Introduction

The classical Shimura–Maaß operators $\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}}+\frac{k}{2iy}$ and $y^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}}$ are well known for taking (elliptic, real-analytic) modular forms of weight $k$ to modular forms of weight $k+2$ and $k-2$ , respectively. In addition, [Reference MaaßMa1, Reference MaaßMa2] consider differential operators which have a similar effect on Siegel modular forms, a work which was generalized in [Reference ShimuraSh2]. The following paper [Reference ShimuraSh3] concerns differential operators on functions on unitary groups which have related properties. All these operators have number-theoretic as well as representation-theoretic (or Lie-algebraic) interpretations, and are therefore the subject of many research papers (see, e.g., the reference [Reference ShimuraSh1], which is strongly related to the case considered in this paper, as well as [Reference ShimuraSh4] for some generalizations of the results of the previously mentioned references or the investigation of invariant differential operators appearing in [Reference ShimuraSh5]).

Our first goal is to define similar operators for modular (or automorphic) forms on another type of Shimura varieties, namely quotients of Grassmannians of vector spaces of signature $(2,b_{-})$ . These are obtained by interpolating the square of the Shimura–Maaß operators from the case $b_{-}=1$ , the multiple Shimura–Maaß operators obtained in the case $b_{-}=2$ , and the operators for Siegel modular forms appearing in the case $b_{-}=3$ . One may use Lie-theoretic considerations in order to establish the existence of such operators, but obtaining their explicit formula in this way is very tedious, because of the change of coordinates between the tube domain model and the transitive free action of an appropriate parabolic subgroup of $\text{SO}^{+}(V)$ . We also remark that [Reference ShimuraSh1] also considers differential operators on automorphic forms on orthogonal groups. However, the operators defined in that reference take automorphic forms of some weight (i.e., a representation of the maximal compact subgroup) $\unicode[STIX]{x1D70C}$ to automorphic forms having weight $\unicode[STIX]{x1D70C}\otimes \unicode[STIX]{x1D702}$ for some $b_{-}$ -dimensional representation $\unicode[STIX]{x1D702}$ , hence in particular take scalar-valued automorphic forms on Grassmannians to vector-valued functions. Moreover, since that reference works with the coordinates arising from the bounded model while we consider the tube domain model (since the explicit formulas for the theta functions are more neatly presented in this model), an appropriate change of coordinates must be employed. It is true that after this change of coordinates, using the natural bilinear form on the tangent space of the Grassmannian in the tube domain model we may indeed obtain differential operators which remain in the scalar-valued realm. Indeed, after some additional normalization we obtain the operators defined in this paper using this method. However, the calculations involved are very delicate, laborious, long, and unenlightening, for which reason we have chosen to state and prove the formulas for the operators directly.

The second goal of this paper is to present two applications of these weight changing operators, in the theory of theta lifts. We recall the generalization, defined in [Reference BorcherdsB], for the Doi–Naganuma lifting first introduced in [Reference Doi and NaganumaDN, Reference NaganumaNg]. This map is given in [Reference BorcherdsB] in terms of a singular theta lift, and takes weakly holomorphic elliptic modular forms to meromorphic modular forms on Grassmannians. On the other hand, [Reference ZemelZe2] defines a similar theta lift, using the same theta functions with polynomials. The first result of this paper states that in the case of an even dimension, a power of our weight raising operator sends the theta lift from [Reference ZemelZe2] to the generalized Doi–Naganuma lifting of [Reference BorcherdsB].

In addition, recall that the theta lift from [Reference BorcherdsB, Section 13] (which is also studied extensively in [Reference BruinierBru] and others) is a real function. No automorphic forms of nonzero weight can be real. As a second application for our operators we define a notion of $m$ -real automorphic forms of positive weight $m$ , and show that in case one applies the theta lift from [Reference BorcherdsB, Section 14] (or from [Reference ZemelZe2]) to a modular form with real Fourier coefficients, the resulting theta lift is $m$ -real.

The first half of the paper contains numerous statements whose proofs are delayed to later sections. We choose this way of presentation since most of the proofs consist of direct calculations, which may divert the reader’s attention from the main ideas. Specifically, the paper is divided into 4 sections. In Section 1 we define the weight raising and weight lowering operators and state their properties. Section 2 presents the images of theta functions with special polynomials under the weight changing operators, and proves the main theorems. Section 3 presents the proofs for the assertions of Section 1, while Section 4 contains the missing proofs of Section 2.

1 Weight changing operators for automorphic forms on orthogonal groups

In this section, we present automorphic forms on complex manifolds arising as orthogonal Shimura varieties of signature $(2,b_{-})$ , introduce the weight raising and weight lowering operators on such forms, and give some of their properties. The proofs of most assertions are postponed to Section 3.

Let $V$ be a real vector space with a nondegenerate bilinear form of signature $(b_{+},b_{-})$ . The pairing of $x$ and $y$ in $V$ is written $(x,y)$ , and $x^{2}$ stands for the norm $(x,x)$ of $x$ . For $S\subseteq V$ , $S^{\bot }$ denotes the subspace of $V$ which is perpendicular to $S$ . The Grassmannian $G(V)$ of $V$ is defined to be the set of all decompositions of $V$ into the orthogonal direct sum of a positive definite space $v_{+}$ and a negative definite space $v_{-}$ . In the case $b_{+}=2$ (which is the only case we consider in this paper), it is shown in [Reference BorcherdsB, Section 13], [Reference BruinierBru, Sections 3.2 and 3.3], or [Reference ZemelZe2, Section 2.2] (among others), that $G(V)$ carries a complex structure and has several equivalent models, which we now briefly present. Let

$Z_{V}\in V_{\mathbb{C}}$ lies in $P$ if and only if $X_{V}$ and $Y_{V}$ are orthogonal and have the same positive norm. $P$ has two connected components (which are interchanged by complex conjugation), and let $P^{+}$ be one component. The map

is surjective, and $C^{\ast }$ acts freely and transitively on each fiber of this map by multiplication. This realizes $G(V)$ as the image of $P^{+}$ in the projective space $\mathbb{P}(L_{\mathbb{C}})$ , which is an analytically open subset of the (algebraic) quadric $Z_{V}^{2}=0$ , yielding a complex structure on $G(V)$ . This is the projective model of $G(V)$ .

Let $z$ be a nonzero vector in $V$ which is isotropic, that is, $z^{2}=0$ . The vector space $K_{\mathbb{R}}=z^{\bot }/\mathbb{R}z$ is nondegenerate and Lorentzian of signature $(1,b_{-}-1)$ . Choosing some $\unicode[STIX]{x1D701}\in V$ with $(z,\unicode[STIX]{x1D701})=1$ and restricting the projection $z^{\bot }\rightarrow K_{\mathbb{R}}$ to $\{z,\unicode[STIX]{x1D701}\}^{\bot }$ gives an isomorphism. We thus write $V$ as $K_{\mathbb{R}}\times \mathbb{R}\times \mathbb{R}$ , in which

A (holomorphic) section $s:G(V)\rightarrow P^{+}$ is defined by the pairing with $z$ being 1. Subtracting $\unicode[STIX]{x1D701}$ from any $s$ -image and taking the $K_{\mathbb{C}}$ -image of the result yields a biholomorphism between $G(V)\cong s\big(G(V)\big)$ and the tube domain $K_{\mathbb{R}}+iC$ , where $C$ is a cone of positive norm vectors in the Lorentzian space $K_{\mathbb{R}}$ . $C$ is called the positive cone, and it is determined by the choice of $z$ and the connected component $P^{+}$ . The inverse biholomorphism takes $Z=X+iY\in K_{\mathbb{C}}$ to

with the real and imaginary parts denoted by $X_{V,Z}$ and $Y_{V,Z}$ , respectively. They are orthogonal and have norm $Y^{2}>0$ . This identifies $G(V)$ with the tube domain model $K_{\mathbb{R}}+iC$ . Taking the other connected component of $P$ corresponds to taking the other cone $-C$ to be the positive cone, and to the conjugate complex structure.

The subgroup $O^{+}(V)$ consisting of elements of $O(V)$ preserving the orientation on the positive definite part acts on $P^{+}$ and $G(V)$ , respecting the projection. Elements of $O(V)\setminus O^{+}(V)$ interchange the connected components of $P$ . The action of $O^{+}(V)$ (and also of the connected component $\text{SO}^{+}(V)$ ) on $G(V)$ is transitive, with the stabilizer $K$ (or $SK\leqslant \text{SO}^{+}(V)$ ) of a point being isomorphic to $\text{SO}(2)\times O(n)$ (resp. $\text{SO}(2)\times \text{SO}(n)$ ). Therefore $G(V)$ is isomorphic to $O^{+}(V)/K$ and to $\text{SO}^{+}(V)/SK$ . Given an isotropic $z$ as above, the action of $O^{+}(V)$ transfers to $K_{\mathbb{R}}+iC$ , and for $M\in O^{+}(V)$ and $Z\in K_{\mathbb{R}}+iC$ we have

$J$ is a factor of automorphy, namely the equality

holds for all $Z\in K_{\mathbb{R}}+iC$ and $M$ and $N$ in $O^{+}(V)$ . For such $M$ we define the slash operator of weight $m$ , and more generally of weight $(m,n)$ , by

The fact that $(Z_{V},\overline{Z_{V}})=2Y^{2}$ and the definition of $J(M,Z)$ yield the equalities

the latter holding for every $m$ , $n$ , $t$ , and function $F$ on $K_{\mathbb{R}}+iC$ (see [Reference BruinierBru, Lemma 3.20] for the first equality in Equation (1), and the second one follows immediately).

The invariant measure on $K_{\mathbb{R}}+iC$ is $\frac{dXdY}{(Y^{2})^{b_{-}}}$ (see [Reference BruinierBru, Section 4.1], but one can also prove this directly, using the generators of $O^{+}(V)$ considered in Section 3 below). Note that this measure depends on the choice of a basis for $K_{\mathbb{R}}+iC$ , but changing the basis only multiplies this measure by a positive global scalar. Let $\unicode[STIX]{x1D6E4}$ be a discrete subgroup of $O^{+}(V)$ of cofinite volume. In most of the interesting cases $\unicode[STIX]{x1D6E4}$ will be either the $O^{+}$ or the $\text{SO}^{+}$ part of the orthogonal group of an even lattice $L$ in $V$ , or the discriminant kernel of such a group. Given $m\in \mathbb{Z}$ , an automorphic form of weight $m$ with respect to $\unicode[STIX]{x1D6E4}$ is defined to be a (complex valued) function $\unicode[STIX]{x1D6F7}$ on $K_{\mathbb{R}}+iC$ for which the equation

holds for all $M\in \unicode[STIX]{x1D6E4}$ and $Z\in K_{\mathbb{R}}+iC$ . Using the standard argument, such a function is equivalent to a function on $P^{+}$ which is $-m$ -homogeneous (with respect to the action of $\mathbb{C}^{\ast }$ ) and $\unicode[STIX]{x1D6E4}$ -invariant, as considered, for example, in [Reference BorcherdsB].

We now consider some differential operators on functions on $K_{\mathbb{R}}+iC$ . Given a basis for $K_{\mathbb{R}}$ , we write $\unicode[STIX]{x2202}_{x_{k}}$ for $\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{k}}$ (for $1\leqslant k\leqslant b_{-}$ ). Similarly, $\unicode[STIX]{x2202}_{y_{k}}$ stands for the coordinates of the imaginary part from $C$ . The notation for the derivatives $\unicode[STIX]{x2202}_{z_{k}}=\frac{1}{2}(\unicode[STIX]{x2202}_{x_{k}}-i\unicode[STIX]{x2202}_{y_{k}})$ and $\unicode[STIX]{x2202}_{\overline{z_{k}}}=\frac{1}{2}(\unicode[STIX]{x2202}_{x_{k}}+i\unicode[STIX]{x2202}_{y_{k}})$ will be shortened further to $\unicode[STIX]{x2202}_{k}$ and $\unicode[STIX]{x2202}_{\overline{k}}$ , respectively.

The operator $I=\sum _{k}x_{k}\unicode[STIX]{x2202}_{x_{k}}$ multiplies a homogeneous function on $K_{\mathbb{R}}$ by its homogeneity degree, and is thus independent of the choice of basis (indeed, it has an intrinsic Lie-theoretic description). The operators

from [Reference NakajimaNa] are intrinsic as well, and they are also invariant under translations in the real part of $K_{\mathbb{R}}+iC$ . If the basis for $K_{\mathbb{R}}$ is orthonormal, that is, orthogonal with the first vector having norm 1 and the rest having norm $-1$ , then the Laplacian of $K_{\mathbb{R}}$ , denoted $\unicode[STIX]{x1D6E5}_{K_{\mathbb{R}}}$ , is defined to be $\unicode[STIX]{x2202}_{x_{1}}^{2}-\sum _{k=2}^{b_{-}}\unicode[STIX]{x2202}_{x_{k}}^{2}$ . It is independent of the choice of the orthonormal basis (though using a basis which is not orthonormal it takes a different form), and it is invariant under the action of $O(K_{\mathbb{R}})$ as well as under translations in $K_{\mathbb{R}}$ . With complex coordinates it has three counterparts,

which we call the holomorphic Laplacian of $K_{\mathbb{C}}$ (of Hodge weight $(2,0)$ ), the anti-holomorphic Laplacian of $K_{\mathbb{C}}$ (of Hodge weight $(0,2)$ ), and the real Laplacian of $K_{\mathbb{C}}$ (of Hodge weight $(1,1)$ ), respectively. These operators have the same invariance and independence properties as $\unicode[STIX]{x1D6E5}_{K_{\mathbb{R}}}$ . Note that the appropriate combinations appearing in [Reference BruinierBru, Reference NakajimaNa] can be identified as our operators $\frac{1}{2}\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{h}$ , $\frac{1}{2}\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{\overline{h}}$ , and $\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{\mathbb{R}}$ , respectively, expressed in a basis which is not orthonormal. We shall indeed discuss and generalize the operators $\unicode[STIX]{x1D6E5}_{1}$ and $\unicode[STIX]{x1D6E5}_{2}$ of [Reference NakajimaNa] in Proposition 1.5 below.

The weight changing operators and their defining property are given in

The different descriptions of $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ coincide by Lemma 3.1 below. Theorem 1.1 has the following standard

In correspondence with Theorem 1.1 and Corollary 1.2 we call $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ the weight raising operator of weight $m$ and the weight lowering operator for automorphic forms on Grassmannians of signature $(2,b_{-})$ , respectively. As already mentioned in the Introduction, these operators may also be given a Lie-theoretic description (see Section 3 for more details). However, the explicit operators appearing in Theorem 1.1 are more useful for our applications.

We shall make use of the operator

which we denote by $|D^{\ast }|^{2}$ . Lemma 2.2 of [Reference ZemelZe2] shows that

is the weight $(m,n)$ Laplacian on $K_{\mathbb{R}}+iC$ , and the weight $m$ Laplacian $\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}$ is just $\unicode[STIX]{x1D6E5}_{m,0}^{(b_{-})}$ (this extends the corresponding assertion of [Reference NakajimaNa], since his operator $\unicode[STIX]{x1D6E5}_{1}$ is our $\unicode[STIX]{x1D6E5}_{0}^{(b_{-})}$ divided by 8). The constants are normalized such that

holds for every $m$ , $n$ , and $t$ (see the remark after Lemma 3.1 below). The relations between $R_{m}^{(b_{-})}$ , $L^{(b_{-})}$ , and the corresponding Laplacians are given by

We recall that an automorphic form of weight $m$ on $K_{\mathbb{R}}+iC$ is said to have eigenvalue $\unicode[STIX]{x1D706}$ if it is annihilated by $\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}+\unicode[STIX]{x1D706}$ (i.e., eigenvalues are of $-\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}$ ). Hence Proposition 1.3 has the following

By evaluating compositions of the weight changing operators one shows

Proposition 1.5 provides another proof to [Reference ZemelZe2, Lemma 2.2] about $\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}$ . It also implies that $\unicode[STIX]{x1D6EF}_{m}^{(b_{-})}$ preserves the spaces of automorphic forms of weight $m$ for all $m\in \mathbb{Z}$ and for every discrete subgroup $\unicode[STIX]{x1D6E4}$ of cofinite volume in $O^{+}(V)$ . It also commutes with $\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}$ , hence preserves eigenvalues of such automorphic forms. By rank considerations, one can probably show that the ring of differential operators which commute with all the slash operators of weight $m$ is generated by $\unicode[STIX]{x1D6E5}_{m}^{(b_{-})}$ and $\unicode[STIX]{x1D6EF}_{m}^{(b_{-})}$ , hence is a polynomial ring in two variables (if $b_{-}>1$ ). This assertion should also follow from [Reference ShimuraSh5, part (3) of Theorem 3.3] (since the rank of the symmetric space $G(V)$ is 2 if $b_{-}>1$ ), though I have not verified this in detail. As $\unicode[STIX]{x1D6E5}_{0}^{(b_{-})}$ is $8\unicode[STIX]{x1D6E5}_{1}$ and $\unicode[STIX]{x1D6EF}_{0}^{(b_{-})}$ is $16\unicode[STIX]{x1D6E5}_{2}$ in the notation of [Reference NakajimaNa], Proposition 1.5 generalizes the main result of that reference to other weights. A similar argument yields results of the same sort for $(m,n)$ , where a possible normalization for $\unicode[STIX]{x1D6EF}_{m,n}^{(b_{-})}$ is $(Y^{2})^{-n}\unicode[STIX]{x1D6EF}_{m-n}^{(b_{-})}(Y^{2})^{n}$ , for which an equality similar to Equation (2) holds. We shall not need these results in what follows.

We now consider compositions of the weight raising operators. The natural $l$ th power of $R_{m}^{(b_{-})}$ is the composition

The general formula for the resulting operator seems too complicated to write as a combination of $\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{h}$ , $D^{\ast }$ , and $\frac{1}{Y^{2}}$ with explicit coefficients. However, we can establish the properties given in the following

The binomial symbol appearing in part (iv) of Proposition 1.6 is the extended binomial coefficient: Indeed, for two nonnegative integers $x$ and $n$ we have

a formula which makes sense for $x\in \mathbb{R}$ (as well as $x$ in any $\mathbb{Q}$ -algebra).

Part (i) of Proposition 1.6 follows immediately from Corollary 1.2. For part (ii) Corollary 1.4 shows that the application of $R_{m+2r}$ (for $0\leqslant r\leqslant l-1$ ) to an eigenfunction adds $4m+8r+4-2b_{-}$ to the eigenvalue, so the assertion follows from evaluating

The proofs of parts (iii) and (iv) are given in Section 3.

We recall that $M=\binom{a~~b}{c~~d}\in \text{SL}_{2}(\mathbb{R})$ defines the holomorphic map

the latter being the factor of automorphy of this action. Modular forms of weight $(k,l)$ (or just weight $k$ if $l=0$ ) with respect to a discrete subgroup $\unicode[STIX]{x1D6E4}$ of $\text{SL}_{2}(\mathbb{R})$ with cofinite volume (with respect to the invariant measure $\frac{dxdy}{y^{2}}$ ) are functions $f:{\mathcal{H}}\rightarrow \mathbb{C}$ which are invariant under the corresponding weight $(k,l)$ slash operators for elements of $\unicode[STIX]{x1D6E4}$ . The weight $(k,l)$ Laplacian is

normalized such that $\unicode[STIX]{x1D6E5}_{k}=\unicode[STIX]{x1D6E5}_{k,0}$ annihilates holomorphic functions and the Laplacians commute with powers of $y$ as in Equation (2). The Shimura–Maaß operators

(note the different normalization from [Reference BruinierBru, Reference ZemelZe2]!) take modular forms of weight $k$ to modular forms of weight $k+2$ and $k-2$ , respectively, or more precisely, satisfy an appropriate commutation relation with the slash operators for all the elements of $\text{SL}_{2}(\mathbb{R})$ . They also change Laplacian eigenvalues (again, with respect to $-\unicode[STIX]{x1D6E5}_{k}$ rather than $\unicode[STIX]{x1D6E5}_{k}$ ): $\unicode[STIX]{x1D6FF}_{k}$ adds $k$ to the eigenvalue, while $y^{2}\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}$ subtracts $k-2$ from it. Moreover, the powers of the Shimura–Maaß operators are given by, for example, [Reference Zagier, Bruinier, van der Geer and HarderZa, equation (56)], stating that

(for arbitrary $k$ , not necessarily integral and nonnegative). Theorem 1.1 and Proposition 1.3 show that our weight changing operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ have similar properties. However, our operators are differential operators of order 2 while the Shimura–Maaß operators are of order 1. This is why the results of Propositions 1.5 and 1.6 are more complicated than the fact that $\unicode[STIX]{x1D6FF}_{k-2}y^{2}\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}$ is just $\frac{\unicode[STIX]{x1D6E5}_{k}}{4}$ , the commutator $[\unicode[STIX]{x1D6FF},y^{2}\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}]_{k}$ being simply $\frac{k}{4}$ , and [Reference Zagier, Bruinier, van der Geer and HarderZa, equation (56)].

Nonetheless, the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ for small values of $b_{-}$ are closely related to the Shimura–Maaß operators. Indeed, for $b_{-}=1$ the group $\text{SO}_{2,1}^{+}$ is $\text{PSL}_{2}(\mathbb{R})$ and the tube domain $K_{\mathbb{R}}+iC$ is just ${\mathcal{H}}$ . We have

(the same assertions hold for the operators involving anti-holomorphic weights). Our operators $R_{m}^{(1)}$ and $L^{(1)}$ are squares of the Shimura–Maaß operators, namely

Note that in this case

in accordance with the rank of the group being 1 rather than 2 (in particular, in the notation of [Reference NakajimaNa] we have $\unicode[STIX]{x1D6E5}_{2}=\frac{\unicode[STIX]{x1D6E5}_{1}}{4}$ in this case).

For $b_{-}>1$ many authors (including [Reference BruinierBru, Reference NakajimaNa]) take the basis for $K_{\mathbb{R}}$ as two elements spanning a hyperbolic plane together with an orthogonal basis of elements of norm $-2$ . In elements of the positive cone $C$ , the first two coordinates are positive. In particular, for $b_{-}=2$ we have $K_{\mathbb{R}}+iC\cong {\mathcal{H}}\times {\mathcal{H}}$ , with $\unicode[STIX]{x1D70F}=x+iy$ and $\unicode[STIX]{x1D70E}=s+it$ being the two coordinates. The group $\text{SO}_{2,2}^{+}$ is an order 2 quotient of $\text{SL}_{2}(\mathbb{R})\times \text{SL}_{2}(\mathbb{R})$ , acting on $G(V)\cong {\mathcal{H}}\times {\mathcal{H}}$ through

It follows that

(which extend to the operators with anti-holomorphic weights as well). Our operators are

In both cases $b_{-}=1$ and $b_{-}=2$ the assertions of this section follow from properties of the Shimura–Maaß operators (note that $Y^{2}$ is $2y^{2}$ for $b_{-}=1$ ). When $b_{-}=2$ the special orthogonal group of a negative definite subspace is also $\text{SO}(2)$ , which makes the theory of automorphic forms more symmetric.

Working with $b_{-}=3$ in this model yields another coordinate $z=u+iv$ . The positivity of $y$ , $t$ , and $yt-v^{2}$ is equivalent to

Hence $K_{\mathbb{R}}+iC$ is identified with the Siegel upper half-plane of degree 2. The group $\text{SO}_{2,3}^{+}$ is $\text{PSp}_{4}(\mathbb{R})$ , with the symplectic action and the factor of automorphy (hence the slash operators) from the theory of Siegel modular forms. In this case

in the notation of [Reference MaaßMa1, Reference MaaßMa2] for degree 2 (for weight $(m,n)$ the latter assertion extends to the modified Laplacian $\widetilde{\unicode[STIX]{x1D6E5}}_{m,n}^{(3)}$ presented in Section 3). The operator $\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{h}$ is also a constant multiple of the operator $\mathbb{D}$ considered, for example, in [Reference Choie and EholzerCE, Reference ChoieCh].

2 Images of Theta Lifts under $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$

For natural $r$ , $s$ , $t$ , and $l$ we define the polynomials

As a function of $\unicode[STIX]{x1D707}\in V$ , the polynomial $P_{r,s,t}(\unicode[STIX]{x1D707},Z)$ , considered, for example, in [Reference ZemelZe2], is homogeneous of degree $(r+t,0)$ with respect to the element of $G(V)$ represented by $Z$ , while $P_{r,s,t}^{(l)}(\unicode[STIX]{x1D707},Z)$ has homogeneity degree $(r+t,2l)$ . Equation (5) of [Reference ZemelZe2] extends from $P_{r,s,t}=P_{r,s,t}^{(0)}$ to the more general polynomials $P_{r,s,t}^{(l)}$ : The equality

holds for every $\unicode[STIX]{x1D707}\in V$ , $Z\in K_{\mathbb{R}}+iC$ , $M\in O^{+}(V)$ , and $r$ , $s$ , $t$ , and $l$ from $\mathbb{N}$ .

We shall assume that $V=L_{\mathbb{R}}$ for some fixed even lattice $L$ (of signature $(2,b_{-})$ ), and consider the theta function of $L$ which is based on the polynomial $P_{r,s,t}^{(l)}$ . This is a (vector-valued) function of $\unicode[STIX]{x1D70F}=x+iy\in {\mathcal{H}}$ and $Z\in K_{\mathbb{R}}+iC$ , which is a sum of expressions of the form

Here $\unicode[STIX]{x1D707}_{\pm }$ are the parts of $\unicode[STIX]{x1D707}\in V$ which lie in the spaces $v_{\pm }$ according to the element of $G(V)$ corresponding to $Z$ , $\unicode[STIX]{x1D6E5}_{v}$ is the Laplacian on $V$ which corresponds to the majorant associated with that element (i.e., to the bilinear form in which the sign on the pairing on $v_{-}$ is inverted to be positive as well), and $\mathbf{e}(w)=e^{2\unicode[STIX]{x1D70B}iw}$ for every complex $w$ . A simple and direct calculation proves

Part (i) of Lemma 2.1 shows that we can write the exponent in Equation (4) as the constant $\mathbf{e}\big(\overline{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x1D707}^{2}}{2}\big)$ (independent of $Z$ ) times $e^{-2\unicode[STIX]{x1D70B}yP_{1,1,1}}$ . Since the differences in the indices in part (ii) of Lemma 2.1 remain the same, $l$ does not affect the weight of modularity of $P_{r,s,t}^{(l)}$ , and $P_{1,1,1}$ is invariant (by Equation (3)), we find that replacing $P$ by $F$ in Equation (3) still yields a valid equation. Let $L^{\ast }=\text{Hom}(L,\mathbb{Z})$ be the dual lattice of $L$ and $L^{\ast }/L$ the (finite) discriminant group of $L$ . Then the theta function $\unicode[STIX]{x1D6E9}_{L,r,s,t}^{(l)}$ is the $\mathbb{C}[L^{\ast }/L]$ -valued function defined by

(this function is $\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D70F},0,0;v,P_{r,s,t}^{(l)})$ in the notation of [Reference BorcherdsB], where $v\in G(L_{\mathbb{R}})$ corresponds to $Z\in K_{\mathbb{R}}+iC$ ). The extension of Equation (3) to $\unicode[STIX]{x1D6E9}_{L,r,s,t}^{(l)}$ shows that $\unicode[STIX]{x1D6E9}_{L,r,s,t}^{(l)}$ is automorphic of weight $(s-r,s-t)$ as a function of $Z\in K_{\mathbb{R}}+iC$ , and [Reference BorcherdsB, Theorem 4.1] shows that as a function of $\unicode[STIX]{x1D70F}\in {\mathcal{H}}$ it is a vector-valued modular form of weight $\big(1+r+t,2l+\frac{b_{-}}{2}\big)$ and the Weil representation $\unicode[STIX]{x1D70C}_{L}$ . The latter is a representation of the metaplectic double cover $\text{Mp}_{2}(\mathbb{Z})$ of $\text{SL}_{2}(\mathbb{Z})$ , which is defined by sending the generators $T$ and $S$ of $\text{Mp}_{2}(\mathbb{Z})$ lying over the elements $\binom{1~~1}{0~~1}$ and $\binom{0~~-1}{1~~~~0}$ of $\text{SL}_{2}(\mathbb{Z})$ , respectively, to

respectively. For the properties of $\unicode[STIX]{x1D70C}_{L}$ see [Reference ZemelZe1], as well as the references cited there. The space $\mathbb{C}[L^{\ast }/L]$ comes with a Hermitian pairing $\langle \cdot ,\cdot \rangle _{\unicode[STIX]{x1D70C}_{L}}$ in which the $e_{\unicode[STIX]{x1D6FE}}$ are orthonormal, and $\unicode[STIX]{x1D70C}_{L}$ is a unitary representation with respect to this pairing. The operation of complex conjugation on $\unicode[STIX]{x1D6E9}_{L,r,s,t}^{(l)}$ interchanges $r$ and $t$ and sends $\unicode[STIX]{x1D70F}$ to $-\overline{\unicode[STIX]{x1D70F}}$ (this is equivalent to multiplying the bilinear form on $V$ by $-1$ , but as we rather stay in the signature $(2,b_{-})$ setting, we prefer this anti-holomorphic operation on $\unicode[STIX]{x1D70F}$ ). It also replaces $\unicode[STIX]{x1D70C}_{L}$ by its dual representation, but we shall consider the effect of complex conjugation only for the automorphy in the $Z$ variable.

We are interested in the action of the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ on theta kernels, and the resulting differential properties of the associated theta lifts. Several proofs will involve comparisons of these actions on theta kernels with the actions of the operators $\unicode[STIX]{x1D6FF}_{k}$ and $y^{2}\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}$ on these theta kernels (multiplied by the appropriate powers of $y$ ). The latter are given (in a more general context) in [Reference ZemelZe2, equations (6a) and (6b)]. As $P_{r,s,t}^{(l)}=P_{r,s,t}(\unicode[STIX]{x1D707}_{-}^{2})^{l}$ , Lemma 2.1 shows that in our case these equations take the form

(where $k=1-\frac{b_{-}}{2}+r+t-2l$ ) and

(note again the different normalization of these operators).

Recall that given a modular form $F$ of weight $1+r+t-\frac{b_{-}}{2}-2l$ and representation $\unicode[STIX]{x1D70C}_{L}$ , possibly with exponential growth at the cusps, its theta lift with respect to the polynomial $P_{r,s,t}^{(l)}$ is defined in [Reference BorcherdsB, Reference ZemelZe2], and others as follows. For $w>1$ let

and assume that

exists for $\Re \unicode[STIX]{x1D70E}\gg 0$ and defines a holomorphic function of $\unicode[STIX]{x1D70E}$ on some right half-plane, which may be extended to a meromorphic function of $\unicode[STIX]{x1D70E}$ for all $\unicode[STIX]{x1D70E}\in \mathbb{C}$ . Then the theta lift $\unicode[STIX]{x1D6F7}_{L,r,s,t}^{(l)}(F,Z)$ is the constant term of the expansion of this meromorphic function at $\unicode[STIX]{x1D70E}=0$ . Now, the modular form $F$ has a Fourier expansion of the sort

where $q^{n}$ denotes $\mathbf{e}(n\unicode[STIX]{x1D70F})$ and the $c_{n,\unicode[STIX]{x1D6FE}}$ are smooth functions of $y=\Im \unicode[STIX]{x1D70F}$ , which vanish unless $n\in \frac{\unicode[STIX]{x1D6FE}^{2}}{2}+\mathbb{Z}$ . The modular forms which are usually considered also satisfy the condition that $c_{n,\unicode[STIX]{x1D6FE}}=0$ unless $n\gg -\infty$ (for $F$ which is holomorphic on ${\mathcal{H}}$ this means at most a pole at the cusp, and not an essential singularity).

The relations between the action of the (classical) Shimura–Maaß operators on the lifted modular form and the action of these operators on the theta kernel used for the theta lift are given in the following

The complex conjugation of $\unicode[STIX]{x1D6E9}_{L,r,s,t}^{(l)}$ in the definition of the theta lift implies that $\unicode[STIX]{x1D6F7}_{L,r,s,t}^{(l)}(Z,F)$ is automorphic of weight $(s-t,s-r)$ . We shall thus consider only the case $r=s$ , where the automorphy in (the corresponding) Equation (3) involves only $J(M,Z)$ and not its complex conjugate. As with $P_{r,s,t}$ , we may omit the superscript $(l)$ in case $l=0$ . In the same manner as in Section 1, we shall postpone most of the (calculational) proofs to Section 4. Only the assertions about theta lifts will be proved here.

The first assertion we are interested in is

We remark that Proposition 2.3 may be formulated in terms of comparing the actions of elements from the universal enveloping algebras of $\mathfrak{sl}_{2}(\mathbb{R})$ and $\mathfrak{so}(V)\cong \mathfrak{so}_{2,b_{-}}$ on the theta kernel. However, unlike [Reference ZemelZe2, Proposition 2.3] (and [Reference BruinierBru, Proposition 4.5]), which compares the action of order 2 elements of both universal enveloping algebras, here the one from the algebra of $\mathfrak{so}_{2,b_{-}}$ has order 2 while the element from $\mathfrak{sl}_{2}$ has order 1.

We can now establish the first property of the theta lift from [Reference ZemelZe2].

We remark that in case the modular form $f$ is a harmonic weak Maaß form the modular form $\widetilde{F}$ from the proof of Theorem 2.4 is again weakly holomorphic. Moreover, in case the image of $f$ under the operator $\unicode[STIX]{x1D709}_{k-2m}$ of [Reference Bruinier and FunkeBF] does not have a pole at the cusp, the theta lift has no additional singularities, and the result of Theorem 2.4 extends to this case. However, in the theta lift $\unicode[STIX]{x1D6F7}_{L,m,m,0}(Z,F)$ itself one can still distinguish the case where $f$ is weakly holomorphic from the one where $F$ is such a harmonic weak Maaß form.

For the weight lowering operator $L^{(b_{-})}$ , we do not have a nice equivalent to Proposition 2.3. However, we do have an interesting result concerning its $m$ th power. We begin with

Lemma 2.5 allows us to establish the following

The case $s=m$ in Proposition 2.6 is of particular importance, as is shown in the following

Automorphic forms of nonzero weight can never be real-valued, because complex conjugation yields an automorphic form with a different weight. However, multiplying the complex conjugate automorphic form by a power of $Y^{2}$ leads to an object which is comparable with the image of our automorphic form under the appropriate power of a weight changing operator, as these two functions do have the same weight. We shall thus say that an automorphic form $\unicode[STIX]{x1D6F7}$ , of positive weight $m$ , is $m$ -real if its image under the $m$ th power of the weight lowering operators $L^{(b_{-})}$ coincides with its complex conjugate multiplied by a positive multiple of $(Y^{2})^{m}$ . We now show that the theta lifts from [Reference ZemelZe2, Theorem 3.9] are $m$ -real, or more generally:

We remark that the choice of $\unicode[STIX]{x1D706}=-\frac{mb_{-}}{2}$ in Theorem 2.8 is not crucial. Any choice of $\unicode[STIX]{x1D706}$ for which the number $\prod _{r=0}^{m-1}\big[r\big(m-r-\frac{b_{-}}{2}\big)-\unicode[STIX]{x1D706}\big]$ is positive will be sufficient for Theorem 2.8 to hold (with the same proof). However, we chose this eigenvalue as it is the eigenvalue of the theta lifts from [Reference ZemelZe2].

3 Proofs of the Properties of $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$

In this section, we include the proofs of the properties of the weight raising and weight lowering operators appearing in Section 1.

We first introduce (following [Reference NakajimaNa]) a convenient set of generators for $O^{+}(V)$ . For $\unicode[STIX]{x1D709}\in K_{\mathbb{R}}$ we define the element $p_{\unicode[STIX]{x1D709}}\in \text{SO}^{+}(V)$ whose action is

Furthermore, given an element $A\in O(K_{\mathbb{R}})$ and a scalar $a\in \mathbb{R}^{\ast }$ such that $a>0$ if $A\in O^{+}(K_{\mathbb{R}})$ and $a<0$ otherwise, we let $k_{a,A}\in O^{+}(V)$ be the element acting as

For any $Z\in K_{\mathbb{R}}+iC$ we have

Note that the relation between $A$ and the sign of $a$ is equivalent to preserving $C$ rather than mapping $Z$ into $K_{\mathbb{R}}-iC$ —it appears that [Reference NakajimaNa] ignored this point. Choose now an element of $G(K_{\mathbb{R}})$ in which the positive definite space is generated by the norm 1 vector $u_{1}$ , and consider the involution $w\in \text{SO}^{+}(K_{\mathbb{R}})$ defined by

( $w$ inverts the positive definite space $\mathbb{R}u_{1}$ ). Its action on $K_{\mathbb{R}}+iC$ is through

The elements $k_{a,A}$ with $(a,A)$ in the index 2 subgroup of $R^{\ast }\times O(K_{\mathbb{R}})$ thus defined and $p_{\unicode[STIX]{x1D709}}$ for $\unicode[STIX]{x1D709}\in K_{\mathbb{R}}$ generate the stabilizer $St_{O^{+}(V)}(\mathbb{R}z)$ of the isotropic space $\mathbb{R}z$ in $O^{+}(V)$ as the semi-direct product of these groups. The fact that adding $w$ to $St_{O^{+}(V)}(\mathbb{R}z)$ generates $O^{+}(V)$ is now easily verified by considering the action on isotropic 1-dimensional subspaces of $V$ .

Some useful relations are derived in the following

We remark that Lemma 3.1 holds for $K_{\mathbb{R}}$ of arbitrary signature (not necessarily Lorentzian), but not negative definite (for $Y^{2}>0$ to be possible).

We remark that the third operator $\unicode[STIX]{x1D6E5}_{K_{\mathbb{C}}}^{\mathbb{R}}$ bears a property similar to Lemma 3.1, which is used implicitly in [Reference ZemelZe2, Section 3] in order to prove Equation (2).

We can now present the

In order to indicate what is the Lie-theoretic interpretation of the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ , we recall the vector $u_{1}$ we used for defining $w$ above, and take a vector $\tilde{u} \in K_{\mathbb{R}}$ of norm $-1$ which is orthogonal to $u_{1}$ (we assume here $b_{-}>1$ , but for $b_{-}=1$ our operators are squares of the order 1 operators $\unicode[STIX]{x1D6FF}_{2m}$ and $y^{2}\unicode[STIX]{x2202}_{\overline{\unicode[STIX]{x1D70F}}}$ , whose Lie-theoretic interpretation is given, for example, in [Reference VerdierVe]). These choices determine the parabolic subgroup of $\text{SO}^{+}(V)$ appearing in the following

Let $K\cong \text{SO}(2)\times \text{SO}(b_{-})$ be the stabilizer, in $\text{SO}^{+}(V)$ , of the element of $G(V)$ represented by $Z=iu_{1}$ , and let $\mathfrak{k}$ be its Lie algebra. The action of a normalized generator of $\mathfrak{so}(2)\subseteq \mathfrak{k}$ on $\mathfrak{so}(V)_{\mathbb{C}}$ decomposes the latter space into the eigenspaces with eigenvalue 0 (this is precisely $\mathfrak{k}$ ) and $\pm i$ (complex conjugate spaces of dimension $b_{-}$ each). Hence the action on the space of products of two elements of $\mathfrak{so}(V)$ (inside its universal enveloping algebra, say) decomposes into eigenspaces with eigenvalues 0, $\pm i$ , and $\pm 2i$ . One verifies that in each of the $\pm 2i$ -eigenspaces, precisely one combination commutes with the part $\mathfrak{so}(b_{-})$ of $\mathfrak{k}$ . As our automorphic forms correspond to functions on $\text{SO}(V)$ on which $\text{SO}(2)\subseteq K$ operates according to a specific character and $\text{SO}(b_{-})$ operate trivially (normalized suitably), these elements (of order 2) of the universal enveloping algebra of $\mathfrak{so}(V)$ lead to weight raising and weight lowering operators. One may then evaluate, using the interplay between the operations of $\mathfrak{k}$ and the Lie algebra of the group $H$ from Proposition 3.2, the action of these operators, and find that they lead to our $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ . However, the change of coordinates between $H_{K_{\mathbb{R}}}$ and $K_{\mathbb{R}}+iC$ in this evaluation is more tedious than one might believe.

We also indicate briefly the connection between our operators and those of [Reference ShimuraSh1]. That reference defines, for every representation $\unicode[STIX]{x1D70C}$ of $\mathbb{C}^{\times }\times GL_{b_{-}}(\mathbb{C})$ (a subgroup of which we identify with the complexification of the compact subgroup $K$ , which is isomorphic to the product $\mathbb{C}^{\times }\times \text{SO}(b_{-},\mathbb{C})$ ), a differential operator that roughly sends (vector-valued) automorphic forms with weight (i.e., representation) $\unicode[STIX]{x1D70C}$ to automorphic forms having representation $\unicode[STIX]{x1D70C}\otimes \unicode[STIX]{x1D714}$ , where $\unicode[STIX]{x1D714}$ is the standard representation of that product on $\mathbb{C}^{b_{-}}$ . This representation space is considered as the holomorphic cotangent space of $G(V)$ , and the operator is, in fact, just the holomorphic differential map $d$ , twisted by the image of a scalar $\unicode[STIX]{x1D702}$ and a matrix $\unicode[STIX]{x1D709}$ (both defined explicitly in [Reference ShimuraSh1]) via the representation $\unicode[STIX]{x1D70C}$ . Starting with the 1-dimensional representation which is the $m$ th power of $\mathbb{C}^{\times }$ (this is the representation associated with our automorphic forms of weight $m$ ) and repeating this operation twice, we obtain an automorphic form with representation involving $\unicode[STIX]{x1D714}^{\otimes 2}$ . The idea is expressing the resulting automorphic form when $\unicode[STIX]{x1D714}$ is identified with $K_{\mathbb{C}}$ , and using the bilinear form on the latter space in order to replace the $\unicode[STIX]{x1D714}^{\otimes 2}$ -valued automorphic forms by scalar-valued ones.

Now, we replace the coordinate denoted $z$ in [Reference ShimuraSh1] by $u=\sqrt{2}z$ , considering it as lying in the complexified space $(v_{-})_{\mathbb{C}}$ associated with some base point for $G(V)$ , and decompose it as some multiple $u_{z}$ of $z_{v_{-}}$ plus a vector $u_{\bot }$ which is perpendicular to $z_{v_{-}}$ . Here $z$ is again the isotropic vector we used for defining $K_{\mathbb{R}}$ . Choosing the positive part of $z$ appropriately (recall that the vector denoted $p(z)$ in [Reference ShimuraSh1] is not presented in the canonical form), we obtain that our norm 0 vector has pairing $1+u^{t}u-2u^{t}z_{v_{-}}$ with $z$ and its positive and negative $K_{\mathbb{C}}$ coordinates are $i(1-u^{t}u)$ and $2u_{\bot }$ , respectively. It follows that the associated element $Z$ of $K_{\mathbb{C}}$ (which can be shown to be in $K_{\mathbb{R}}+iC$ ) satisfies $(Z+ie_{+})^{2}=\frac{-4}{1+u^{t}u-2u^{t}z_{-}}$ (where $e_{+}$ is the generator of the positive part of $K_{\mathbb{R}}$ ), so that the inverse map sends $Z$ to the vector obtained by multiplying the positive part of $-2\frac{Z+ie_{+}}{(Z+ie_{+})^{2}}$ by $i$ , and adding $z_{v_{-}}$ to the result. Given an automorphic form $F$ of weight $m$ on $G(V)$ , a very lengthy, tedious, and involved calculation gives us the expression for the $\unicode[STIX]{x1D714}^{\otimes 2}$ -valued automorphic form obtained from $F$ under the operator mentioned in the previous paragraph, and after applying the pairing we obtain an expression closely related to $(Z+ie_{+})^{2m}R_{m}^{(b_{-})}[(Z+ie_{+})^{-2m}F]$ . Indeed, the expression denoted by $\unicode[STIX]{x1D702}$ in [Reference ShimuraSh1] becomes $\frac{16Y^{2}}{|(Z+ie_{+})^{2}|^{2}}$ using our variable, so that multiplying by $\unicode[STIX]{x1D702}^{m}$ before applying the operator and by $\unicode[STIX]{x1D702}^{-m}$ afterward corresponds to the operation involving $Y^{2m}$ appearing in the definition of $R_{m}^{b_{-}}$ , as well as the additional operation with $(Z+ie_{+})^{2m}$ . However, the details of this calculation are very long as well, and therefore we have chosen to state and prove Theorem 1.1 more directly.

For calculational purposes it turns out convenient to introduce the operator

on which complex conjugation interchanges the indices $m$ and $n$ . The operator

will also show up, so we denote it $\widetilde{(D^{\ast })^{2}}$ . We now turn to the

Our next task is the

Finally, we come to the

4 Actions on theta kernels—proofs

The main technical lemma, which will be required for the evaluations in most of the following proofs, is based on

Another useful evaluation appears in the following

For $\unicode[STIX]{x1D707}\in L_{\mathbb{R}}$ and $\unicode[STIX]{x1D70F}=x+iy\in {\mathcal{H}}$ we denote the vector $\sqrt{2\unicode[STIX]{x1D70B}y}\unicode[STIX]{x1D707}$ by $\tilde{\unicode[STIX]{x1D707}}$ . Its norm is $2\unicode[STIX]{x1D70B}y\unicode[STIX]{x1D707}^{2}$ , and after choosing an element of $G(L_{\mathbb{R}})$ , it decomposes into $\tilde{\unicode[STIX]{x1D707}}_{+}$ (of norm $2\unicode[STIX]{x1D70B}y\unicode[STIX]{x1D707}_{+}^{2}$ ) and $\tilde{\unicode[STIX]{x1D707}}_{-}$ (whose norm is $2\unicode[STIX]{x1D70B}y\unicode[STIX]{x1D707}_{-}^{2}$ ). We now prove

We now turn to proving assertions concerning the images of theta lifts (or complex conjugates of theta functions), having only holomorphic weights of automorphy, under the operators $R_{m}^{(b_{-})}$ and $L^{(b_{-})}$ . This was seen to boil down to the operation on the function $F_{r,s,t}^{(l)}$ from Equation (4), with $\unicode[STIX]{x1D70F}$ replaced by $-\overline{\unicode[STIX]{x1D70F}}$ , under the additional assumption $s=t$ . The exponent was seen, using part (i) of Lemma 2.1, to be $\mathbf{e}\big(-\unicode[STIX]{x1D70F}\frac{\unicode[STIX]{x1D707}^{2}}{2}\big)e^{-\tilde{\unicode[STIX]{x1D707}}_{+}^{2}}$ , where the first multiplier is a constant (i.e., independent of $Z$ ). The polynomial part is evaluated in

We allow the index $n-k$ appearing in Part (i) here to be negative, with the natural extension of the definition of $P_{r,s,t}$ to negative $r$ . We remark that the expressions obtained in this part are just the generalized Laguerre polynomials $L_{k}^{(n-k)}$ , multiplied by the exponents, and normalized appropriately.

As $\tilde{\unicode[STIX]{x1D707}}_{-}^{2}=\tilde{\unicode[STIX]{x1D707}}^{2}-\tilde{\unicode[STIX]{x1D707}}_{+}^{2}$ , Lemma 4.4 implies that the dependence of the expression $(2\unicode[STIX]{x1D70B}y)^{n+l}P_{n-k,0,0}F_{k,n,n}^{(l)}(-\overline{\unicode[STIX]{x1D70F}},Z,\unicode[STIX]{x1D707})$ (or the corresponding theta function) on the variable $Z$ is only through the quantity $\tilde{\unicode[STIX]{x1D707}}_{+}^{2}$ . For convenience, we gather these results in the following

We can now present the

We now turn to the

We go on to the

Finally, we come to the