On differential operators on automorphic forms and invariant pluri-harmonic polynomials

Tomoyoshi Ibukiyama

Several mathematicians has given holomorphic differential operators D on automorphic forms F on bounded symmetric domains D such that the restriction of DF on some bounded domain Δ of lower dimension in D is also automorphic (cf. [5], [7], [11], [12], [2], [3]). The aim of this paper is to give a characterization of such holomorphic linear differential operators with constant coefficients in symplectic cases by using harmonic polynomials on ℝ^{n}, or pluri-harmonic polynomials on matrix arguments with some invariance property.

For any natural number m, we denote by H_{m} the Siegel upper half space of degree m. We shall treat the following two kind of pairs (D, Δ) of domains:

(I) D = H_{n} and Δ = H_{n_{1}} × ⋯ × H_{n_{r}} with n = n_{1} + ⋯ + n_{r},

(II) D = (H_{n})^{r} and Δ = H_{n} (diagonally embedded in D). The case (I) contains the cases treated by Eichler-Zagier [7] (n = r = 2), Böcherer-Satoh-Yamazaki [2](n ≥ 2, r = 2 with certain vector valued weights), Ibukiyama-Zagier [8](n = r ≥ 2), and the case (II) contains the Rankin-Cohen operators in Cohen [5], and Choie-Eholzer [4], Eholzer-Ibukiyama [6] (r = 2, n = 1, 2, ≥ 3, respectively), and Satoh’s operators in [12]. These two cases are in a sense “dual” to each other. Our theorems assert that, in the above two cases, everything is reduced to the representation theory of the orthogonal groups. Our differential operators D are expressed as Q(\frac{∂}{∂Z}) , where Q are obtained from pluri-harmonic polynomial maps which are characterized by some invariance property (cf. §1, Theorems 1 and 2). The polynomial maps Q will be given explicitly in several cases in section 3. It is plausible that whole theory might be generalized for any dual reductive pairs or any tube domains.

This paper is organized as follows: In section 1, after reviewing some standard notions, we shall state our main results Theorem 1 and 2. In section 2, we shall give proofs of these theorems. In section 3, we shall give several explicit examples of the differential operators we are concerned with.

This work is motivated by Zagier’s interest in differential operators and triple L function. During his stay in Kyushu University in fall in 1990, he convinced me earnestly that this problem is very important. Without his strong interest in this problem, this work would not have been done. The author would like to thank him for this point and also for several discussions. He also thanks the Max Planck Institute for Mathematics for the kind hospitality while he was revising the paper.

1 Main results

1.1 Notation

First, we shall fix some notation. For any natural integer n, let Sp(n, ℝ) be the usual split symplectic group of size 2n. This group acts on H_{n} in the usual way by

gZ = (AZ + B)(CZ + D)^{−1},

where g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ). Now, let (τ, V_{τ}) be an irreducible finite dimensional

rational representation of GL(n, ℂ). For any Z ∈ H_{n} and any g = ( \begin{matrix}A & B \\ C & D \end{matrix} ), we

write j_{τ}(g, Z) = τ(CZ + D). For any V_{τ}-valued function F on H_{n}, and any element g ∈ Sp(n, ℝ), we write

F |_{τ}[g] = j_{τ}(g, Z)^{−1}F (gZ).

In particular, when τ is a power of the determinant, that is, when τ(CZ + D) = det(CZ + D)^{k} for some natural number k, we write F |_{k}[g] = F |_{τ}[g] as usual. More generally, for a fixed natural number r, let n_{1}, …, n_{r} be natural numbers and for each j with 1 ≤ j ≤ r, let (τ_{i}, V_{τ_{i}}) be an irreducible representation of GL(n_{i}, ℂ). Take an irreducible representation (τ, V_{τ}) of GL(n_{1}, ℂ)×⋯×GL(n_{r}, ℂ) defined by τ = τ_{1}⊗⋯⊗τ_{r} and V_{τ} = V_{τ_{1}}⊗⋯⊗V_{τ_{r}}. For any g = (g_{1}, …, g_{r}) ∈ Sp(n_{1}, ℝ)×⋯×Sp(n_{r}, ℝ), and Z = (Z_{1}, …, Z_{r}) ∈ H_{n_{1}} × ⋯ × H_{n_{r}}, we write

j_{τ}(g, Z) = τ_{1}(C_{1}Z_{1} + D_{1}) ⊗ ⋯ ⊗ τ_{r}(C_{r}Z_{r} + D_{r}),

where we write g_{i} = ( \begin{matrix}A_{i} & B_{i} \\ C_{i} & D_{i} \end{matrix} ) for each i with 1 ≤ i ≤ r. For any V_{τ}-valued function

F (Z) on Z ∈ H_{n_{1}} × ⋯ × H_{n_{r}} and any g ∈ Sp(n_{1}, ℝ) × ⋯ × Sp(n_{r}, ℝ), we write

(F |_{τ}[g])(Z) = j_{τ}(g, Z)^{−1}F (Z).

When τ_{i} = det^{k_{i}} for each i with 1 ≤ i ≤ r, we write F |_{k_{1},...,k_{r}}[g] = F |_{τ}[g]. We say that a function on H_{n_{1}} ×⋯×H_{n_{r}} is C^{∞}, if it is a C^{∞} funtion as a function of ∑_{j=1}^{r} n_{j}(n_{j} +1) components of real symmetric matrices X_{j} and Y_{j}, where Z_{j} = X_{j} + iY_{j} ∈ H_{n_{j}} (1 ≤ j ≤ r). For any discrete subgroup Γ of Sp(n_{1}, ℝ)×⋯×Sp(n_{r}, ℝ) with covolume finite, we say that a C^{∞} function F transforms like a modular form of weight τ, if F |_{τ}[γ] = F for any γ ∈ Γ. Now, we fix a natural number n and r, and choose natural numbers n_{1}, …, n_{r} so that n_{1} + ⋯ + n_{r} = n. For any matrix M = (m_{kl})_{1≤k,l≤n} ∈ M_{n}(ℂ), and any pair of integers (i, j) with 1 ≤ i, j ≤ r, we denote by M_{ij} the submatrix (m_{kl}) of M with n_{1} + ⋯ + n_{i−1} + 1 ≤ k ≤ n_{1} + ⋯ + n_{i} and n_{1} + ⋯ + n_{j−1} + 1 ≤ l ≤ n_{1} + ⋯ + n_{j} . That is, we write M = (M_{ij})_{1≤i,j≤r} by submatrices. For the sake of simplicity, we sometimes write M_{i} = M_{ii} for each i with 1 ≤ i ≤ r. Now, we put

G(n_{1}, …, n_{r}) = Sp(n_{1}, ℝ) × Sp(n_{2}, ℝ) × ⋯ × Sp(n_{r}, ℝ)

and we regard it as a subgroup of Sp(n, ℝ) by the “diagonal embedding” ι, where the group homomorphism ι of G(n_{1}, …, n_{r}) into Sp(n, ℝ) is defined by

ι(g_{1}, …, g_{r}) = ( \begin{matrix}(A_{ij}) & (B_{ij}) \\ (C_{ij}) & (D_{ij}) \end{matrix} ) ∈ Sp(n, ℝ)

with g_{i} = ( \begin{matrix}A_{i} & B_{i} \\ C_{i} & D_{i} \end{matrix} ) and A_{ij} = B_{ij} = C_{ij} = D_{ij} = 0 for pairs (i, j) with i ≠ j.

1.2 Pluri-harmonic polynomials

We review pluri-harmonic polynomials of matrix arguments. Let m and d be natural integers and let P be a ℝ-valued polynomial of md variables x_{ij} (1 ≤ i ≤ m, 1 ≤ j ≤ d) regarded as a function of X = (x_{iν}) ∈ M_{m,d}(R). For each i, j with 1 ≤ i, j ≤ m, denote by Δ_{i,j} the following differential operator:

Δ_{i,j} = ∑_{ν=1}^{d} \frac{∂^{2}}{∂x_{iν}∂x_{jν}} .

A polynomial P (X) on M_{m,d}(ℝ) is called harmonic if ∑_{i=1}^{m} Δ_{i,i}P = 0, and it is said to be pluri-harmonic if Δ_{i,j}P = 0 for each pair (i, j) with 1 ≤ i, j ≤ m. This is equivalent to say that P (AX) is harmonic for any A ∈ GL(m, ℂ) (cf. Kashiwara and Vergne [9]). We denote by H_{m,d} the space of all pluri-harmonic polynomials on M_{m,d}. The group GL(m, ℂ) × O(d) acts on M_{m,d} by: P (X) → P (^{t}AXB) for A ∈ GL(m, ℂ) and B ∈ O(d). The irreducible decomposition of H_{m,d} has been given in Kashiwara and Vergne [9]. They have shown there for example that each irreducible component has multiplicity one, and if an irreducible representation τ ⊗ λ of GL(n, ℂ) × O(d) appears in H_{m,d}, then the representation τ of GL(n, ℂ) is uniquely determined by the representation λ of O(d) and vice versa. (cf. Kashiwara and Vergne loc. cit. (5.7),(6.14).) We denote this τ by τ(λ). It is also clear that pluri-harmonic polynomials in each irreducible representation space in H_{m,d} are homogeneous. Now, assume that an irreducible representation τ ⊗ λ appears as an irreducible component of H_{m,d}. We denote by V_{τ} the representation space of τ. Then, pluri-harmonic polynomial maps P of M_{m,d} to V_{τ} such that P (aX) = τ(a)P (X) form a vector space of dimension dim λ. We denote this vector space by H_{m,d}(τ, λ) = H(τ, λ).

1.3 Invariant pluri-harmonic polynomials and Theorems

We first treat the case (I) in the introduction. Fix n_{1}, …, n_{r} with n = n_{1} + ⋯ + n_{r} as in the introduction. Also put D = H_{n} and Δ = H_{n_{1}} + ⋯ + H_{n_{r}}. We fix a natural number d. Let λ_{i} be an irreducible representation of O(d) which appears in H_{n_{i},d}. We take the tensor representation λ = λ_{1} ⊗ ⋯ ⊗ λ_{r} of the direct product O(d)^{r} of r copies of O(d). We can take the representation space of λ as the tensor product V (λ) = H(τ(λ_{1}), λ_{1})⊗⋯⊗H(τ(λ_{r}), λ_{r}). We fix a basis of V (λ). Then, each element of V (λ) is regarded as a vector each of whose component is a polynomial P (X_{1}, …, X_{r}) of X_{i} ∈ M_{n_{i},d} (1 ≤ i ≤ r) such that P is pluri-harmonic for each X_{i} (1 ≤ i ≤ r). Besides, the above polynomial P is a linear combination of products P_{ν_{1}}(X_{1}) ⋯ P_{ν_{r}}(X_{r}) of polynomials P_{ν_{i}} such that each P_{ν_{i}} is homogeneous and pluri-harmonic. We denote by Δ(O(d)) the image of the diagonal embedding of O(d) into O(d)^{r}. Now, assume that the restriction of λ to Δ(O(d)) contains the trivial representation of Δ(O(d)). (The multiplicity of the trivial representation is not necessarily one, even if it is not zero.) We denote by H^{inv} = H_{n_{1},...,n_{r}}^{inv}(λ) the isotypic component of the trivial representation of λ|Δ(O(d)). This space consists of polynomial maps P (X_{1}, …, X_{r}) of (X_{1}, …, X_{r}) ∈ M_{n_{1},d} × ⋯ × M_{n_{r},d} to V = V (τ(λ_{1})) ⊗ ⋯ ⊗ V (τ(λ_{r})) such that the following three conditions are satisfied.

(1) P (X_{1}, …, X_{r}) is pluri-harmonic for each X_{i} (1 ≤ i ≤ r).

(2) P (X_{1}g, …, X_{r}g) = P (X_{1}, …, X_{r}) for each g ∈ O(d).

(3) P (a_{1}X_{1}, …, a_{r}X_{r}) = (τ_{1}(a_{1})⊗⋯⊗τ_{r}(a_{r}))P (X_{1}, …, X_{r}) for each a_{i} ∈ GL(n_{i}, ℂ) (1 ≤ i ≤ r). Now, denote by X the n × d -matrix defined by:

X = ( X_{1} ; X_{2} ; ⋮ ; X_{r} ) ,

where X_{i} ∈ M_{n_{i},d} as before. From now on, we assume that d ≥ n. Then, by virtue of Weyl [13], any O(d)-invariant polynomial of the components of X is a polynomial of the components of the n × n symmetric matrix X^{t}X, and n(n + 1)/2 different components of X^{t}X are algebraically independent. Hence, under the assumption that d ≥ n, for each element P ∈ H^{inv}, there exists a unique polynomial map Q of the set of n × n symmetric matrices to V such that P (X_{1}, …, X_{r}) = Q(X^{t}X). We call Q the associated polynomial of P. For Z = (z_{kl})_{1≤k,l≤n} ∈ H_{n}, we put

\frac{∂}{∂Z} = (\frac{1 + δ_{kl}}{2} \frac{∂}{∂z_{kl}} ) .

For the case (I) in the introduction, we consider the commutation relation below. For each i = 1, , , .r, we take a finite dimensional irreducible representation (τ_{i}, V_{τ_{i}}) of GL(n_{i}, ℂ) and put V = ⊗_{i=1}^{r}V_{τ_{i}}. We take an even natural number d = 2k and put τi^{′} = τ_{i} ⊗ det^{k}. We put n = ∑_{i=1}^{r} n_{j} as before. We take a polynomial map Q of the set of n × n symmetric matrices to V, and put D = Q(\frac{∂}{∂Z}).

Condition 1 For any C^{∞} function F on H_{n}, and any element g = (g_{1}, ..., g_{r}) ∈ G(n_{1}, ..., n_{r}), we get

(Res_{Δ}(DF))|_{τ′_{1}⊗···⊗τ′_{r}}[g] = Res_{Δ}(D(F |_{k}[g])).

(Here we denote by Res_{Δ} the restriction of maps to Δ.) Theorem 1 Suppose that d ≥ n. Let Q be a polynomial map from the set of n × n symmetric matrices to V. Then the associated differential operators D = Q(\frac{∂}{∂Z}) satisfies the commutation relation, Condition 1, if and only if the polynomial P on M_{n,d} defined by P (X) = Q(X^{t}X) belongs to H^{inv} = H_{n_{1},...,n_{r}}^{inv}(λ_{1} ⊗ ⋯ ⊗ λ_{r}), where λ_{i} is such that τ_{i} = τ(λ_{i}) for each i.

Corollary 1 Assumptions and notation being as above, assume that a V -valued C^{∞} function F on H_{n} transforms like a modular form of weight k with respect to a discrete subgroup Γ of Sp(n, ℝ). Then, the function Res_{Δ}(DF) transforms like a modular form on Δ of weight det^{k_{1}} τ_{1} ⊗ ⋯ ⊗ det^{k_{r}} τ_{r} with respect to Γ ∩ G(n_{1}, …, n_{r}). In particular, if F is a holomorphic modular form, then Res_{Δ}(DF) is also a holomorphic modular form.

Next, we shall treat the case (II) in the introduction. Put D = H_{n}^{r} and Δ = H_{n}, and regard Δ as a subset of D through the diagonal embedding. For each i with 1 ≤ i ≤ r, we fix natural numbers d_{i} such that d_{i} ≥ n. Put d = d_{1} +⋯+d_{r}. We regard the group K = O(d_{1}) × ⋯ × O(d_{r}) as a subgroup of O(d) through the “diagonal” embedding:

O(d_{1}) × ⋯ × O(d_{r}) ∋ (k_{1}, … k_{r}) ⟶ ( \begin{matrix}k_{1} & 0 & ⋯ & ⋯ & 0 \\ 0 & k_{2} & 0 & ⋯ & 0 \\ ⋮ & 0 & ... & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & 0 \\ 0 & 0 & ⋯ & 0 & k_{r} \end{matrix} ) ∈ O(d).

Take a representation τ ⊗ λ of GL(n, ℂ) × O(d) which appears in H_{n,d}. We denote by H^{inv} = H^{inv}(d_{1}, …, d_{r}; λ) the subspace of H(τ, λ) consisting of pluri-harmonic polynomial maps P in H(τ, λ) which is invariant by the action of K, namely P (X_{1}k_{1}, ..., X_{r}k_{r}) = P (X_{1}, ..., X_{r}) for all k_{i} ∈ O(d_{i})) (1 ≤ i ≤ r). If the restriction of λ to K has the trivial representation as an irreducible component of multiplicity l, then the dimension of H^{inv} is l. For each i with 1 ≤ i ≤ r, denote the n × d_{i} matrix argument by X_{i}. By virtue of H. Weyl [13] , for each element P ∈ H^{inv}, there exists a polynomial map Q such that Q(X_{1}^{t}X_{1}, …, X_{r}^{t}X_{r}) = P (X_{1}, …, X_{r}). We call Q the associated map of P.

Now for the case (II), we consider the commutation relation below. Let (τ, V) be a finite dimensional irreducible representation of GL(n, ℂ). Let Q be a polynomial map of r product of the space of n × n symmetric matrices to V. Define a differential operator D by D = Q(\frac{∂}{∂Z_{1}} , …, \frac{∂}{∂Z_{r}}). We assume that each d_{i} (1 ≤ i ≤ r) is even and put d_{i} = 2k_{i} and d = 2k. We regard Sp(n, ℝ) as a subgroup of Sp(n, ℝ)^{r} through the diagonal embedding.

Condition 2 For any C^{∞} function F (Z_{1}, ..., Z_{r}) on H_{n}^{r} and any element g ∈ Sp(n, ℝ), we get the following commutation relation.

Res_{Δ}(D(F |_{k_{1},...,k_{r}}[g])) = (Res_{Δ}(DF))|_{det^{k} ⊗τ}[g]),

where Res_{Δ} is the restriction of the maps to Δ.

Theorem 2 Notation being as above, we assume that d_{i} ≥ n for each i. We take X_{i} ∈ M_{n,d_{i}} (1 ≤ i ≤ r) and define a map P by P (X_{1}, ..., X_{r}) = Q(X_{1}^{t}X_{1}, ..., X_{r}^{t}X_{r}). Then D satisfies the above commutation relation, Condition 2, if and only if P is a pluri-harmonic polynomial map belonging to H^{inv} = H^{inv}(d_{1}, ..., d_{r}; λ), where τ = τ(λ).

Corollary 2 Notation and assumptions being the same as above, we get the following results:

(1) If a function F on H_{n}^{r} transforms like a modular forms of weight (k_{1}, …, k_{r}) with respect to some discrete subgroup Γ of Sp(n, R)^{r}, then Res_{Δ}(DF) transforms like a modular form of weight det^{k} ⊗τ with respect to Γ ∩ Δ(Sp(n, ℝ)), where Δ(Sp(n, ℝ)) is the image of Sp(n, ℝ) in Sp(n, ℝ)^{r} under the diagonal embedding ι.

(2) Let Γ be a discrete sybgroup of Sp(n, ℝ). For each i with 1 ≤ i ≤ r, let F_{i}(Z_{i}) be a ℂ-valued modular form on H_{n} of weight k_{i} with respect to Γ. Take the product F (Z_{1}, …, Z_{r}) = F_{1}(Z_{1}) × ⋯ × F_{r}(Z_{r}). Then, Res_{Δ}(DF) is a modular form on H_{n} of weight det^{k} ⊗τ with respect to Γ.

Remark. Of course, these theorems can be applied also for modular embeddings, if we change the coordinates according to the embedding. Also we can get differential operators on Jacobi forms, using the Fourier-Jacobi expansion.

2 Proofs

2.1

In this section, we prove Theorem 1 and 2. The idea of the proofs of both theorems is based on the following two facts. Firstly, for any natural number n and d with d ≥ n, let Y be a n × d matrix whose components are algebraically independent nd variables over ℂ. For each i with 1 ≤ i ≤ n, denote by y_{i} the i-th row vector of Y. Then, by virtue of the invariant theory on the orthogonal groups (H.Weyl, loc.cit.), the n(n + 1)/2 inner products (in the usual sense) (y_{i}, y_{j}) (1 ≤ i ≤ j ≤ n) are algebraically independent over ℂ. Secondly, we have the following transfomation formula for harmonic polynomials and besides we can show a lemma which asserts the converse.

Lemma 1 (Kashiwara-Vergne [9]) For any pluri-harmonic polynomial P (X) on M_{n,d} and Z ∈ H_{n} and any Y ∈ M_{n,d}, we get

∫_{M_{n,d}} e^{iT r(^{t}XY)}e^{\frac{i}{2} T r(^{t}XZX)}P (X)dX = (2π)^{nd/2}(det(Z/i))^{− \frac{d}{2}} e^{\frac{i}{2} T r(^{t}Y (−Z^{−1})Y)}P (−Z^{−1}Y),

where dX is the usual Lebesgue measure on M_{n,d}(ℝ), and by det(Z/i)^{\frac{d}{2}} we mean the branch which is 1 for Z = i1_{n}.

Lemma 2 We fixed any homogeneous polynomials P_{1}(X) and P_{2}(X) of variable X ∈ ℝ^{N}. We assume that the following equality

∫_{ℝ^{N}} e^{i^{t}XY} e^{− \frac{^{t}XX}{2}} P_{1}(X)dX = (2π)^{N/2}e^{− \frac{^{t}Y Y}{2}} P_{2}(iY)

is satisfied for any vector Y = (y_{1}, ..., y_{N}) ∈ ℝ^{N}. Then we get P_{1}(X) = P_{2}(X) and besides this is a harmonic polynomial.

Proof of Lemma 2. This lemma seems more or less known (cf. Kashiwara and Vergne loc. cit. p.4), but we give here a proof for the reader’s convenience. We write the total degree of P_{i}(X) by deg(P_{i}). For X =^{t} (x_{1}, ..., x_{N}) ∈ ℝ^{N}, write P_{1}(X) as

P_{1}(X) = ∑_{α} c(α)x^{α},

where α runs over multi-indices (α_{1}, ..., α_{N}) such that α_{1} + ⋯ + α_{N} = degP_{1} with α_{i} ≥ 0, and we write x^{α} = ∏_{i=1}^{N} x_{i}^{α_{i}}. We calculate the left hand side. First we get

−\frac{^{t}XX}{2} + i^{t}XY = −\frac{1}{2} ∑_{j=1}^{N}(x_{j} − iy_{j})^{2} − \frac{^{t}Y Y}{2} .

Hence, for a fixed Y, by changing the variable X into X + iY, we get

∫_{ℝ^{N} +iY} e^{− \frac{^{t}XX}{2}} P_{1}(X + iY)dX = (2π)^{N/2}P_{2}(iY)

Here the integrand is holomorphic with respect to X and e(−^{t}XX/2) tends rapidly to zero when X tends to ±∞. Hence the integral remains unchanged if we replace the path ℝ^{N} + iY of the integral by ℝ^{N}. Now we can calculate the integral term by term. First of all, we have

∫_{−∞}^{∞} e^{− \frac{1}{2} x^{2}}x^{l}dx = \{ \begin{matrix}2^{\frac{l+1}{2}} Γ(\frac{l+1}{2}) & if l is even, \\ 0 & if l is odd. \end{matrix}

Hence, we get

∑_{α} c(α) ∫_{ℝ^{N}} ∏_{j=1}^{N} e^{− \frac{1}{2} x_{j}^{2}}(x_{j} + iy_{j})^{α_{j}} ∏_{j=1}^{N} dx_{j}

= ∑_{α} c(α) ∑_{j=1}^{N} ∑_{β_{j}=0,β_{j}:even}^{α_{j}} ( \begin{matrix}α_{j} \\ β_{j} \end{matrix} ) (iy_{j})^{α_{j}−β_{j}}2^{(β_{j}+1)/2}Γ(\frac{β_{j} + 1}{2}).

By our assumption, this is equal to (2π)^{N/2}P_{2}(iY). Since we assumed this equality for all Y ∈ ℝ^{N}, this is a polynomial identity. By definition, the right hand side is homogeneous with respect to y_{j}. On the left hand side, the terms of highest degree with respect to Y come from those with β_{j} = 0 for all j. We get 2^{1/2}Γ(1/2) = (2π)^{1/2}. Hence, comparing the terms of highest degree with respect to Y of both sides, we get P_{1}(iY) = P_{2}(iY). Besides, the terms of degP − 2 of the left hand side with respect to Y , that is, those terms with β_{j} = 2 for one j and β_{l} = 0 for all the other l (1 ≤ j, l ≤ N) vanishes identically. Also we have 2^{3/2}Γ(3/2) = 2^{1/2}Γ(1/2) = (2π)^{1/2}. Hence we get

∑_{α} ∑_{j=1}^{N} α_{j}(α_{j} − 1)c(α)(iy_{j})^{α_{j}−2} ∏_{l=1,l≠j}^{N}(iy_{l})^{α_{l}} = 0,

where the sum is over indices such that ∑_{j=1}^{N} α_{j} = degP. This means that P_{1}(X) is harmonic. q.e.d.

2.2 Proof of Theorem 1

First for a polynomial map Q of M_{n,d} to V associated to an invariant pluri-harmonic polynomial P, we put D = Q(\frac{∂}{∂Z}). We show that D satisfies the commutation relation in Theorem 1. Let F be as in Theorem 1. Then, by definition, F |_{k}[g] = det(CZ + D)^{−k}F (gZ). We must calculate Res_{Δ}(D(det(CZ + D)^{−k}F (gZ))) for any g ∈ G = G(n_{1}, n_{2}, …, n_{r}). Obivously, it is sufficient to show the commutation relation for generators of G. It can be easily shown that the group G is generated by the following three kinds of elements.

(1) u(B) = ( \begin{matrix}1_{n} & B \\ 0 & 1_{n} \end{matrix} ), where B ∈ M_{n}(ℝ), ^{t}B = B, and B_{ij} = 0 if i ≠ j.

(2) d(A) = ( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ), where A ∈ GL(n, ℝ) and A_{ij} = 0 if i ≠ j.

(3) ρ = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ).

Among the above elements, the commutation relation for elements of type (1) is trivial. As for elements of type (2), if we put W = AZ^{t}A, then we get \frac{∂}{∂Z} = ^{t}A\frac{∂}{∂W} A, and by our choice of Q, we get Q(\frac{∂}{∂Z}) = τ_{1}(^{t}A_{1}) ⊗ ⋯ ⊗ τ_{r}(^{t}A_{r})Q(\frac{∂}{∂W}). Hence we get

Q(\frac{∂}{∂Z})(F (AZ^{t}A)(det A)^{k}) = (⊗_{j=1}^{r} det(A_{j})^{d/2}τ(^{t}A^{−1})^{−1})(Q(\frac{∂}{∂Z})F)(AZ^{t}A).

To show the relation for ρ, we need some tricks. We denote by μ multi-index (μ_{ij})_{1≤i≤j≤n}. We put |μ | = ∑_{1≤i≤j≤n} μ_{ij} and denote by Dμ the following differential operator

Dμ = ∏_{1≤i≤j≤n} \frac{∂^{μ_{ij}}}{∂z_{ij}^{μ_{ij}}} .

Then, it is clear that there exist V -valued functions Qμ (Z) which does not depend on F such that

D(det(−Z)^{−k}F (−Z^{−1})) = ∑ ; μ Qμ (Z)(Dμ F)(−Z^{−1}),

where μ runs over multi-indices such that |μ | ≤ deg(Q). So, the rough idea to prove the commutation relation is to find a function F such that Dμ F are linearly independent at any point Z (or over the ring of holomorphic functions) and satisfies the commutation relation. If we put F_{0}(Z) = exp(\frac{i}{2}T r(^{t}Y ZY)) for a matrix Y of size n × d of independent variables, then

(Dμ F_{0})(Z) = (\frac{i}{2})^{|μ |} ∏_{1≤i≤j≤n}(y_{i}, y_{j})^{μ_{ij}}F_{0}(Z),

and the polynomials yμ = ∏_{1≤i≤j≤n}(y_{i}, y_{j})^{μ_{ij}} (μ : multi-indices) are linearly independent over ℂ. So, it is enough to show the relation only for F_{0}. We calculate the action of D on this F_{0}. By Lemma 1, we get

D(F_{0}|_{k}[ρ]) = i^{nd/2}(2π)^{−nd/2} ∫_{M_{n,d}} e^{iT r(^{t}XY)}D(e^{\frac{i}{2} T r(^{t}XZX)})dX
= i^{nd/2}(\frac{i}{2})^{degQ}(2π)^{−nd/2} ∫_{M_{n,d}} e^{iT r(^{t}XY)}e^{\frac{i}{2} T r(^{t}XZX)}Q(X^{t}X)dX.

We write Z_{0} = (Z_{ij}) where Z_{ij} = 0 for all i ≠ j and Z_{ii} = Z_{i}. Then, by the Fubini Theorem and iterated use of Lemma 1, we get

Res_{Δ}(D(F_{0}|_{k}[ρ]))

= i^{nd/2}(2π)^{−nd/2}(\frac{i}{2})^{deg Q} ∫_{M_{n,d}} e^{i ∑_{i=1}^{r} T r(^{t}X_{i}Y_{i})}e^{\frac{i}{2} ∑_{i=1}^{r} T r(^{t}X_{i}Z_{i}X_{i})}Q(X^{t}X)dX_{1}dX_{2} ⋯ dX_{r}
= (\frac{i}{2})^{deg Q} ∏_{i=1}^{r}(det(−Z_{i})^{−d/2}e^{\frac{i}{2} T r(^{t}Y_{i}(−Z_{i}^{−1})Y_{i})})P (−Z_{0}^{−1}Y))
= ((\frac{i}{2})^{deg Q} ∏_{i=1}^{r}(det(−Z_{i})^{− \frac{d}{2}} e^{\frac{i}{2} T r(^{t}Y (−Z_{i}^{−1})Y)}) × (τ_{1}(−Z_{1}^{−1}) ⊗ ⋯ ⊗ τ_{r}(−Z_{r}^{−1})P (Y))
= ∏_{i=1}^{r} det(−Z_{i})^{− \frac{d}{2}} × (τ_{1}(−Z_{1}^{−1}) ⊗ ⋯ ⊗ τ_{r}(−Z_{r}^{−1})) × (DF_{0})(−Z_{0}^{−1})).

Now we prove the converse. Let Q be a polynomial map of n × n symmetric matrices Y to V_{τ}, and put D = Q(\frac{∂}{∂Z}). Assuming that D satisfies the commutation relation, we shall show that Q is an associated polynomial of invariant pluri-harmonic polynomial P. First, taking the element of type (1) above and F = F_{0}, we see that

Q(^{t}AY A) = (τ_{1}(^{t}A_{1}) ⊗ ⋯ ⊗ τ_{r}(^{t}A_{r}))Q(Y).

A fortiori, if we put P (X_{1}, ..., X_{r}) = Q(X^{t}X), then this P is homogeneous for each X_{i}. We shall show that P is pluri-harmonic for each X_{i}. Taking the above relation into account, the commutation relation for this Q and ρ ∈ G and F_{0} is written as

∫_{M_{n,d}(ℝ)} e^{iTr(^{t}X_{j}Y_{j})}e^{\frac{i}{2}Tr(^{t}X_{j}Z_{j}X_{j})}P (X)dX
= ∏_{j=1}^{r}(2π)^{n_{j}d/2} det(Z_{j}/i)^{−d/2}e^{\frac{i}{2}Tr(^{t}Y_{j}(−Z_{j}^{−1})Y_{j})}P (−Z_{1}^{−1}Y_{1}, ..., −Z_{r}^{−1}Y_{r}).

We put Z_{j} = iα_{j}^{2} for some positive definite real symmetric n×n matrices α_{j} (1 ≤ j ≤ r). So changing the variable X_{j} into α_{j}^{−1}X_{j}, we get

∫_{M_{n,d}(ℝ)} e^{iTr(^{t}X_{j}(α_{j}^{−1}Y_{j}))}e^{− \frac{1}{2}Tr(^{t}X_{j}X_{j})}P (α_{1}^{−1}X_{1}., , , .α_{r}^{−1}X_{r})dX

= P (iα_{1}^{−1}(α_{1}^{−1}Y_{1}), ..., iα_{r}^{−1}(α_{r}^{−1}Y_{r})) ∏_{j=1}^{r}((2π)^{n_{j}d/2} det(α_{j})^{−d}e^{− \frac{1}{2}Tr(^{t}(α_{j}^{−1}Y_{j})(α_{j}^{−1}Y_{j})}).

Hence, by iterated use of Lemma 2, we can show that P (α_{1}^{−1}Y_{1}, ..., α_{r}^{−1}Y_{r}) is harmonic for each Y_{j} for any α_{j}. Hence P (X_{1}, ..., X_{r}) is pluri-harmonic for each X_{j} (1 ≤ j ≤ r). Thus, Theorem 1 is proved.

2.3 Proof of Theorem 2

Define a function F_{1}(Z_{1}, …, Z_{r}) on H_{n}^{r} by

F_{1}(Z_{1}, …, Z_{r}) = e^{\frac{i}{2} ∑_{j=1}^{r} T r(^{t}Y_{j}Z_{j}Y_{j})},

where Y_{j} ∈ M_{n,d_{j}} (1 ≤ j ≤ r) are matrices of independent variables. Let P and Q be the polynomial maps in Theorem 2 and assume that P ∈ H^{inv}. The differential operator D is defined by D = Q(\frac{∂}{∂Z_{1}} , …, \frac{∂}{∂Z_{r}}). Then, the commutation relation for the

elements of the form ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ Sp(n, ℝ) is obvious as before. As for the inversion

ρ(Z) = −Z^{−1} (Z ∈ H_{n}), we get by Lemma 1 that

D(F_{1}|_{k_{1},...,k_{r}}[ρ]) = i^{nd/2}(2π)^{−nd/2} ∫_{M_{n,d}} e^{iT r(^{t}XY)}(DF_{1})(Z_{1}, …, Z_{r})dX
= i^{nd/2}(2π)^{−nd/2}(\frac{i}{2})^{deg Q} ∫_{M_{n,d}} e^{iT r(^{t}XY)}Q(X_{1}^{t}X_{1}, …, X_{r}^{t}X_{r})F_{1}(Z_{1}, …, Z_{r})dX.

Denoting the variable on H_{n} by Z, we get F_{1}(Z, ..., Z) = exp(\frac{i}{2}T r(^{t}Y ZY)), where we put Y = (Y_{1}, ..., Y_{r}). Applying Lemma 1 to P (X) = Q(X_{1}^{t}X_{1}, ..., X_{r}^{t}X_{r}), we get

Res_{Δ}(D(F |_{k_{1},...,k_{r}}[ρ]))

= i^{nd/2}(\frac{i}{2})^{deg Q}P (−Z^{−1}Y_{1}, …, −Z^{−1}Y_{r})(det Z/i)^{− \frac{d}{2}} F_{1}(−Z^{−1}, …, −Z^{−1})

= (\frac{i}{2})^{deg P} det(−Z)^{− \frac{d}{2}} (τ(−Z^{−1}))P (Y_{1}, …, Y_{r})F_{1}(−Z^{−1}, …, −Z^{−1})

= (det Z/i)^{− \frac{d}{2}} (τ(−Z^{−1}))(DF_{1})(−Z^{−1}, …, −Z^{−1}).

On the other hand, Dμ F_{1} (μ : multi-indices) are linearly independent over ℂ. Hence, for any C^{∞} function F on H_{n}, we get

Res_{Δ}(D(F |_{k_{1},...,k_{r}}[ρ])) = det(−Z)^{− \frac{d}{2}} τ(−Z^{−1})F (−Z^{−1}, …, −Z^{−1}).

The proof of the converse is almost similar to that of Theorem 1, or even easier, if we use F_{1}. So, we omit the details here. Hence, Theorem 2 is proved.

3 Examples of invariant pluri-harmonic polynomials

In section 1 and 2, we have shown that “invariant” pluri-harmonic polynomials give the differential operators which satisfy the commutation relation and vice versa. Here remains two problems.

(1) When do there exist “invariant” pluri-harmonic polynomials?

(2) How can one describe the ”invariant” pluri-harmonic polynomials explicitly? In this section, we give partial answers to these questions.

3.1 Case I

3.1.1 The case r = 2

Assume that representation λ_{1} or λ_{2} of O(d) appear in H_{n_{1},d} or H_{n_{2},d}, respectively. We get H_{n_{1},n_{2}}^{inv}(λ_{1}⊗λ_{2}) ≠ ∅, if and only if λ_{1} is equivalent to λ_{2}, and besides, dim H_{n_{1}.n_{2}}^{inv}(λ⊗ λ) = 1 in this case. This fact is easily shown by Schur’s Lemma.

The case n = 2, n_{1} = n_{2} = 1 In this case, if d ≥ 3, then H^{inv} ≠ 0 if and only if λ_{1} = λ_{2} = the ν-th spherical representation ρ_{ν}, that is, the representation of O(d) whose representation space is the space of harmonic polynomials of d variables of homogeneous degree ν. Now, for vectors x, y ∈ ℝ^{d}, denote by (x, y) the usual inner product and put n(x) = (x, x), n(y) = (y, y). Define the homogeneous polynomials G_{d}^{ν}(s, m) of degree 2ν of 2 variables as follows:

\frac{1}{(1 − 2st + mt^{2})^{\frac{(d−2)}{2}}} = ∑_{ν=0}^{∞} G_{d}^{ν}(s, m)t^{ν}.

The polynomials G_{d}^{ν}(s, 1) are called the Gegenbauer polynomials. In particular, when d = 3, these polynomials are the Legendre polynomials P_{ν}(s). For each ν and each d, the polynomial G_{d}^{ν}((x, y), n(x)n(y)) of total degree 2ν of 2d variables gives the basis of one dimensional space H_{1,1}(ρ_{ν} ⊗ ρ_{ν}). So, if F is a Siegel modular form of degree 2 of weight k = d/2, where we assume d is even, then (G_{2k}^{ν}(\frac{1}{2} \frac{∂}{∂z_{12}} , \frac{∂^{2}}{∂z_{11}∂z_{22}} )F (Z))_{z_{12}=0} is a tensor product of modular forms of one variables of weight k + ν. This fact is known (cf. Eichler and Zagier [7]).

Spherical case When n, n_{1} and n_{2} are general, then H^{inv} ≠ 0 for the representation λ which is not necessarily spherical. But, in the case that λ = ρ_{2l} for some l, the description of H_{n_{1},n_{2}}^{inv}(ρ_{2l} ⊗ ρ_{2l}) is easy. In fact, let u_{i} (1 ≤ i ≤ n) be independent variables. For each i with 1 ≤ i ≤ n, denote by x_{i} a d-dimensional variable vector and put X =^{t} (x_{1}, …, x_{n}). For each multi-index m = (m_{1}, …, m_{n}), denote by P_{m}(X) the coefficient of ∏_{i=1}^{n} u_{i}^{m_{i}} of the polynomial

G_{d}^{ν}((∑_{i=1}^{n_{1}} u_{i}x_{i}, ∑_{j=n_{1}+1}^{n} u_{j}x_{j}) , n(∑_{i=1}^{n_{1}} u_{i}x_{i})n( ∑_{j=n_{1}+1}^{n} u_{j}x_{j})).

Then, P (X) = (P_{m}(X))_{m} is a pluri-harmonic polynomial map which gives the basis of H_{n_{1},n_{2}}^{inv}(ρ_{2l} ⊗ρ_{2l}). In this case, for Siegel modular forms on H_{n} of weight k, DF gives an automorphic form on H_{n_{1}} × H_{n_{2}} of weight (det^{k} Sym_{n_{1}}^{2l}) ⊗ (det^{k} Sym_{n_{2}}^{2l}), where Sym_{n_{i}}^{2l} is the symmetric tensor representation of GL_{n_{i}} for each i = 1, or 2. This operator is the one which appears in Böcherer-Satoh-Yamazaki [2].

The case n = 4, n_{1} = n_{2} = 2 . We assume that d ≥ 4. We treat the case τ(λ) = det^{ν}, Then we get operators to map Siegel modular form F on H_{4} of weight k to a modular form Res_{Δ}(DF) on H_{2} × H_{2} of homogeneous weight k + ν. This corresponds to the case λ = (ν, ν, 0, …, 0)_{+} in Kashiwara-Vergne’s notation ([9] p.27), where the notation (ν, ν, 0, …, 0) is regarded as the Young diagram of SO(d) which corresponds to the restriction of the representation λ of O(d) to SO(d). For the sake of simplicity, we write H_{2,2}^{inv}(λ ⊗ λ) = H^{inv}(ν). We have already explained that dimH^{inv}(ν) = 1. For each non-negative integer ν, the space H^{inv}(ν) consists of (ℂvalued) pluri-harmonic polynomials P (X, Y) of 4d -variables (X, Y ∈ M_{2,d})such that P (AX, BY) = det(A)^{ν} det(B)^{ν}P (X, Y) and that P (Xg, Y g) = P (X, Y) for each g ∈ O(d). Now, we describe the basis of H^{inv}(ν). Put

f_{1} = f_{1}(X, Y) = det(X^{t}Y),
f_{2} = f_{2}(X, Y) = det(X^{t}X) det(Y^{t}Y),
f_{3} = f_{3}(X, Y) = det ( \begin{matrix}X^{t}X & X^{t}Y \\ Y^{t}X & Y^{t}Y \end{matrix} ) .

Let t be a variable and put

Δ_{0}(X, Y, t) = 1 − 2f_{1}t + f_{2}t^{2}, and
R(X, Y, t) = (Δ_{0}(X, Y, t) + \sqrt{Δ_{0}(X, Y, t)^{2} − 4f_{3}t^{2}})/2.

Then, the each coefficient G_{ν}(X, Y) of t^{ν} of the following formal power series

G(X, Y, t) = ∑_{ν=0}^{∞} G_{ν}(X, Y)t^{ν} = \frac{1}{R(X, Y, t)^{\frac{d−5}{2}} \sqrt{Δ_{0}(X, Y, t)^{2} − 4f_{3}t^{2}}}

gives the basis of H^{inv}(ν). For example,

G_{1}(X, Y) = (d − 3) det(X^{t}Y)
G_{2}(X, Y) = \frac{(d − 1)(d − 3)}{2}f_{1}^{2} − \frac{d − 3}{2}f_{2} + \frac{d − 1}{2}f_{3}

Since the calculation is slightly complicated, we sketch the proof. We must show that G(X, Y, t) is pluri-harmonic for each X and Y. If we write X^{t}X = R = (r_{ij}), Y^{t}Y = S = (s_{ij}) and X^{t}Y = T = (t_{ij}), then as for X, we get

Δ_{11} = 2d\frac{∂}{∂r_{11}} + 4r_{11} \frac{∂^{2}}{∂r_{11}^{2}} + 4r_{12} \frac{∂^{2}}{∂r_{11}∂r_{12}} + r_{22} \frac{∂^{2}}{∂r_{12}^{2}}

+s_{11} \frac{∂^{2}}{∂t_{11}^{2}} + 2s_{12} \frac{∂^{2}}{∂t_{11}∂t_{12}} + s_{22} \frac{∂^{2}}{∂t_{12}^{2}}

+4t_{11} \frac{∂^{2}}{∂r_{11}∂t_{11}} + 4r_{12} \frac{∂^{2}}{∂r_{11}∂t_{12}} + 2t_{21} \frac{∂^{2}}{∂r_{12}∂t_{11}} + 2t_{22} \frac{∂^{2}}{∂r_{12}∂t_{12}} .

Since it is complicated to show the harmonicity of G(X, Y, t) directly, we use the following formula of power series expansion.

\frac{1}{\sqrt{1 − 4t}((1 + \sqrt{1 − 4t})/2)^{s}} = ∑_{n=1}^{∞} ( \begin{matrix}2n + s \\ n \end{matrix} ) t^{n}

To apply this in our case, we can show the following identity.

Δ_{11}(\frac{f_{3}^{n}}{Δ_{0}(X, Y, t)^{\frac{d−3}{2} +2n}})

= (t_{22}^{2}s_{11} − 2t_{22}t_{21}s_{12} + t_{21}^{2}s_{22} − r_{22} det(S))

×((4n + d − 3)(4n + d − 1)\frac{f_{3}^{n}X^{2}}{Δ_{0}(X, Y, t)^{\frac{d−3}{2} +2n+2}} − n(4n + 2d − 10)\frac{f_{3}^{n−1}}{Δ_{0}(X, Y, t)^{\frac{d−3}{2} +2n}}),

Taking both formula above into account, we see Δ_{11}(G(X, Y, t)) = 0. In a similar way, we can show Δ_{ij}(G(X, Y, t)) = 0 and the pluri-harmonicity of G(X, Y, t) for Y follows from the symmetry.

More generally, in the case r = 2 and n = 2n_{1} = 2n_{2}, the polynomial det(X^{t}Y) gives always an invariant harmonic polynomial corresponding to τ(g) = det(g). As for this case, some related and a little more complicated operators were also introduced by Böcherer [1] in fairly different formulation.

Next, we treat the case where n = 4, n_{1} = n_{2} = 2 and λ = (2, 1, 0, …, 0). In this case, the base of H^{inv} is given by the vector ^{t}(v_{1}, v_{2}), where

v_{1} = d(x_{1}, y_{1}) det(X^{t}Y) + | \begin{matrix}n(x_{1}) & (x_{1}, y_{1}) & (x_{1}, y_{2}) \\ (x_{1}, x_{2}) & (x_{2}, y_{1}) & (x_{2}, y_{2}) \\ (y_{1}, x_{1}) & n(y_{1}) & (y_{1}, y_{2}) \end{matrix} |

v_{2} = d(x_{2}, y_{2}) det(X^{t}Y) + | \begin{matrix}n(x_{2}) & (x_{2}, y_{2}) & (x_{2}, y_{1}) \\ (x_{1}, x_{2}) & (x_{1}, y_{2}) & (x_{1}, y_{1}) \\ (y_{2}, x_{2}) & n(y_{2}) & (y_{1}, y_{2}) \end{matrix} |

The weight of Res_{Δ}(DF) is (det ·Sym^{1}) ⊗ (det ·Sym^{1}).

3.1.2 The case r = 3

n = 3, n_{1} = n_{2} = n_{3} = 1 This case is treated in [8] in detail. Here, λ_{1}, λ_{2}, λ_{3} are automatically spherical, if we assume H^{inv} ≠ 0. Hence, denote them by ρ_{a_{1}}, ρ_{a_{2}} and ρ_{a_{3}}. Then, H(ρ_{a_{1}}, ρ_{a_{2}}, ρ_{a_{3}}) ≠ ∅, if and only if a_{1} = ν_{2}+ν_{3}, a_{2} = ν_{3}+ν_{1}, a_{3} = ν_{1}+ν_{2} for some non-negative integers ν_{1}, ν_{2}, ν_{3}. Besides, under this condition, dim H(ρ_{a_{1}}, ρ_{a_{2}}, ρ_{a_{3}}) = 1. As for the generating function of the bases and many other properties, see [8].

3.2 The case II

Spherical representation Take a spherical representation ρ_{ν} of SO(d). When d_{1} + ⋯ + d_{r} = d, the restriction Resρ_{ν} of ρ_{ν} to SO(d_{1}) × ⋯ × SO(d_{r}) contains the trivial representation as an irreducible component, if and only if ν is even. If ν = 2l for some integer l, then the multiplicity of the trivial representation of Resρ_{ν} is equal

to ( \begin{matrix}r + l − 2 \\ l \end{matrix} ) . In particular, if r = 2, then the multiplicity is one. If n = 1 and

r = 2, then for modular forms f of weight k_{1} and g of weight k_{2} on H_{1}, our operators Res_{Δ}D(f(z_{1})f(z_{2})) are nothing but Cohen’s operators in [5]. When n =2 and l = 2, our operators are Satoh’s operators in [12]. The case where n, l general and r = 2 will be written in Eholzer and Ibukiyama [6].

We assume that n = 1 now. Each element of H^{inv}(d_{1}, d_{2}; ρ_{ν}) is a harmonic polynomial P of x ∈ ℝ^{d}, and a function of m_{1} = n(x_{1}), …, m_{r} = n(x_{r}), where x_{i} ∈ ℝ^{d_{i}} (1 ≤ i ≤ r) and x = ^{t}(x_{1}, …, x_{r}). Hence, writing the condition ΔP = 0 by the variables m_{i} (1 ≤ i ≤ r), we get the following differential equations:

\frac{1}{2}Δf = ∑_{i=1}^{r} d_{i} \frac{∂f}{∂m_{i}} + 2 ∑_{i=1}^{r} m_{i} \frac{∂^{2}f}{∂m_{i}^{2}} = 0

Take a solution f(m_{1}, …, m_{r}) of the homogeneous degree l with respect to m_{1},…,m_{r} and write it by the coefficients as follows:

f(m_{1}, …, m_{r}) = ∑_{μ_{1}+···+μ_{r}=l} C(μ_{1}, …, μ_{r})m_{1}^{μ_{1}} ⋯ m_{r}^{μ_{r}}.

Then, we have the following difference equation:

(1+μ_{r})(d_{r}+2μ_{r})C(μ_{1}, …, μ_{r−1}, μ_{r}+1) = − ∑_{i=1}^{r−1}(d_{i}+2μ_{i})C(μ_{1}, …, μ_{i−1}, μ_{i}+1, μ_{i+1}, …, μ_{r})

When r = 2, by the above relation, we get Cohen’s operator explicitly.

The case r = 2 and n general After this work had been finished, an explicit formula for generalized Cohen’s operator was considered by Choie and Eholzer [4] in case n = 2 for ℂ-valued Siegel modular forms of degree 2. This was generalized to the case of general n by Eholzer and Ibukiyama [6]. So, we would like to add some explanation on this. We take the representation (τ, λ) of GL(n) × O(2k) on the pluriharmonic polynomials such that τ(g) = det(g)^{v}. In our setting, this case corresponds to the differential operators on the product of two ℂ-valued Siegel modular forms of weight k_{1} and k_{2} of degree n to ℂ-valued Siegel modular forms of weight k_{1} +k_{2} +v. We consider the restriction of this representation (τ, λ) to {1} × (O(d_{1}) × O(d_{2})). Under the assumption that d_{1} ≥ n and d_{2} ≥ n, the multiplicity of the trivial representation in this restriction is one. Hence using this invariant pluri-harmonic polynomial, we can construct the desired differential operators. If d_{1}, d_{2} ≥ 2n, the assertion for multiplicity is almost a direct consequence of Koike and Terada [10] p. 115 Corollary 2.6. The details will be given in [6], as well as the method to get an explicit operators in this case.

References

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[2] S. Böcherer, T. Satoh and T. Yamazaki. On the pullback of a differential operator and its application to vector valued Eisenstein series, Comm. Math. Univ. St. Pauli 41 (1992), 1-22.

[3] S. Böcherer and R. Schulze-Pillot, On the central critical value of the triple product L-function, MSRI Preprint No. 058-95.

[4] Y. Choie and W. Eholzer, Rankin-Cohen operators for Jacobi and Siegel forms, preprint DAMTP-96-106(alg-geom/9611032).

[5] H. Cohen. Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), 271-285.

[6] W. Eholzer and T. Ibukiyama. Examples of invariant pluri-hamrmonic polynoimals and Rankin-Cohen type differential operators, MPI Preprint.

[7] M. Eichler and D. B. Zagier. The theory of Jacobi forms. Progress in Math.55, Birkhäuser, 1985.

[8] T. Ibukiyama and D. Zagier, Higher spherical polynomials, in preparation.

[9] M. Kashiwara and M. Vergne. On the Segal-Shale-Weil representation and harmonic polynomials. Invent. Math. 44 (1978), 1-47.

[10] K. Koike and I. Terada, Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank, Advances in Math. 79 (1990), 104-135.

[11] M. Harris, Maass operators and Eisenstein series, Math. Ann. 258(1981/82), 135-144.

[12] T. Satoh. On certain vector valued Siegel modular forms of degree two. Math. Ann. 274 (1986), 335-352.

[13] H. Weyl. The classical groups, Princeton Univ. Press 1946.

Department of mathematics Graduate School of mathematics Osaka University Machikaneyama 1-16 Toyonaka, Osaka, 560 Japan ibukiyam@math.wani.osaka-u.ac.jp