobi forms. They all extend,
_mutatis mutandis_ , to the case of $\mathcal{O}$-lattices and Hermitian
Jacobi forms. Here are the important details to note when changing the setting
to the Hermitian one:

  * •

The isomorphism from Lemma 3.1 there extends to the representation of
$\operatorname{U}(1,1)(\mathbb{Z})$ constructed in Proposition 2.3, by adding
conjugation to all the entries and hence inverting the scalar matrices from
$\mu(\mathcal{O})$. This can then be used to construct skew-holomorphic
Hermitian Jacobi forms, via the other operator from Proposition 2.9.

  * •

The over-lattices of an $\mathcal{O}$-lattice, for which one considers the
arrow operators (these are introduced here in Equation (10) below), can be an
even projective $\mathcal{O}$-lattice containing $M$, or obtained by extending
scalars to a larger order in $\mathbb{K}$, or combining these operations.
Lemma 1.11 and Corollary 1.12 transfer it to orthogonal over-lattices over
$\mathbb{Z}$.

  * •

Orthogonal direct sums of $\mathcal{O}$-lattices yields an
$\mathcal{O}$-lattice if and only if both lattices are over the same order
$\mathcal{O}$ in $\mathbb{K}$ (this was already exemplified in Remarks 1.10
and 1.13). The module structure of Hermitian Jacobi forms over the ring of
scalar-valued modular forms holds equally well.

  * •

A primitive sub-lattice of an $\mathcal{O}$-lattice is an
$\mathcal{O}$-lattice, since it is a direct summand of a projective
$\mathcal{O}$-module.

  * •

The theta contraction from [Ma], or the restriction from [Ze5], extend to the
unitary setting via embeddings like that from [Ho2], namely via the embedding
of the space from Equation (2) into that from Equation (1).

  * •

The isomorphism $\iota$ from Proposition 3.9 of [Ze3] should respect the
action of $\mathcal{O}$, or equivalently of the endomorphism $J$. Then $v$ and
$w$ there are from the joint $\operatorname{Gr}^{J}$-space.

In addition, as in Remark 2.9 of [Ze3], one can fix the modular form $F$ and
investigate the properties of
$(\tau,\zeta,\omega,v)\mapsto\Phi_{M,v}^{F}(\tau,\zeta,\omega)$ as a function
of variables from the product $\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\times\operatorname{Gr}^{J}(V_{\mathbb{R}})$ (in
particular, it is invariant under the diagonal action on the last three
variables of the stable unitary group of $M$, acting trivially on $D_{M}$).

The theta functions from Section 4 of [Bor] may also involve characteristics,
and Section 4 of [Ze3] presents the basic properties of Jacobi theta functions
with characteristics. Since such theta functions are, in fact, used implicitly
in [Xi] and more explicitly in [BRZ] (at least in the holomorphic, positive
definite setting), we define them here and state their functional equations.
With $M$, $\tau$, $\zeta$, $\omega$, and $v$ as above, we take two elements
$\alpha$ and $\beta$ in $M_{\mathbb{R}}$, and define the _Hermitian
Jacobi–Siegel theta function with characteristics $\binom{\alpha}{\beta}$_ to
be the expression
$\Theta_{M}\big{(}\tau,\zeta,\omega;\binom{\alpha}{\beta};v\big{)}=\sum_{\gamma\in
D_{M}}\theta_{M+\gamma}\big{(}\tau,\zeta,\omega;\binom{\alpha}{\beta};v\big{)}\mathfrak{e}_{\gamma}$
in which the scalar-valued component is the sum over $\lambda\in M+\gamma$ of

$\mathbf{e}\big{(}\tau|(\lambda+\beta)_{v_{+}}|^{2}+\overline{\tau}|(\lambda+\beta)_{v_{-}}|^{2}+\langle\zeta,\lambda+\beta\rangle+\langle\lambda+\beta,\omega\rangle\big{)}-\big{\langle}\lambda+\tfrac{\beta}{2},\alpha\big{\rangle}-\big{\langle}\alpha,\lambda+\tfrac{\beta}{2}\big{\rangle}\big{]}.$
(9)

Combining the proofs of Theorem 4.1 from [Ze3] with Lemma 2.4 (whose proof
relates the theta function from Equation (9) to the one with characteristics
defined in Equation (16) of [Ze3], with the variable $\zeta+\omega\in
M\otimes_{\mathbb{Z}}\mathbb{C}$) as in the proofs of Proposition 2.7 and
Theorem 2.8, as well as Proposition 2.9 for the last assertion, yields the
following result.

######  Theorem 3.8.

The expression

$\Theta_{M}\big{(}\tau,\zeta+\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}},\omega+\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}};\textstyle{\binom{\alpha}{\beta}};v\big{)}$

is equal, when $\sigma$ and $\nu$ are elements of $M$, to

$\mathbf{e}\big{(}-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle+\langle\nu,\beta\rangle+\langle\beta,\nu\rangle+\langle\sigma,\alpha\rangle+\langle\alpha,\sigma\rangle\big{)}$

times $\Theta_{M}\big{(}\tau,\zeta,\omega;\binom{\alpha}{\beta};v\big{)}$.
When we view an element $A\in\operatorname{U}(1,1)(\mathbb{Z})$ as a $2\times
2$ matrix with entries in $\mathcal{O}$, which can thus act on elements of $M$
hence of $M_{\mathbb{R}}$, we have the equality

$\Theta_{M}\Big{(}A\tau,\tfrac{\zeta_{v_{+}}}{j(A,\tau)}+\tfrac{\zeta_{v_{-}}}{\overline{j^{c}(A,\tau)}},\tfrac{\omega_{v_{+}}}{j^{c}(A,\tau)}+\tfrac{\omega_{v_{-}}}{\overline{j(A,\tau)}};\textstyle{\binom{\alpha}{\beta}};v\Big{)}=$
$=j(A,\tau)^{b_{+}}\overline{j(A,\tau)}^{b_{-}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega;\textstyle{\binom{\alpha}{\beta}};v).$

The theta functions from Equation (9) are also annihilated by all the
differential operators mentioned in Proposition 2.9.

Note that while the proof of Theorem 4.1 of [Ze3] suffices (via Lemma 2.4) for
establishing the second equality in Theorem 3.8 when the matrix $A$ is in
$\operatorname{SU}(1,1)(\mathbb{Z})=\operatorname{SL}_{2}(\mathbb{Z})$, the
considerations from the proof of Theorem 2.8 apply here too (with $\beta$ also
multiplied by $\overline{\xi}$, as is $\alpha$ for preserving the pairings
involved) for extending it to the full representation of
$\operatorname{SU}(1,1)(\mathbb{Z})$, as defined in Proposition 2.3. As in
Proposition 4.2 of that reference, the functions on $\mathcal{H}\times
M_{\mathbb{C}}\times M_{\overline{\mathbb{C}}}$ that satisfy the first
equality from Theorem 3.8 and are holomorphic in $\zeta$ and $\omega$ are
$[\cdot,\cdot]_{M}$-pairings of the theta function from Equation (9) with
smooth $\mathbb{C}[D_{\overline{M}}]$-valued functions (that are holomorphic
if and only if the pairing is pseudo-holomorphic), but the pairings are no
longer Jacobi forms when $\binom{\alpha}{\beta}\neq\binom{0}{0}$. The
similarity with the theta functions with characteristics as in [FZ] and
others, and the relation with the arrow operator as in Proposition 4.4 there,
are valid here equally well.

##  4 Vector–Valued Jacobi Forms

The vector-valued modular and Jacobi forms (either Hermitian and orthogonal)
will be based on Weil representations associated with discriminant forms, that
have to be appropriately related to $D_{M}$ for our lattice $M$.

We thus formally define the objects with which we shall work.

######  Definition 4.1.

A _discriminant form_ is finite Abelian group $D$ with a
$\mathbb{Q}/\mathbb{Z}$-valued quadratic form, denoted by
$\gamma\mapsto\frac{\gamma^{2}}{2}$, such that the associated bilinear pairing
$(\cdot,\cdot):D\times D\to\mathbb{Q}/\mathbb{Z}$ is non-degenerate in the
sense that the homomorphism that it defines from $D$ to
$\operatorname{Hom}(D,\mathbb{Q}/\mathbb{Z})$ is injective hence bijective.
For a subgroup $H\subseteq D$, we denote by $H^{\perp}$ the set of elements of
$D$ pairing trivially with $H$. A subgroup $H\subseteq D$ is called
_isotropic_ if $\frac{\gamma^{2}}{2}=0$ for all $\gamma\in H$.

If $H$ is an isotropic subgroup of $D$ as in Definition 4.1, then $H$ pairs
trivially with itself, namely we have $H\subseteq H^{\perp}$, and the quotient
$\Delta:=H^{\perp}/H$ is non-degenerate with the induced bilinear pairing.
However, not every $H$ that is contained in $H^{\perp}$ is isotropic, and only
the quotients arising from isotropic subgroups $H$ carry well-defined
$\mathbb{Q}/\mathbb{Z}$-valued quadratic forms, and are thus discriminant
forms themselves.

######  Remark 4.2.

Note that discriminant forms like $D=D_{M}$ for an $\mathcal{O}$-lattice $M$
carry additional structure, namely an endomorphism induced by $J$ that squares
to $-d\operatorname{Id}_{D}$ and satisfies condition $(iii)$ of Lemma 1.5,
whence also the conditions $(ii)$, $(iv)$, and $(v)$ there for the pairings.
This structure carries over to $\Delta$ in case $J$ preserves $H$, and
commutes with operations like the arrow operators defined below. However, this
will play no role in what follows.

For such $D$, $H$, and $\Delta$ there are two arrow operators, which we denote
by $\uparrow^{D}_{\Delta}=\uparrow_{H}:\mathbb{C}[\Delta]\to\mathbb{C}[D]$ and
$\downarrow^{D}_{\Delta}=\downarrow_{H}:\mathbb{C}[D]\to\mathbb{C}[\Delta]$,
and their actions on the natural bases is given by

$\uparrow_{H}\mathfrak{e}_{\gamma}:=\sum_{\delta\in H^{\perp},\
\delta+H=\gamma}\mathfrak{e}_{\delta}\qquad\mathrm{and}\qquad\downarrow_{H}\mathfrak{e}_{\delta}:=\begin{cases}\mathfrak{e}_{\delta+H},&\mathrm{when\
}\delta\in H^{\perp},\\\ 0,&\mathrm{if\ }\delta\not\in H^{\perp}.\end{cases}$
(10)

As is shown in [Ma] and considered further in [Ze5] (but has already been
known before), both maps are homomorphisms of representations of
$\operatorname{SL}_{2}(\mathbb{Z})$ (or of its metaplectic cover in case the
signature is odd) when the spaces are considered as the representation spaces
of the corresponding Weil representations (a fact that was used in [Ze4] and
others for investigating reducibility and related questions), and they are
dual to one another when considering pairings like $[\cdot,\cdot]_{M}$ from
above.

One of the most natural settings in which these operators show up is when $M$
is a primitive sublattice of another lattice $N$, of higher rank. Then if
$M^{\perp}$ stands for the orthogonal complement of $M$ in $N$, then $N$
contains $M\oplus M^{\perp}$, the discriminant form of this direct sum is
$D_{M}\oplus D_{M^{\perp}}$, and the isotropic subgroup $N/(M\oplus
M^{\perp})$ of this direct sum intersects each of the defining summands
trivially. Since the lattice $M$ (and not only $D_{M}$) is part of the data,
but the lattices $M^{\perp}$ and $N$ are not, we make the following
definition.

######  Definition 4.3.

Let $D$ be a any discriminant form. Then a subgroup $H$ of $D_{M}\oplus D$ is
called _horizontal_ if $H\cap D_{M}=H\cap D=\\{0\\}$ inside $D_{M}\oplus D$.

We shall use, for elements $\alpha\in D_{M}\oplus D$, the notations
$\alpha_{M}$ and $\alpha_{D}$ for the images of $\alpha$ under the natural
projections onto $D_{M}$ and $D$ respectively. For horizontal isotropic
subgroups we now prove the following simple lemma, which is mainly used for
introducing our notation.

######  Lemma 4.4.

The following assertions hold:

  1. $(i)$

Let $H_{M}\subseteq D_{M}$ be any subgroup, and assume that we have an
injective homomorphism $\iota:H_{M}\to D$. Then
$H:=\\{\gamma_{M}+\iota(\gamma_{M})|\gamma\in H_{M}\\}$ is a horizontal
subgroup of $D_{M}\oplus D$, and every horizontal subgroup is obtained in this
way.

  2. $(ii)$

An element $\alpha\in D_{M}\oplus D$ lies in $H^{\perp}$ for $H$ from part
$(i)$ if and only if the equality
$(\alpha_{M},\gamma_{M})=-\big{(}\alpha_{D},\iota(\gamma_{M})\big{)}$ holds
for every $\gamma\in H$.

  3. $(iii)$

The group $H$ is isotropic if and only if the equality
$\frac{\iota(\gamma_{M})^{2}}{2}=-\frac{\gamma_{M}^{2}}{2}$ holds for any
$\gamma_{M}\in H_{M}$.

###### Proof.

For every non-zero element $\gamma\in H$, we know that $\gamma_{M}\neq 0$ as
well as $\gamma_{D}=\iota(\gamma_{M})\neq 0$ (as $\iota$ is injective),
yielding the horizontality from Definition 4.3. Conversely, let $H\subseteq
D_{M}\oplus D$ be horizontal, and let $H_{M}$ denote the image of $H$ under
the projection to $D_{M}$. The fact that $H$ intersects $D$ trivially (by
Definition 4.3) implies the projection map from $H$ to $H_{M}$ is injective
hence bijective, and we can thus invert it. The map $\iota$ is then the
composition of this inverted map with the projection onto $D$, and as this
projection is also injective when restricted to $H$ (Definition 4.3 implies
that since $H$ intersects $D_{M}$ trivially as well), we deduce that $\iota$
is also injective. This proves part $(i)$.

Part $(ii)$ then follows from the evaluation of the pairing of
$\alpha=\alpha_{M}+\alpha_{D}$ and $\gamma=\gamma_{M}+\iota(\gamma_{M})\in H$
and the orthogonality of $D_{M}$ and $D$ inside their direct sum, and this
orthogonality is also applied in the direct and straightforward calculation
that is required for part $(iii)$. This proves the lemma. ∎

Using Lemma 4.4 and our notations for the projections, whenever a horizontal
subgroup $H$ of $D_{M}\oplus D$ is given, we will use $H_{M}$ and $H_{D}$ for
its images in $D_{M}$ and $D$ respectively, and the elements $\gamma\in H$,
$\gamma_{M}$, and $\gamma_{D}=\iota(\gamma_{M})$ will be related as in the
proof of that lemma.

For considering the quotient $\Delta_{H}:=H^{\perp}/H$ that arises from a
horizontal isotropic subgroup $H$ of $D_{M}\oplus D$, we shall need the
following description.

######  Lemma 4.5.

Let $H\subseteq D_{M}\oplus D$ be a horizontal subgroup, and fix a set
$R\subseteq D_{M}$ of representatives for the quotient $D_{M}/H_{M}$.

  1. $(i)$

Any element $\alpha\in D_{M}\oplus D$ can be uniquely written as a sum
$\beta+\gamma$, with $\gamma\in H$ and $\beta_{M}\in R$.

  2. $(ii)$

When $H$ is isotropic, the condition from part $(ii)$ of Lemma 4.4 holds for
$\alpha$ if and only if it holds for $\beta$.

  3. $(iii)$

In the isotropic case, the set $H^{\perp}_{R}$ of $\beta\in D_{M}\oplus D$ for
which $\beta_{M}\in R$ and the condition from part $(ii)$ of Lemma 4.4 is
satisfied map bijectively onto $\Delta_{H}$ in the natural projection. We
denote the image of such $\beta$ by $\overline{\beta}$.

  4. $(iv)$

The element $\mathfrak{e}_{\alpha}$ associated with $\alpha\in D_{M}\oplus D$,
decomposed as in part $(i)$, is sent by $\downarrow_{H}$ to
$\mathfrak{e}_{\overline{\beta}}$ in case the condition from part $(ii)$ of
Lemma 4.4 holds for $\alpha$ (or equivalently $\beta$), and to 0 otherwise.

###### Proof.

Part $(i)$ is a direct consequence from the fact that adding an element
$\gamma\in H$ changes the $D_{M}$-coordinate by $\gamma_{M}\in H_{M}$, and the
assumption on $R$. As $H$ is isotropic, elements $\gamma\in H$ lie in
$H^{\perp}$ and hence satisfy the condition in question, and part $(ii)$
follows. Part $(iii)$ is deduced via parts $(i)$ and $(ii)$ from the
definition of $\Delta_{H}$ as a quotient, and so is part $(iv)$ in view of the
definition of $\downarrow_{H}$ in Equation (10). This proves the lemma. ∎

######  Remark 4.6.

In case $H_{M}$, or equivalently $H_{D}$, is non-degenerate, the papers [Ma]
and [Ze5] give explicit formulae involving the arrow operator $\downarrow_{H}$
using the resulting orthogonal decomposition of $D_{M}$ and $D$. This
corresponds to taking the set $R$ in Lemma 4.5 to be $H_{M}^{\perp}$ (it is a
set of representatives for $D_{M}/H_{M}$ precisely when $H_{M}$ is non-
degenerate), and then there is a corresponding set for $D/H_{D}$ (which we
shall not need). We will, however, work more generally, without assuming
anything on $H_{M}$.

Considering $D_{M}\oplus D$ as a discriminant form, Definition 2.5 produces
the corresponding operators with any index in $D_{M}\oplus D$. In particular,
the operators having index from $D_{M}\subseteq D_{M}\oplus D$ act on
$\mathbb{C}[D_{M}\oplus D]$, which may be seen as an extension of the action
of the operators from Definition 2.5 (for $D_{M}$) to an action on this larger
space. We will need an induced action of the operators from Definition 2.5 on
$\mathbb{C}[\Delta_{H}]$.

######  Lemma 4.7.

The operators associated with elements of $M^{*}$ whose $D_{M}$-images are in
$H_{M}^{\perp}$ induce a well-defined action on $\mathbb{C}[\Delta_{H}]$.

###### Proof.

The subgroup $H_{M}^{\perp}$ of $D_{M}$, viewed as embedded into $D_{M}\oplus
D$, is contained in $H^{\perp}$, and clearly intersects the horizontal
subgroup $H$ trivially. It therefore admits an isomorphic copy also in
$\Delta_{H}$. Definition 2.5 then yields operators on $\mathbb{C}[\Delta_{H}]$
for every element of $\Delta_{H}$, and the desired operators are those arising
from this image of $H_{M}^{\perp}$ inside $\Delta_{H}$. This proves the lemma.
∎

Using the induced operators from Lemma 4.7, we can now consider pairings with
theta functions. Since we work with the Hermitian ones, we assume that $M$ is
a Hermitian $\mathcal{O}$-lattice, producing all the objects defined above.
However, recall that we represent the basis for $\mathbb{C}[\Delta_{H}]$
through the set $H^{\perp}_{R}$ from Lemma 4.5, which is based on a set of
representatives $R\subseteq D_{M}$ for $D_{M}/H_{M}$. As this set need not
respect the addition from $H_{M}^{\perp}$, we shall be using the following
notation and its properties.

######  Lemma 4.8.

Let a set $R\subseteq D_{M}$ of representatives for $D_{M}/H_{M}$ be given.

  1. $(i)$

For $\delta\in D_{M}$ and $\sigma\in M^{*}$, there exists a unique pair of
elements $r_{\sigma}(\delta)\in R$ and $h_{\sigma}(\delta)\in H$ such that
$\delta+(\sigma+M)\in D_{M}$ equals
$r_{\sigma}(\delta)+h_{\sigma}(\delta)_{M}$.

  2. $(ii)$

We have the equalities $r_{-\sigma}\big{(}r_{\sigma}(\delta)\big{)}=\delta$
and $h_{-\sigma}\big{(}r_{\sigma}(\delta)\big{)}=-h_{\sigma}(\delta)$.

###### Proof.

Part $(i)$ follows directly from the fact that $R$ is a set of representatives
for $D_{M}/H_{M}$ and the projection from $H$ onto $H_{M}$ is bijective. For
part $(ii)$ we just note that the defining equation from part $(i)$ yields the
equality of $r_{\sigma}(\delta)-(\sigma+M)$ and $\delta-
h_{\sigma}(\delta)_{M}$, and we apply the uniqueness from part $(i)$ with
$\delta$ replaced by $r_{\sigma}(\delta)$ and $\sigma$ inverted. This proves
the lemma. ∎

We can now state and prove the following result.

######  Proposition 4.9.

Let $F:\mathcal{H}\to\mathbb{C}[D]$ be any smooth function. Then the function
$\Phi_{M,v}^{F,H}:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}[\Delta_{H}]$ taking
$(\tau,\zeta,\omega)$ to
$\downarrow_{H}\big{(}\Theta_{M}(\tau,\zeta,\omega;v)\otimes F(\tau)\big{)}$
is holomorphic in the variables $\zeta$ and $\omega$ and satisfies, for every
$\sigma$ and $\nu$ from $M^{*}$ with $\sigma+M$ and $\nu+M$ in
$H_{M}^{\perp}$, the equality

$\Phi_{M,v}^{F,H}(\tau,\zeta+\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}},\omega+\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}})=$
$=\mathbf{e}\big{(}-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle\big{)}\chi_{\nu}t_{\sigma}\Phi_{M,v}^{F,H}(\tau,\zeta,\omega).$

###### Proof.

By writing $F=\sum_{\delta\in D}f_{\delta}\mathfrak{e}_{\delta}$, the tensor
product $\Theta_{M}\otimes F$ (with the appropriate values) is, under the
natural identification of $\mathbb{C}[D_{M}]\otimes_{\mathbb{C}}\mathbb{C}[D]$
with $\mathbb{C}[D_{M}\oplus D]$, the $\mathbb{C}[D_{M}\oplus D]$-valued
function given by $\sum_{\alpha\in D_{M}\oplus
D}\theta_{\alpha_{M}}f_{\alpha_{D}}\mathfrak{e}_{\alpha}$. Then its
$\downarrow_{H}$-image $\Phi_{M,v}^{F,H}$ is given, by the parts of Lemma 4.5,
by

$\Phi_{M,v}^{F,H}(\tau,\zeta,\omega)=\textstyle{\sum_{\beta\in
H^{\perp}_{R}}\big{[}\sum_{\gamma\in
H}\theta_{\beta_{M}+\gamma_{M}}(\tau,\zeta,\omega;v)f_{\beta_{D}+\gamma_{D}}(\tau)\big{]}\mathfrak{e}_{\overline{\beta}}}.$
(11)

For the periodicity formula, Proposition 2.7 and its proof show that changing
$\zeta$ and $\omega$ by the parameters arising from $\sigma$ and $\nu$ yields
the asserted exponent times the expression

$\textstyle{\sum_{\beta\in H^{\perp}_{R}}\big{[}\sum_{\gamma\in
H}\mathbf{e}\big{(}(\beta_{M}+\gamma_{M},\nu+M)\big{)}\theta_{\beta_{M}+\gamma_{M}+(\sigma+M)}(\tau,\zeta,\omega;v)f_{\beta_{D}+\gamma_{D}}(\tau)\big{]}\mathfrak{e}_{\overline{\beta}}}.$

But if $\nu+M\in H_{M}^{\perp}$ then the pairing in this exponent reduces to
$(\beta_{M},\nu+M)$, or equivalently $(\beta,\nu+M)$ (or also
$(\overline{\beta},\nu+M)$), independently of $\gamma$. Moreover, if
$\sigma+M\in H_{M}^{\perp}$ then we write $\beta_{M}+\gamma_{M}+(\sigma+M)$ as
$r_{\sigma}(\beta_{M})+\gamma_{M}+h_{\sigma}(\beta_{M})_{M}$ via part $(i)$ of
Lemma 4.8, and thus $\beta_{D}+\gamma_{D}$ becomes the sum of
$\beta_{D}-h_{\sigma}(\beta_{M})_{D}$ and the corresponding element
$\gamma_{D}+h_{\sigma}(\beta_{M})_{D}$ of $H_{D}$. By taking $\tilde{\beta}$
to be with $\tilde{\beta}_{M}=r_{\sigma}(\beta_{M})$ and
$\tilde{\beta}_{D}=\beta_{D}-h_{\sigma}(\beta_{M})_{D}$, and
$\tilde{\gamma}:=\gamma+h_{\sigma}(\beta_{M})$, we deduce from part $(ii)$ of
that Lemma 4.8 that $\beta_{M}=r_{-\sigma}(\tilde{\beta}_{M})$ and
$\beta_{D}=\tilde{\beta}_{D}-h_{-\sigma}(\tilde{\beta}_{M})_{D}$. We can thus
take the sum to be over $\tilde{\beta}\in H^{\perp}_{R}$ and
$\tilde{\gamma}\in H$, with the theta function
$\theta_{\tilde{\beta}_{M}+\tilde{\gamma}_{M}}$ and the coefficient
$f_{\tilde{\beta}_{D}+\tilde{\gamma}_{D}}$, and the associated vector
$\mathfrak{e}_{\overline{\beta}}$ being the one associated with the
$\Delta_{H}$-image of $\tilde{\beta}-(\sigma+M)$. As this combines with the
effect of the exponents $(\overline{\beta},\nu+M)$ to yield the effect of
$\chi_{\nu}t_{\sigma}$ from Lemma 4.7 on
$\Phi_{M,v}^{F,H}(\tau,\zeta,\omega)$, the desired formula follows. This
proves the proposition. ∎

######  Remark 4.10.

When $\sigma$ and $\nu$ are in $M$, the relation from Proposition 4.9 amounts
to the scalar-valued components of
$\Phi=\Phi_{M,v}^{F,H}=\sum_{\beta\in\Delta_{H}}\Phi_{\beta}\mathfrak{e}_{\beta}$
satisfying the one from Equation (6). Therefore we can, by Corollary 3.2,
write every smooth function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}[\Delta_{H}]$ that is holomorphic in the
variables from $M_{\mathbb{C}}$ and from $M_{\overline{\mathbb{C}}}$ and
satisfies this relation (given also in Equation (12) below) as
$\Phi(\tau,\zeta,\omega)=\sum_{\beta\in\Delta_{H}}[\Theta_{M}(\tau,\zeta,\omega;v),F_{\beta}(\tau)]_{M}\mathfrak{e}_{\beta}$
for smooth functions $F_{\beta}:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$,
which are determined by $\Phi$.

We can now define our vector-valued Hermitian Jacobi forms.

######  Definition 4.11.

Let an $\mathcal{O}$-lattice $M$ from Definition 1.9, another discriminant
form $D$, a horizontal isotropic subgroup $H\subseteq D_{M}\oplus D$ with
image $H_{M}\subseteq D_{M}$, weights $k$ and $l$ from $\mathbb{Z}$, a
subgroup $\Gamma\subseteq\operatorname{U}(1,1)(\mathbb{Z})$ of finite index,
and an element $v\operatorname{Gr}^{J}(M_{\mathbb{R}})$ from Equation (2) be
given. We then set $\Delta_{H}:=H^{\perp}/H$, and define a _Hermitian Jacobi
form of weight $(k,l)$, index $(M,H,v)$, and representation
$\rho_{\Delta_{H}}$ with respect to $\Gamma$_ be a function
$\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}[\Delta_{H}]$, that is smooth and has
the following properties. The first one is the usual holomorphicity in the
variables from $M_{\mathbb{C}}$ and $M_{\overline{\mathbb{C}}}$. For the
second one, recall from Lemma 4.7 the operators on $\mathbb{C}[\Delta_{H}]$
that are induced from those from Definition 2.5, and then for
$\tau\in\mathcal{H}$, $\zeta\in M_{\mathbb{C}}$, $\omega\in
M_{\overline{\mathbb{C}}}$, and $\sigma$ and $\nu$ from $M^{*}$ with
$D_{M}$-images in $H_{M}^{\perp}$, the equality

$\Phi(\tau,\zeta+\tau\sigma_{v_{+},\mathbb{C}}+\overline{\tau}\sigma_{v_{-},\mathbb{C}}+\nu_{\mathbb{C}},\omega+\tau\sigma_{v_{+},\overline{\mathbb{C}}}+\overline{\tau}\sigma_{v_{-},\overline{\mathbb{C}}}+\nu_{\overline{\mathbb{C}}})=$
$=\mathbf{e}\big{(}-\tau|\sigma_{v_{+}}|^{2}-\overline{\tau}|\sigma_{v_{-}}^{2}|-\langle\zeta,\sigma\rangle-\langle\sigma,\omega\rangle\big{)}\chi_{\nu}t_{\sigma}\Phi(\tau,\zeta,\omega)$
(12)

has to be satisfied. For the third one, let $j(A,\tau)$ and $j^{c}(A,\tau)$ be
as in Theorem 2.8, and then for such $\tau$, $\zeta$, and $\omega$, and for
$A\in\Gamma$, we require the equality

$\Phi\Big{(}A\tau,\tfrac{\zeta_{v_{+}}}{j(A,\tau)}+\tfrac{\zeta_{v_{-}}}{\overline{j^{c}(A,\tau)}},\tfrac{\omega_{v_{+}}}{j^{c}(A,\tau)}+\tfrac{\omega_{v_{-}}}{\overline{j(A,\tau)}}\Big{)}=$
$=j(A,\tau)^{k}\overline{j(A,\tau)}^{l}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta_{v_{+}},\omega_{v_{+}}\rangle}{j(A,\tau)}+\tfrac{\overline{j_{A}^{\prime}}\langle\zeta_{v_{-}},\omega_{v_{-}}\rangle}{\overline{j(A,\tau)}}\Big{)}\rho_{\Delta_{H}}(A)\Phi(\tau,\zeta,\omega).$
(13)

The fourth one is the requirement that for each $\beta\in\Delta_{H}$, the
components of the associated function
$F_{\beta}:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$ from Remark 4.10 grow
at most linearly exponentially towards any cusp of $\Gamma$. The condition for
$\Phi$ to be _pseudo-holomorphic_ is as in Definition 7.

We can now prove our main result for vector-valued Hermitian Jacobi forms.

######  Theorem 4.12.

Any smooth function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}[\Delta_{H}]$ that satisfied Equation
(12) and the holomorphicity condition from Definition 13 is of the form
$\Phi_{M,v}^{F,H}$ from Proposition 4.9 for a smooth function
$F:\mathcal{H}\to\mathbb{C}[D]$, which is then unique. In this case $\Phi$ is
a Hermitian Jacobi form of weight $(k,l)$, index $(M,H,v)$, and representation
$\rho_{\Delta_{H}}$ with respect to a finite index subgroup
$\Gamma\subseteq\operatorname{U}(1,1)(\mathbb{Z})$ if and only $F$ is a
modular form of weight $(k-b_{+},l-b_{-})$ and representation $\rho_{D}$ with
respect to $\Gamma$, as in Equation (8). Moreover, $F$ is weakly holomorphic
if and only if $\Phi$ is pseudo-holomorphic, and then $l=b_{-}$.

###### Proof.

We shall be using the representing set $H^{\perp}_{R}$ from Lemma 4.5 for
expressing the elements of $\Delta_{H}$, where $R\subseteq D_{M}$ is some set
of representatives for $D_{M}/H_{M}$. Then if $\overline{\beta}\in\Delta_{H}$
is the image of $\beta\in H^{\perp}_{R}$ then Remark 4.10 allows us, via
Corollary 3.2, to write the component $\Phi_{\overline{\beta}}$ as the
$[\cdot,\cdot]_{M}$-pairing of $\Theta_{M}$ with a unique smooth function
$F_{\beta}:\mathcal{H}\to\mathbb{C}[D_{\overline{M}}]$. By expanding this
pairing, we obtain that we can write

$\textstyle{\Phi(\tau,\zeta,\omega)=\sum_{\beta\in
H^{\perp}_{R}}\big{[}\sum_{\alpha\in
D_{M}}\theta_{\alpha}(\tau,\zeta,\omega;v)\tilde{f}_{\beta,\alpha}(\tau)\big{]}\mathfrak{e}_{\overline{\beta}}},$

where $\\{\tilde{f}_{\beta,\alpha}\\}_{\alpha\in D_{M}}$ are the components of
$F_{\beta}$ for any $\beta\in H^{\perp}_{R}$.

We now consider $\sigma=0$ but any $\nu\in M^{*}$ with $\nu+M\in
H_{M}^{\perp}$ in Equation (12), and compare it with the corresponding
property of $\theta_{\alpha}$ from Proposition 2.7. The action of $\chi_{\nu}$
in Equation (12) implies that the summand associated with $\beta\in
H^{\perp}_{R}$ and $\alpha\in D_{M}$ has to be multiplied by
$\mathbf{e}\big{(}(\nu+M,\beta_{M})\big{)}$, while via Proposition 2.7 it has
to be multiplied by $\mathbf{e}\big{(}(\nu+M,\alpha)\big{)}$. As both sides
have to be equal and the theta expansions are unique, we deduce that this
summand can be non-zero, i.e., we have $f_{\beta,\alpha}\neq 0$, only when
$\mathbf{e}\big{(}(\nu+M,\beta_{M})\big{)}=\mathbf{e}\big{(}(\nu+M,\alpha)\big{)}$
for every such $\nu$. But the pairing with the elements $\nu+M\in
H_{M}^{\perp}$ determine the element of $D_{M}$ up to translations by $H_{M}$,
meaning that $\alpha$ has to be of the form $\beta_{M}+\gamma_{M}$ for
$\gamma\in H$. By writing each $\beta\in H^{\perp}_{R}$ as the sum of
$\beta_{M}\in R$ and $\beta_{D}\in D$ (which satisfy the equality from part
$(ii)$ of Lemma 4.4), it follows that our expansion becomes

$\textstyle{\Phi(\tau,\zeta,\omega)=\sum_{\beta\in
H^{\perp}_{R}}\big{[}\sum_{\gamma\in
H}\theta_{\beta_{M}+\gamma_{M}}(\tau,\zeta,\omega;v)\hat{f}_{\beta_{M}+\beta_{D},\gamma}(\tau)\big{]}\mathfrak{e}_{\overline{\beta_{M}+\beta_{D}}}},$
(14)

where $\hat{f}_{\beta,\gamma}$, for $\beta=\beta_{M}+\beta_{D}$, is the
function $\tilde{f}_{\beta,\beta_{M}+\gamma_{M}}$ from above.

We now turn to $\sigma\in M^{*}$ in Equation (12) and Proposition 2.7, and
observe that the exponents showing up in both of them are the same. We thus
take $\sigma$ with $\sigma+M\in H_{M}^{\perp}$ (for preserving $H^{\perp}$ via
the proof of Lemma 4.7 and for making Equation (12) applicable), replace
$\zeta$ and $\omega$ in Equation (14) by the resulting variables from the left
hand sides of Equation (12) and Proposition 2.7, and apply them. After
canceling the common exponent, changing the summation indices on the left hand
side, and expanding the resulting right hand side via Equation (14), we obtain
the equality

$t_{\sigma}\sum_{\begin{subarray}{c}\tilde{\beta}\in H^{\perp}_{R}\\\
\tilde{\gamma}\in
H\end{subarray}}\theta_{\tilde{\beta}_{M}+\tilde{\gamma}_{M}}\hat{f}_{\tilde{\beta}_{M}+\tilde{\beta}_{D},\tilde{\gamma}}\mathfrak{e}_{\overline{\tilde{\beta}_{M}+\tilde{\beta}_{D}}}=\sum_{\begin{subarray}{c}\beta\in
H^{\perp}_{R}\\\ \gamma\in
H\end{subarray}}\theta_{\beta_{M}+\gamma_{M}+(\sigma+M)}\hat{f}_{\beta_{M}+\beta_{D},\gamma}\mathfrak{e}_{\overline{\beta_{M}+\beta_{D}}},$

with the usual variables $\tau$, $\zeta$, $\omega$, and $v$ suppressed.

Now, as in the proof of Proposition 4.9, we replace $\tilde{\beta}_{M}$ on the
right hand side by $\tilde{\beta}_{M}=r_{\sigma}(\beta_{M})$ from part $(i)$
of Lemma 4.8, set $\tilde{\gamma}:=\gamma+h_{\sigma}(\beta_{M})$, and define
$\tilde{\beta}_{D}=\beta_{D}-h_{\sigma}(\beta_{M})_{D}$, so that the theta
function there becomes $\theta_{\tilde{\beta}_{M}+\tilde{\gamma}_{M}}$ like on
the left hand side. Moreover, via part $(ii)$ of that lemma we deduce that the
vector $\mathfrak{e}_{\overline{\beta_{M}+\beta_{D}}}$ from the right hand
side is the one obtained by subtracting $\sigma+M$ from the index of
$\mathfrak{e}_{\overline{\tilde{\beta}_{M}+\tilde{\beta}_{D}}}$, namely it
equals the $t_{\sigma}$-image of the latter vector. Since the theta expansion
in Remark 4.10 and Corollary 3.2 is unique, we obtain an equality in the
coefficients from both sides as functions of $\tau$, which by our
substitutions amounts, for any $\beta\in H^{\perp}_{R}$, $\gamma\in H$, and
$\sigma\in M^{*}$ with $\sigma+M\in H_{M}^{\perp}$, to the equality

$\hat{f}_{\beta_{M}+\beta_{D},\gamma}=\hat{f}_{r_{\sigma}(\beta_{M})+\beta_{D}-h_{\sigma}(\beta_{M})_{D},\gamma+h_{\sigma}(\beta_{M})}.$
(15)

We now construct the smooth $\mathbb{C}[D]$-valued function
$F=\sum_{\alpha_{D}\in D}f_{\alpha_{D}}$ as follows. Since the pairing on
$D_{M}$ is non-degenerate, the map from $D_{M}$ to the quotient
$\operatorname{Hom}(H_{M},\mathbb{Q}/\mathbb{Z})$ or
$\operatorname{Hom}(D_{M},\mathbb{Q}/\mathbb{Z})$ is surjective, and in
particular there exists $\alpha_{M}\in D_{M}$ such that the equality
$(\alpha_{M},\gamma_{M})=-\big{(}\alpha_{D},\iota(\gamma_{M})\big{)}$ from
part $(ii)$ of Lemma 4.4 holds for any $\gamma\in H$, so that
$\alpha:=\alpha_{M}+\alpha_{D}\in D_{M}\oplus D$ lies in $H^{\perp}$. We thus
choose such an $\alpha_{M}$, and note that any other choice is obtained by
subtracting some element of $H_{M}^{\perp}$. We express $\alpha$ as
$\beta+\gamma$ as in part $(i)$ of Lemma 4.5, namely with $\beta_{M}\in R$ and
$\gamma\in H$, and we set $f_{\alpha_{D}}$ to be $\hat{f}_{\beta,\gamma}$. We
observe that if we replace $\alpha_{M}$ by $\alpha_{M}-(\sigma+M)$ for some
$\sigma\in M^{*}$ with $H_{M}^{\perp}$ (which will produce all the other
choices for $\alpha_{M}$) then $\beta_{M}$ will be taken to
$r_{\sigma}(\beta)_{M}$, $\gamma$ will be sent to
$\gamma+h_{\sigma}(\beta_{M})$, and $\beta_{D}$ will thus become
$\beta_{D}-h_{\sigma}(\beta_{M})_{D}$. As the function arising from this
choice equals the one from our initial choice by Equation (15), we deduce that
$f_{\alpha_{D}}$ is well-defined.

Finally, the construction implies that given $\beta\in H^{\perp}_{R}$ and
$\gamma\in H$ is $f_{\alpha_{D}}$, the function $\hat{f}_{\beta,\gamma}$ is
$f_{\alpha_{D}}$ for $\alpha:=\beta+\gamma$, namely
$f_{\beta_{D}+\gamma_{D}}$. By substituting this equality into the right hand
side of Equation (14) and comparing with Equation (11), we deduce that our
$\Phi$ equals $\Phi_{M,v}^{F,H}$ with $F=\sum_{\alpha_{D}\in
D}f_{\alpha_{D}}$, as desired. The fact that $F$ is a modular form if and only
if $\Phi=\Phi_{M,v}^{F,H}$ is a Hermitian Jacobi form is established using the
uniqueness of our theta expansion like in the proof of Theorem 3.4, and the
relation between the holomorphicity of $F$ and the pseudo-holomorphicity of
$\Phi$ is deduced as in Proposition 3.6, since the operators from Proposition