2.9 act in the same manner on scalar-valued functions and on vector-valued
functions with constant bases. This completes the proof of the theorem. ∎

######  Remark 4.13.

The proofs of Proposition 4.9 and Theorem 4.12 can be significantly simplified
in the case where $H_{M}$ is non-degenerate as a subgroup of $D_{M}$, where
formulae like those mentioned in Remark 4.6 can also be obtained. Indeed, in
this case we can take $R$ to be just the orthogonal complement
$H_{M}^{\perp}$, where in Lemma 4.8 we have
$r_{\sigma}(\delta)=\delta+(\sigma+M)$ and $h_{\sigma}(\delta)=0$ for every
$\delta\in R$ and $\sigma$ with $D_{M}$-image in $H_{M}^{\perp}$, and the
expressions in these proofs become much more natural. This case also has the
advantage that $D$ can be constructed directly from $\Delta$, $D_{M}$, and
$H_{M}^{\perp}$, a construction that does not seem easy, or even possible, in
other situations. In particular, the case where $H=\\{0\\}$ produces Jacobi
forms that are just tensor products $\Theta_{M}\otimes F$ for modular forms
$F$, and if $D=D_{\overline{M}}$ and $H$ is the natural diagonal subgroup,
with $H^{\perp}=H$, Proposition 4.9 and Theorem 4.12 reproduce Corollary 3.2
and Theorem 3.4, because the pairing $[\Theta_{M},F]$ is the same as
$\downarrow_{H}(\Theta_{M}\otimes F)$ for this $H$. However, in applications,
like those from [BRZ] in the holomorphic setting to be expressed explicitly
below, this construction has to be carried out with horizontal subgroups with
arbitrary projections, so we needed to work in this generality. We also note
that in applications in the Hermitian setting, typically $D$ and the isotropic
subgroup $H$ are assumed to carry the $J$-structure mentioned in Remark 4.2,
but this does not affect any part of the construction or the results.

All of the constructions involving discriminant forms only used their
structure involving the $\mathbb{Q}/\mathbb{Z}$-valued quadratic form, which
can be obtained from the natural construction of the discriminant form
associated with an even orthogonal lattice over $\mathbb{Z}$. Since [Ze3] only
considered scalar-valued Jacobi forms, we give the definition of vector-valued
Jacobi forms in this setting as well, and state the corresponding theorem.

So let $L$ be an even lattice over $\mathbb{Z}$, with quadratic form
$\lambda\mapsto\frac{\lambda^{2}}{2}$ and induced pairing $(\cdot,\cdot)$, of
some signature $(b_{+},b_{-})$, with discriminant $D_{L}$. Given $v$ in the
associated Grassmannian from Equation (1) of [Ze3] (or equivalently Equation
(2) here), and elements $\tau\in\mathcal{H}$ and $\zeta\in L_{\mathbb{C}}$,
let

$\Theta_{L}(\tau,\zeta;v):=\textstyle{\sum_{\gamma\in
D_{L}}\big{[}\sum_{\lambda\in
L+\gamma}\mathbf{e}\big{(}\tau\lambda_{v_{+}}^{2}/2+\overline{\tau}\lambda_{v_{-}}^{2}/2+(\lambda,\zeta)\big{)}\big{]}\mathfrak{e}_{\gamma}}$
(16)

be the associated theta function from Equation (7) of [Ze3]. It satisfies not
only the periodicity property from Proposition 1.1 there, but the extended one
given in Proposition 2.7, using the operators from Definition 2.5. We write
$\operatorname{Mp}_{2}(\mathbb{Z})$ for the double cover of
$\operatorname{SL}_{2}(\mathbb{Z})$ that is used for modular forms of half-
integral weight, with $\rho_{L}$ the representation of
$\operatorname{Mp}_{2}(\mathbb{Z})$ on $\mathbb{C}[D_{L}]$ as defined in
Equation (5) of [Ze3] (or Equation (3) here).

Let $D$ be another discriminant form, take a horizontal isotropic subgroup $H$
of $D_{L}\oplus D$, and let $H_{L}$ and $H_{D}$ be the projections of $H$ as
usual, with $\Delta_{H}:=H^{\perp}/H$ the associated quotient as above. Then
Lemma 4.7 holds here equally well, when the elements $\nu$ and $\sigma$ from
$L^{*}$ are such that their images in $D_{L}$ are in $H_{L}^{\perp}$, and the
vector-valued Jacobi forms in this setting are the following functions.

######  Definition 4.14.

Consider a finite index subgroup $\Gamma$ of
$\operatorname{Mp}_{2}(\mathbb{Z})$, two weights $k$ and $l$ from
$\frac{1}{2}\mathbb{Z}$. Then a _Jacobi form of weight $(k,l)$, index
$(L,H,v)$, and representation $\rho_{\Delta_{H}}$ with respect to $\Gamma$_ is
a smooth function $\Phi:\mathcal{H}\times L_{\mathbb{C}}\to\mathbb{C}$ for
which the following four conditions are satisfied: First, we require the
equation

$\Phi(\tau,\zeta+\tau\sigma_{v_{+}}+\overline{\tau}\sigma_{v_{-}}+\nu)=\mathbf{e}\big{(}-\tau\sigma_{v_{+}}^{2}/2-\overline{\tau}\sigma_{v_{-}}^{2}/2-(\sigma,\zeta)\big{)}\chi_{\nu}t_{\sigma}\Phi(\tau,\zeta)$
(17)

for any $\tau\in\mathcal{H}$, $\zeta\in L_{\mathbb{C}}$, and elements $\sigma$
and $\nu$ in $L^{*}$ for which $\sigma+L$ and $\nu+L$ are in $H_{L}^{\perp}$;
Second, the we demand the equality

$\Phi\Big{(}A\tau,\tfrac{\zeta_{v_{+}}}{j(A,\tau)}+\tfrac{\zeta_{v_{-}}}{\overline{j(A,\tau)}}\Big{)}=j(A,\tau)^{k}\overline{j(A,\tau)}^{l}\mathbf{e}\bigg{(}\tfrac{j_{A}^{\prime}\zeta_{v_{+}}^{2}}{2j(A,\tau)}+\tfrac{j_{A}^{\prime}\zeta_{v_{-}}^{2}}{2\overline{j(A,\tau)}}\bigg{)}\rho_{\Delta_{H}}(A)\Phi(\tau,\zeta)$
(18)

to hold for any $\tau$ and $\zeta$ and for any $A\in\Gamma$, with the half-
integral powers of $j$ determined by the metaplectic data of $A$ if necessary;
Third, for any fixed $\tau\in\mathcal{H}$, the function
$\zeta\mapsto\Phi(\tau,\zeta)$ is holomorphic on $L_{\mathbb{C}}$; And fourth,
the functions deduced from the theta expansions of all the coefficients of
$\Phi$ via Corollary 2.2 of [Ze3] are bounded, towards any cusp of $\Gamma$,
by the exponential of a linear function of the height there. We saw that
$\Phi$ is _pseudo-holomorphic_ when it is annihilated by the operator $4\pi
i\partial_{\overline{\tau}}-\Delta_{v_{-}}^{h}$ from Proposition 1.3 of [Ze3]
(or Proposition 2.9 here), and _skew-holomorphic_ if it is annihilated by the
operator $4\pi i\partial_{\tau}-\Delta_{v_{+}}^{h}$ defined there.

The main result concerning the Jacobi forms from Definition 18 is the
following analogue of Proposition 4.9 and Definition 13.

######  Theorem 4.15.

Given a smooth function $\Phi:\mathcal{H}\times
L_{\mathbb{C}}\to\mathbb{C}[\Delta_{H}]$, Equation (17) and the holomorphicity
in Definition 18 are equivalent to the existence of a smooth function
$F:\mathcal{H}\to\mathbb{C}[D]$, which is then determined by $\Phi$, such that
$\Phi(\tau,\zeta)=\downarrow_{H}\big{(}\Theta_{L}(\tau,\zeta;v)\otimes
F(\tau)\big{)}$ for every $\tau\in\mathcal{H}$ and $\zeta\in L_{\mathbb{C}}$,
where $\Theta_{L}$ is the function from Equation (16). In this case Equation
(18) for $\Phi$ is equivalent to $F$ satisfying Equation (8) with the weight
$\big{(}k-\frac{b_{+}}{2},l-\frac{b_{-}}{2}\big{)}$ and the representation
$\rho_{D}$, and $\Phi$ is a Jacobi form of weight $(k,l)$ and index $(L,H,v)$,
and representation $\rho_{\Delta_{H}}$ with respect to $\Gamma$ if and only if
$F$ is modular of weight $\big{(}k-\frac{b_{+}}{2},l-\frac{b_{-}}{2}\big{)}$
and representation $\rho_{D}$. The Jacobi form is pseudo-holomorphic if and
only if $F$ is weakly holomorphic, a case which can only occur when
$l=\frac{b_{-}}{2}$.

The proof of Theorem 4.15 goes just like that of Theorem 4.12, with the
orthogonal notions. The maps from Lemma 3.1 and Corollary 3.2 of [Ze3] also
apply to the discriminants $D$ and $\Delta_{H}$, yielding an analogous result
to Theorem 4.15, in which the holomorphicity of $F$ is equivalent to the
corresponding Jacobi form (in this construction) being skew-holomorphic.

##  5 Relations with Arrow Operators

Recall from Proposition 3.4 of [Ze3] that if the modular form $F$ producing a
scalar-valued Jacobi form $\Phi$ is the $\uparrow$-operator image of another
modular form $G$, then $\Phi$ can also be obtained from $G$ using the theta
function of the associated over-lattice (this works equally well for Hermitian
$\mathcal{O}$-lattices, as mentioned above). We shall need similar relations
in the vector-valued setting. We shall work with an orthogonal
$\mathbb{Z}$-lattice $L$, but as above, all the results in this section hold
for any Hermitian $\mathcal{O}$-lattice $M$ as well, provided that all the
subgroups involved in the calculations, and hence all the resulting quotients,
are closed under the action of $\mathcal{O}$.

Our calculations will be based on a certain combination of arrow operators.
For defining it, we shall need some notation. Let $A$ be any discriminant
form, and take two isotropic subgroups $H$ and $I$ of $A$. Then the subgroup
$I+(H\cap I^{\perp})$ is also isotropic (as so are $I$ and $H\cap I^{\perp}$,
and they are perpendicular by definition), and as for subgroups $Y$ and $Z$ of
$A$ one has $(Y^{\perp})^{\perp}=Y$, $(Y+Z)^{\perp}=Y^{\perp}\cap Z^{\perp}$,
and $(Y\cap Z)^{\perp}=Y^{\perp}+Z^{\perp}$, the subgroup perpendicular to
ours is $I^{\perp}\cap(H^{\perp}+I)$, which equals $(I^{\perp}\cap
H^{\perp})+I$ since $I\subseteq I^{\perp}$.

We shall be using the following simple observation below.

######  Lemma 5.1.

The quotients $|H|/|H\cap I^{\perp}|$ and $|I|/|I\cap H^{\perp}|$ have the
same size. This size is $|I|$ when $I\cap H^{\perp}=\\{0\\}$.

###### Proof.

We restrict the pairing on $A$ to a pairing from $H\times I$ to
$\mathbb{Q}/\mathbb{Z}$. Since this pairing is trivial when the first element
lies in $H\cap I^{\perp}$ or when second one in is $I\cap H^{\perp}$, we
obtain an induced pairing on the product of $H/(H\cap I^{\perp})$ and
$I/(I\cap H^{\perp})$, the sizes of which are the asserted quotients.
Moreover, these intersections are precisely the subgroups of elements for
which the pairing with the other group is trivial, which implies that in the
quotients, the pairing is now perfect, and embeds each group with the dual of
the other. As the dual group of a finite group has the same size of the group
itself, these embeddings yield two opposite inequalities between the sizes of
these groups, whence the desired equality (and indeed, for finite groups such
embeddings are isomorphisms when the pairing is perfect). The second assertion
is an immediate special case of the first. This proves the lemma. ∎

We shall also use the simplest case of Lemma 5.1 in which both sizes are 1.
This amounts to the equivalence of the statements $I\cap H^{\perp}=I$,
$I\subseteq H^{\perp}$, the pairing between $H$ and $I$ is trivial,
$H\subseteq I^{\perp}$, and $H\cap I^{\perp}=H$.

Both $H$ and $I+(H\cap I^{\perp})$ contain $H\cap I^{\perp}$, with the
quotient $\overline{I}$ resulting from the second inclusion being isomorphic
to $I/(I\cap H)$ (since $I\subseteq I^{\perp}$ again). We set
$\overline{A}:=(H^{\perp}+I)/(H\cap I^{\perp})$, $B:=H^{\perp}/H$, and
$\overline{B}:=[(I^{\perp}\cap H^{\perp})+I]/[I+(H\cap I^{\perp})]$, and then
$B$ and $\overline{B}$ can also be obtained from $\overline{A}$ using the
isotropic subgroups $H/(H\cap I^{\perp})$ and $\overline{I}$ respectively. We
shall need the following operator.

######  Lemma 5.2.

The subgroup $[H+(I^{\perp}\cap H^{\perp})]/H\subseteq B$ admits a natural
surjective map $p$ onto $\overline{B}$, which preserves the quadratic values
in $\mathbb{Q}/\mathbb{Z}$ and whose kernel is $[H+(I\cap H^{\perp})]/H$. The
combination $\downarrow_{H/(H\cap
I^{\perp})}\uparrow_{\overline{I}}:\mathbb{C}[\overline{B}]\to\mathbb{C}[B]$
is the injective map taking $\mathfrak{e}_{\alpha}$ for
$\alpha\in\overline{B}$ to $\sum_{\beta\in[H+(I^{\perp}\cap H^{\perp})]/H,\
p\beta=\alpha}\mathfrak{e}_{\beta}$.

###### Proof.

The Isomorphism Theorem from basic Group Theory identifies our subgroup of $B$
with the subgroup $(I^{\perp}\cap H^{\perp})/(H\cap
I^{\perp})\subseteq\overline{A}$. As $I^{\perp}\cap H^{\perp}$ is
perpendicular to both $I$ and $H\cap I^{\perp}$, our subgroup of
$\overline{A}$ is contained in $\overline{I}^{\perp}$, and we define $p$ to be
the composition of the isomorphism from above with the projection of
$\overline{I}^{\perp}$ onto $\overline{B}=\overline{I}^{\perp}/\overline{I}$.
The image of this map is the image, modulo the subgroup $I+(H\cap I^{\perp})$
defining $\overline{B}$, of the sum of $I^{\perp}\cap H^{\perp}$ from the
numerator of the subgroup of $\overline{A}$ and the numerator $I+(H\cap
I^{\perp})$ of $\overline{I}$, which together yield the numerator
$(I^{\perp}\cap H^{\perp})+I$ of $\overline{A}$, proving that $p$ is indeed
surjective. We now note that the kernel of the projection from $(I^{\perp}\cap
H^{\perp})/(H\cap I^{\perp})$ onto $\overline{B}$ is its intersection with
$\overline{I}$, which is thus $[(I\cap H^{\perp})+(H\cap I^{\perp})]/(H\cap
I^{\perp})$ (because $H\cap I^{\perp}\subseteq I^{\perp}\cap H^{\perp}$), and
transferring this kernel by the isomorphism from Group Theory indeed produces
the asserted kernel.

It remains to establish the formula for $\downarrow_{H/(H\cap
I^{\perp})}\uparrow_{\overline{I}}\mathfrak{e}_{\alpha}$ for any
$\alpha\in\overline{B}$, since the surjectivity of $p$ and the fact that any
$\beta$ in the subgroup of $B$ only shows up in the image of
$\downarrow_{H/(H\cap I^{\perp})}\uparrow_{\overline{I}}\mathfrak{e}_{p\beta}$
immediately imply the injectivity of our combination of arrow operators. But
indeed, we have
$\uparrow_{\overline{I}}\mathfrak{e}_{\alpha}=\sum_{\delta\in\overline{I}^{\perp},\
\delta+\overline{I}=\alpha}\mathfrak{e}_{\delta}$, where
$\overline{I}^{\perp}\subseteq\overline{A}$ consists of elements that must be
perpendicular to the image of $I$ in $\overline{A}$ and is thus contained in
$I^{\perp}/(H\cap I^{\perp})$. Applying $\downarrow_{H/(H\cap I^{\perp})}$ to
this sum restricts the set of $\delta$’s associated with each such $\alpha$ to
those in $H^{\perp}/(H\cap I^{\perp})$ and takes every such $\delta$ to its
$B$-image $\delta+H/(H\cap I^{\perp})$. But these are thus precisely the set
of $\delta$ from the subgroup of $(I^{\perp}\cap H^{\perp})/(H\cap I^{\perp})$
of $\overline{A}$ that are sent to $\alpha$ by our surjective map, and
$\delta\mapsto\delta+H$ is precisely the inversion of the isomorphism from the
previous paragraph, yielding the asserted formula. The fact that $p$ preserves
the quadratic values follows either from the naturality of the maps in the
previous paragraph, or from the fact that the arrow operators from Equation
(10) preserve the action of $T$ via the Weil representations. This proves the
lemma. ∎

We deduce the following consequence.

######  Corollary 5.3.

Let $A$, $H$, $I$, $\overline{A}$, $B$, and $\overline{B}$ be as above.

  1. $(i)$

If $I\cap H=\\{0\\}$ and $I\subseteq H^{\perp}$ then $B$ equals $\overline{A}$
and contains a isotropic subgroup $I_{B}$ that is isomorphic to $I$, such that
$\overline{B}$ is the associated quotient and the operator from Lemma 5.2 is
just $\uparrow_{I_{B}}$.

  2. $(ii)$

When $I\cap H^{\perp}=\\{0\\}$, there is an injection
$\iota_{\overline{B}}:\overline{B}\to B$ of discriminant forms, such that the
combination from Lemma 5.2 just takes $\mathfrak{e}_{\alpha}$ to
$\mathfrak{e}_{\iota_{\overline{B}}\alpha}$.

###### Proof.

If $I\subseteq H^{\perp}$ then $H\subseteq I^{\perp}$, so that the group
$H\cap I^{\perp}$ used for defining $\overline{A}$ equals $H$ and thus
$\overline{A}=B$. Then $B$ contains the isotropic group $I_{B}:=\overline{I}$
from above, which was seen to be isomorphic to $I/(I\cap H)$ and thus to $I$
under our assumption, and we saw that $\overline{B}$ was obtained from
$\overline{A}=B$ using the isotropic subgroup $\overline{I}=I_{B}$. Since
$\uparrow_{\overline{I}}$ clearly equals $\uparrow_{I_{B}}$ and the operator
$\downarrow_{H/(H\cap I^{\perp})}=\downarrow_{H/H}$ it trivial, this
establishes part $(i)$.

For part $(ii)$ note that our assumption makes the kernel of the map $p$ from
Lemma 5.2, which is isomorphic to $(I\cap H^{\perp})/(I\cap H)$ (as
$H\subseteq H^{\perp}$), trivial, so that $p$ becomes an isomorphism in this
case, which was seen to preserve the $\mathbb{Q}/\mathbb{Z}$-quadratic
structure as well. By defining $\iota_{\overline{B}}$ to be the composition of
the inverse isomorphism with the inclusion of the subgroup from Lemma 5.2 into
$B$, the assertion follows directly from the formula given in that lemma. This
proves the corollary. ∎

Under the condition from part $(ii)$ of Corollary 5.3, we shall allow
ourselves the slight abuse of notation by using $\iota_{\overline{B}}$ also
for the map from $\mathbb{C}[\overline{B}]$ to $\mathbb{C}[B]$ taking
$\mathfrak{e}_{\alpha}$ to $\mathfrak{e}_{\iota_{\overline{B}}\alpha}$, namely
the operator from Lemma 5.2 reduces to just $\iota_{\overline{B}}$ in this
case.

In our case the discriminant form $A$ is $D_{L}\oplus D$, and the group $H$ is
a horizontal one as in Definition 4.3 (so that $B$ is $\Delta_{H}$ from
above), thus yielding the isomorphic images $H_{L}\subseteq D_{L}$ and
$H_{D}\subseteq D$, inside which $H_{L}^{\perp}$ and $H_{D}^{\perp}$ are the
respective perpendicular group. We shall only consider isotropic subgroups $I$
that are contained in $D$, for which $I\cap H=\\{0\\}$ by horizontality, and
then we have $I\cap H^{\perp}=I\cap H_{D}^{\perp}$ and $H\cap I^{\perp}$ maps
isomorphically onto $H_{D}\cap I^{\perp}$ in restriction of the projection
from $A$ onto $D$ (we write just $(H\cap I^{\perp})_{L}$ for its image in
$D_{L}$). In this setting we have many additional groups, as follows.

######  Definition 5.4.

Write $I_{D}$ for the subgroup $I\cap H_{D}\subseteq H_{D}$, let $I_{H}$ and
$I_{L}$ be its isomorphic images in $H$ and in $H_{L}$ respectively, and set
$\overline{H}:=(H\cap I^{\perp})/I_{H}$. Denote by $\Lambda$ the over-lattice
of $L$ for which $\Lambda/L=I_{L}$, let $D_{\Lambda}$ be the discriminant
group $I_{L}^{\perp}/I_{L}$ for $I_{L}^{\perp}\subseteq D_{L}$, and with
$I^{\perp}\subseteq D$, denote $I^{\perp}/I$ by $\overline{D}$.

We shall need the following relation arising from Definition 5.4.

######  Lemma 5.5.

The group $\overline{H}$ can be identified with a horizontal isotropic
subgroup of $D_{\Lambda}\oplus\overline{D}$. Its projections are
$\overline{H}_{\Lambda}=(H\cap I^{\perp})_{L}/I_{L}$ and
$\overline{H}_{\overline{D}}=(H_{D}\cap I^{\perp})/I_{D}$.

###### Proof.

Any element of $(H\cap I^{\perp})_{L}$ is $\gamma_{L}$ for some $\gamma\in
H\cap I^{\perp}$, and we write an element of $I_{L}$ as $\delta_{L}$ with
$\delta\in I_{H}$ via the isomorphism $H\to H_{L}$ and the notation from
Definition 5.4. We write $\gamma_{D}$ and $\delta_{D}$ for their images in
$H_{D}\cap I^{\perp}$ and $I_{D}\subseteq I$, meaning that
$(\gamma_{D},\delta_{D})=0$, and since $(\gamma,\delta)=0$ as well (since
$\gamma$ and $\delta$ are both in the isotropic subgroup $H$), and the latter
pairing is the sum of the pairings of the projections (as such is the pairings
of elements of $D_{L}\oplus D$), we deduce that $(\gamma_{L},\delta_{L})=0$ as
well. This proves that $(H\cap I^{\perp})_{L}\subseteq I_{L}^{\perp}$, and
thus has an image in $D_{\Lambda}$. Similarly, $H_{D}\cap I^{\perp}$ is
contained in $I^{\perp}$ and thus has an image in $\overline{D}$, meaning that
we have a natural map from $H\cap I^{\perp}$ to
$D_{\Lambda}\oplus\overline{D}$. The fact that the kernel of the map to
$D_{\Lambda}$ is the inverse image $I_{H}$ of $I_{L}$ (by which we divided to
obtain $D_{\Lambda}$) in $H$ and the kernel of the map to $\overline{D}$ is
the inverse image $I_{H}$ of $I\cap H_{D}=I_{D}$ in $H$ implies that this map
factors through an injective map from $\overline{H}$, as defined in Definition
5.4, to $D_{\Lambda}\oplus\overline{D}$, and that the image of $\overline{H}$
is indeed horizontal as in Definition 4.3. The expressions for the images of
$\overline{H}$ also follow from this description. This proves the lemma. ∎

We shall thus henceforth identify $\overline{H}$ with its image in
$D_{\Lambda}\oplus\overline{D}$, and with $\overline{H}^{\perp}\subseteq
D_{\Lambda}\oplus\overline{D}$, and as above we denote the discriminant group
$\overline{H}^{\perp}/\overline{H}$ resulting from Lemma 5.5 by
$\overline{\Delta}_{\overline{H}}$.

######  Remark 5.6.

While $H_{L}$, and even its subgroup $(H\cap I^{\perp})_{L}$, contain $I_{L}$
(which allowed us to obtain the formula for $\overline{H}_{\Lambda}$ in Lemma
5.5), it is not necessarily true for the complement $H_{L}^{\perp}\subseteq
D_{L}$ showing up in the periodicity property for the index $(L,H)$. This
group is contained in $I_{L}^{\perp}$, and its image in $D_{\Lambda}$ is
$(H_{L}^{\perp}+I_{L})/I_{L}\cong H_{L}^{\perp}/(I_{L}\cap H_{L}^{\perp})$.
The same argument from the proof of Lemma 5.5 shows that an element of
$I_{L}\subseteq H_{L}$ is perpendicular to all of $H_{L}$ if and only if its
image in $I_{D}=I\cap H_{D}$ is perpendicular to all of $H_{D}$, so that the
size of this image, which is clearly contained in
$\overline{H}_{\Lambda}^{\perp}\subseteq D_{\Lambda}$, is
$|H_{L}^{\perp}|/|I\cap H_{D}\cap H_{D}^{\perp}|$, which we compare with the
size of $\overline{H}_{\Lambda}^{\perp}$ in Remark 5.9 below.

We will need an analogue of Lemma 4.5 that will respect the relation with $I$
as well. Recall that for any subgroup $I\subseteq D$, the elements of
$D_{L}\oplus D$ that are perpendicular to $I$ are precisely those that lie in
$D_{L}\oplus I^{\perp}$.

######  Lemma 5.7.

Given a horizontal subgroup $H\subseteq D_{L}\oplus D$ and a subgroup
$I\subseteq D$, let $R_{I}\subseteq D_{L}$ be a set of representatives for
$D_{L}/(H\cap I^{\perp})_{L}$.

  1. $(i)$

Any element of $D_{L}\oplus D$ such that its $H$-coset intersects $D_{L}\oplus
I^{\perp}$ has a unique presentation as $\beta+\gamma$, in which $\gamma$ lies
in $H$ and $\beta$ is with $\beta_{L}\in R_{I}$ and $\beta_{D}\in I^{\perp}$.

  2. $(ii)$

If $H$ is isotropic, then the map $\beta\mapsto\overline{\beta}:=\beta+H$ is a
bijection from the set $H^{\perp}_{R_{I},I}$ consisting of
$\beta\in(D_{L}\oplus I^{\perp})\cap H^{\perp}$ with $\beta_{L}\in R_{I}$ onto
its image in $\Delta_{H}$, and adding $\gamma\in H$ to $\beta$ does not affect
the $H^{\perp}$ condition.

  3. $(iii)$

In case $I$ is also isotropic, the values of the pairings of $\beta_{L}\in
R_{I}$ with $I_{L}$ depend only on its image in $D_{L}/(H\cap
I^{\perp})_{L}\oplus I^{\perp}$ (and not on the choice of the set of
representatives $R_{I}$). Moreover, if $\beta$ lies in $H^{\perp}_{R_{I},I}$
then $\beta_{L}\in R_{I}$ is perpendicular to $I_{L}$.

  4. $(iv)$

In the latter case, the set $\overline{R}:=\\{\beta_{L}+I_{L}|\beta_{L}\in
R_{I}\cap I_{L}^{\perp}\\}\subseteq D_{\Lambda}$ is a set of representatives
for $D_{\Lambda}/\overline{H}_{\Lambda}$ which is in bijection with $R_{I}\cap
I_{L}^{\perp}$. The map taking $\beta\in H^{\perp}_{R_{I},I}$ to
$\beta+I_{L}+I\in D_{\Lambda}\oplus\overline{D}$ is surjection from
$H^{\perp}_{R_{I},I}$ onto the set $\overline{H}^{\perp}_{\overline{R}}$
associated via Lemma 4.5 with $\overline{H}\subseteq
D_{\Lambda}\oplus\overline{D}$ and $\overline{R}$, such that the map $p$ from
Lemma 5.2 takes $\beta+H$ to $(\beta+I_{L}+I)+\overline{H}$.

###### Proof.

For part $(i)$, the assumption on the intersection allows us to write our
element as $\tilde{\beta}+\tilde{\gamma}$ with $\tilde{\gamma}\in H$ and
$\tilde{\beta}\in D_{L}\oplus I^{\perp}$. Any other possibility of expressing
our element as such a sum is obtained by changing $\tilde{\beta}$ and
$\tilde{\gamma}$ by the same element, which must be in $H$ but also in
$I^{\perp}$, hence part $(i)$ follows from $R_{I}$ being a set of
representatives modulo the isomorphic image $(H\cap I^{\perp})_{L}$ of the
intersection $H\cap I^{\perp}$ in $D_{L}$. The second assertion in part $(ii)$
is immediate (as for part $(ii)$ of Lemma 4.5), and the first one follows
directly from part $(i)$ (with the image being in $\Delta_{H}$ by the
$H^{\perp}$ condition). We now recall from the proof of Lemma 5.5 that when
$I$ and $H$ are isotropic, the defining group $(H\cap I^{\perp})_{L}$ for
$R_{I}$ is perpendicular to $I_{L}$ (i.e., we have $(H\cap
I^{\perp})_{L}\subseteq I_{L}^{\perp}$), yielding the first assertion in part
$(iii)$. The fact that for $\beta\in H^{\perp}_{R_{I},I}$ and $\gamma\in
I_{H}$ we have $(\beta,\gamma)=0$ (since $\beta\in H^{\perp}$ and $\gamma\in
H$) and also $(\beta_{D},\gamma_{D})=0$ (because $\beta_{D}\in I^{\perp}$ and
$\gamma_{D}\in I_{D}\subseteq I$) proves the second assertion there as well.
Intersecting with $I_{L}^{\perp}$ thus gives a set of representatives for the
quotient $I_{L}^{\perp}/(H\cap I^{\perp})_{L}$, and as this quotient is
isomorphic to $D_{\Lambda}/\overline{H}_{\Lambda}$ by Lemma 5.5, the first
assertion in part $(iv)$ follows. Noting that $H^{\perp}_{R_{I},I}$ is
contained in $I^{\perp}\cap H^{\perp}$, this set can also serve as a set of
representatives for $(I^{\perp}\cap H^{\perp})/(H\cap I^{\perp})$ from the
proof of Lemma 5.2, which are all perpendicular to $I_{L}\oplus I$ and, in the
quotient modulo the latter group, to $\overline{H}$. Thus projecting modulo
this direct sum, as in the construction of the map $p$ from Lemma 5.2, yields
the set of $\alpha\in D_{\Lambda}\oplus\overline{D}$ that are perpendicular to
$\overline{H}$ and satisfy $\alpha_{\Lambda}\in\overline{R}$, which were seen
in Lemma 4.5 to represent $\overline{\Delta}_{\overline{H}}$, as desired. This
proves the lemma. ∎

We can now state our extension of Proposition 3.4 of [Ze3], in which we recall
that the Grassmannian associated with $L$ is the same as the one arising from
any over-lattice $\Lambda$ of $L$ (a fact that clearly holds also when $L$ and
$\Lambda$ are replaced by an $\mathcal{O}$-lattice $M$ and an appropriate
over-lattice, with Hermitian structures inducing the orthogonal ones, and the
Grassmannian is the Hermitian one from Equation (2)), and that $\overline{I}$
stands for the group $[I+(H\cap I^{\perp})]/(H\cap I^{\perp})$, which is
isomorphic to $I$ since $I\cap H=\\{0\\}$.

######  Proposition 5.8.

Assume that the smooth function $F:\mathcal{H}\to\mathbb{C}[D]$ from
Proposition 4.9 is $\uparrow_{I}G$ for a smooth function
$G:\mathcal{H}\to\mathbb{C}[\overline{D}]$. Then the associated function
$\Phi_{L,v}^{F,H}=\downarrow_{H}(\Theta_{L}\otimes F)$ equals
$\downarrow_{H/(H\cap
I^{\perp})}\uparrow_{\overline{I}}\Phi_{\Lambda,v}^{G,\overline{H}}$, where
$\Phi_{\Lambda,v}^{G,\overline{H}}$ is similarly defined to be
$\downarrow_{\overline{H}}(\Theta_{\Lambda}\otimes G)$ and the combination of
the arrow operators is the one from Lemma 5.2.

As the formulation and result of Proposition 5.8 may be a bit complicated in
general, we present four simpler and clearer cases in Corollary 5.10 below,
from which the general case can, in fact, be constructed via Remark 5.12
below.

###### Proof.

Recall from the proof of Proposition 4.9 that by taking any set of
representatives $\tilde{R}$ inside $H^{\perp}\subseteq D_{L}\oplus D$ for
$\Delta_{H}$, the parts of Lemma 4.5 express $\Phi_{L,v}^{F,H}$ as
$\sum_{\beta\in\tilde{R}}\big{[}\sum_{\gamma\in
H}\theta_{\beta_{M}+\gamma_{L}}f_{\beta_{D}+\gamma_{D}}\big{]}\mathfrak{e}_{\overline{\beta}}$,
with $\overline{\beta}=\beta+H$ for every $\beta\in\tilde{R}$ (Equation (11)
does this with the set $H^{\perp}_{R}$, but we will soon need a different
one). The assumption that $F=\uparrow_{I}G$ means, via the definition in
Equation (10), that $f_{\beta_{D}+\gamma_{D}}$ equals
$g_{\beta_{D}+\gamma_{D}+I}$ when $\beta_{D}+\gamma_{D}\in I^{\perp}$, and 0
otherwise. Therefore the coefficient $\mathfrak{e}_{\overline{\beta}}$ in
$\Phi_{L,v}^{F,H}$ may be non-zero only if the coset $\beta+H$ intersects
$D_{L}\oplus I^{\perp}$, so that by Lemma 5.7 we can take, for the
corresponding subset of $\Delta_{H}$, the set $H^{\perp}_{R_{I},I}$ as the
associated part of our $\tilde{R}$, where $R_{I}\subseteq D_{L}$ is any fixed
set of representatives for $D_{L}/(H\cap I^{\perp})_{L}$. Moreover, as for
$\beta\in H^{\perp}_{R_{I},I}$ and $\gamma\in H$ we have
$\beta_{D}+\gamma_{D}\in I^{\perp}$ precisely when $\gamma\in H\cap
I^{\perp}$, the formula that we get is

$\Phi_{L,v}^{F,H}=\textstyle{\sum_{\beta\in
H^{\perp}_{R_{I},I}}\big{[}\sum_{\gamma\in H\cap
I^{\perp}}\theta_{\beta_{L}+\gamma_{L}}g_{\beta_{D}+\gamma_{D}+I}\big{]}\mathfrak{e}_{\beta+H}}.$
(19)

Now, part $(iii)$ of Lemma 5.7 shows that if $\beta\in H^{\perp}_{R_{I},I}$
then $\beta_{L}\in I_{L}^{\perp}$, and recalling from the proof of Lemma 5.5
that $\gamma_{L}$ is also in $I_{L}^{\perp}$ for any $\gamma\in H\cap
I^{\perp}$, we deduce that Equation (19) contains only theta functions with
indices from $I_{L}^{\perp}$. Moreover, changing $\gamma$ by an element of
$I_{H}\subseteq H\cap I^{\perp}$ leaves the index $\beta_{D}+\gamma_{D}+I$ of
$g$ invariant, and since $\Theta_{\Lambda}$ equals
$\downarrow_{I_{L}}\Theta_{L}$ (see, e.g., Lemma 1.5 of [Ze5]), the sum of the
resulting theta functions produces the corresponding component of
$\Theta_{\Lambda}$. We can thus take the internal sum to be over
$\overline{H}$, and part $(iv)$ of Lemma 5.7 shows that if we write
$p(\beta+H)\in\overline{\Delta}_{\overline{H}}$, for $\beta\in
H^{\perp}_{R_{I},I}$, as $\alpha+\overline{H}$ for
$\alpha\in\overline{H}^{\perp}_{\overline{R}}$ (which decomposes as
$\alpha_{\Lambda}+\alpha_{\overline{D}}$), then the multiplier of
$\mathfrak{e}_{\overline{\beta}}$ in Equation (19) is
$\sum_{\gamma\in\overline{H}}\theta_{\alpha_{\Lambda}+\gamma_{\Lambda}}g_{\alpha_{\overline{D}}+\gamma_{\overline{D}}}$.
But as this is the coefficient of $\mathfrak{e}_{\alpha+\overline{H}}$ in
Equation (11) for $\Phi_{\Lambda,v}^{G,\overline{H}}$, and Lemma 5.2 combines
with part $(iv)$ of Lemma 5.7 to show that $\downarrow_{H/(H\cap
I^{\perp})}\uparrow_{\overline{I}}\mathfrak{e}_{\alpha+\overline{H}}$ equals
$\sum_{\beta\in H^{\perp}_{R_{I},I},\
p(\beta+H)=\alpha+\overline{H}}\mathfrak{e}_{\beta+H}$, the sum from Equation
(19) indeed equals $\downarrow_{H/(H\cap
I^{\perp})}\uparrow_{\overline{I}}\Phi_{\Lambda,v}^{G,\overline{H}}$, as
desired. This proves the proposition. ∎

Recall from Theorem 4.15 that $F$ is a modular form if and only if
$\Phi_{L,v}^{F,H}$ is a Jacobi form as in Definition 7, and similarly $G$ is a
modular form if and only if $\Phi_{\Lambda,v}^{G,\overline{H}}$ is a Jacobi
form. This corresponds, via Proposition 5.8, to the fact that the arrow
operators take modular forms to modular forms and Jacobi forms to Jacobi
forms.

######  Remark 5.9.

Note that the dimension of the space $\mathbb{C}[\Delta_{H}]$ associated with
$\Phi_{L,v}^{F,H}$ is $|D_{L}|\cdot|D|/|H|^{2}$, while that of
$\mathbb{C}[\overline{\Delta}_{\overline{H}}]$ and
$\Phi_{\Lambda,v}^{G,\overline{H}}$ is
$|D_{\Lambda}|\cdot|\overline{D}|/|\overline{H}|^{2}$. Moreover, we have
$|D_{\Lambda}|=|D_{L}|/|I_{L}|^{2}$, $|\overline{D}|=|D|/|I|^{2}$, and
$|\overline{H}|=|H\cap I^{\perp}|/|I_{H}|$, with $|I_{H}|=|I_{L}|$. As
applying $\uparrow_{\overline{I}}$ to
$\mathbb{C}[\overline{\Delta}_{\overline{H}}]$ multiplies the dimension of the
space by $|\overline{I}|^{2}=|I|^{2}$, and $\downarrow_{H/(H\cap I^{\perp})}$
divides the dimension by $|H|^{2}/|H\cap I^{\perp}|^{2}$, the dimensions
indeed match. In addition, the scalar-valued components of $\Phi_{L,v}^{F,H}$
satisfy the periodicity relations from Proposition 1.1 of [Ze3] for every
$\sigma$ and $\nu$ in the larger lattice $\Lambda$ (as so do those of
$\Phi_{\Lambda,v}^{G,\overline{H}}$), and not only $L$, and the more general
periodicity property holds for $\sigma$ and $\nu$ in
$\overline{H}_{\Lambda}^{\perp}\subseteq D_{\Lambda}$. As
$|\overline{H}_{\Lambda}|=|\overline{H}|$, its size is
$|D_{\Lambda}|/|\overline{H}|$, which is the size $|D_{L}|/|H|$ of
$H_{L}^{\perp}\subseteq D_{L}$ multiplied by the quotient $|H|/|H\cap
I^{\perp}|\cdot|I_{H}|$, which also equals $|I|/|I\cap H^{\perp}|\cdot|I_{H}|$
by Lemma 5.1. Comparing with the size of the image of $H_{L}^{\perp}$ in
$\overline{H}_{\Lambda}^{\perp}$ given in Remark 5.6, we deduce that this
subgroup has index $|I|\cdot|I_{D}\cap H^{\perp}|/|I\cap
H^{\perp}|\cdot|I_{H}|$ in $\overline{H}_{\Lambda}^{\perp}$. For more explicit
descriptions in some special cases, see Remark 5.11 below.

We now illustrate Proposition 5.8 by presenting some interesting special
cases, which will also be useful later. We recall that $I\subseteq D$
intersects the horizontal subgroup $H$ trivially, so in case $I$ and $H$ are
perpendicular to one another, part $(i)$ of Corollary 5.3 produces a subgroup
$I_{\Delta}\subseteq\Delta_{H}$ that is an isomorphic image of $I$. In case
$I$ intersects $H^{\perp}$ trivially, we denote the embedding from part $(ii)$
of that corollary by
$\iota_{\overline{\Delta}}:\overline{\Delta}_{\overline{H}}\to\Delta_{H}$,
using again the same notation for the resulting map from
$\mathbb{C}[\overline{\Delta}_{\overline{H}}]$ to $\mathbb{C}[\Delta_{H}]$.

######  Corollary 5.10.

Assume that $F=\uparrow_{I}G$ as in Proposition 5.8.

  1. $(i)$

If $I\subseteq H_{D}\cap H_{D}^{\perp}$ then $\overline{H}$ equals
$H/I_{H}\subseteq D_{\Lambda}\oplus\overline{D}$ with $I_{H}\cong I$, and we
have $\Phi_{L,v}^{F,H}=\uparrow_{I_{\Delta}}\Phi_{\Lambda,v}^{G,H/I_{H}}$.

  2. $(ii)$

When $I\subseteq H_{D}^{\perp}$ but $I\cap H_{D}=\\{0\\}$, the group $H$ has
an isomorphic image in $D_{L}\oplus\overline{D}$, and we get
$\Phi_{L,v}^{F,H}=\uparrow_{I_{\Delta}}\Phi_{L,v}^{G,H}$.

  3. $(iii)$

In case $I\subseteq H_{D}$ and $I\cap H_{D}^{\perp}=\\{0\\}$, the subgroup
$I_{H}$ is again isomorphic to $I$, and we obtain
$\Phi_{L,v}^{F,H}=\iota_{\overline{\Delta}}\Phi_{\Lambda,v}^{G,\overline{H}}$.

  4. $(iv)$

Assuming that $I\cap H_{D}=I\cap H_{D}^{\perp}=\\{0\\}$, there is an
isomorphic image of $H\cap I^{\perp}$ inside $D_{L}\oplus\overline{D}$, which
produces the equality
$\Phi_{L,v}^{F,H}=\iota_{\overline{\Delta}}\Phi_{L,v}^{G,H\cap I^{\perp}}$.

###### Proof.

In case $(i)$ we have $I=I_{D}$ and $H\subseteq I^{\perp}$, so that $H\cap
I^{\perp}=H$ and $\overline{H}=H/I_{H}$. Part $(i)$ of Corollary 5.3 shows
that $\Delta_{H}$ contains the subgroup $I_{\Delta}\cong I$, which produces
the subquotient $\overline{\Delta}_{\overline{H}}$, and the $\uparrow$
operator is trivial, so that the formula from Proposition 5.8 takes the
asserted form. For case $(ii)$, note that $I_{D}=I_{H}=I_{L}=\\{0\\}$ (so that
in particular $\Lambda=L$) and again $H\cap I^{\perp}=H$, implying that
$\overline{H}$ is an isomorphic copy of $H$ inside $D_{L}\oplus\overline{D}$.
The result thus follows from Proposition 5.8 via part $(i)$ of Corollary 5.3
again. Moving to case $(iii)$, we again have $I=I_{D}$ but the index of $H\cap
I^{\perp}$ in $H$ is $|I|$ (because $I$ injects into
$\operatorname{Hom}(H,\mathbb{Q}/\mathbb{Z})$, and $\operatorname{Hom}(H\cap
I^{\perp},\mathbb{Q}/\mathbb{Z})$ is the resulting quotient), as and part
$(ii)$ of Corollary 5.3 shows that the combination of operators from Lemma 5.2
reduces to the injection $\iota_{\overline{\Delta}}$, we deduce the desired
formula from Proposition 5.8 once again. Finally, the situation from case
$(iv)$ implies that $I_{D}=I_{H}=I_{L}=\\{0\\}$ again and $H\cap I^{\perp}$,
which again has index $|I|$ in $H$, has $\overline{H}\subseteq
D_{L}\oplus\overline{D}$ as its isomorphic image. Using part $(ii)$ of
Corollary 5.3 once more and Proposition 5.8 yet again produces the required
result. This proves the corollary. ∎

######  Remark 5.11.

The cardinalities $|D_{\Lambda}|$, $|\overline{D}|$, and $|\overline{H}|^{2}$
are obtained, in case $(i)$ of Corollary 5.10, from $|D_{L}|$, $|D|$, and
$|H|^{2}$ by division by $|I|^{2}$, and $\uparrow_{I_{\Delta}}$ multiplies the
dimension by $|I|^{2}$, exemplifying the matching from Remark 5.9. The group
$\overline{H}_{\Lambda}^{\perp}$ with respect to which the vector-valued
periodicity holds in this case is of size $|H_{L}^{\perp}|/|I|$, as is the
image of $H_{L}^{\perp}$ given in Remark 5.6, and indeed the index in the
former remark is 1. In case $(ii)$ we have the same group $D_{L}$ as well as
$H$ (up to isomorphism), again with $|\overline{D}|=|D|/|I|^{2}$ and the
dimension extension by $|I|^{2}$ via $\uparrow_{I_{\Delta}}$. The periodicity
properties are the same, as so are $L$ and $H$, and indeed both
$\overline{H}_{\Lambda}^{\perp}$ and the image of $H_{L}^{\perp}$ in it via
Remark 5.6 have the same size as $H_{L}^{\perp}$ itself (with the index from
Remark 5.9 again being 1). Turning to case $(iii)$, here division by $|I|^{2}$
produces $|D_{\Lambda}|$, $|\overline{D}|$, and $|\overline{H}|$ from
$|D_{L}|$, $|D|$, and $|H|$, the sizes of $\Delta_{H}$ and
$\overline{\Delta}_{\overline{H}}$ are the same, and indeed
$\iota_{\overline{\Delta}}$ is an isomorphism. Also here
$\overline{H}_{\Lambda}^{\perp}$ and the subgroup arising from $H_{L}^{\perp}$
are of size $|H_{L}^{\perp}|$, and the index is 1 yet again. Finally, case
$(iv)$ yields $D_{L}$ again, with $|\overline{D}|$, and $|\overline{H}|^{2}$
being $|D|$ and $|H|$ divided by $|I|^{2}$ respectively, and again the
relation uses the isomorphism $\iota_{\overline{\Delta}}$. Here
$H_{L}^{\perp}$ injects into $\overline{H}_{\Lambda}^{\perp}=(H\cap
I^{\perp})_{L}$, but the latter contains the former with the index $|I|$ of
$H\cap I^{\perp}$ in $H$, which is indeed the index showing up in Remark 5.9
in this case.

The size comparisons in cases $(iii)$ and $(iv)$ of Corollary 5.10, as given
in Remark 5.11, also indicate that in this case $\iota_{\overline{\Delta}}$ is
an isomorphism between $\overline{\Delta}_{\overline{H}}$ and $\Delta_{H}$,
with the subgroup $H_{L}^{\perp}$ being mapped onto
$\overline{H}_{\Lambda}^{\perp}$ in the former case (and onto a subgroup of
index $|I|$ in it in the latter). The scalar-valued case from Proposition 3.4
of [Ze3], which is obtained when $D=D_{\overline{L}}$,
$\overline{D}=D_{\overline{\Lambda}}$, and $H$ and $\overline{H}$ are the
associated diagonal subgroups, is contained in case $(iii)$ of Corollary 5.10,
including the trivial groups $H_{L}^{\perp}$ and
$\overline{H}_{\Lambda}^{\perp}$ which are matched by Remark 5.11 in this
case.

######  Remark 5.12.

If $J$ is a subgroup of $I$, then the operator $\uparrow_{I}$, taking $G$ to
$F$ in Proposition 5.8, decomposes as the composition
$\uparrow_{J}\uparrow_{I/J}$, through the intermediate subquotient between $D$
and $\overline{D}$. The formula from that proposition thus decomposes
correspondingly, a fact that can be checked directly but is an immediate
consequence of the proposition itself via the uniqueness of theta expansions
observed above. This can be used in order to construct the general case in
Proposition 5.8 from the special cases given in Corollary 5.10 (with the
details about $H$ being as described in Remark 5.11 in each stage).

The function $\Phi_{L,v}^{F,H}$ that is obtained from
$\Phi_{\Lambda,v}^{G,\overline{H}}$ via the arrow operators as in Proposition
5.8 inherits periodicity properties that are sometimes finer than those given
in Proposition 4.9 (or Equation (17) in the orthogonal case). Indeed, in cases
$(i)$ and $(iii)$ of Corollary 5.10 the scalar-valued components have the
stronger periodicity property concerning $\sigma$ and $\nu$ from $\Lambda$
(rather than just $L$), while case $(iv)$ there introduces a periodicity
property with respect to $(H\cap I^{\perp})_{L}^{\perp}$, which can be larger
than $H_{L}^{\perp}$ (as Remark 5.11 shows). We now show that these enhanced
periodicity properties characterize functions that arise in this way. This
complements Theorem 4.15 (as well as Theorem 4.12 in the Hermitian case), but
note that here we have the image of the connecting operator as well.

We begin with the scalar-valued periodicity.

######  Theorem 5.13.

Assume that the scalar-valued components of $\Phi_{L,v}^{F,H}$ satisfy the
periodicity property for elements of $\Lambda$, where $\Lambda$ is an over-
lattice of $L$ such that the isotropic subgroup $I_{L}:=\Lambda/L\subseteq
D_{L}$ is contained in $H_{L}$, and let $I=I_{D}$ be the corresponding
subgroup of $H_{D}\subseteq D$, with $I_{H}\subseteq H$ as usual. Then
$F=\uparrow_{I}G$ for a unique function
$G:\mathcal{H}\to\mathbb{C}[\overline{D}]$, where $\overline{D}=I^{\perp}/I$
for $I^{\perp}\subseteq D$, and $\Phi_{L,v}^{F,H}$ is obtained from
$\Phi_{\Lambda,v}^{G,\overline{H}}$ as in Proposition 5.8, where
$\overline{H}=(H\cap I^{\perp})/I_{H}$ as in Definition 5.4.

###### Proof.

As adding $\nu\in\Lambda\subseteq L^{*}$ to the argument of $\Theta_{L}$
multiplies the component associated with some element $\beta_{L}+\gamma_{L}\in
D_{L}$ by $\mathbf{e}\big{(}(\nu+L,\beta_{L}+\gamma_{L})\big{)}$, the fact
that $\Phi_{L,v}^{F,H}$ remains invariant under this operation implies that in
its expansion as in Equation (11) (or its orthogonal analogue), only terms
$\theta_{\beta_{L}+\gamma_{L}}$ in which $\beta_{L}+\gamma_{L}\in
I_{L}^{\perp}$ may show up. The same argument from the proof of Lemma 5.5
shows that if $\beta+\gamma\in H^{\perp}\subseteq I_{H}^{\perp}$ and
$\beta_{L}+\gamma_{L}\in I_{L}^{\perp}$ then $\beta_{D}+\gamma_{D}\in
I^{\perp}$, meaning first that $f_{\beta_{D}+\gamma_{D}}=0$ unless
$\beta_{D}+\gamma_{D}\in I^{\perp}$, and second that we may present
$\Phi_{L,v}^{F,H}$ using the set of representatives $H^{\perp}_{R_{I},I}$ from
Lemma 5.7, and with $\gamma$ taken only from $H\cap I^{\perp}$.

For the periodicity property associated with $\sigma\in\Lambda$, recall that
its effect, except for the usual exponent, simply takes each theta function
$\theta_{\beta_{L}+\gamma_{L}}$ to $\theta_{\beta_{L}+\gamma_{L}+(\sigma+L)}$,
and note that in our situation, $\sigma+L\in I_{L}$ is contained in $(H\cap
I^{\perp})_{L}=H_{L}\cap I_{L}^{\perp}$. When comparing it with the expression
resulting from the assumption that the scalar-valued components of
$\Phi_{L,v}^{F,H}$ satisfy this periodicity property, we deduce that for every
$\beta\in H^{\perp}_{R_{I},I}$, the equality

$\textstyle{\sum_{\gamma\in H\cap
I^{\perp}}\theta_{\beta_{L}+\gamma_{L}+(\sigma+L)}f_{\beta_{D}+\gamma_{D}}=\sum_{\gamma\in
H\cap I^{\perp}}\theta_{\beta_{L}+\gamma_{L}}f_{\beta_{D}+\gamma_{D}}}$

must hold. Replacing $\gamma$ on the left hand side by $\gamma-\sigma_{H}$,
where $\sigma_{H}\in I_{H}\subseteq H\cap I^{\perp}$ is the element with
$(\sigma_{H})_{L}=\sigma+L$, we obtain the same theta functions on both sides,
and the uniqueness of theta expansions implies that changing the index of $f$
by $(\sigma_{H})_{D}$, namely by an arbitrary element of $I$, does not change
the function. We may therefore write $f_{\beta_{D}+\gamma_{D}}$, for
$\beta_{D}+\gamma_{D}\in I^{\perp}$, as $g_{\beta_{D}+\gamma_{D}+I}$ for some
function $G:\mathcal{H}\to\mathbb{C}[\overline{D}]$ (which is clearly unique),
and obtain that $F=\uparrow_{I}G$ as desired. Our expression for
$\Phi_{L,v}^{F,H}$ also becomes the one from Equation (19) in the proof of
Proposition 5.8 in this case, so that $\Phi_{L,v}^{F,H}$ is indeed the image
of $\Phi_{\Lambda,v}^{G,\overline{H}}$, with this $\overline{H}$, under the
corresponding operator. This proves the theorem. ∎

Note that the proof of Theorem 5.13 did not require any assumption on whether
$I$ is contained in $H_{D}^{\perp}$ intersects it trivially, so that it
corresponds to any mixture of cases $(i)$ and $(iii)$ in Corollary 5.10,
combined via Remark 5.12 to yield the general combination (with indeed the
groups $\overline{H}$ and $\overline{H}_{\Lambda}^{\perp}$ being the ones
constructed from $H$ and $H_{L}^{\perp}$, as Remark 5.11 shows).

We now leave the lattice $L$ and the scalar-valued periodicity properties as
in Definition 18, but increase the validity of Equation (17) to a larger
subgroup. We recall that this equation is based on $\Delta_{H}$ containing the
isomorphic copy $\\{\delta_{L}+H|\delta_{L}\in H_{L}^{\perp}\subseteq
H^{\perp}\subseteq D_{L}\oplus D\\}$ of $H_{L}^{\perp}$, in which $\delta_{L}$
can be recognized as the projection onto $D_{L}$ of a particular lift $\delta$
of its $\Delta_{H}$-image to $H^{\perp}\subseteq D_{L}\oplus D$. Our way of
extending $H_{L}^{\perp}$ is as follows.

######  Lemma 5.14.

Assume that there is a subgroup $K_{L}\subseteq D_{L}$, which contains
$H_{L}^{\perp}$, and which admits a lift into $H^{\perp}\subseteq D_{L}\oplus
D$ whose composition with the projection onto $D_{L}$ is
$\operatorname{Id}_{K_{L}}$, such that the composition with the projection
onto $\Delta_{H}$ identifies $K_{L}$ with a subgroup of $\Delta_{H}$, with the
$\mathbb{Q}/\mathbb{Z}$-quadratic structure, which contains the image of
$H_{L}^{\perp}$. Then $K_{L}/H_{L}^{\perp}$ embeds into $D$ as an isotropic
subgroup $I$ which intersects both $H_{D}$ and $H_{D}^{\perp}$ trivially, and
$K_{L}$ can be reproduced from $I$ and $H$ as the group $(H\cap
I^{\perp})_{L}^{\perp}$.

###### Proof.

The lift $\delta$ of $\delta_{L}\in K_{L}$ to $H^{\perp}\subseteq D_{L}\oplus
D$ must be of the form $\delta_{L}+\delta_{D}$ with
$\delta_{D}=\tilde{\kappa}\delta_{L}$ for a group homomorphism
$\tilde{\kappa}:K_{L}\to D$. The fact that an element $\delta_{L}\in
H_{L}^{\perp}\subseteq K_{L}$ lifts to $\delta_{L}+0$ for $0\in D$ implies
that $H_{L}^{\perp}\subseteq\ker\tilde{\kappa}$, so that $\tilde{\kappa}$
induces a map $\kappa:K_{L}/H_{L}^{\perp}$ into $D$. As the lift respects the
$\mathbb{Q}/\mathbb{Z}$-quadratic structure, and $D_{L}\oplus D$ is an
orthogonal direct summand, we deduce that $\kappa(K_{L}/H_{L}^{\perp})$, which
will be our subgroup $I$, is isotropic.

Assume now that $\delta_{D}=\tilde{\kappa}\delta_{L}$ lies in $H_{D}^{\perp}$
for some $\delta_{L}\in K_{L}$, with $\delta$ being the lift
$\delta_{L}+\delta_{D}\in D_{L}\oplus D$, and take any $\gamma\in H$. The fact
that $(\delta,\gamma)=0$ for $\delta\in H^{\perp}$ and $\gamma\in H$, and that
$(\delta_{D},\gamma_{D})=0$ since $\delta_{D}\in H_{D}^{\perp}$ and
$\gamma_{D}\in H_{D}$, implies that $(\delta_{L},\gamma_{L})=0$, and as
$\gamma\in H$ is arbitrary, and with it so is $\gamma_{L}\in H_{L}$, we deduce
that $\delta_{L}\in H_{L}^{\perp}$. In particular, with $\delta_{D}=0$ this
shows that $\ker\tilde{\kappa}=H_{L}^{\perp}$ and $\kappa$ is injective (so
that $I\cong K_{L}/H_{L}^{\perp}$), but also that $I\cap
H_{D}^{\perp}=\\{0\\}$.

Consider now the case that $\delta_{D}=\tilde{\kappa}\delta_{L}$ is in
$H_{D}$, and write $\delta_{H}$ for the corresponding element of $H$, which we
write as $\delta_{H,L}+\delta_{D}$ with $\delta_{H,L}$ the $H_{L}$-image of
$\delta_{H}$. Then the $\Delta_{H}$-image of $\delta$, hence of $\delta_{L}\in
K_{L}$, coincides with that of $\delta-\delta_{H}$, which is the sum of
$\delta_{L}-\delta_{H,L}\in D_{L}$ and $0\in D$ and lies in $H^{\perp}$, so
that $\delta_{L}-\delta_{H,L}\in H_{L}^{\perp}$ and is taken to the same
element of $\Delta_{H}$. As the map from $K_{L}$ to $\Delta_{H}$ is injective
by hypothesis, we obtain that $\delta_{L}=\delta_{L}-\delta_{H,L}\in
H_{L}^{\perp}$, and again $\delta_{D}$ vanishes, which proves that $I\cap
H_{D}=\\{0\\}$ as well. Altogether, $\kappa$ embeds $K_{L}/H_{L}^{\perp}$ into
$D$ as the isotropic subgroup $I$, with the desired properties.

Now, Lemma 5.1 combines with the fact that $I\cap H^{\perp}=\\{0\\}$ to show
that $H$ contains $H\cap I^{\perp}$ with index $|I|$, hence the same relation
holds between their isomorphs $H_{L}$ and $(H\cap I^{\perp})_{L}$, which
implies that $(H\cap I^{\perp})_{L}^{\perp}$ contains $H_{L}^{\perp}$ with the
same index inside the discriminant for $D_{L}$. As $K_{L}$ also contains
$H_{L}^{\perp}$ with the same index (indeed, $I$ is isomorphic to the
quotient), it has the same size as $(H\cap I^{\perp})_{L}^{\perp}$, and it
suffices to prove that $K_{L}$ is contained in the latter subgroup. But
indeed, for any $\delta_{L}\in K_{L}$, embedded as
$\delta_{L}+\tilde{\kappa}\delta_{L}\in H^{\perp}\subseteq D_{L}\oplus D$, its
pairing with $\gamma\in H$ was seen to vanish, and if $\gamma$ lies in $H\cap
I^{\perp}$ then $(\gamma_{D},\tilde{\kappa}\delta_{L})=0$ as well, yielding
the orthogonality of $\delta_{L}$ to any $\gamma_{L}$ for $\gamma$ in that
intersection. This completes the proof of the lemma. ∎

Recalling from the proof of Lemma 4.7 that the operators on
$\mathbb{C}[\Delta_{H}]$ that arise from $H_{M}^{\perp}$ (or $H_{L}^{\perp}$
in our setting) are obtained via Definition 2.5 through the embedding of this
group into $\Delta_{H}$ considered also in Lemma 5.14, we can thus state and
prove the following result.

######  Theorem 5.15.

Let $K_{L}\subseteq D_{L}$ be as in Lemma 5.14, with $I\subseteq D$ the
corresponding isotropic subgroup, and set $\overline{D}:=I^{\perp}/I$ as
usual. Assume that the function $\Phi_{L,v}^{F,H}$ associated with
$F:\mathcal{H}\to\mathbb{C}[\Delta_{H}]$ via Proposition 4.9 satisfies the
periodicity from Equation (17), in which $\sigma$ and $\nu$ from $L^{*}$ can
have arbitrary $D_{L}$-images in $K_{L}$, and where $t_{\sigma}$ and
$\chi_{\nu}$ are defined on $\Phi_{L,v}^{F,H}$ via this embedding of $K_{L}$
into $\Delta_{H}$. Then there is a unique function
$G:\mathcal{H}\to\mathbb{C}[\overline{D}]$ such that $F=\uparrow_{I}G$, and
$\Phi_{L,v}^{F,H}$ is $\iota_{\overline{\Delta}}\Phi_{L,v}^{G,H\cap
I^{\perp}}$ as in case $(iv)$ of Corollary 5.10.

###### Proof.

We begin by considering an element $\nu\in L^{*}$ with $\nu+L\in K_{L}$, and
comparing the resulting periodicity relations for the theta components
$\theta_{\beta_{L}+\gamma_{L}}$ and using $\chi_{\nu}$ for this $\nu$ by
assumption. We recall that for the simple case of $\nu+L\in H_{L}^{\perp}$,
the action of $t_{\nu}$ on the basic vector
$\mathfrak{e}_{\beta}=\mathfrak{e}_{\beta_{L}}\otimes\mathfrak{e}_{\beta_{D}}$
of $\mathbb{C}[D_{L}\oplus D]$ multiplies it by
$\mathbf{e}\big{(}(\nu+L,\beta_{L})\big{)}$, and the pairing is that $\nu+L\in
H_{L}^{\perp}\subseteq D_{L}$ as contained in $D_{L}\oplus D$, with
$\beta=\beta_{L}+\beta_{D}$. Our assumption means that if $\nu+L$ is a more
general element of $K_{L}$, which is embedded in $H^{\perp}\subseteq
D_{L}\oplus D$, then the action of $\chi_{\nu}$ multiplies this vector
$\mathfrak{e}_{\beta}$ by $\mathbf{e}\big{(}(\nu+L,\beta)\big{)}$ (which for
$\beta\in H^{\perp}$ can be evaluated using the image of $\nu+L\in K_{L}$ in
$\Delta_{H}$), with the pairing being the sum of $(\nu+L,\beta_{L})$ and
$\big{(}\tilde{\kappa}(\nu+L),\beta_{D}\big{)}$, with $\tilde{\kappa}:K_{L}\to
D$ the map defined in the proof of Lemma 5.14.

So take now $\beta\in H^{\perp}\subseteq D_{L}\oplus D$, and consider the
vector $\mathfrak{e}_{\beta+H}\in\mathbb{C}[\Delta_{H}]$ and the coefficient
$\sum_{\gamma\in H}\theta_{\beta_{L}+\gamma_{L}}f_{\beta_{D}+\gamma_{D}}$
multiplying it in $\Phi_{L,v}^{F,H}$. The translation of the argument by our
$\nu$ multiplies the theta function by the exponent of
$(\nu+L,\beta_{L}+\gamma_{L})$, and when we compare it with the action of
$\chi_{\nu}$ on this vector, as evaluated above, we deduce that only summands
associated with $\gamma\in H$ for which $(\nu+L,\gamma_{L})$ equals
$\big{(}\tilde{\kappa}(\nu+L),\beta_{D}\big{)}$ can occur, where
$\tilde{\kappa}(\nu+L)$ or equivalently $\kappa[(\nu+L)+H_{L}^{\perp}]$ for
$(\nu+L)+H_{L}^{\perp}\in K_{L}/H_{L}^{\perp}$, runs over $I$ as $\nu$ varies
over the elements of $L^{*}$ with $D_{L}$-images in $K_{L}$.

We now observe that as $I\cap H^{\perp}=\\{0\\}$ by Lemma 5.14, the proof of
Lemma 5.1 produces a perfect pairing from $I\times[H/(H\cap I^{\perp})]$ to
$\mathbb{Q}/\mathbb{Z}$, so that for every such $\beta$ we have an element
$\alpha\in H$ such that $(\delta_{D},\alpha_{D})=(\delta_{D},\beta_{D})$ for
every $\delta_{D}\in I$. We can thus represent $\beta+H\in\Delta_{H}$ by
$\beta+\delta\in D_{L}\oplus I^{\perp}$, noting that for an element of
$D_{L}\oplus I^{\perp}$ the set of $\gamma$ from the previous paragraph
becomes the subgroup $H\cap I^{\perp}$ (via the argument from the proof of
Lemma 5.5 once again). We thus take the set of representatives
$H^{\perp}_{R_{I},I}$ from Lemma 5.7, and deduce that for each $\beta$ in that
set, the coefficient in front of $\mathfrak{e}_{\beta+H}$ in
$\Phi_{L,v}^{F,H}$ is $\sum_{\gamma\in H\cap
I^{\perp}}\theta_{\beta_{L}+\gamma_{L}}f_{\beta_{D}+\gamma_{D}}$, recalling
that all the indices $\beta_{D}+\gamma_{D}$ of $f$ that show up here lie in
$I^{\perp}$.

We now turn to the periodicity property arising from $\sigma\in L^{*}$ with
$\sigma+L\in K_{L}$ (which is indeed the group $(H\cap I^{\perp})_{L}^{\perp}$
by Lemma 5.14), and to the action of $t_{\sigma}$ for such $\sigma$ as defined
here. The argument from the proofs of Proposition 4.9 and Theorem 4.12
combines with the set of representatives in our setting to yield the equality

$t_{\sigma}\sum_{\begin{subarray}{c}\tilde{\beta}\in H^{\perp}_{R_{I},I}\\\
\tilde{\gamma}\in H\cap
I^{\perp}\end{subarray}}\theta_{\tilde{\beta}_{L}+\tilde{\gamma}_{L}}f_{\tilde{\beta}_{D}+\tilde{\gamma}_{D}}\mathfrak{e}_{\tilde{\beta}_{L}+\tilde{\beta}_{D}+H}=\sum_{\begin{subarray}{c}\beta\in
H^{\perp}_{R_{I},I}\\\ \gamma\in H\cap
I^{\perp}\end{subarray}}\theta_{\beta_{L}+\gamma_{L}+(\sigma+M)}f_{\beta_{D}+\gamma_{D}}\mathfrak{e}_{\beta_{L}+\beta_{D}+H}$

and we employ once again the notation from Lemma 4.8, for the set of
representatives $R_{I}$ for $D_{L}/(H\cap I^{\perp})_{L}$ (the
perpendicularity condition from part $(iii)$ of Lemma 5.7 it trivial, since
$I_{L}=\\{0\\}$ in our case).

We make again the substitution $\tilde{\beta}_{L}=r_{\sigma}(\beta_{L})$ and
$\tilde{\gamma}:=\gamma+h_{\sigma}(\beta_{L})$ from that lemma with these
parameters, but note that $t_{\sigma}$ subtracts the image of $\sigma+L\in
K_{L}$ from $\beta$, and that image is the sum of $\sigma+L\in D_{L}$ and
$\tilde{\kappa}(\sigma+L)\in I\subseteq I^{\perp}\subseteq D$. We subtract
$h_{-\sigma}(\tilde{\beta}_{L})\in H\cap I^{\perp}\subseteq H$ to remain in
the same coset, and the representing element of $H^{\perp}$ is the sum of
$h_{-\sigma}(\tilde{\beta}_{L})\in R_{I}\subseteq D_{L}$ and the element
$\tilde{\beta}_{D}-h_{-\sigma}(\tilde{\beta}_{L})_{D}-\tilde{\kappa}(\sigma+L)$
from $I^{\perp}\subseteq D$. Part $(ii)$ of Lemma 4.8 implies that the first
expression is $\beta_{L}$ back again, and we set $\tilde{\beta}_{D}$ to be
$\beta_{D}+h_{-\sigma}(\tilde{\beta}_{L})_{D}+\tilde{\kappa}(\sigma+L)$, or
equivalently $\beta_{D}-h_{\sigma}(\beta_{L})_{D}+\tilde{\kappa}(\sigma+L)$
(by the same lemma), so that we get the same vector
$\mathfrak{e}_{\beta_{L}+\beta_{D}+H}$ on both sides. Using the uniqueness of
theta expansions once again, and the fact that
$\tilde{\beta}_{D}+\tilde{\gamma}_{D}$ equals $\beta_{D}+\gamma_{D}$ plus the
arbitrary element $\tilde{\kappa}(\sigma+L)\in I$, we deduce that
$f_{\beta_{D}+\gamma_{D}}$ remains invariant under changing the index from
$D$, which lies in $I^{\perp}$, by an element of $I$.

We can thus write $f_{\beta_{D}+\gamma_{D}}$ for an index from $I^{\perp}$ as
$g_{\beta_{D}+\gamma_{D}+I}$ for function
$G:\mathcal{H}\to\mathbb{C}[\overline{D}]$, that is unique as usual, and
obtain that $F=\uparrow_{I}G$. We also observe that if
$\beta=\beta_{L}+\beta_{D}\in H^{\perp}_{R_{I},I}$ and $\alpha+(H\cap
I^{\perp})=p(\beta+H)$ for $\alpha=\alpha_{L}+\alpha_{\overline{D}}\in
H^{\perp}_{R_{I}}\subseteq D_{L}\oplus\overline{D}$ then
$\alpha_{L}=\beta_{L}\in R_{I}$ and $\alpha_{\overline{D}}=\beta_{D}+I$, so
that the theta function is $\theta_{\alpha_{L}+\gamma_{L}}$ and it multiplies
$g_{\alpha_{\overline{D}}+\gamma_{\overline{D}}}$. Moreover, $\beta_{D}$ is
the unique element of $I^{\perp}$ with $\overline{D}$-image
$\alpha_{\overline{D}}$ and such that $\beta_{L}+\beta_{D}$, with
$\beta_{L}=\alpha_{L}$, lies in $H^{\perp}$, which defines $\beta+H$ for this
$\beta$ to be $\iota_{\overline{\Delta}}\big{(}\alpha+(H\cap
I^{\perp})\big{)}$. Altogether we deduce that $\Phi_{L,v}^{F,H}$ is indeed
$\iota_{\overline{\Delta}}\Phi_{L,v}^{G,H\cap I^{\perp}}$, like Proposition
5.8 and case $(iv)$ of Corollary 5.10 imply. This completes the proof of the
theorem. ∎

Like with Proposition 5.8, Theorems 5.13 and 5.15 only deal with the
periodicity properties, and the uniqueness in the constructions imply that
$\Phi_{L,v}^{F,H}$ is a Jacobi form of some weight with respect to some group
$\Gamma$ if and only if the corresponding functions arising from some $G$ have
the same property (with the appropriate representations), as indeed $F$ is
modular with the associated parameters if and only if $G$ is. However, as we
saw, the modularity property plays no role in these results. We remark again
that when $L$ is an $\mathcal{O}$-lattice $M$, with the quadratic structure
arising from the Hermitian one on $M$, then all of the results of this section
extend to this setting, provided that all the groups involved in the
constructions are preserved under the action of $\mathcal{O}$.

######  Remark 5.16.

In some sense, expressing a function as $\Phi_{L,v}^{F,H}$, or as
$\Phi_{M,v}^{F,H}$ in the Hermitian setting, implies that it satisfies at
least the periodicity properties arising from $L$ for the scalar-valued
components and from the inverse image of $H_{L}^{\perp}$ inside $D_{L}$ with
the action. Theorems 5.13 and 5.15 can be seen as saying that the precise
periodicity properties (the scalar-valued and the vector-valued ones
respectively) that are satisfied by $\Phi_{L,v}^{F,H}$ (or $\Phi_{M,v}^{F,H}$)
detect what are the most natural lattice $\Lambda\subseteq L$ and group
$\overline{H}_{\Lambda}^{\perp}$ (containing the image of $H_{L}^{\perp}$)
that really define this function. This will be important for defining vector-
valued Jacobi forms with matrix index below. Note that case $(ii)$ of
Corollary 5.10 does not affect the periodicity properties at all, and simply
presents an interplay between the $\uparrow$ operators on $G$ and on $\Phi$.

##  6 Holomorphic Hermitian Jacobi Forms and Jacobi Forms of Matrix Index

The main motivation and most applications for considering Jacobi forms with
lattice index are clearer and more tangible when the lattice is positive
definite. Since (as far as I know) no reference gathers their properties in
this particularly important special case, we write the simpler expressions and
properties that are obtained from our general ones in this case.

So let $M$ be an $\mathcal{O}$-lattice, which we now assume to be positive
definite of some rank $b_{+}$. Then the sets from Equations (1) and (2)
consist of one point, so that the variable $v$ can be omitted. In addition,
the projections onto the negative definite space $v_{-}=\\{0\\}$ vanish (as
does the Laplacian operator $\Delta_{v_{-}}^{h}$), and projecting onto the
positive definite space $v_{+}=M_{\mathbb{R}}$ leaves every vector invariant
(with $\Delta_{v_{+}}^{h}$ being just $\Delta_{M_{\mathbb{R}}}^{h}$).
Therefore the definition in Equation (4) and the properties from Propositions
2.7 and 2.9 and Theorem 2.8 combine to the following statement, in which we
recall again the operators from Definition 2.5.

######  Theorem 6.1.

The theta function

$\Theta_{M}(\tau,\zeta,\omega):=\textstyle{\sum_{\lambda\in
M^{*}}\mathbf{e}\big{(}\tau|\lambda}|^{2}+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}\mathfrak{e}_{\lambda+M}$

is a holomorphic function from $\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}$ to $\mathbb{C}[D_{M}]$ that is a solution to the
holomorphic heat equation and satisfies the periodicity equation

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

for any $\sigma$ and $\nu$ from $M^{*}$, as well as, for every
$A\in\operatorname{U}(1,1)(\mathbb{Z})$, the equality

$\Theta_{M}\Big{(}A\tau,\tfrac{\zeta}{j(A,\tau)},\tfrac{\omega}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{b_{+}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta,\omega\rangle}{j(A,\tau)}\Big{)}\rho_{M}(A)\Theta_{M}(\tau,\zeta,\omega).$

In this setting, from Definition 7 we deduce that a _Hermitian Jacobi form of
weight $k\in\mathbb{Z}$ and index $M$ with respect to
$\Gamma\subseteq\operatorname{U}(1,1)(\mathbb{Z})$_ (of finite index) is a
function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}$, holomorphic in all the variables
except perhaps the first one (in which it is smooth), for which the equality

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

holds for every two elements $\sigma$ and $\nu$ from $M$, as does the equality

$\Phi\Big{(}A\tau,\tfrac{\zeta}{j(A,\tau)},\tfrac{\omega}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta,\omega\rangle}{j(A,\tau)}\Big{)}\Phi(\tau,\zeta,\omega)$

for every $A\in\Gamma$, plus some growth conditions. Via Remarks 3.5 and 3.7
we see that every such Jacobi form has a Fourier expansion of the sort

$\Phi(\tau,\zeta,\omega)=\textstyle{\sum_{m\in\frac{1}{N}\mathbb{Z}}\sum_{\lambda\in
M^{*}}\tilde{c}_{m,\lambda}(y)\mathbf{e}\big{(}m\tau+\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle\big{)}}$
(20)

(for some $N$ depending on $\Gamma$), with the coefficients
$\tilde{c}_{m,\lambda}(y)$ being constant if and only of $\Phi$ is holomorphic
also in $\tau\in\mathcal{H}$, a situation where the growth condition amounts
to $\tilde{c}_{m,\lambda}=0$ unless $m\gg-\infty$ (such functions are called
_weakly holomorphic Hermitian Jacobi forms_). The Fourier coefficients satisfy
the relation
$\tilde{c}_{m,\lambda}=\tilde{c}_{m+\langle\sigma,\lambda\rangle+\langle\lambda,\sigma\rangle+|\sigma|^{2},\lambda+\sigma}$
for every $\sigma\in M$ (regardless of holomorphicity), and we say that a
weakly holomorphic Hermitian Jacobi form is a _weak Hermitian Jacobi form_ ,
or a _holomorphic Hermitian Jacobi form_ , or a _Hermitian Jacobi cusp form_ ,
if a non-zero Fourier coefficient $\tilde{c}_{m,\lambda}\neq 0$ can occur only
if $m\geq 0$ (resp. $m\geq|\lambda|^{2}$, resp. $m>|\lambda|^{2}$). In the
coordinates used when the index is a matrix, the Fourier expansion takes the
form presented in Remark 6.12 below.

Theorem 3.4 and Proposition 3.6, with some additional details involving
holomorphicity, now read as follows.

######  Theorem 6.2.

The Hermitian Jacobi forms of weight $k\in\mathbb{Z}$ and index $M$ with
respect to $\Gamma$ are precisely the functions on $\mathcal{H}\times
M_{\mathbb{C}}\times M_{\overline{\mathbb{C}}}\to\mathbb{C}$ that are obtained
as pairings
$\Phi_{M}^{F}(\tau,\zeta,\omega):=[\Theta_{M}(\tau,\zeta,\omega),F(\tau)]_{M}$
of $\Theta_{M}$ with modular forms $F$ of weight $k-b_{+}$ and representation
$\rho_{\overline{M}}$ with respect to $\Gamma$, as in Equation (8). The
Hermitian Jacobi form $\Phi_{M}^{F}$ and the associated modular form $F$ are
weakly holomorphic, holomorphic, and cuspidal together.

As with other types of Jacobi forms, the weakly holomorphic modular form $F$
that is associated with a weak Hermitian Jacobi form $\Phi:=\Phi_{M}^{F}$ via
Theorem 6.2 need not be holomorphic, but rather its components multiplying
basis vectors $\mathfrak{e}_{\gamma}^{*}$ for anisotropic $\gamma\in D_{M}$
are allowed to have poles of very small order at the cusps.

We shall also be needing the vector-valued objects, for which we assume that
$D$, $H$, and $\Delta_{H}$ are given as in Definition 13, and $k$ and $\Gamma$
are as above. Then a _Hermitian Jacobi form of weight $k$, index $(M,H)$, and
representation $\rho_{\Delta_{H}}$ with respect to $\Gamma$_ is defined to be
a smooth function $\Phi:\mathcal{H}\times M_{\mathbb{C}}\times
M_{\overline{\mathbb{C}}}\to\mathbb{C}[\Delta_{H}]$, which is holomorphic when
the first variable is fixed, and which satisfies the equality

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

whenever $\sigma$ and $\nu$ are $M^{*}$ and $\sigma+M$ and $\nu+M$ lie in
$H_{M}^{\perp}$, as well as

$\Phi\Big{(}A\tau,\tfrac{\zeta}{j(A,\tau)},\tfrac{\omega}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\langle\zeta,\omega\rangle}{j(A,\tau)}\Big{)}\rho_{\Delta_{H}}(A)\Phi(\tau,\zeta,\omega)$

whenever $A\in\Gamma$, and the usual growth conditions on the resulting
Fourier coefficients as functions on $\mathcal{H}$. When $\Phi$ is also
holomorphic in $\tau$, it is a weakly holomorphic Jacobi form (with these
parameters), and one can extend the definitions of weak, holomorphic, and
cuspidal Jacobi forms (or Jacobi cusp forms) to the vector-valued setting in
an analogous manner, using the resulting Fourier expansions of the scalar-
valued components and assumptions required for non-vanishing.

In this setting, Theorem 4.12 transforms into the following result.

######  Theorem 6.3.

Vector-valued Hermitian Jacobi forms $\Phi$ of weight $k$, index $(M,H)$, and
representation $\rho_{\Delta_{H}}$ with respect to $\Gamma$ are in a bijective
correspondence with modular forms $F$ of weight $k-b_{+}$ and representation
$\rho_{D}$ with respect to $\Gamma$, with the map $F\mapsto\Phi$ given by
$\Phi(\tau,\zeta,\omega):=\downarrow_{H}\big{(}\Theta_{M}(\tau,\zeta,\omega)\otimes
F(\tau)\big{)}$. This bijection preserves weak holomorphicity, holomorphicity,
and cuspidality.

As before, taking $D=D_{\overline{M}}$ and $H$ the diagonal subgroup in
Theorem 6.3 reproduces Theorem 6.2.

Since [BRZ] will also be using the holomorphic version of Theorem 4.15, we
give it here as well. Let now $L$ be an positive definite even lattice of rank
$b_{+}$ over $\mathbb{Z}$, take $D$, $H$ with the projections $H_{L}$ and
$H_{D}$, and $\Delta_{H}$ as above. The theta function

$\Theta_{L}(\tau,\zeta):=\textstyle{\sum_{\gamma\in
D_{L}}\big{[}\sum_{\lambda\in
L+\gamma}\mathbf{e}\big{(}\tau\frac{\lambda^{2}}{2}+(\lambda,\zeta)\big{)}\big{]}\mathfrak{e}_{\gamma}}$

(with no Grassmannian variable) then satisfies the usual, holomorphic
modularity of weight $\frac{b_{+}}{2}$ and representation $\rho_{L}$ with
respect to $\operatorname{Mp}_{2}(\mathbb{Z})$ and periodicity with respect to
$L^{*}\times L^{*}$ (with the operators $\chi_{\nu}$ and $t_{\sigma}$ as
usual), and take a weight $k\in\frac{1}{2}\mathbb{Z}$ and a subgroup
$\Gamma\subseteq\operatorname{Mp}_{2}(\mathbb{Z})$. Then, via Definition 18, a
_Jacobi form of weight $k$, index $(L,H)$, and representation
$\rho_{\Delta_{H}}$ with respect to $\Gamma$_ is a smooth function
$\Phi:\mathcal{H}\times L_{\mathbb{C}}\to\mathbb{C}$ which is holomorphic in
$\zeta\in L_{\mathbb{C}}$, satisfies

$\Phi(\tau,\zeta+\tau\sigma+\nu)=\mathbf{e}\big{(}-\tau\frac{\sigma^{2}}{2}-(\sigma,\zeta)\big{)}\chi_{\nu}t_{\sigma}\Phi(\tau,\zeta)$

for any $\sigma$ and $\nu$ in $L^{*}$ with $D_{L}$-images in $H_{L}^{\perp}$
and

$\Phi\big{(}A\tau,\tfrac{\zeta}{j(A,\tau)}\big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\zeta^{2}}{2j(A,\tau)}\Big{)}\rho_{\Delta_{H}}(A)\Phi(\tau,\zeta)$

wherever $A\in\Gamma$, and an appropriate growth condition. Weak
holomorphicity, and holomorphicity are defined as usual, as are weak Jacobi
forms and Jacobi cusp forms with these parameters.

The associated special case of Theorem 4.15 can now be given.

######  Theorem 6.4.

The Jacobi forms $\Phi$ of weight $k$, index $(L,H)$, and representation
$\rho_{\Delta_{H}}$ with respect to $\Gamma$ are in one-to-one correspondence
with modular forms $F$ of weight $k-\frac{b_{+}}{2}$ and representation
$\rho_{D}$ with respect to $\Gamma$, which to $F$ associates the Jacobi form
$\Phi:(\tau,\zeta)\mapsto\downarrow_{H}\big{(}\Theta_{L}(\tau,\zeta)\otimes
F(\tau)\big{)}$. In this correspondence, $\Phi$ is weakly holomorphic if and
only if $F$ is, and the same for holomorphicity and cuspidality.

Once again, if $D=D_{\overline{M}}$ and $H$ is the diagonal subgroup, then
Theorem 6.3 yields the holomorphic case of Theorem 2.5 of [Ze3], and the known
cases of theta expansions of scalar-valued Jacobi forms with positive definite
lattice index. The analogues of Proposition 5.8 and Theorems 5.13 and 5.15 in
the holomorphic special case (also in the Hermitian setting) look the same as
the general case, and we shall not rewrite them here.

The classical Jacobi forms, from [EZ] and others, had integers as their
indices, with some higher rank ones having symmetric matrices as their indices
(see, e.g., [Zi]), and the $\zeta$-variable was replaced by a complex number
$z$ (or a vector of complex numbers $\\{z_{i}\\}_{i=1}^{b_{+}}$ for some rank
$b_{+}$). These matrices (or integers, as $1\times 1$ matrices) were later
realized to be just the Gram matrices of a basis of positive definite
orthogonal $\mathbb{Z}$-lattice, that now serves as a more natural index as in
[A], [Mo], and in other contexts in [Boy], where $\zeta$ was the complex
linear combination of the chosen basis elements, with the coefficients
$z_{i}$, $1\leq i\leq b_{+}$ (or just $z$ when $b_{+}=1$). Note that in
Fourier–Jacobi expansions of Siegel modular forms, the Fourier–Jacobi
coefficients are Jacobi forms with matrix (or integer) index, and when
establishing modularity results of various objects, as in [Zh], [BRa], or the
orthogonal part of [BRZ], indeed this interpretation is the required one.
Since [Xi] and the unitary part of [BRZ] need Hermitian Jacobi forms also of
matrix index, We will now carry out the same transfer for defining those
formally. We shall stick to the positive definite setting, as when the pairing
gets involved with the decomposition associated with an element $v$ in some
Grassmannian, more complicated notation is required, and the simpler special
case suffices for the applications.

Since we shall work with a basis over $\mathcal{O}$, we thus take the
$\mathcal{O}$-lattice $M$ to be a _free_ module over $\mathcal{O}$, and we fix
some basis $M=\bigoplus_{i=1}^{b_{+}}\mathcal{O}\varepsilon_{i}$. Let $S$ be
the matrix whose $ij$-entry is
$\langle\varepsilon_{j},\varepsilon_{i}\rangle\in\mathcal{D}^{-1}$, which is
Hermitian and thus its diagonal entries are in $\frac{1}{2}\mathbb{Z}$. The
fact that $M$ is assumed to be even and positive definite implies that the
diagonal entries of $S$ are integral and that $S$ is a positive definite
matrix. We let $\pi$ be the map identifying a vector $a\in\mathbb{K}^{b_{+}}$
with the element $\sum_{i=1}^{b_{+}}a_{i}\varepsilon_{i}$ of
$V:=M\otimes_{\mathbb{Z}}\mathbb{Q}=M\otimes_{\mathcal{O}}\mathbb{K}$, which
thus identifies $\mathcal{O}^{b_{+}}$ with $M$.

Now, as $\frac{1}{\sqrt{-d}}$ generates $\mathcal{D}^{-1}$ over $\mathcal{D}$,
we say that the _dual basis_ for $\\{\varepsilon_{i}\\}_{i=1}^{b_{+}}$ are the
elements $\\{\epsilon_{i}\\}_{i=1}^{b_{+}}\subseteq M_{\mathbb{Q}}$ such that
$\langle\epsilon_{j},\varepsilon_{i}\rangle=\frac{\delta_{ij}}{\sqrt{-d}}$.
Therefore the transition matrix from the basis to its dual is
$\frac{1}{\sqrt{-d}}S^{-t}=\frac{1}{\sqrt{-d}}\overline{S}^{-1}$ (recall that
$S$ is Hermitian and invertible), and thus $M^{*}$ is spanned (freely) over
$\mathcal{O}$ by
$\big{\\{}S^{-t}\varepsilon_{i}\big{/}\sqrt{-d}\big{\\}}_{i=1}^{b_{+}}$,
namely the dual lattice $M^{*}$ is the image of
$S^{-t}\mathcal{O}^{b_{+}}\big{/}\sqrt{-d}$ under the map $\pi$ from the
previous paragraph.

We shall be using, for our variables, the convention that the vector $\zeta$
is $\sum_{i=1}^{b_{+}}z_{i}\varepsilon_{i,\mathbb{C}}$ with some
$z\in\mathbb{C}^{b_{+}}$, and
$\omega=\sum_{i=1}^{b_{+}}w_{i}\varepsilon_{i,\overline{\mathbb{C}}}$ for
$w\in\mathbb{C}^{b_{+}}$. We shall use the notation $r^{*}$ for the conjugate
transpose of any vector $r\in\mathbb{C}^{b_{+}}$, and all this converts
Theorem 6.1 into the following form.

######  Theorem 6.5.

Consider the
$\mathbb{C}\big{[}\big{(}S^{-t}\mathcal{O}^{b_{+}}\big{/}\sqrt{-d}\big{)}/\mathcal{O}^{b_{+}}\big{]}$-valued
function $\Theta_{S}$ on the product
$\mathcal{H}\times\mathbb{C}^{b_{+}}\times\mathbb{C}^{b_{+}}$, which is
defined by

$\Theta_{S}(\tau,z,w):=\textstyle{\sum_{a\in
S^{-t}\mathcal{O}^{b_{+}}/\sqrt{-d}}\mathbf{e}(\tau
a^{*}Sa+a^{*}Sz+w^{t}Sa)\mathfrak{e}_{a+\mathcal{O}^{b_{+}}}}.$

This function is holomorphic and satisfies the heat equation associated with
$S$, its scalar-valued periodicity law is

$\Theta_{S}(\tau,z+\tau
u+v,w+\tau\overline{u}+\overline{v})=\mathbf{e}\big{(}-\tau
u^{*}Su-u^{*}Sz-w^{t}Su\big{)}\Theta_{S}(\tau,z,w)$

wherever $u$ and $v$ are in $\mathcal{O}^{b_{+}}$, and for
$A\in\operatorname{U}(1,1)(\mathbb{Z})$ it satisfies

$\Theta_{S}\Big{(}A\tau,\tfrac{z}{j(A,\tau)},\tfrac{w}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{b_{+}}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\cdot
w^{t}Sz}{j(A,\tau)}\Big{)}\rho_{M}(A)\Theta_{S}(\tau,z,w).$

Note that for the vector $a$ in the sum from Theorem 6.5, as well as in $u$
representing the translationn by $\sigma\in M$, the action of $\mathcal{O}$
involves the endomorphism $J$, meaning that $a$ and $u$ have to be conjugated
(hence taken to $a^{*}$) when it represents $\lambda\in M^{*}$ in the second
variable of the Hermitian pairing (including for
$|\lambda|^{2}=\langle\lambda,\lambda\rangle$), while for $w$ the extension by
scalars is $\mathbb{C}$-linear also in this second variable, leaving $w$
unconjugated (which is also required for the holomorphicity). Also, recall
from Corollary 1.3 that in the basis $\varepsilon_{i,\mathbb{C}}$ associated
with the coordinates from $z$, the action of $\mathbb{K}$ is linear, while
using the basis $\varepsilon_{i,\overline{\mathbb{C}}}$, it is conjugate
linear, explaining the conjugation in the second variable in the periodicity
equation in that theorem (the scalars $j(A,\tau)$ and $j^{c}(A,\tau)$ in the
modularity equation are not affected, as they do not involve the action of
$\mathbb{K}$). We remark that $\Theta_{S}$ depends also on the order
$\mathcal{O}$, despite it not appearing in our notation.

We can now adapt the definition of Jacobi forms to have matrix index as well.
For $\Gamma$ as above, we say that
$\Phi:\mathcal{H}\times\mathbb{C}^{b_{+}}\times\mathbb{C}^{b_{+}}\to\mathbb{C}$
is a _Hermitian Jacobi form of weight $k$ and index $S$ with respect to
$\Gamma$_ in case it is holomorphic in the variables from
$\mathbb{C}^{b_{+}}\times\mathbb{C}^{b_{+}}$, satisfies the equality

$\Phi(\tau,z+\tau u+v,w+\tau\overline{u}+\overline{v})=\mathbf{e}\big{(}-\tau
u^{*}Su-u^{*}Sz-w^{t}Su\big{)}\Phi(\tau,z,w)$

for any $u$ and $v$ in $\mathcal{O}^{b_{+}}$ (with $\tau\in\mathcal{H}$ and
$z$ and $w$ from $\mathbb{C}^{b_{+}}$), as well as

$\Phi\Big{(}A\tau,\tfrac{z}{j(A,\tau)},\tfrac{w}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\cdot
w^{t}Sz}{j(A,\tau)}\Big{)}\Phi(\tau,z,w)$

for any $A\in\Gamma$ (and the growth conditions). The Fourier expansion from
Equation (20) thus involves the coefficients $\tilde{c}_{m,r}$ for $r\in
S^{-t}\mathcal{O}^{b_{+}}/\sqrt{-d}$ (which are constants if and only if
$\Phi$ is holomorphic) and the exponents $\mathbf{e}(m\tau+r^{*}Sz+w^{t}Sr)$,
the equality of the coefficients becomes
$\tilde{c}_{m,r}=\tilde{c}_{m+u^{*}Sr+r^{*}Su+u^{*}Su,r+u}$ for any
$u\in\mathcal{O}^{b_{+}}$. The definitions of weakly holomorphic, weak,
holomorphic, and cuspidal Hermitian Jacobi forms of index $S$ take the obvious
analogous forms.

We can now state the matrix index analogue of Theorem 6.2.

######  Theorem 6.6.

The map taking a modular form $F$ of weight $k-b_{+}$ and representation
$\rho_{\overline{M}}$ with respect to $\Gamma$ to the function
$\Phi_{S}^{F}:\mathcal{H}\times\mathbb{C}^{b_{+}}\times\mathbb{C}^{b_{+}}\to\mathbb{C}$
defined by $\Phi_{S}^{F}(\tau,z,w):=[\Theta_{S}(\tau,z,w),F(\tau)]_{M}$ is a
linear bijection from such modular forms onto the space of Hermitian Jacobi
forms of weight $k$ and index $S$ with respect to $\Gamma$, which preserves
weak holomorphicity, usual holomorphicity, and cuspidality.

For appropriate rank 1 lattices, Theorem 6.6 reproduces some of the results of
[Hav1], [Hav2], [De], and others (at least in the integral weight case).

The positive definite Hermitian matrix $S$ in Theorems 6.5 and 6.6 represented
the pairings of elements from some even $\mathcal{O}$-lattice $M$, and were
thus taken from $\mathcal{D}^{-1}$ (with diagonal entries from $\mathbb{Z}$).
Applications like those from [Xi] and [BRZ] require, however, Jacobi forms
whose index is a Hermitian matrix $S$ which does not necessarily satisfy these
integrality conditions. These will no longer exist as scalar-valued Jacobi
forms, and will have to be constructed as vector-valued Jacobi forms.
Moreover, in these papers the Hermitian matrix does not have to be positive
definite, but can be positive semidefinite as well.

So let $S$ be any positive semidefinite Hermitian matrix of size $c_{+}\times
c_{+}$ over $\mathbb{K}$, consider the order $\mathcal{O}\subseteq\mathbb{K}$
as fixed, and let $\tilde{V}$ be a vector space over $\mathbb{K}$ with basis
$\\{\tilde{\varepsilon}_{i}\\}_{i=1}^{c_{+}}$, which we endow with the
Hermitian form defined by
$\langle\tilde{\varepsilon}_{i},\tilde{\varepsilon}_{j}\rangle=S_{ji}$. This
Hermitian form is thus positive semidefinite, in which the elements of
$\tilde{V}$ that are perpendicular to all of $\tilde{V}$ are the combinations
$\sum_{i=1}^{c_{+}}a_{i}\tilde{\varepsilon}_{i}$ for vectors $a$ that are
annihilated by $S$ as a linear operator. We thus define $V$ to be the
resulting quotient, denote its dimension by $b_{+}$ and the image of
$\tilde{\varepsilon}_{i}\in\tilde{V}$ by $\varepsilon_{i}\in V$, and write
$\pi:\mathbb{K}^{c_{+}}\to V$ for the map taking $a\in\mathbb{K}^{c_{+}}$ to
$\sum_{i=1}^{c_{+}}a_{i}\varepsilon_{i}$ once again, which is now surjective
with kernel equal to $\ker S$.

Set $M_{S}:=\sum_{i=1}^{c_{+}}\mathcal{O}\varepsilon_{i}\subseteq V$ (it is no
longer an $\mathcal{O}$-lattice in the sense of Definition 1.9 since it may be
not projective and the pairings from $M_{S}$ need not necessarily by in
$\mathcal{D}^{-1}$ anymore), and let $M_{S}^{*}$ be its dual $\\{\lambda\in
V|\langle\lambda,\tilde{M}_{S}\rangle\subseteq\mathcal{D}^{-1}\\}$. Then, if
$M$ is any even $\mathcal{O}$-lattice, as in Definition 1.9, that is contained
in $M_{S}^{*}$, then $M_{S}$ is contained in $M^{*}$, namely we have
$\varepsilon_{i}\in M^{*}$ for every $1\leq i\leq c_{+}$. For such $M$ we set
$\Omega:=\pi^{-1}(M)$ and $\Omega^{*}:=\pi^{-1}(M^{*})$, with the latter
containing $\mathcal{O}^{c_{+}}$ since
$\pi(\mathcal{O}^{c_{+}})=M_{S}\subseteq M^{*}$ by our assumption, and let
$\overline{\Omega}:=\Omega/\ker S$ and $\overline{\Omega}^{*}:=\Omega^{*}/\ker
S$ be the associated quotients. Then the quotients
$D_{\Omega}:=\Omega^{*}/\Omega$,
$D_{\overline{\Omega}}:=\overline{\Omega}^{*}/\overline{\Omega}$, and $D_{M}$
are all isomorphic (the latter via $\pi$), and we set $D_{M,S}$ to be the
image $(M_{S}+M)/M$ of $M_{S}$ in $D_{M}$, as well as $D_{\Omega,S}$ its
isomorphic image in $D_{\Omega}$, namely
$(\mathcal{O}^{c_{+}}+\Omega)/\Omega$.

The orthogonal pairing takes the $\pi$-images of two elements $a$ and $b$ of
$\Omega^{*}$ (and more generally, in $\mathbb{K}^{c_{+}}$) to
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(a^{*}Sb)\in\mathbb{Q}$, and by
projecting to $\mathbb{Q}/\mathbb{Z}$ the result depends only on the class of
$b$ (and also just that of $a$) in $D_{\Omega}$. We can thus write
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(a^{*}S\gamma)$ for the resulting
well-defined expression in $\mathbb{Q}/\mathbb{Z}$ for $a\in\Omega^{*}$ and
$\gamma\in D_{\Omega}$, and similarly
$\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(\delta^{*}S\gamma)\in\mathbb{Q}/\mathbb{Z}$
when $\delta$ is also from $D_{\Omega}$, which represents the pairing of their
$D_{M}$-images.

The same considerations leading to Theorem 6.5, including the substitutions
$\zeta=\sum_{i=1}^{c_{+}}z_{i}\varepsilon_{i,\mathbb{C}}$ and
$\omega=\sum_{i=1}^{c_{+}}w_{i}\varepsilon_{i,\overline{\mathbb{C}}}$ for
vectors $z$ and $w$ from $\mathbb{C}^{c_{+}}$, then transforms the theta
function $\Theta_{M}$ into the function

$\Theta_{S,\Omega}(\tau,z,w):=\textstyle{\sum_{\gamma\in
D_{\Omega}}\big{[}\sum_{a\in\gamma+\Omega}\mathbf{e}(\tau
a^{*}Sa+a^{*}Sz+w^{t}Sa)\big{]}\mathfrak{e}_{\gamma}}$ (21)

(note that both sides indeed remain invariant when one changes $z$ or $w$ by
an element of the appropriate complexification of $\ker S$). The map taking
$\mathcal{O}^{c_{+}}$ to its image in $D_{\Omega,S}\subseteq D_{\Omega}\cong
D_{M}$ (with the quadratic structure) combines with the operators from
Definition 2.5 to produce the operators
$t_{u}:\mathfrak{e}_{\gamma}\mapsto\mathfrak{e}_{\gamma-(u+\Omega)}$ and
$\chi_{v}:\mathfrak{e}_{\gamma}\mapsto\mathbf{e}\big{(}\operatorname{Tr}^{\mathbb{K}}_{\mathbb{Q}}(v^{*}S\gamma)\big{)}\mathfrak{e}_{\gamma}$
for every $u$ and $v$ from $\mathcal{O}^{c_{+}}$. Then the periodicity from
Theorem 6.1, restricted to $\mathcal{O}^{c_{+}}\subseteq\Omega^{*}$, becomes

$\Theta_{S,\Omega}(\tau,z+\tau
u+v,w+\tau\overline{u}+\overline{v})=\mathbf{e}\big{(}-\tau
u^{*}Su-u^{*}Sz-w^{t}Su\big{)}\chi_{v}t_{u}\Theta_{S,\Omega}(\tau,z,w)$ (22)

for any such $u$ and $v$ (this includes the scalar-valued periodicity for the
sublattice $\Omega\cap\mathcal{O}^{c_{+}}=\pi^{-1}(M\cap M_{S}^{*})$, but not
necessarily for all of $\Omega$ and $M$). The theta function
$\Theta_{S,\Omega}$ also has the modularity property from Theorem 6.5, with
the representation $\rho_{M}$ associated with our $\mathcal{O}$-lattice $M$.

Note that in the case considered in that theorem, the module $M_{S}$ is an
even free $\mathcal{O}$-lattice and is thus contained in $M_{S}^{*}$, the
lattice $M$ is $M_{S}$, and thus $\Omega$ (which also equals
$\overline{\Omega}$, and $c_{+}=b_{+}$, since $S$ is positive definite now) is
$\mathcal{O}^{b_{+}}$. Hence the function $\Theta_{S,\Omega}$ from Equation
(21) is $\Theta_{S}$ defined there, and Equation (22) reduces to the scalar-
valued periodicity property from that theorem, since $t_{u}$ and $\chi_{v}$
become trivial for $u$ and $v$ in $\mathcal{O}^{b_{+}}=\Omega$.

Consider now any other discriminant form $D$ and a horizontal isotropic
subgroup $H$ in $D_{\Omega}\oplus D$, such that the projection $H_{\Omega}$ of
$H$ to $D_{\Omega}$ is perpendicular to $D_{\Omega,S}$. Then Lemma 4.7 implies
that $t_{u}$ and $\chi_{v}$ remain well-defined on the associated quotient
$\Delta_{H}:=H^{\perp}/H$ for any $u$ and $v$ in $\mathcal{O}^{c_{+}}$.
Recalling that for any $a\in\mathcal{O}^{c_{+}}$, its image in $D_{\Omega}$
must have quadratic value $a^{*}Sa+\mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$, we may
define vector-valued Hermitian Jacobi forms with index $S$ as follows.

######  Definition 6.7.

Let $\Delta$ be a discriminant form, admitting a group homomorphism
$\psi:\mathcal{O}^{c_{+}}\to\Delta$ such that $\psi(a)=0$ in case $Sa=0$ and
$|\psi a|^{2}=a^{*}Sa+\mathbb{Z}$ for any $a\in\mathcal{O}^{c_{+}}$, and let
$\Gamma$ be a finite index subgroup of $\operatorname{U}(1,1)(\mathbb{Z})$.
Then a _Hermitian Jacobi form of weight $k$, representation $\rho_{\Delta}$,
and index $(S,\psi)$_ is a smooth function
$\Phi:\mathcal{H}\times\mathbb{C}^{c_{+}}\times\mathbb{C}^{c_{+}}\to\mathbb{C}[\Delta]$
which satisfies the periodicity property

$\Phi(\tau,z+\tau u+v,w+\tau\overline{u}+\overline{v})=\mathbf{e}\big{(}-\tau
u^{*}Su-u^{*}Sz-w^{t}Su\big{)}\chi_{\psi v}t_{\psi u}\Phi(\tau,z,w)$ (23)

for every $u$ and $v$ in $\mathcal{O}^{c_{+}}$, the modularity property

$\Phi\Big{(}A\tau,\tfrac{z}{j(A,\tau)},\tfrac{w}{j^{c}(A,\tau)}\Big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\cdot
w^{t}Sz}{j(A,\tau)}\Big{)}\rho_{\Delta}(A)\Phi(\tau,z,w),$ (24)

and the exponential growth condition on the resulting Fourier coefficients.

Before we state the result for the Hermitian Jacobi forms from Definition 24,
we consider the orthogonal analogue, for which establishing the result is
simpler, for reasons to be explained just before the result itself, which
appears as Theorem 6.11 below.

In a similar manner, if $S$ is any positive semidefinite symmetric matrix over
$\mathbb{Q}$, of size $c_{+}\times c_{+}$, then we can endow the vector space
$\tilde{V}:=\bigoplus_{i=1}^{c_{+}}\mathbb{Q}\tilde{\varepsilon}_{i}$ by the
positive definite quadratic form taking
$\sum_{i=1}^{c_{+}}a_{i}\tilde{\varepsilon}_{i}$ for $a\in\mathbb{Q}^{c_{+}}$
to $a^{t}Sa$. Then we get
$(\tilde{\varepsilon}_{i},\tilde{\varepsilon}_{j})=2S_{ji}$, yielding a
positive semidefinite symmetric bilinear form whose kernel again coincides
with that of $S$, and we define the corresponding quotient $V$ of dimension
$b_{+}$, the images $\varepsilon_{i}$, $1\leq i\leq c_{+}$, and the surjection
$\pi:\mathbb{Q}^{c_{+}}\to V$ as above. We again set
$L_{S}=\sum_{i=1}^{c_{+}}\mathbb{Z}\varepsilon_{i}\subseteq V$ and its dual
$L_{S}^{*}$ there, and given an even lattice $L\subseteq L_{S}^{*}\subseteq V$
we again define $\Omega:=\pi^{-1}(L)$ as contained in
$\Omega^{*}:=\pi^{-1}(L^{*})$, and the quotients
$\overline{\Omega}:=\Omega/\ker S$ and $\overline{\Omega}^{*}:=\Omega^{*}/\ker
S$, with the isomorphic quotients $D_{\Omega}$, $D_{\overline{\Omega}}$, and
$D_{L}$. Observing that the pairing of $\pi(a)$ with $\pi(b)$ is now
$2a^{t}Sb$, and expressing the variable $\zeta\in V_{\mathbb{C}}$ of any
orthogonal Jacobi theta function (such as the one from Equation (16), but for
positive definite $L$) as $\sum_{i=1}^{c_{+}}z_{i}\varepsilon_{i}$ for some
$z\in\mathbb{C}^{c_{+}}$, we can give the orthogonal analogue of Definition
24.

######  Definition 6.8.

Consider a discriminant form $\Delta$, a subgroup $\Gamma$ of finite index in
$\operatorname{SL}_{2}(\mathbb{Z})$ (or in its metaplectic cover), and take a
map $\psi:\mathbb{Z}^{c_{+}}\to\Delta$ satisfying $\frac{(\psi
a)^{2}}{2}=a^{t}Sa+\mathbb{Z}$ for every $a\in\mathbb{Z}^{c_{+}}$ and
vanishing on elements of $\ker S$. We then define a _Jacobi form of weight
$k$, representation $\rho_{\Delta}$, and index $(S,\psi)$_ to be a smooth
function $\Phi:\mathcal{H}\times\mathbb{C}^{c_{+}}\to\mathbb{C}[\Delta]$ for
which the equality

$\Phi(\tau,z+\tau u+v)=\mathbf{e}\big{(}-\tau
u^{t}Su-2u^{t}Sz\big{)}\chi_{\psi v}t_{\psi u}\Phi(\tau,z)$ (25)

holds for any to elements $u$ and $v$ from $\mathbb{Z}^{c_{+}}$ as does the
equality

$\Phi\big{(}A\tau,\tfrac{z}{j(A,\tau)}\big{)}=j(A,\tau)^{k}\mathbf{e}\Big{(}\tfrac{j_{A}^{\prime}\cdot
z^{t}Sz}{j(A,\tau)}\Big{)}\rho_{\Delta}(A)\Phi(\tau,z)$ (26)

for $A\in\Gamma$, with an exponential growth condition on the Fourier
coefficients.

The Jacobi forms from Definition 26 are slightly related, in the positive
definite case, to those investigated in [W].

Let $\psi$ be as in Definition 24, and set $\Omega_{\mathbb{Z}}:=\ker\psi$,
$\Omega:=\Omega_{\mathbb{Z}}+\ker S\subseteq\mathbb{Q}^{c_{+}}$ (this equals
$\Omega_{\mathbb{Z}}$ in the positive definite case, but not otherwise),
$\overline{\Omega}=\Omega/\ker S$, and
$L:=\pi(\Omega)=\pi(\Omega_{\mathbb{Z}})$. The fact that $\psi$ preserves the
quadratic value modulo $\mathbb{Z}$ implies that $L$ is an even lattice in
$V$, and since changing an element of $\mathbb{Z}^{c_{+}}$ by $\Omega$ does
not affect the quadratic value (by the same assumption), we deduce that
$L_{S}=\pi(\mathbb{Z}^{c_{+}})$ is contained in $L^{*}$. We define
$\Omega^{*}:=\pi^{-1}(L^{*})$ and $\overline{\Omega}:=\Omega^{*}/\ker S$, with
the quotients $D_{\Omega}\cong D_{\overline{\Omega}}\cong D_{L}$ (thus
inheriting the quadratic structure), and then $\psi$ identifies the quotient
$D_{\Omega,S}:=\mathbb{Z}^{c_{+}}/\Omega_{\mathbb{Z}}\subseteq D_{\Omega}$
with a subgroup of $\Delta$. Set $H_{\Omega}:=D_{\Omega,S}^{\perp}\subseteq
D_{\Omega}$, so that $D_{\Omega,S}$ is $H_{\Omega}^{\perp}$, and
$H_{\Omega}^{\perp}$ is embedded in $\Delta$.

Now, note that Definition 26 involves only the discriminant form $\Delta$ and
not $D$ (as does Definition 24), and the construction of $D$ from $\Delta$,
$D_{L}\cong D_{\Omega}$, and the image $H_{L}$ of $H_{\Omega}$ is not always
clear (see Remark 4.13). We define $\Theta_{S,\Omega}$ in this setting to be
the orthogonal analogue $\sum_{a\in\Omega^{*}}\mathbf{e}(\tau
a^{t}Sa+2a^{t}Sz)\mathfrak{e}_{a+\Omega}$ of the theta function from Equation
(21) (which satisfies the periodicity condition comparing
$\Theta_{S,\Omega}(\tau,z+\tau u+v)$ with $\mathbf{e}(-\tau
u^{t}Su-2u^{t}Sz)\chi_{v}t_{u}\Theta_{S,\Omega}(\tau,z)$ for any $u$ and $v$
in $\mathbb{Z}^{c_{+}}$), and obtain the following result.

######  Theorem 6.9.

Assume that there exists a discriminant form $D$, with a subgroup $H_{D}$ that
is isomorphic to $H_{\overline{L}}$ such that if $H$ is the isotropic
horizonal subgroup of $D_{\Omega}\oplus D$ resulting from this identification
as in Lemma 4.4, then the associated quotient $\Delta_{H}:=H^{\perp}/H$ is
isomorphic with $\Delta$ in a way that commutes with the embedddings of
$H_{L}^{\perp}=H_{\Omega}^{\perp}$ into $\Delta_{H}$ and into $\Delta$. Then
any Jacobi form $\Phi$ of weight $k$, representation $\rho_{\Delta}$, and
index $(S,\psi)$ with respect to $\Gamma$, as in Definition 26, is given by
$\Phi(\tau,z)=\downarrow_{H}\big{(}\Theta_{\Omega,S}(\tau,z)\otimes
F(\tau)\big{)}$ for a unique modular form $F$ of weight $k-\frac{b_{+}}{2}$
and representation $\rho_{D}$ with respect to $\Gamma$, which is holomorphic
if and only if $\Phi$ is.

###### Proof.

The fact that $\psi$ is trivial on elements of $(\ker S)_{\mathbb{Z}}:=\ker
S\cap\mathbb{Z}^{c_{+}}$ in Definition 26 implies that Equation (25) there
yields the invariance of $\Phi$, for fixed $\tau\in\mathcal{H}$, under
translations from the full lattice $(\ker S)_{\mathbb{Z}}\oplus\tau(\ker
S)_{\mathbb{Z}}$ inside the complex vector space $(\ker S)_{\mathbb{C}}:=(\ker
S)\otimes_{\mathbb{Q}}\mathbb{C}$. As $\Phi$ is holomorphic in $z$, this
implies that $\Phi(\tau,z)$ does not change when $z$ is altered by an element
of this vector space, meaning that its dependence on $z$ can be given in terms
of $z+(\ker S)_{\mathbb{C}}$, or equivalently its isomorph $V_{\mathbb{C}}$
via the complexification of $\pi$. We thus write
$\Phi(\tau,z)=\tilde{\Phi}(\tau,\zeta)$ for
$\zeta=\sum_{i=1}^{c_{+}}z_{i}\varepsilon_{i}$ and some smooth function
$\tilde{\Phi}$ which is holomorphic in $\zeta$.

By substituting $\sigma=\sum_{i=1}^{c_{+}}u_{i}\varepsilon_{i}$ and
$\nu=\sum_{i=1}^{c_{+}}v_{i}\varepsilon_{i}$, and noting that $t_{\sigma}$ and
$\chi_{\nu}$ act as $t_{\psi u}$ and $\chi_{\psi v}$ respectively, Equation
(25) from Definition 26 for $\Phi$ becomes Equation (17) inside Definition 18
for the function $\tilde{\Phi}$ and the positive definite lattice $L$.
Moreover, the fact that $\frac{\zeta^{2}}{2}$ equals $z^{t}Sz$ compares
Equation (26) for $\Phi$ with Equation (18) for $\tilde{\Phi}$. This implies
that $\tilde{\Phi}$ is a Jacobi form of weight $k$, index $(L,H)$ with our
$H$, and representation $\rho_{\Delta}$, to which we can apply Theorem 4.15 by
the assumption on the existence of $D$. The fact that the theta function
$\Theta_{L}(\tau,\zeta)$ from that theorem, evaluated at our $\zeta$, is just
$\Theta_{S,\Omega}(\tau,z)$ establishes the desired result. This proves the
theorem. ∎

######  Remark 6.10.

Some Jacobi forms $\Phi$ may have finer periodicity properties, thus
increasing $\Omega$ to a larger sublattice in $\mathbb{Q}^{c_{+}}$ via Theorem
5.13 (up to some arrow operators), and in some situations it may be more
convenient to work with the maximal one from Remark 5.16. Note that decreasing
the subgroup $H_{L}^{\perp}$ as in Theorem 5.15 means increasing the image
$D_{\Omega,S}$ of $\mathbb{Z}^{c_{+}}$ in $D_{\Omega}$, producing functions
that may be better described in terms of another map $\psi$, hence a different
matrix $S$.

In order to establish an analogue of Theorem 6.9 for the Hermitian Jacobi
forms from Definition 24, we can use the same construction, but we need the
image $\overline{\Omega}$ of the kernel $\Omega_{\mathcal{O}}$ of $\psi$
inside $\mathcal{O}^{c_{+}}$, or of $\Omega:=\Omega_{\mathcal{O}}+\ker S$,
modulo $\ker S$ to be an $\mathcal{O}$-module, and a projective one (then it
will be an $\mathcal{O}$-lattice as in Definition 1.9). The module condition
can be obtained by assuming that $\Delta$ carries an $\mathcal{O}$-structure
as in Remark 4.2, and $\psi$ to be a map of $\mathcal{O}$-modules. The
projectivity is now immediate in case $\mathcal{O}=\mathcal{O}_{\mathbb{K}}$,
but not at all clear otherwise. Since we need $\psi$ to embed $D_{\Omega,S}$
into $\Delta$, we cannot replace $\Omega$ by a subgroup that may have this
properties in a straightforward manner. This is why we make the additional
assumptions in the following result.

######  Theorem 6.11.

Take $S$ and $\psi$ as in Definition 24 for which the quotient
$\overline{\Omega}$ of $\ker\psi$ modulo its intersection with $\ker S$ is a
projective $\mathcal{O}$-module, and let $D$ be a discriminant form satisfying
the assumptions from Theorem 6.9. In this case, for any Hermitian Jacobi form
$\Phi$ of weight $k$, representation $\rho_{\Delta}$, and index $(S,\psi)$
with respect to $\Gamma$ there exists a unique modular form $F$, holomorphic
precisely when $\Phi$ is and having weight $k-b_{+}$ and representation
$\rho_{D}$ with respect to $\Gamma$, such that the equality
$\Phi(\tau,z,w)=\downarrow_{H}\big{(}\Theta_{\Omega,S}(\tau,z,w)\otimes
F(\tau)\big{)}$, with $\Theta_{\Omega,S}$ from Equation (21), holds for every
$\tau\in\mathcal{H}$ and $z$ and $w$ from $\mathbb{C}^{c_{+}}$.

###### Proof.

We follow the proof of Theorem 6.9. Equation (23) and the triviality
assumption on $\psi$ yields the invariance of $\Phi$ under translating the
pair $(z,w)$ by the sum of $\\{(a,\overline{a})|a\in(\ker S)_{\mathbb{Z}}\\}$
and $\tau\\{(a,\overline{a})|a\in(\ker S)_{\mathbb{Z}}\\}$ (via Corollary
1.3), which is a full lattice inside $(\ker S)_{\mathbb{C}}\oplus(\ker
S)_{\overline{\mathbb{C}}}$. We can thus write $\Phi(\tau,z,w)$ as
$\tilde{\Phi}(\tau,\zeta,\omega)$ with
$\zeta=\sum_{i=1}^{c_{+}}z_{i}\varepsilon_{i,\mathbb{C}}$ and
$\omega=\sum_{i=1}^{c_{+}}w_{i}\varepsilon_{i,\overline{\mathbb{C}}}$ for some
smooth $\tilde{\Phi}$. Using the same substitutions (and Corollary 1.3 again),
Equations (23) and (24) for $\Phi$ translate into Equations (12) and (13) in
Definition 13 for $\tilde{\Phi}$ (the latter uses the equality
$\langle\zeta,\omega\rangle=w^{t}Sz$ obtained via Proposition 1.6), so that
the assumptions on $\overline{\Omega}$ and $D$ show that $\tilde{\Phi}$ is a
Hermitian Jacobi form of weight $k$, index $(M,H)$ for $M:=\pi(\ker\psi)$ and
our $H$, and representation $\rho_{\Delta}$. Theorem 4.12 and the fact that
substituting our $\zeta$ and $\omega$ inside the theta function from Equation
(4) for our positive definite $\mathcal{O}$-lattice $M$ produces
$\Theta_{S,\Omega}(\tau,z,w)$ from Equation (21) yields the desired assertion.
This proves the theorem. ∎

Of course, Remark 6.10 holds equally well in the Hermitian setting considered
in Theorem 6.11.

We conclude with the following formula, which is also useful for several
applications of orthogonal and Hermitian Jacobi forms of matrix index.

######  Remark 6.12.

The Fourier expansion from Equation (20) for $\zeta$ and $\omega$ given in
terms of $z$ and $w$ from $\mathbb{C}^{c_{+}}$, can be written in terms of
coordinates from $\mathbb{K}$, also in the vector-valued setting considered in
Definition 24 and Theorem 6.11. In these coordinates, the expression
$\langle\zeta,\lambda\rangle+\langle\lambda,\omega\rangle$ will be
$r^{*}z+w^{t}r$ for some $r\in\mathbb{K}^{c_{+}}$ representing the coordinates
of $\lambda$ in the appropriate dual space, and the fact that the Jacobi form
$\Phi$ was seen in the proof of Theorem 6.11 to be unaffected by changing $z$
or $w$ by elements of $\ker S$, we deduce that only vectors
$r\in\mathbb{K}^{c_{+}}$ such that $r^{t}\ker S=\\{0\\}$, or equivalently
$r^{*}\ker S=\\{0\\}$, may be considered (indeed, this is the dual condition
to the division by $\ker S$ resulting from $\pi$). We thus extend the notation
from Theorem 6.5 and write $S^{-t}\mathcal{O}^{c_{+}}\big{/}\sqrt{-d}$ for the
set of elements $r\in\mathbb{K}^{c_{+}}$ satisfying
$S^{t}r\in(\mathcal{D}^{-1})^{c_{+}}$ and $r^{t}\ker S=\\{0\\}$ (this will
indeed produce $S^{-t}\mathcal{O}^{c_{+}}\big{/}\sqrt{-d}$ in the positive
definite setting, when $S$ can be inverted), and the Fourier expansion thus
takes the form

$\Phi(\tau,z,w)=\textstyle{\sum_{\beta\in\Delta}\sum_{m\in\frac{\beta^{2}}{2}+\mathbb{Z}}\sum_{r\in
S^{-t}\mathcal{O}^{c_{+}}/\sqrt{-d}}\tilde{c}_{m,r,\beta}(y)\mathbf{e}(m\tau+r^{*}z+w^{t}r)\mathfrak{e}_{\beta}}$

(with some conditions relating the possible indices $r$ that may show up for
every $\beta\in\Delta$). Similarly, for orthogonal Jacobi forms of index a
positive semidefinite symmetric matrix $S$ of that size, we denote by
$S^{-1}\mathbb{Z}^{c_{+}}$ the set of $r\in\mathbb{Q}^{c_{+}}$ for which
$Sr\in\mathbb{Z}^{c_{+}}$ and $r^{t}\ker S=\\{0\\}$ (again yielding the
meaning of this notation for invertible, positive definite $S$), yielding the
Fourier expansion

$\Phi(\tau,z)=\textstyle{\sum_{\beta\in\Delta}\sum_{m\in\frac{\beta^{2}}{2}+\mathbb{Z}}\sum_{r\in
S^{-1}\mathbb{Z}^{c_{+}}}\tilde{c}_{m,r,\beta}(y)\mathbf{e}(m\tau+r^{t}z)\mathfrak{e}_{\beta}}.$

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