Relating Siegel cusp forms to Siegel–Maaß forms

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 Γ⊂SL2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \subset \mathrm {SL}_{2}({\mathbb {R}})$$\end{document} and the space of certain Maaß forms of weight k for Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} to an isomorphism between the space of Siegel cusp forms of weight k for a subgroup Γ⊂Spn(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})$$\end{document}, which is commensurable with the Siegel modular group Spn(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Sp}_{n}({\mathbb {Z}})$$\end{document}, and a suitable space of Siegel–Maaß forms of weight k for Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}.


Introduction
Let H := {z = x + iy ∈ C | y > 0} denote the upper half-plane and ⊂ SL 2 (R) a Fuchsian subgroup of the first kind, which acts by fractional linear transformations on H. Let S k ( ) denote the space of cusp forms of weight k for and let H k ( ) denote the space of real-analytic automorphic forms of weight k for , on which the Maaß Laplacian k := y 2 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 − iky ∂ ∂x of weight k acts. Then, it is well-known that there is an isomorphism of C-vector spaces, induced by the assignment f → y k/2 f , where the right-hand side consists of the Maaß forms in H k ( ) with eigenvalue k/2(1−k/2) of k . This identification of two types of automorphic forms for 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 to bounds for the heat kernel for k on the quotient space \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 Sym n (C) be the set of complex symmetric (n×n)-matrices, we let H n := {Z = X +iY ∈ Sym n (C) | Y > 0} denote the Siegel upper half-space of degree n ≥ 1 and we let ⊂ Sp n (R) denote a subgroup acting by generalized fractional linear transformations on H n , which is commensurable with the Siegel modular group Sp n (Z). Then, let S n k ( ) denote the space of Siegel cusp forms of weight k and degree n for and let H n k ( ) be the space of real-analytic automorphic forms of weight k and degree n for , 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 C-vector spaces, induced by the assignment f → 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 H n k ( ) with eigenvalue nk(n − k + 1)/4) of 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 k (see [6]). The most straightforward proof of the symmetry of k is obtained by constructing a suitable SL 2 (R)-invariant differential form using the raising or the lowering operators on H, and then integrating it over the quotient space \H (see [1], p. 135). Generalizations of all these operators as well as their transformation behaviour under the action of the symplectic group Sp n (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 k of weight k. We provide a complete proof of the symmetry of k in Theorem 5.1, where we construct the appropriate Sp n (R)-invariant differential form on 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 to bounds for the heat kernel for the Siegel-Maaß Laplacian k on the quotient space \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 α , β , and α,β 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 α,β can also be obtained via a somewhat different analysis on H n × R/2πZ 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 α and β can also be derived from the alternative method in [4]. However, for the transformation behaviours of K α and β , 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 α,β . Finally in Sect. 5, piecing together Maaß's results, we construct the appropriate Sp n (R)-invariant differential form on H n to show the symmetry of the Siegel-Maaß Laplacian k , and then use it to show the generalization (1.3) of the isomorphism (1.1).

Basic notations and definitions
For n ∈ N >0 and a commutative ring R, let M n (R) denote the set of (n × n)-matrices with entries in R and Sym n (R) the set of symmetric matrices in M n (R). The Siegel upper half-space H n of degree n is then defined by The symplectic group Sp n (R) of degree n is defined by where J n ∈ M 2n (R) is the skew-symmetric matrix J n := 0 1 n −1 n 0 with 1 n denoting the identity matrix of M n (R). The group Sp n (R) acts by the symplectic action on H n . Under this action 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 dZ transforms as The arclength ds 2 n and the volume form dμ n on H n in terms of Z = (z j,k ) 1≤j≤k≤n ∈ H n are given by From Eqs. (2.2) and (2.3) it is obvious that the arclength ds 2 n and the volume form dμ n on H n given by the above equations are invariant under the symplectic action. Corresponding to this metric, we have the Laplace-Beltrami operator on H n , called the Siegel Laplacian, which is also invariant under the symplectic action.

Definition 2.1 Let
⊂ Sp n (R) be a subgroup commensurable with Sp n (Z), i.e., the intersection ∩ Sp n (Z) is a finite index subgroup of as well as of Sp n (Z). We let γ j ∈ Sp n (Z) (j = 1, . . . , h) denote a set of representatives for the left cosets of ∩ Sp n (Z) in Sp n (Z). Then, a Siegel modular form of weight k and degree n for is a function f : H n −→ C satisfying the following conditions: Moreover, a Siegel modular form f as above is called a Siegel cusp form of weight k and degree n for if condition (iii) above is strengthened to the condition

Remark 2.2
The sets of Siegel modular forms and Siegel cusp forms of weight k and degree n for have the structure of C-vector spaces, which we denote by M n k ( ) and S n k ( ), respectively, and which turn out to be finite dimensional. Moreover, the space S n k ( ) is equipped with the so-called Petersson inner product given by making S n k ( ) into a hermitian inner product space.

Siegel-Maaß Laplacian of weight (α, β)
In this section, we will recall from [3] various differential operators acting on smooth complex valued functions defined on H n . In particular, we will define the Siegel-Maaß Laplacian of weight (α, β), where α, β ∈ R. Letting α = k/2 and β = −k/2 will then lead us to the Siegel-Maaß Laplacian k mentioned in formula (1.2) in the introduction. We point out that the Siegel Laplacian mentioned in the previous section and the Siegel-Maaß Laplacian k are related by the formula with the symmetric (n × n)-matrix ∂/∂X of partial derivatives being defined below.
Given Z = X + iY ∈ H n , we start by introducing the following symmetric (n × n)matrices of partial derivatives: where δ j,k is the Kornecker delta symbol.
Definition 3.1 Following Maaß [3], we define, using the above notations, for arbitrary real numbers α, β ∈ R, the following (n × n)-matrices of differential operators acting on smooth complex valued functions on H n : Next, we want to expand α,β and α,β in terms of Z, Z, ∂/∂Z, and ∂/∂Z. For that we need the following lemma.
(ii) Assuming that ∂C/∂Z = 0 and ∂D/∂Z = 0, we have Proof Since part (ii) follows from part (i) by conjugation, we prove only (i). Let be a matrix depending on Z and Z such that the product (CZ + D) t makes sense. Then, writing the (j, k)-th entry of the matrix ∂/∂Z(CZ + D) t as the sum 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.

Corollary 3.3
For Z ∈ H n , the following operator identities hold: Proof As ∂Z/∂Z = 0, we can choose C = 1 n and D = −Z in Lemma 3.2 (i), from which the first claimed formula follows. The second formula follows analogously.
Using the above corollary, one can expand α,β and α,β as Then, α,β and α,β are related by the identity

Transformation behaviour of Maaß operators
Recall that the symplectic action of γ = A B C D ∈ Sp n (R) on the point Z ∈ H n is given by to avoid cumbersome notation, we will use in this paper sometimes the shorthand Z γ := γ 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 γ α , γ β , γ α,β obtained by replacing Z, Z by Z γ , Z γ in K α , β , α,β , respectively, as they operate on smooth complex valued functions defined on H n . We begin by investigating how the matrix differential operators ∂/∂Z and ∂/∂Z transform under the symplectic action of γ on Z. From equation (2.3), we know that the differential dZ transforms like Therefore, as the differential of a smooth function ϕ : H n −→ C depending only on Z is given by dϕ = tr(∂ϕ/∂Z dZ), we have i.e., the operator ∂/∂Z transforms as (4.1) By conjugation, the operator ∂/∂Z transforms as Next we need to know how to differentiate det(Z − Z) and det(CZ + D) with respect to Z, which we carry out in the next two lemmas.

Lemma 4.1 The matrix identity
Proof By writing Im(Z) = Y = (y j,k ) 1≤j≤k≤n ∈ Sym n (R) as before, we obtain from equation (140) in subsection 2.8.2 of [5] the equality which is equivalent to the identity stated in the lemma.

Lemma 4.2 Let γ = A B
C D ∈ Sp n (R). Then, the matrix identity Proof By writing Z = (z j,k ) 1≤j≤k≤n ∈ Sym n (C), we compute, using the chain rule of differentiation By equation (49)  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.
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 γ = A B
C D ∈ Sp n (R) and ϕ : H n −→ C be a smooth function. Then, the operator K γ α obtained by replacing Z ∈ H n in K α by Z γ = γ Z is related to the operator K α by the identity Proof From the definition of K γ α , we have Then, expanding ∂/∂Z γ 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 − Z) gives Now writing (ZC t − ZC t ) = (CZ + D) t − (CZ + D) t and using the definition of K α on the right-hand side of the above equation, we have Therefore, multiplying on the left by (CZ + 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. Proof Since K β = − β , the required identity follows from Proposition 4.3 by complex conjugation.

Proposition 4.5 Let γ = A B
C D ∈ Sp n (R) and ϕ : H n −→ C be a smooth function. Then, the operator γ α,β obtained by replacing Z ∈ H n in α,β by Z γ = γ Z is related to the operator α,β 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 α,β . Alternatively, the claimed transformation formula can also be found on p. 120 of [4].  in Sp n (Z). We then let V n α,β ( ) denote the space of all functions ϕ : H n −→ C satisfying the following conditions: 1, . . . , h). Remark 4.8 For ϕ ∈ V n α,β ( ), we set whenever it is defined. In this way we obtain the Hilbert space We note that in order to enable ϕ < ∞, the exponent N ∈ 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 α,β acts on the Hilbert space H n α,β ( ).

Definition 4.9
Let ⊂ Sp n (R) be a subgroup commensurable with Sp n (Z). The elements of the Hilbert space H n α,β ( ) are called automorphic forms of weight (α, β) and degree n for . Moreover, if ϕ ∈ H n α,β ( ) is an eigenform of α,β , it is called a Siegel-Maaß form of weight (α, β) and degree n for .

Corollary 4.10 Let
⊂ Sp n (R) be a subgroup commensurable with Sp n (Z) and ϕ ∈ H n α,β ( ). Then, we have for all γ = A B C D ∈ Proof The proof is an immediate consequence of Propositions 4.3-4.5 and the definition of the Hilbert space H n α,β ( ).

Symmetry of the Siegel-Maaß Laplacian of weight (α, β)
Let dZ := (dz j,k ) 1≤j,k≤n denote the (n × n)-matrix consisting of differential forms of degree 1 and let [dZ] := 1≤j≤k≤n dz j,k denote the differential form of degree n(n + 1)/2 at Z ∈ H n . We introduce an (n × n)-matrix {dZ} consisting of differential forms of degree (n(n + 1)/2 − 1), namely where j,k is defined by in case j ≤ k and j,k = k,j in case j > k with the sign ε j,k = ±1 determined by dz j,k ∧ j,k = [dZ]. It is easy to see that dZ ∧ {dZ} = 1 2 (n + 1)[dZ]1 n . Next we shall use these differential forms to show that the Siegel-Maaß Laplacian α,β acts as a symmetric operator on the Hilbert space H n α,β ( ).
Proof We start by considering the differential form Let γ = A B C D ∈ . Then, the transformation formulas show that ω(Z γ ) = ω(Z) for all γ ∈ , i.e., ω(Z) is a -invariant differential form on H n , and hence can be considered as a differential form on the quotient space \H n . Since the automorphic forms ϕ, ψ are real-analytic, the differential form ω is a smooth differential form. Therefore, by Stokes' theorem, we have As ϕ, ψ are compactly supported, the integral on the right-hand side of the above equation vanishes, which gives 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 n j,k,l=1 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 = β ϕ(Z)(Z − 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 . By the first operator identity in Corollary 3.3, we have the matrix identity which gives, upon rearrangement of the scalar quantities, the identity Rearranging terms on the right-hand side of the last expression, leads to n j,k,l=1 Identifying the operator (Z − Z)∂/∂Z + β1 n on the right-hand side of the above equation as − β and the operator (Z − Z)∂/∂Z + (α − (n + 1)/2)1 n as K α−(n+1)/2 , we can rewrite the right-hand side of the above equation as which, by definition of α,β , is equal to tr α,β − β(α − (n + 1)/2)1 n ϕ(Z)ψ(Z) − tr β ϕ(Z) β ψ(Z) .
In total, we get n j,k,l=1 Thus, substituting n j,k,l=1 (∂/∂Z) l,j (ρ 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 ) α+β dμ 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 α,β .
To prove the second part of the corollary, we observe that the above equation shows that the equality λ = nβ(α − (n + 1)/2) is equivalent to Since we have ∂ det(Y ) the above equality becomes In total, this shows that ∂f /∂Z = 0, i.e., the function f is holomorphic, if and only if β ϕ(Z) = 0, which, by the previous argument, is equivalent to ϕ ∈ H n α,β ( ) being a Siegel-Maaß form with eigenvalue λ = β(α − (n + 1)/2).
Furthermore, as the function ϕ ∈ H n α,β ( ) has the transformation behaviour Finally, letting γ j = A j B j C j D j ∈ Sp n (Z) (j = 1, . . . , h) be a set of representatives for the left cosets of ∩ Sp n (Z) in Sp n (Z), Remark 4.8 shows that given Y 0 ∈ Sym n (R) with Y 0 > 0, the quantities | det(C j Z + D j ) −α det(C j Z + D j ) −β ϕ(γ j Z)| have to be bounded in the region {Z = X + iY ∈ H n | Y ≥ Y 0 }. Therefore, if β < 0, this implies that given Y 0 ∈ Sym n (R) with Y 0 0, the quantities | det(C j Z + D j ) −(α−β) f (γ j Z)| = | det(C j Z + D j ) −α det(C j Z + D j ) −β det(Im(γ j Z)) β ϕ(γ j Z)| will become arbitrarily small in the region {Z = X + iY ∈ H n | Y ≥ Y 0 }. In other words, f is indeed a Siegel cusp form of weight α − β and degree n for . With all this, the proof of the corollary is complete.