Abstract
In this paper we generalize a well-known isomorphism between the space of cusp forms of weight k for a Fuchsian subgroup of the first kind \(\Gamma \subset \mathrm {SL}_{2}({\mathbb {R}})\) and the space of certain Maaß forms of weight k for \(\Gamma \) to an isomorphism between the space of Siegel cusp forms of weight k for a subgroup \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\), which is commensurable with the Siegel modular group \(\mathrm {Sp}_{n}({\mathbb {Z}})\), and a suitable space of Siegel–Maaß forms of weight k for \(\Gamma \).
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1 Introduction
Let \({\mathbb {H}}:=\{z=x+i y\in {\mathbb {C}}\,\vert \,y>0\}\) denote the upper half-plane and \(\Gamma \subset \mathrm {SL}_{2}({\mathbb {R}})\) a Fuchsian subgroup of the first kind, which acts by fractional linear transformations on \({\mathbb {H}}\). Let \({\mathcal {S}}_{k}(\Gamma )\) denote the space of cusp forms of weight k for \(\Gamma \) and let \({\mathcal {H}}_{k}(\Gamma )\) denote the space of real-analytic automorphic forms of weight k for \(\Gamma \), on which the Maaß Laplacian
of weight k acts. Then, it is well-known that there is an isomorphism
of \({\mathbb {C}}\)-vector spaces, induced by the assignment \(f\mapsto y^{k/2}f\), where the right-hand side consists of the Maaß forms in \({\mathcal {H}}_{k}(\Gamma )\) with eigenvalue \(k/2(1-k/2)\) of \(\Delta _{k}\). This identification of two types of automorphic forms for \(\Gamma \) has various useful applications. For example, in the article [2], the isomorphism (1.1) was crucial in relating the sup-norm bound problem for cusp forms of weight k for \(\Gamma \) to bounds for the heat kernel for \(\Delta _{k}\) on the quotient space \(\Gamma \backslash {\mathbb {H}}\).
In this paper, we attempt a generalization of the isomorphism (1.1) to the Siegel modular setting, which, to our surprise, we could not find in the literature. Letting \(\mathrm {Sym}_{n}({\mathbb {C}})\) be the set of complex symmetric \((n\times n)\)-matrices, we let \({\mathbb {H}}_{n}:=\{Z=X+iY\in \mathrm {Sym}_{n}({\mathbb {C}})\,\vert \,Y>0\}\) denote the Siegel upper half-space of degree \(n\ge 1\) and we let \(\Gamma \subset \mathrm {Sp}_{n} ({\mathbb {R}})\) denote a subgroup acting by generalized fractional linear transformations on \({\mathbb {H}}_{n}\), which is commensurable with the Siegel modular group \(\mathrm {Sp}_{n}({\mathbb {Z}})\). Then, let \({\mathcal {S}}_{k}^{n}(\Gamma )\) denote the space of Siegel cusp forms of weight k and degree n for \(\Gamma \) and let \({\mathcal {H}}_{k}^{n}(\Gamma )\) be the space of real-analytic automorphic forms of weight k and degree n for \(\Gamma \), on which the Siegel–Maaß Laplacian
of weight k acts. As the main result of this paper, we show in Corollary 5.4 the isomorphism
of \({\mathbb {C}}\)-vector spaces, induced by the assignment \(f\mapsto \det (Y)^{k/2}f\), thereby generalizing the isomorphism (1.1) to the Siegel modular setting. The right-hand side of (1.3) now consists of the Siegel–Maaß forms in \({\mathcal {H}}_{k}^{n}(\Gamma )\) with eigenvalue \(nk(n-k+1)/4)\) of \(\Delta _{k}\).
In case \(n=1\), the isomorphism (1.1) is obtained as a by-product of the proof of the symmetry of the Maaß Laplacian \(\Delta _{k}\) (see [6]). The most straightforward proof of the symmetry of \(\Delta _{k}\) is obtained by constructing a suitable \(\mathrm {SL}_{2}({\mathbb {R}}) \)-invariant differential form using the raising or the lowering operators on \({\mathbb {H}}\), and then integrating it over the quotient space \(\Gamma \backslash {\mathbb {H}}\) (see [1], p. 135). Generalizations of all these operators as well as their transformation behaviour under the action of the symplectic group \(\mathrm {Sp}_{n}({\mathbb {R}})\) to the Siegel modular setting have been provided by Maaß in [3]. However, in spite of all these crucial ingredients being around for a long time, we could not find in the literature a precise proof of the symmetry of the Siegel–Maaß Laplacian \(\Delta _{k}\) of weight k. We provide a complete proof of the symmetry of \(\Delta _{k}\) in Theorem 5.1, where we construct the appropriate \(\mathrm {Sp}_{n}({\mathbb {R}})\)-invariant differential form on \({\mathbb {H}}_{n}\), which, while calculationally a bit cumbersome, is conceptually a rather straightforward piecing-together of Maaß’s calculations. Our main result in Corollary 5.4 is then a consequence of Theorem 5.1.
As indicated above, the generalization of the isomorphism (1.1) will perspectively allow us, among others, to relate the sup-norm bound problem for Siegel cusp forms of weight k and degree n for \(\Gamma \) to bounds for the heat kernel for the Siegel–Maaß Laplacian \(\Delta _ {k}\) on the quotient space \(\Gamma \backslash {\mathbb {H}}_{n}\).
This paper is organized as follows: In Sect. 2, we provide a quick summary of the basics of the Siegel upper half-space and Siegel modular forms. In the subsequent two Sects. 3 and 4, we introduce and discuss the transformation behaviour of the Maaß operators \(K_{\alpha }, \Lambda _ {\beta }\), and \(\Omega _{\alpha ,\beta }\) in the Siegel modular setting that is central to our analysis. This material is already present in Chapter 15 of [3], but due to sub-optimal typesetting, at places, it is hard to decipher. The transformation behaviour of \(\Omega _{\alpha ,\beta }\) can also be obtained via a somewhat different analysis on \({\mathbb {H}}_{n}\times {\mathbb {R}}/2\pi {\mathbb {Z}}\) given in Chapter 8 of Maaß’s later book [4]. As the two analyses turn out to be essentially equivalent, the transformation behaviours of \(K_{\alpha }\) and \(\Lambda _{\beta }\) can also be derived from the alternative method in [4]. However, for the transformation behaviours of \(K_{\alpha }\) and \(\Lambda _{\beta }\), we stick to Maaß’s original analysis in [3] as it is more direct and we redo these calculations for the reader’s convenience, but to keep the exposition short, we refer to [4] for the transformation behaviour of \(\Omega _{\alpha ,\beta }\). Finally in Sect. 5, piecing together Maaß’s results, we construct the appropriate \(\mathrm {Sp}_{n}({\mathbb {R}})\)-invariant differential form on \({\mathbb {H}}_{n}\) to show the symmetry of the Siegel–Maaß Laplacian \(\Delta _{k}\), and then use it to show the generalization (1.3) of the isomorphism (1.1).
2 Basic notations and definitions
For \(n\in {\mathbb {N}}_{>0}\) and a commutative ring R, let \(\mathrm {M}_{n}(R)\) denote the set of \((n\times n)\)-matrices with entries in R and \(\mathrm {Sym}_{n}(R)\) the set of symmetric matrices in \(\mathrm {M}_{n}(R)\). The Siegel upper half-space \({\mathbb {H}}_{n}\) of degree n is then defined by
The symplectic group \(\mathrm {Sp}_{n}({\mathbb {R}})\) of degree n is defined by
where \(J_{n}\in \mathrm {M}_{2n}({\mathbb {R}})\) is the skew-symmetric matrix
with \(\mathbb {1}_{n}\) denoting the identity matrix of \(\mathrm {M}_{n}({\mathbb {R}})\). The group \(\mathrm {Sp}_{n}({\mathbb {R}})\) acts by the symplectic action
on \({\mathbb {H}}_{n}\). Under this action \(\mathrm {Im}(Z)\) transforms as
which, on taking determinants on both sides, gives
Similarly, taking matrix-differentials on both sides of the symplectic action (2.1), it is easy to see that under this action, the matrix-differential form \(\mathrm {d}Z\) transforms as
The arclength \(\mathrm {d}s^{2}_{n}\) and the volume form \(\mathrm {d}\mu _{n}\) on \({\mathbb {H}}_{n}\) in terms of \(Z=(z_{j,k})_{1\le j\le k\le n}\in {\mathbb {H}}_{n}\) are given by
From Eqs. (2.2) and (2.3) it is obvious that the arclength \(\mathrm {d}s^{2}_{n}\) and the volume form \(\mathrm {d}\mu _{n}\) on \({\mathbb {H}}_{n}\) given by the above equations are invariant under the symplectic action. Corresponding to this metric, we have the Laplace–Beltrami operator \(\Delta \) on \({\mathbb {H}}_{n}\), called the Siegel Laplacian, which is also invariant under the symplectic action.
Definition 2.1
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\), i.e., the intersection \(\Gamma \cap \mathrm {Sp}_{n}({\mathbb {Z}})\) is a finite index subgroup of \(\Gamma \) as well as of \(\mathrm {Sp}_{n}({\mathbb {Z}})\). We let \(\gamma _ {j}\in \mathrm {Sp}_{n}({\mathbb {Z}})\) (\(j=1,\ldots ,h\)) denote a set of representatives for the left cosets of \(\Gamma \cap \mathrm {Sp}_{n}({\mathbb {Z}})\) in \(\mathrm {Sp}_{n}({\mathbb {Z}})\). Then, a Siegel modular form of weight k and degree n for \(\Gamma \) is a function \(f:{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) satisfying the following conditions:
-
(i)
f is holomorphic;
-
(ii)
\(f(\gamma Z)=\det (CZ+D)^{k}f(Z)\) for all \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \);
-
(iii)
given \(Y_{0}\in \mathrm {Sym}_{n}({\mathbb {R}})\) with \(Y_{0}>0\), the quantities \(\det (C_{j}Z+D_{j})^{-k}f(\gamma _{j} Z)\) are bounded in the region \(\{Z=X+iY\in {\mathbb {H}}_{n}\,\vert \,Y\ge Y_{0}\}\) for the set of representatives \(\gamma _{j}=\big ({\begin{matrix}A_{j}&{}B_{j}\\ C_{j}&{}D_{j}\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {Z}})\) (\(j=1,\ldots ,h\)).
Moreover, a Siegel modular form f as above is called a Siegel cusp form of weight k and degree n for \(\Gamma \) if condition (iii) above is strengthened to the condition
-
(iii’)
given \(Y_{0}\in \mathrm {Sym}_{n}({\mathbb {R}})\) with \(Y_{0}\gg 0\), the quantities \(\det (C_{j}Z+D_{j})^{-k}f(\gamma _{j} Z)\) become arbitrarily small in the region \(\{Z=X+iY\in {\mathbb {H}}_{n}\,\vert \,Y\ge Y_{0}\}\) for the set of representatives \(\gamma _{j}=\big ({\begin{matrix}A_{j}&{}B_{j}\\ C_ {j}&{}D_{j}\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {Z}})\) (\(j=1,\ldots ,h\)).
Remark 2.2
The sets of Siegel modular forms and Siegel cusp forms of weight k and degree n for \(\Gamma \) have the structure of \({\mathbb {C}}\)-vector spaces, which we denote by \({\mathcal {M}}_{k}^{n}(\Gamma )\) and \({\mathcal {S}}_{k}^{n}(\Gamma )\), respectively, and which turn out to be finite dimensional. Moreover, the space \({\mathcal {S}}_{k}^{n}(\Gamma )\) is equipped with the so-called Petersson inner product given by
making \({\mathcal {S}}_{k}^{n}(\Gamma )\) into a hermitian inner product space.
3 Siegel–Maaß Laplacian of weight \((\alpha ,\beta )\)
In this section, we will recall from [3] various differential operators acting on smooth complex valued functions defined on \({\mathbb {H}}_ {n}\). In particular, we will define the Siegel–Maaß Laplacian of weight \((\alpha ,\beta )\), where \(\alpha ,\beta \in {\mathbb {R}}\). Letting \(\alpha =k/2\) and \(\beta =-k/2\) will then lead us to the Siegel–Maaß Laplacian \(\Delta _{k}\) mentioned in formula (1.2) in the introduction. We point out that the Siegel Laplacian \(\Delta \) mentioned in the previous section and the Siegel–Maaß Laplacian \(\Delta _{k}\) are related by the formula
with the symmetric \((n\times n)\)-matrix \(\partial /\partial X\) of partial derivatives being defined below.
Given \(Z=X+iY\in {\mathbb {H}}_{n}\), we start by introducing the following symmetric \((n\times n)\)-matrices of partial derivatives:
where \(\delta _{j,k}\) is the Kornecker delta symbol.
Definition 3.1
Following Maaß [3], we define, using the above notations, for arbitrary real numbers \(\alpha ,\beta \in {\mathbb {R}}\), the following \((n \times n)\)-matrices of differential operators acting on smooth complex valued functions on \({\mathbb {H}}_{n}\):
Next, we want to expand \(\Omega _{\alpha ,\beta }\) and \({\widetilde{\Omega }}_{\alpha ,\beta }\) in terms of \(Z,\overline{ Z},\partial /\partial Z\), and \(\partial /\partial {{\overline{Z}}}\). For that we need the following lemma.
Lemma 3.2
Let \(C,D:{\mathbb {H}}_{n}\longrightarrow \mathrm {M}_{n}({\mathbb {C}})\) be smooth matrix valued functions depending on Z and \({\overline{Z}}\). Then, the following matrix operator identities hold:
(i) Assuming that \(\partial C/\partial Z=0\) and \(\partial D/\partial Z=0\), we have
(ii) Assuming that \(\partial C/\partial {\overline{Z}}=0\) and \(\partial D/\partial {\overline{Z}}=0\), we have
Proof
Since part (ii) follows from part (i) by conjugation, we prove only (i). Let \(\Phi \) be a matrix depending on Z and \({\overline{Z}}\) such that the product \((CZ+D)^{t}\Phi \) makes sense. Then, writing the (j, k)-th entry of the matrix \(\partial /\partial Z(CZ+D)^{t}\Phi \) as the sum
and noting that \(\partial Z/\partial z_{j,l}=(1-\delta _{j,l})E_{j,l}+E_{l,j}\), where \(E_{j,k}\in \mathrm {M}_{n}({\mathbb {C}})\) is the matrix with its (j, k)-th entry being 1 and the remaining entries being 0, elementary calculations lead us to the operator identity
which is what we needed to prove. \(\square \)
Corollary 3.3
For \(Z\in {\mathbb {H}}_{n}\), the following operator identities hold:
Proof
As \(\partial {\overline{Z}}/\partial Z=0\), we can choose \(C=\mathbb {1}_{n}\) and \(D=-{\overline{Z}}\) in Lemma 3.2 (i), from which the first claimed formula follows. The second formula follows analogously. \(\square \)
Using the above corollary, one can expand \(\Omega _{\alpha ,\beta }\) and \({\widetilde{\Omega }}_{\alpha ,\beta }\) as
Then, \(\Omega _{\alpha ,\beta }\) and \({\widetilde{\Omega }}_{\alpha ,\beta }\) are related by the identity
Definition 3.4
The operator \(\Delta _{\alpha ,\beta }:=-{{\,\mathrm{\mathrm {tr}}\,}}(\Omega _{\alpha ,\beta })=-{{\,\mathrm{\mathrm {tr}}\,}}({\widetilde{\Omega }}_{\alpha ,\beta })\) is called the Siegel–Maaß Laplacian of weight \((\alpha ,\beta )\).
4 Transformation behaviour of Maaß operators
Recall that the symplectic action of \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\) on the point \(Z\in {\mathbb {H}}_{n}\) is given by
to avoid cumbersome notation, we will use in this paper sometimes the shorthand \(Z^{\gamma }:=\gamma Z\). In this section, we will then study the transformation behaviour of the Maaß operators introduced in Definition 3.1 by expressing the operators \(K^{\gamma }_ {\alpha }\), \(\Lambda ^{\gamma }_{\beta }\), \(\Omega ^{\gamma }_{\alpha ,\beta }\) obtained by replacing \(Z,{\overline{Z}}\) by \(Z^{\gamma },{\overline{Z}}^{\gamma }\) in \({K}_{\alpha }\), \({\Lambda }_{\beta }\), \({\Omega }_{\alpha ,\beta }\), respectively, as they operate on smooth complex valued functions defined on \({\mathbb {H}}_{n}\).
We begin by investigating how the matrix differential operators \(\partial /\partial Z\) and \(\partial /\partial {\overline{Z}}\) transform under the symplectic action of \(\gamma \) on Z. From equation (2.3), we know that the differential \(\mathrm {d}Z\) transforms like
Therefore, as the differential of a smooth function \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) depending only on Z is given by \(\mathrm {d}\varphi ={{\,\mathrm{\mathrm {tr}}\,}}(\partial \varphi /\partial Z\,\mathrm {d}Z)\), we have
i.e., the operator \(\partial /\partial Z\) transforms as
By conjugation, the operator \(\partial /\partial {\overline{Z}}\) transforms as
Next we need to know how to differentiate \(\det (Z-{\overline{Z}})\) and \(\det (CZ+D)\) with respect to Z, which we carry out in the next two lemmas.
Lemma 4.1
The matrix identity
holds.
Proof
By writing \(\mathrm {Im}(Z)=Y=(y_{j,k})_{1\le j\le k\le n}\in \mathrm {Sym}_{n}({\mathbb {R}})\) as before, we obtain from equation (140) in subsection 2.8.2 of [5] the equality
Now since \(((1+\delta _{j,k})/2)(2-\delta _{j,k})=1\), we have \(\partial \det (Y)/\partial Y=\det (Y)Y^{-1}\), which is equivalent to the identity stated in the lemma. \(\square \)
Lemma 4.2
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\). Then, the matrix identity
holds.
Proof
By writing \(Z=(z_{j,k})_{1\le j\le k\le n}\in \mathrm {Sym}_{n}({\mathbb {C}})\), we compute, using the chain rule of differentiation
By equation (49) in subsection 2.1.2 of [5], we note
Then, plugging (4.4) into (4.3), we arrive at
Now, entrywise partial differentiation with respect to the entries \(z_{j,k}\) of Z followed by an elementary calculation with taking care of the ensuing Kronecker delta symbols leads us to the matrix identity
which is what we needed to prove. \(\square \)
Lemmas 4.1 and 4.2 prepare the groundwork for calculating the transformation behaviour of the Maaß operators, which we undertake one by one in the subsequent three propositions.
Proposition 4.3
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\) and \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) be a smooth function. Then, the operator \(K^{\gamma }_{\alpha }\) obtained by replacing \(Z\in {\mathbb {H}}_{n}\) in \({K}_{\alpha }\) by \(Z^ {\gamma }=\gamma Z\) is related to the operator \({K}_{\alpha }\) by the identity
Proof
From the definition of \(K^{\gamma }_{\alpha }\), we have
Then, expanding \(\partial /\partial Z^{\gamma }\) by means of equation (4.1) gives
Now, focusing on the second line of the above equality and using Lemma 4.2, we get
Multiplying the above equation from the left by \((Z-{\overline{Z}})\) gives
Now writing \((ZC^{t}-{\overline{Z}}C^{t})=(CZ+D)^{t}-(C{\overline{Z}}+D)^{t}\) and using the definition of \(K_{\alpha }\) on the right-hand side of the above equation, we have
Therefore, multiplying on the left by \((C{\overline{Z}}+D)^{-t}\) and on the right by \((CZ+D)^{t}\), we obtain
Combining the last equality with Eq. (4.5), leads to the identity
which is what we had set out to prove. \(\square \)
Proposition 4.4
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\) and \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) be a smooth function. Then, the operator \(\Lambda ^{\gamma }_{\beta }\) obtained by replacing \(Z\in {\mathbb {H}}_{n}\) in \({\Lambda }_{\beta }\) by \(Z^{\gamma }=\gamma Z\) is related to the operator \({\Lambda }_{\beta }\) by the identity
Proof
Since \({\overline{K}}_{\beta }=-\Lambda _{\beta }\), the required identity follows from Proposition 4.3 by complex conjugation. \(\square \)
Proposition 4.5
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\) and \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) be a smooth function. Then, the operator \(\Omega ^{\gamma }_{\alpha ,\beta }\) obtained by replacing \(Z\in {\mathbb {H}}_{n}\) in \({\Omega }_{\alpha , \beta }\) by \(Z^{\gamma }=\gamma Z\) is related to the operator \({\Omega }_{\alpha ,\beta }\) by the identity
Proof
The proof of this proposition can be obtained by suitably combining Propositions 4.3 and 4.4 according to the definition of the operator \(\Omega _{\alpha ,\beta }\). Alternatively, the claimed transformation formula can also be found on p. 120 of [4]. \(\square \)
Remark 4.6
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\) and \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) be a smooth function. Taking traces on both sides of Eq. (4.6) leads to the following transformation behaviour of the Siegel–Maaß Laplacian \(\Delta _{\alpha ,\beta }\)
Now, if the smooth function \(\varphi \) satisfies the functional equation
the transformation behaviour of \(\Delta _{\alpha ,\beta }\) leads to the identity
Definition 4.7
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\), i.e., the intersection \(\Gamma \cap \mathrm {Sp}_{n}({\mathbb {Z}})\) is a finite index subgroup of \(\Gamma \) as well as of \(\mathrm {Sp}_{n}({\mathbb {Z}})\). We let \(\gamma _{j}\in \mathrm {Sp}_{n} ({\mathbb {Z}})\) (\(j=1,\ldots ,h\)) denote a set of representatives for the left cosets of \(\Gamma \cap \mathrm {Sp}_{n}({\mathbb {Z}})\) in \(\mathrm {Sp}_{n}({\mathbb {Z}})\). We then let \({\mathcal {V}}_{\alpha ,\beta }^{n}(\Gamma )\) denote the space of all functions \(\varphi :{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) satisfying the following conditions:
-
(i)
\(\varphi \) is real-analytic;
-
(ii)
\(\varphi (\gamma Z)=\det (CZ+D)^{\alpha }\det (C{\overline{Z}}+D)^{\beta }\varphi (Z)\) for all \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \);
-
(iii)
given \(Y_{0}\in \mathrm {Sym}_{n}({\mathbb {R}})\) with \(Y_{0}>0\), there exist \(M\in {\mathbb {R}}_{>0}\) and \(N\in {\mathbb {N}}\) such that the inequalities
$$\begin{aligned} \vert \det (C_{j}Z+D_{j})^{-\alpha }\det (C_{j}{\overline{Z}}+D_{j})^{-\beta }\varphi (\gamma _{j} Z)\vert \le M{{\,\mathrm{\mathrm {tr}}\,}}(Y)^{N} \end{aligned}$$hold in the region \(\{Z=X+iY\in {\mathbb {H}}_{n}\,\vert \,Y\ge Y_{0}\}\) for the set of representatives \(\gamma _{j}=\big ({\begin{matrix}A_{j}&{}B_{j}\\ C_{j}&{} D_{j}\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {Z}})\) (\(j=1,\ldots ,h\)).
Remark 4.8
For \(\varphi \in {\mathcal {V}}_{\alpha ,\beta }^{n}(\Gamma )\), we set
whenever it is defined. In this way we obtain the Hilbert space
equipped with the inner product
We note that in order to enable \(\Vert \varphi \Vert <\infty \), the exponent \(N\in {\mathbb {N}}\) in part (iii) of Definition 4.7 has to be 0. Moreover, we note that Remark 4.6 shows that the Siegel–Maaß Laplacian \(\Delta _{\alpha ,\beta }\) acts on the Hilbert space \({\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\).
Definition 4.9
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\). The elements of the Hilbert space \({\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) are called automorphic forms of weight \((\alpha ,\beta )\) and degree n for \(\Gamma \). Moreover, if \(\varphi \in {\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) is an eigenform of \(\Delta _{\alpha ,\beta }\), it is called a Siegel–Maaß form of weight \((\alpha , \beta )\) and degree n for \(\Gamma \).
Corollary 4.10
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\) and \(\varphi \in {\mathcal {H}}_{\alpha , \beta }^{n}(\Gamma )\). Then, we have for all \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \)
Proof
The proof is an immediate consequence of Propositions 4.3–4.5 and the definition of the Hilbert space \({\mathcal {H}}_{\alpha , \beta }^{n}(\Gamma )\). \(\square \)
5 Symmetry of the Siegel–Maaß Laplacian of weight \((\alpha ,\beta )\)
Let \(\mathrm {d}Z:=(\mathrm {d}z_{j,k})_{1\le j,k\le n}\) denote the \((n\times n)\)-matrix consisting of differential forms of degree 1 and let \([\mathrm {d}Z]:= \bigwedge _{1\le j\le k\le n}\mathrm {d}z_{j,k}\) denote the differential form of degree \(n(n+1)/2\) at \(Z\in {\mathbb {H}}_{n}\). We introduce an \((n\times n)\)-matrix \( \{\mathrm {d}Z\}\) consisting of differential forms of degree \((n(n+1)/2-1)\), namely
where \(\varpi _{j,k}\) is defined by
in case \(j\le k\) and \(\varpi _{j,k}=\varpi _{k,j}\) in case \(j>k\) with the sign \(\varepsilon _{j,k}=\pm 1\) determined by \({\mathrm {d}}z_{j,k}\wedge \varpi _{j,k}=[{\mathrm {d}}Z]\). It is easy to see that
Let now \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {R}})\). Since we have \(\mathrm {d}Z^{\gamma }=(CZ+D)^{-t} \,\mathrm {d}Z\,(CZ+D)^{-1}\) and \([\mathrm {d}Z^{\gamma }]=\det (CZ+D)^{-(n+1)}[\mathrm {d}Z]\), we derive from the relation
that the matrix \(\{\mathrm {d}Z\}\) has the transformation behaviour
Next we shall use these differential forms to show that the Siegel–Maaß Laplacian \(\Delta _{\alpha ,\beta }\) acts as a symmetric operator on the Hilbert space \({\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\).
Theorem 5.1
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\) and let \(\varphi ,\psi \in {\mathcal {H}}_{\alpha , \beta }^{n}(\Gamma )\) be compactly supported. Then, we have the formula
In particular, this formula establishes the relation
which shows that the Siegel–Maaß Laplacian \(\Delta _{\alpha ,\beta }\) acts as a symmetric operator on the Hilbert space \({\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\).
Proof
We start by considering the differential form
Let \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \). Then, the transformation formulas
show that \(\omega (Z^{\gamma })=\omega (Z)\) for all \(\gamma \in \Gamma \), i.e., \(\omega (Z)\) is a \(\Gamma \)-invariant differential form on \({\mathbb {H}}_{n}\), and hence can be considered as a differential form on the quotient space \(\Gamma \backslash {\mathbb {H}}_{n}\). Since the automorphic forms \(\varphi ,\psi \) are real-analytic, the differential form \(\omega \) is a smooth differential form. Therefore, by Stokes’ theorem, we have
As \(\varphi ,\psi \) are compactly supported, the integral on the right-hand side of the above equation vanishes, which gives
As we shall see, by explicitly computing \(\mathrm {d}\omega (Z)\), the vanishing of the above integral will lead to the formula claimed in the theorem.
For the computation of \(\mathrm {d}\omega (Z)\), we set \(\rho :=\det (Z-{\overline{Z}})^{\alpha +\beta -(n+1)}{\overline{\psi }}(Z)\), \(P:=\Lambda _{\beta }\varphi (Z)\), and \(Q:=(Z-{\overline{Z}})\). Then, we obtain
Taking exterior derivatives on both sides leads to
Now a term by term differentiation in the last expression on the right-hand side of the above equation allows us to write it as the sum of the three traces
which we calculate one by one next.
(i) We begin by considering
which, by Lemma 4.1, calculates to
Now multiplying both sides of the above equation on the right by \(P\,Q=\Lambda _{\beta }\varphi (Z)(Z-{\overline{Z}})\) and taking the trace gives
which, upon rearranging the terms inside the traces on the right-hand side by cyclically permuting them, becomes
(ii) Next, we consider the second trace
in Eq. (5.3), which, again through rearrangement of the terms inside the trace by a cyclical permutation, takes the form
(iii) Finally, we consider the third trace
in Eq. (5.3). By the first operator identity in Corollary 3.3, we have the matrix identity
which gives, upon rearrangement of the scalar quantities, the identity
Now, adding up Eqs. (5.4)–(5.6), it follows from Eq. (5.3) that
Rearranging terms on the right-hand side of the last expression, leads to
Identifying the operator \((Z-{\overline{Z}})\partial /\partial Z+\beta \mathbb {1}_{n}\) on the right-hand side of the above equation as \(-{\overline{\Lambda }}_{\beta }\) and the operator \((Z-{\overline{Z}})\partial /\partial Z+(\alpha -(n+1)/2)\mathbb {1}_{n}\) as \(K_{\alpha -(n+1)/2}\), we can rewrite the right-hand side of the above equation as
which, by definition of \({\widetilde{\Omega }}_{\alpha ,\beta }\), is equal to
In total, we get
Thus, substituting \(\sum \limits _{j,k,l=1}^{n}(\partial /\partial Z)_{l,j}(\rho \,p_{j,k}\,q_{k,l})\) back into equation (5.2), we arrive at
Now, noting that the volume form
is just a constant multiple of \(\det (Y)^{\alpha +\beta }\mathrm {d}\mu _{n}(Z)\), it follows readily from the vanishing result (5.1) that
which is the claimed formula.
Using the latter formula, we compute
which proves the claimed symmetry of the Siegel–Maaß Laplacian \(\Delta _{\alpha ,\beta }\). \(\square \)
Corollary 5.2
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\) and let \(\varphi \in {\mathcal {H}}_{\alpha , \beta }^{n}(\Gamma )\) be a Siegel–Maaß form of weight \((\alpha ,\beta )\) and degree n for \(\Gamma \). Then, if \(\varphi \) is an eigenform of \(\Delta _{\alpha , \beta }\) with eigenvalue \(\lambda \), we have \(\lambda \in {\mathbb {R}}\) and \(\lambda \ge n\beta (\alpha -(n+1)/2)\).
Furthermore, \(\varphi \) has eigenvalue \(\lambda =\beta (\alpha -(n+1)/2)\) if and only if \(\varphi (Z)=\det (Y)^{-\beta }f(Z)\), where \(f:{\mathbb {H}}_{n}\longrightarrow {\mathbb {C}}\) is a holomorphic function satisfying
for all \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \). Moreover, if \(\beta <0\), then f is a Siegel cusp form of weight \(\alpha - \beta \) and degree n for \(\Gamma \).
Proof
Since \(\varphi \in {\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) is an eigenform of \(\Delta _{\alpha ,\beta }\) with eigenvalue \(\lambda \), i.e., we have \((\Delta _{\alpha ,\beta }+ \lambda \,\mathrm {id})\varphi =0\), we compute using Theorem 5.1
This immediately implies that \(\lambda \in {\mathbb {R}}\). Furthermore, since \({{\,\mathrm{\mathrm {tr}}\,}}(\vert \Lambda _{\beta }\varphi (Z)\vert ^{2})\ge 0\), we conclude that
To prove the second part of the corollary, we observe that the above equation shows that the equality \(\lambda =n\beta (\alpha -(n+1)/2)\) is equivalent to
Since \({{\,\mathrm{\mathrm {tr}}\,}}(\vert \Lambda _{\beta }\varphi (Z)\vert ^{2})\ge 0\), the above integral vanishes if and only if \({{\,\mathrm{\mathrm {tr}}\,}}(\vert \Lambda _{\beta }\varphi (Z)\vert ^{2})=0\). Now, as the matrix
is similar to the complex symmetric matrix
as we have the relation \(\Lambda _{\beta }\varphi (Z)=Y^{1/2}S(Z)Y^{-1/2}\), the matrix \(\vert \Lambda _{\beta }\varphi (Z)\vert ^{2}\) becomes similar to the positive semidefinite hermitian matrix \(S(Z){\overline{S}}(Z)\), which is diagonalizable with non-negative real eigenvalues. Therefore, the condition \({{\,\mathrm{\mathrm {tr}}\,}}(S(Z){\overline{S}}(Z))= {{\,\mathrm{\mathrm {tr}}\,}}(\vert \Lambda _{\beta }\varphi (Z)\vert ^{2})=0\) is equivalent to the vanishing of all the eigenvalues of \(S(Z){\overline{S}}(Z)\), which is equivalent to the vanishing of S(Z) and hence of \(\Lambda _{\beta }\varphi (Z)\). All in all, this proves that the equality \(\lambda =n\beta (\alpha -(n+1)/2)\) is equivalent to the vanishing condition \(\Lambda _{\beta }\varphi =0\).
Continuing, we now set \(f(Z):=\det (Y)^{\beta }\varphi (Z)\), and compute
Since we have
the above equality becomes
In total, this shows that \(\partial f/\partial {\overline{Z}}=0\), i.e., the function f is holomorphic, if and only if \(\Lambda _{\beta }\varphi (Z)=0\), which, by the previous argument, is equivalent to \(\varphi \in {\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) being a Siegel–Maaß form with eigenvalue \(\lambda =\beta (\alpha -(n+1)/2)\).
Furthermore, as the function \(\varphi \in {\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) has the transformation behaviour
for all \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \), the function \(f(Z)=\det (Y)^{\beta }\varphi (Z)=\det ({{\,\mathrm{Im}\,}}(Z))^{\beta }\varphi (Z)\) has the transformation behaviour
as claimed.
Finally, letting \(\gamma _{j}=\big ({\begin{matrix}A_{j}&{}B_{j}\\ C_{j}&{}D_{j}\end{matrix}}\big )\in \mathrm {Sp}_{n}({\mathbb {Z}})\) (\(j=1,\ldots ,h\)) be a set of representatives for the left cosets of \(\Gamma \cap \mathrm {Sp}_{n}({\mathbb {Z}})\) in \(\mathrm {Sp}_{n}({\mathbb {Z}})\), Remark 4.8 shows that given \(Y_{0}\in \mathrm {Sym}_{n}({\mathbb {R}})\) with \(Y_{0}>0\), the quantities
have to be bounded in the region \(\{Z=X+iY\in {\mathbb {H}}_{n}\,\vert \,Y\ge Y_{0}\}\). Therefore, if \(\beta <0\), this implies that given \(Y_{0}\in \mathrm {Sym}_{n} ({\mathbb {R}})\) with \(Y_{0}\gg 0\), the quantities
will become arbitrarily small in the region \(\{Z=X+iY\in {\mathbb {H}}_{n}\,\vert \,Y\ge Y_{0}\}\). In other words, f is indeed a Siegel cusp form of weight \(\alpha - \beta \) and degree n for \(\Gamma \).
With all this, the proof of the corollary is complete. \(\square \)
Remark 5.3
For \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\) and \(\alpha =k/2\), \(\beta =-k/2\) with \(k\in {\mathbb {N}}_{>0}\), we denote the Hilbert space \({\mathcal {H}}_{\alpha ,\beta }^{n}(\Gamma )\) simply by \({\mathcal {H}}_{k}^{n}(\Gamma )\). Similarly, we write for the operator \(\Omega _{\alpha ,\beta }\) simply \(\Omega _{k}\), which becomes
Finally, we write for the operator \(\Delta _{\alpha ,\beta }\) simply \(\Delta _{k}\) and call it the Siegel–Maaß Laplacian of weight k; it is given as
We note that the transformation behaviour of a Siegel–Maaß form \(\varphi \) of weight k and degree n for \(\Gamma \) takes the form
where \(\gamma =\big ({\begin{matrix}A&{}B\\ C&{}D\end{matrix}}\big )\in \Gamma \).
In the last corollary, we summarize the main results about Siegel–Maaß forms of weight k and degree n for \(\Gamma \).
Corollary 5.4
Let \(\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})\) be a subgroup commensurable with \(\mathrm {Sp}_{n}({\mathbb {Z}})\) and let \(\varphi \in {\mathcal {H}}_{k}^{n} (\Gamma )\) be a Siegel–Maaß form of weight k and degree n for \(\Gamma \). Then, if \(\varphi \) is an eigenform of \(\Delta _{k}\) with eigenvalue \(\lambda \), we have \(\lambda \in {\mathbb {R}}\) and
with equality attained if and only if the function \(\varphi \) is of the form \(\varphi (Z)=\det (Y)^{k/2}f(Z)\) for some Siegel cusp form \(f\in {\mathcal {S}}_{k}^{n}(\Gamma )\) of weight k and degree n for \(\Gamma \). In other words, there is an isomorphism
of \({\mathbb {C}}\)-vector spaces, induced by the assignment \(f\mapsto \det (Y)^{k/2}f\).
Proof
The proof is an immediate consequence of Corollray 5.2 by setting \(\alpha =k/2\) and \(\beta =-k/2\). \(\square \)
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Acknowledgements
We thank Valentin Blomer and Anilatmaja Aryasomayajula for inspiring discussions related to the material presented here. We also thank the anonymous referee for his/her helpful comments.
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The first author acknowledges support from the DFG Cluster of Excellence MATH+. In parts, the material of this manuscript is contained in the doctoral dissertation of the second author completed under the supervision of the first author at the Humboldt-Universität zu Berlin, during which he acknowledges support from the university as well as from the Berlin Mathematical School. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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Kramer, J., Mandal, A. Relating Siegel cusp forms to Siegel–Maaß forms. Res. number theory 8, 57 (2022). https://doi.org/10.1007/s40993-022-00351-8
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DOI: https://doi.org/10.1007/s40993-022-00351-8