ON HOLOMORPHIC DIFFERENTIAL OPERATORS EQUIVARIANT FOR THE INCLUSION OF Sp(n, R) IN U (n, n) ¨ SIEGFRIED BOCHERER AND SOUMYA DAS

Abstract. We construct differential operators on holomorphic functions defined on the Hermitian half-space of degree n which are equivariant under the action of Sp(n, R) thereby increasing the weight by ν. For this, we construct and use as a tool the ν-th Rankin–Cohen bracket for holomorphic functions on the Hermitian half-space of degree n, which is equivariant under the action of SU (n, n). Explicit combinatorial description of the differential operators is given for n = 2, arbitrary ν; and for arbitrary n, ν = 1. We also treat the vector-valued case, where a twisted automorphy factor shows up naturally for equivariance under the group U (n, n). Further, our differential operators are unique upto scalars after restriction of variables to the Siegel half-space.

1. Introduction Linear holomorphic differential operators acting on Hermitian Symmetric spaces and changing the weight of modular forms are quite rare, e.g., on the ordinary complex upper half-plane ∂ which changes the weight from 0 to 2. In many cases one can use the the only example is ∂z (nonholomorphic) Maaß-operators as a substitute. The situation becomes better, if one asks for equivariance only for a proper subgroup, or even weaker, one asks for equivariance for that subgroup only after restricting the function in question to a subdomain (which should be invariant under that subgroup). This topic was studied from several viewpoints for the situation relevant for the doubling method for Siegel modular forms: There one considers an embedding S Sp(n, R) × Sp(n, R) ֒→ Sp(2n, R) and looks for holomorphic differential operators on H2n (the Siegel upper half-space of degree 2n) compatible with the action of the two copies of Sp(n, R) and changing weights. The weaker version is to restrict the functions at the same time to the S . The former operators can be iterated, whereas the latter ones are subdomain HnS × HnS ⊂ H2n more easily accessible. For all of this we refer to several works of Ibukiyama and the first author, see [B1, B3, EI, I]. The purpose of our paper is to do a similar investigation for the unitary group U (n, n) and the real symplectic group naturally embedded in U (n, n). This corresponds to embedding the Siegel upper half-space HnS into Hn . In one of our main results, namely, Theorem 5.1 we write down explicitly such a differential operator changing the weight by one. To obtain our result, we follow the lines of the notes [B3] where we gave a new approach for the Siegel case using RankinCohen type operators. Rankin-Cohen operators were introduced in [Co] and were generalised to other types of modular forms, see [B2, CE, EI] for example. So, we first introduce Rankin-Cohen 2000 Mathematics Subject Classification. Primary 11F60; Secondary 11F46. Key words and phrases. Differential Operators, Hermitian modular form, Siegel modular form, Restriction, Harmonic polynomial. 1

2

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

type operators for holomorphic functions on Hn which are equivariant under SU (n, n) thereby increasing the weight by 2. These are bilinear holomorphic differential operators constructed by means of minors of matrices of differentials. The transformation properties of such minors seem not to be available in the literature and we have to write down some preparatory material on them. We follow [Fr] in this regard. Some subtlety concerning the automorphy factors in question comes up, if we want to include vector-valued Rankin-Cohen operators or if we want automorphy properties in the scalar-valued case not only within SU (n, n) but for the full unitary group U (n, n). This is the content of sect. 4.2. We get the operator of Theorem 5.1 by fixing a quite elementary function as one of the arguments of the bilinear Rankin-Cohen operator. This operator can be iterated and after restriction of variables from Hn to HnS we get weight increasing holomorphic differential operators Dkν,0 from Hermitian modular forms to Siegel modular forms. In sect. 7, we focus on the case n = 2. We write down the explicit combinatorial formula for the coefficients of these differential operators increasing the weight from k to k + ν by any ν ≥ 0 after restriction Also, we prove that Dkν,0 carries the Hermitian Maaß space of weight k to it’s Siegel counterpart of weight k + ν (see Theorem 7.4). We further explain how our differential operators can be used to define similar operators for Hermitian Jacobi forms. See sect. 7.4 for this. For an arbitrary even n, we determine the explicit combinatorial form of the differential operator raising the weight by 1 in sect. 8. In the final section we shortly describe the relation of our differential operators to pluriharmonic polynomials (analogues of Gegenbauer polynomials) and Hermitian Theta series. The space of such pluriharmonic polynomials invariant under U (k, C) is known to be one dimensional by representation theory. (See [GW].) This shows that our differential operators are unique up to constants. This paper is written in the spirit of [B3, EI, I]. We are aware that our operators have an interpretation within the universal enveloping algebra attached to the ambient Lie group U (n, n). Our main point is however, to write down the operators explicitly on the symmetric domain. From a representation-theoretic point of view the embeddings Sp(n) × Sp(n) ֒→ Sp(2n) and Sp(n) ֒→ SU (n, n) seem to be quite different from each other. In the investigation of our differential operators however, several considerations run parallel to the symplectic case and it would be desirable to have a uniform treatment. Acknowledgement. Most of the paper was written during the two stays of the first author at the School of Mathematics, Tata Institute of Fundamental Research, Mumbai. He thanks the Tata Institute for it’s hospitality and for an excellent research environment. We would like to sincerely thank D. Prasad, A. Krieg and R. Schulze-Pillot for several helpful discussions on the topics of this paper.

2. Notation and Preliminaries We define the symplectic and unitary groups of degree n as ¯ ′ JM = J}, Sp(n, R) := {M ∈ GL(2n, R) | M ′ JM = J}, U (n, n) := {M ∈ GL(2n, C) | M

ON HOLOMORPHIC DIFFERENTIAL OPERATORS

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 −In

and for a matrix M , we denote it’s transpose by M ′ . It is well known that where J = I0n 0 Sp(n, R) ⊂ SL(2n, R). For the unitary group, we further define SU (n, n) = U (n, n) ∩ SL(n, C) = {M ∈ U (n, n) | det M = 1}. We mostly follow standard notation throughout the paper: M (n, R) denotes, as usual, the space of n × n matrices over a commutative ring R; for A ∈ M (n, C), A∗ := A¯′ ; A is Hermitian if A = A∗ and is positive definite (resp. semi–definite) if ξ ∗ Aξ > 0 (resp. ≥ 0) for all ξ ∈ C\{0}. We define next the Pfaffian of an alternating matrix A, which is a polynomial in the entries of A by Pf(A)2 = det A, (Pf(A) ∈ C[(ai,j )], A = (ai,j )).  0 a For example, the Pfaffian of the matrix −a 0 is a.

We recall the Siegel and Hermitian upper half-spaces of degree n on which most of the holomorphic functions in this paper live: ¯ HnS := {Z = Z ′ ∈ M (n, C) | (Z − Z)/2i > 0}, Hn := {Z ∈ M (n, C) | (Z − Z ∗ )/2i > 0}.

It is well known that the groups Sp(n, R) (resp. U (n, n)) act on HnS (resp. Hn ) as biholomorphic automorphisms via: Z 7→ M < Z >= (AZ + B)(CZ + D)−1 ,

A B ) ∈ Sp(n, R) (resp. U (n, n)). For HS , the space Bihol(HS ) of the all biholomorwhere M = ( C n n D phic automorphisms equals Sp(n, R)/{±1}; however for Hn , we have

Bihol(Hn ) = (U (n, n) ∪ {Z 7→ M < Z ′ >| M ∈ U (n, n)}) /{±1}. √ Let now K denote an imaginary quadratic field of discriminant −DK , i.e., K = Q(i DK ). × Denote the ring of integers of K by OK and the order of the unit group OK of OK by wK . The inverse different of K is denoted by i # OK := √ OK . DK We denote by Γn (Z) the Siegel modular group and by Γn (OK ) the Hermitian modular group of degree n defined by Γn := Γn (Z) := Sp(n, R) ∩ M (2n, Z), Γn (OK ) := U (n, n) ∩ M (2n, OK ). Given an integer k, the vector space of Siegel (resp. Hermitian) modular forms of degree n and weight k consists of all holomorphic functions f : HnS → C (resp. f : Hn → C) satisfying f (Z) = f |k M (Z) ∀Z ∈ HnS , M ∈ Γn (resp. ∀Z ∈ Hn , M ∈ Γn (OK )). where f |k M (Z) := det(CZ +D)−k f (M < Z >); with the usual boundedness condition for n = 1. The vector space of Siegel (resp. Hermitian w.r.t. K) modular forms of degree n and weight k is denoted by [Γn , k] (resp. [Γn (OK ), k]). Let now L denote either Q or an imaginary quadratic field K with discriminant −DK . Each modular form in [Γn (OL ), k] has a Fourier expansion of the form: X f (Z) = (2.1) af (T )e (trT Z) , T ∈PSym(n,OL )

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

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where e(z) := e2πiz (z ∈ C) and # PSym(n, OL ) := {T ∈ M (n, C) | T = T ∗ ≥ 0, tµ,µ ∈ Z, tµ,ν ∈ OL }

is the dual lattice of the lattice of OL -integral Hermitian matrices with respect to the trace form tr. We also define Symn := {T ∈ M (n, C) | T = T ∗ } and PSymn := {T ∈ M (n, C) | T = T ∗ , T > 0}. For T ∈ PSymn , we denote by T 1/2 ∈ PSymn to be the unique square root of T . 3. Transformation properties on Hermitian upper half-space This is mainly a transcription of a part of Section III.6 in E. Freitag’s book [Fr] for the case of Hermitian upper half spaces. It is concerned with transformation properties of minors of matrix differentials. The Hermitian case is (to our knowledge) only written down for the determinant case (see [Re]) and by Shimura [Sh] for the matrix differential itself. 3.1. Freitag’s calculus in the Hermitian case. We first consider differential operators on the space PSymn of positive definite Hermitian matrices of size n. We collect them in a matrix   ∂ , Y = (yi,j ) ∈ PSymn . ∂ := ∂yij Note that ∂ii is then concerned with a real variable whereas for i 6= j the derivative ∂ij = concerned with a complex variable and ∂ji =

∂ ∂yij

is

∂ ∂y ij .

For a matrix A of size n, for 0 ≤ h ≤ n and two subsets a = {a1 < a2 < · · · < ah }, b = {b1 < b2 < · · · < bh } of N := {1, . . . , n} we denote by | A |ab the determinant of the h × h-matrix whose rows and colums are given by a and b in their natural order. We collect these   minors in a matrix A[h] of order nh × nh ; their entries are labelled by sets a and b as above. In particular, we will investigate properties of differential operators defined by ∂ [h] or | ∂ |ab . Further, A[−h] := (A−1 )[h] , for A invertible. We use the operation ⊓ as in [Fr] and we refer to loc. cit. for its basic properties. In this subsection, we follow Freitag’s exposition [Fr] quite closely (we always quote the corresponding statement of Freitag’s book). We decided however, to write down details here, because of the subtlety of the twisted automorphy factor coming up. Lemma 3.1. (3.1)

Z



Proof. ([Fr, Hilfssatz 6.5]) Starting with Z

C

one proceecds as in loc. cit.



det(P P )e−πtr(P ·P ) dP = C(n,n)

zz e−πzz dz =

n! . πn

1 , π 

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Lemma 3.2. Let a, b be subsets of N = {1, . . . , n} with p elements and c, d be subsets of N = 1, . . . , n with q elements. Then ( Z p! ′ if a = c, b = d c −πtr(P P ) a πp | P |b | P |d e dP = (n,n) 0 if otherwise. C Proof. ([Fr, Hilfssatz 6.6]) If there exists ν ∈ a, ν ∈ / c, then the integral is zero, as can be seen by substituting the ν-th row by its negative (the same applies for the set of columns). In the R remaining case, use (3.1) and C e−πww dw = 1.  Lemma 3.3. Let A be an arbitrary complex matrix of size n, 0 ≤ h ≤ n Z X p! hn − q  ′ ′ ′ 1p ⊓ (A · A )[q] . (P + A)[h] (P + A )[h] e−πtr(P P ) dP = p p π p (n,n) C p+q=h

Proof. ([Fr, Hilfssatz 6.7] We consider each component of this matrix valued integral separately and we write these components in terms of minors of P and A and use the lemma above.  Lemma 3.4. Assume that Y is Hermitian, positive definite. Z Z ′ ′ e−πtr(P ·Y ·P ) dP = e−πtr(P ·Y ·P ) dP = det(Y )−n . C(n,n)

C(n,n) t

Proof. ([Fr, Hilfsatz 6.8]) Use U ∈ C(n,n) with U · U = Y to make a substitution with Jacobian det(Y )n (when viewed as transformation in a space over R) to reduce it to the case Y = 1n .  Proposition 3.5. Assume that Y is Hermitian, s ∈ C, ∂ [h] det(Y )s = Dh (s) det(Y )s Y ′[−h] with Dh (s) = s(s + 1) . . . (s + h − 1) =

Γ(s + h) . Γ(s)

Proof. ([Fr, Satz 6.9]) It suffices to prove it for infinitely many s. We first prove it for s = −n, using the lemma above: Z ′ [h] [h] −n h ∂ det(Y ) = (−π) (P · P ′ e−πtr(P Y P ) dP. t

The substitution P 7→ U −1 with Y = U U yields Z ′ h −n [−h] (P · P ′ )[h] e−πtr(P P ) (U ′ )[−h] dP. (−π) det(Y ) U The integral (or its transposed one) was computed in lemma 6.7. We get ∂ [h] det(Y )−n = (−1)h

n! [−h] U (U ′ )[−h] = Dh (−n) det(Y )−n (Y ′ )[−h] . (n − h)!

Using a product formula for ∂ [h] , we observe that if the proposition is true for s1 and s2 , then it is also true for s1 + s2 ; one needs here the combinatorial formula X h Dh (s1 + s2 ) = Dp (s1 )Dq (s2 ). p p+q=h



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The key property to be proved is the transformation property under Y 7−→ Y −1 : Theorem 3.6. For 0 ≤ h ≤ n we consider the operator matrix D := (Y ′ )[h] ∂ [h] and the correˆ defined by D(f ˆ ) := D(fˆ)(Y −1 ) with fˆ(Y ) := f (Y −1 ). Then sponding operator D, ˆ = (−1)h det(Y )h−1 D∗ det(Y )−h+1 D ′ with D∗ := Y [h] · ∂ ′[h] .

Remark 3.7. It is important to note that the operator D∗ , defined above by f 7−→ Y [h] · ∂ ′[h] f is different from the operator defined by f 7−→ ∂ [h] Y ′[h] f .

′

Proof. ([Fr, Satz 6.10]): The theorem should be viewed as an equality between operators acting on C ∞ -functions on PSymn . It is enough to prove it for polynomial functions (use Taylor’s formula). Using integration and differentiation with respect to T , we get all polynomial functions on PSymn from functions of type f (Y ) := det(Y )n eπtr(T ·Y ) with T Hermitian, positive definite. We prove the theorem for such “test functions” f by comparing ˆ )(Y ) L := D(f and R := (−1)h det(Y )h−1 Y [h] (∂ ′ )[h] det(Y )−h+1 f. Using a product formula for ∂ [h] we get R = (−1)h f (Y )Y [h]

X

p+q=h

πq

  h Dp (n − h + 1)Y [−p] ⊓ T [q] . p 1

To compute L, we do the substitution P 7−→ P + Y −1 T 2 in Lemma 3.4. This gives f (Y −1 )

= =

−1

det(Y )−n etr(T Y ) Z 1 ∗ 1 ∗ e−πtr(P Y P +T 2 P +P T 2 ) dP.

We apply ∂ [h] on both sides and obtain Z 1 ∗ ∗ 1 ∂ [h] fˆ(Y ) = (−π)h (P P ′ )[h] e−πtr(P Y P +T 2 P +P T 2 )) dP. 1

1

1

The substitution P 7−→ Y − 2 (P − Y − 2 T 2 ) yields after simplification in the exponential, Z [h] ∗ 1 1′ 1 −1 1 1 (−π)h f (Y −1 )(Y 2 )[h] (P − Y − 2 T 2 ) ((P − Y − 2 T 2 )′ )[h] e−πtr(P P ) dP · (Y − 2 )[h] . Therefore ∂ [h] fˆ(Y ) = (−π)h (Y

− 21 [h]

)

1′

· I · (Y − 2 )[h] ,

where (following Lemma 3.3) Z ∗ 1 1 1 1 I ′ = (P − Y − 2 T 2 )[h] ((P − Y − 2 T 2 )∗ )[h] e−πtr(P P ) dP =

X p! hn − q  1 1 1p ⊓ (Y − 2 · T · Y − 2 )[q] πp p p

p+q=h

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Then ∂

[h]

(fˆ)(Y −1 )

[q] 1 X p! hn − q  1 [h]  1 1 [h] ′2 ′ ′2 2 Y T Y Y ′2 = (−π) f (Y ) · 1 ⊓ Y p πp p p p+q=h X p! hn − q  [h] −[p] [q] h Y′ ·Y′ ⊓ T ′ (Y ′ )[h] . = (−π) f (Y ) p π p p h

p+q=h

After an easy identification of the elementary factors, we see that   (Y ′ )[−h] ∂ [h] fˆ (Y −1 ) = R′ .



We can reformulate the result above for ∂ [h] (fˆ) in the following: Corollary 3.8. −[h] ∂ [h] (fˆ) = (−1)h Y ′ det(Y )−h+1 {∂ [h] (f · det(Y )−h+1 )}(Y −1 ) · (Y ′ )−[h] .

These transformation properties also hold for functions defined on spaces of nondegenerate complex matrices (which should be invariant under the transformations Z −→ Z −1 or Z 7−→ −Z −1 ). Of course the ∂ij should now denote the complex derivatives for the (i, j)-coordinates of these matrices. The appropriate formulation for applications to Hermitian modular forms is (with an arbitrary integral weight k and applying Corollary 3.8 for G(Z) := F (Z) · det(Z)k−h+1 ): Proposition 3.9. For a holomorphic function on the Hermitian upper half-space we have   ∂ [h] (F (−Z −1 ) det(Z)−k ) = det(Z)−h+1 Z −[h] ∂ [h] (G) (−Z −1 ) · Z [−h]   X h = det(Z)−k Dα (k − h + 1)Z ′[−h] × (−1)α α α+β=h

(Z

′[−α]

⊓ (∂ [β] F )(−Z −1 ))Z ′−[h] .

There are a few weights k, where the above simplifies considerably because of the vanishing of almost all coefficients, thereby generalising the result of [Re] where the case h = n was treated. Recall that for an n × n matrix A, A[n] = det A. Putting k = n − 1 in Proposition 3.9 we get, Corollary 3.10. det(∂) (F |n−1 J) = (det(∂)F ) |n+1 J, i.e., one gets a simple formula in the scalar-valued case: the operator det(∂) changes the weight from n − 1 to n + 1. 4. Rankin-Cohen brackets for SU (n, n) and U (n, n) 4.1. Rankin-Cohen brackets for SU (n, n), changing weights by 2.

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Theorem 4.1. There exists a bilinear holomorphic differential operator [ , ] = [ , ]k1 ,k2 on Hn satisfying [F |k1 g, G |k2 g] = [F, G] |k1 +k2 +2 g given explictly by [F, G] =

X

a+b=n

with

for all g ∈ SU (n, n)

A(a, b) ∂ [a] F ⊓ ∂ [b] G

  n Da (k2 − n + 1) · Db (k1 − n + 1). A(a, b) = (−1) a a

Remark 4.2. There are several ways to prove the existence of such an operator. One might try to adopt the methods of Ibukiyama or Ibukiyama-Eholzer to the unitary situation. Another approach would be to use Maass non-holomorphic differential operators and holomorphic projection as in [B2]. An approach via Hermitian Jacobi forms is used by Kim [K] for degree 2 and arbitrary change of weights. Here we use a quite direct approach, which also gives us explict formulas; this approach only works for the simple case of changing weights by 2. Remark 4.3. In the normalization chosen above, the operator [ , ]k1 ,k2 is different from zero unless the conditions 0 ≤ k1 ≤ n − 1 or 0 ≤ k2 ≤ n − 1 are satisfied. In particular, it is nonzero, if k1 ≥ n or k2 ≥ n. Proof. We use the transformation formulas from the previous section and note that (as in the proof of Theorem 3.6) it is sufficient to prove the theorem for the special test functions F (Z) = exp(tr(T ′ Z)),

G(Z) = exp(tr(S ′ Z)).

We first check the automorphy property under the transformation J, defined by J < Z >:= −Z . With unknown coefficients A(a, b) we then need to have −1

[F |k1 J, G |k2 J] = X

a+b=n

X  a  b  Dα (k1 − a + 1)Dγ (k2 − b + 1) α γ α+β=a γ+δ=b     ′ ′ −1 Z ′[−a] (Z ′[α] ⊓ T [β] Z ′[−a] ⊓ Z ′[−b] (Z ′[γ] ⊓ S [δ] )Z ′[−b] e−(S +T )Z .

A(a, b) det(Z)−k1 −k2

X

This should then be equal to X ′ ′ −1 A(a, b)T [a] ⊓ S b] det(Z)−k1 −k2 −2 e−(S +T )Z . a+b=n

We first observe that     Z ′[−a] (Z ′[α] ⊓ T [β] Z ′[−a] ⊓ Z ′[−b] (Z ′[γ] ⊓ S [δ] )Z ′[−b]     = det(Z)−2 (Z ′[α] ⊓ T [β] ⊓ (Z ′[γ] ⊓ S [δ] ) = det(Z)−2 Z ′[α+γ] ⊓ T [β] ⊓ S [δ] .

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We do a resummation according to the value of α + γ = t with 0 ≤ t ≤ n, so we consider    X X X a b A(a, b) Dα (k1 − a + 1)Dγ (k2 − b + 1) det(Z)−2 Z ′[t] ⊓ T [β] ⊓ S [δ] . α γ t α+γ=t β+δ=n−t

Then the requested transformation property is satisfied if for fixed t and fixed β, δ we have (    X 0 if t > 0 a b A(a, b) Dα (k1 − a + 1)Dγ (k2 − b + 1) = α γ A(β, δ) if t = 0 α+γ=t The condition for t = 0 is automatically satisfied. The case t = 1 provides a recursion formula for A(a, b). The contributions are: • for α = 1 (with γ = 0, β + γ = n − 1, a = 1 + β, b = δ): • for α = 0 (with γ = 1, β + γ = n − 1, a = β, b = 1 + δ):

A(1 + β, δ)(1 + β)(k1 − β); A(β, δ + 1)(δ + 1)(k2 − δ).

We get a recurrence relation A(β + 1, δ) = − which gives

(δ + 1)(k2 − δ) · A(β, δ + 1) (β + 1)(k1 − β)

  n Da (k2 − n + 1) A(0, n). A(a, b) = (−1) a Da (k1 − a + 1) a

We multiply all this with Db (k1 − n + 1), using the identity Db (k1 − n + 1) · Da (k1 − a + 1) = Dn (k1 − n + 1) and we may choose a normalization now by putting   n Da (k2 − n + 1) · Db (k1 − n + 1). A(a, b) := (−1)a a With fixed t, β, δ satisfying t + β + γ = n, a = α + β, b = γ + δ we have to consider     X n a b α+β Da (k2 − n + 1)Db (k1 − n + 1)Dα (k1 − a + 1)Dγ (k2 − b + 1). (−1) α+β α β α+γ=t We use the elementary identities Da (k2 − n + 1) · Dγ (k2 − b + 1) = Dα+β+γ (k2 − n + 1) Db (k1 − n + 1) · Dα (k1 − a + 1) = Dα+β+γ (k1 − n + 1); these expressions are then constants for our sum and we are left with X (α + β)! (γ + δ)! n! (−1)α+β (α + β)!(γ + δ)! α!β! γ!δ! α+γ=t = (−1)β

n! X 1 . (−1)α β!δ! α+γ=t α!γ!

This is indeed equal to zero for t ≥ 1 and hence completes the proof.



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4.2. On equivariance for U (n, n) and vector-valued Rankin-Cohen brackets. The group U (n, n) is generated by J, the translations ! ! In t g∗ 0 {u(t) = | t ∈ Symn } and {ι(g) = | g ∈ GL(n, C)}, 0 In 0 g −1 with the subgroup SU (n, n) being generated by the same matrices, but the g ∈ GL(n, C) being subject to the condition det(g) = det(g ∗ ). The latter fact follows easily form the observation, that for g ∈ GL(n, C) the matrix ! 1/n  g∗ 0 1 det(g ∗ ) det(g −1) 0 g −1 is in SU (n, n). The transformation properties for the translations are evident and for the elements ι(g) they can be checked for the individual summands ∂ [a] F ⊓ ∂ [b] G: For general g (and the same test functions as before) we have to compare det(g)k1 +k2 +2 S [a] ⊓ T [b] and g T g ′ )[a] ⊓ (¯ g Sg ′ )[b] = det(g)k1 +k2 +1 det(¯ g )S [a] ⊓ T [b] . det(g)k1 +k2 (¯ This is fine as long as we stick to SU (n, n), but is troublesome for the bigger group U (n, n).

4.2.1. Twisted automorphy factor of U (n, n). To understand the transformation properties for the group U (n, n) (and to include vector-valued situations as well), we introduce a twisted automorphy ˜ ∈ U (n, n) factor: As one can check for the generators above, for each M ∈ U (n, n) there is a M such that ˜ < Z >′ ), M < Z ′ >= (M namely ^ ! 0 In = −In 0

0 −In

In 0

!

,

g = u(t′ ) = u(t¯), u(t)

g = ι(g). ι(g)

˜ is a group automorphism of U (n, n), explicitly described for arbitrary M by Then M 7−→ M taking complex conjugates of its entries. Moreover, ˜ , Z ′) J ∗ (M, Z) := J(M A B) ∈ defines a new automorphy factor. Here, as usual, we put J(M, Z) = CZ + D for M = ( C D U (n, n).

Then we see that (by checking the case ι(g) for g ∈ GL(n, C)) the correct transformation formula for the Rankin-Cohen brackets is [F |k1 M, G |k2 M ]k1 ,k2 (Z) = [F, G]k1 ,k2 (M < Z >) det J(M, Z)−k1 −k2 −1 det J ∗ (M, Z)−1 for all M ∈ U (n, n).

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4.2.2. Vector valued Rankin-Cohen brackets. There is a natural generalization of the RankinCohen operator from above: For 0 ≤ h ≤ n we define a bilinear operator [ , ]k1 ,k2 ,h mapping   pairs of functions on Hn to functions with values in the space of nh × nh matrices. We put [F, G]k1 ,k2 ,h =

X

a+b=h

with

Ah (a, b) ∂ [a] F ⊓ ∂ [b] G

  h Da (k2 − h + 1)Db (k1 − h + 1). Ah (a, b) := (−1) a a

One shows (in the same way as above for the special case h = n) that this operator satisfies [F |k1 M, G |k2 M ]k1 ,k2 ,h = [F, G]k1 ,k2 ,h |ρ M

∀M ∈ U (n, n),

where the slash operator on the right hand side is defined for matrix-valued functions H : Hn −→ M(n) (C) by h

(H |ρ M )(Z) = det J(M, Z)−k1 −k2 +1 J ∗ (M, Z)[−h] H(g < Z >)(J(M, Z)′ )[−h] A B ) ∈ U (n, n). for all M = ( C D

In particular, these operators map pairs of scalar-valued Hermitian modular forms to vectorvalued modular forms for a “mixed automorphy factor” involving J and J ∗ . The proof goes along exactly the same lines as in the scalar case and will be omitted. Remark 4.4. The same kind of calculation as above also works for the Siegel case, where the scalar-valued situation was treated by Eholzer and Ibukiyama in a different way [EI], relying on the results of [I] and on properties of pluriharmonic polynomials. The problem with twisted automorphy factors does not come up in the Siegel case. We can then get in this way vector-valued modular forms for automorphy factors detk1 +k2 ⊗τh , where τh is the irreducible representation of GL(n, C) of weight (2, 2, . . . , 2, 0, . . . 0). We omit details. It seems that in the literature one can find only the cases where a weight change is done by Symr ([EI]) or tensors of it with powers of determinants. (See [EI, Mi].) 5. Differential operator with invariance property for Sp(n) Theorem 5.1. For even n there is a differential operator Dk on Hn satisfying Dk (F |k g) = (Dk F ) |k+1 g,

∀ g ∈ Sp(n, R).

This operator is a polynomial in the ∂ij with polynomials in the entries of Z − Z ′ as nonconstant coefficients. When viewed as a polynomial in ∂ij and the zij −zji , it has coefficients in Z, depending polynomially on the weight k. The operator is nonzero for all weights k ≥ n. Remark 5.2. (1) The statement above is then automatically true for all weights s ∈ C, the ambiguity of the branch of log does not matter, because it appears on both sides. (2) The properties are similar to those of differential operators compatible with the embedding Sp(n, R) × Sp(n, R) ֒→ Sp(2n, R), see [B1]. (3) Note that such operators do not exist for n odd, as can be seen by looking at the action of g = −12n ∈ Sp(n, R).

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

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(4) The advantage of these operators is that we can iterate them: For ν ∈ N we put Dkν := Dk+ν−1 ◦ . . . Dk+1 ◦ Dk . Proof. First we introduce a function on Hn , which behaves nicely under the action of Sp(n, R). For Z ∈ Hn , the matrix Z − Z t is alternating, we may therefore consider its Pfaffian. We define on Hn the function ℓ

φℓ : Z 7−→ (Pf(Z − Z ′ )) .

(5.1) Using the observation

Pf(−Z −1 + Z ′−1 ) = det(−Z) Pf(Z − Z ′ ).  n for weight −l. Keeping in mind that we see that φl behaves well under the action of J = 01nn −1 0n J and the real translations generate Sp(n, R), we get the crucial property φl |−l g = φl

(5.2)

∀g ∈ Sp(n, R).

We use the Rankin-Cohen bracket to define D by Dk (F ) := [F, φ1 ]k,−1 . Clearly, this operator has the requested transformation properties. The assertion about the coefficients of the differential operator is equally clear. Furthermore, Proposition 8.4 ensures the nonvanishing of Dkν for k ≥ 0.  Remark 5.3.

(1) We can write Dk as Dk (F ) =

X

a+b=n, b≤ n 2

    A(a, b) ∂ [a] F ⊓ ∂ [b] φ1 .

Note that the condition b ≤ n2 in the summation comes from the simple fact that ∂ [b] (φ1 ) = 0 for b > n2 . (2) The operator Dkν consists of monomials of degree r in the derivatives ∂ij and degree s in the entries of Z − Z t with r − s = n2 · ν (with the entries of Z − Z t written in front of the derivatives!). For ν = 1 this can be seen directly from the definition; this property then carries over to the iterated operators. (3) There is also a vector-valued analogue of this operator by using the Rankin-Cohen bracket [∗, φ1 ]k,−1,h

(0 ≤ h ≤ n)

with equivariance properties for Sp(n, R). The iteration of this operator is more delicate (it needs Rankin-Cohen brackets starting from vector valued cases) and therefore we do not go into the details here. We will describe a quite explicit form of Dk in Section 8. But first, we would like to achieve one of our goals of this paper, viz., to produce holomorphic differential operators from [Γn (OK ), k] to [Γn , k + ν] for any ν ≥ 0. This is done in the next section.

ON HOLOMORPHIC DIFFERENTIAL OPERATORS

13

6. Restriction to Siegel upper half space

In this section n is always even. We now define a differential operator (via restriction of variables): Dkν,0 F := Dkν F|Z=Z ′ . This maps holomorphic functions F on Hn to holomorphic functions on the Siegel upper half-space HnS ⊂ Hn and it satisfies   Dkν,0 (F |k g) = Dkν,0 F |k+ν g (∀g ∈ Sp(n, R)). For ν = 0 this is just the restriction map from Hn to HnS ; also it is clear, that Dkν,0 maps Hermitian modular forms of weight k to Siegel modular forms of weight k + ν. We note here that n ≥ 2, so the boundedness at infinity is guaranteed by Koecher’s principle. Remark 6.1. By representation-theoretic considerations about pluriharmonic polynomials defined by means of the Dkν,0 , we will later show that these operators are unique up to scalars. Proposition 6.2. For ν > 0, the operators Dkν,0 map Hermitian modular forms to Siegel cusp forms. Proof. Let Dkν,0 F = f . Due to the compatiblity of Dkν,0 with the action of





gt 0 0 g −1 = t01 00r

for g ∈  do not GL(n, Q) it is sufficient to show that nonzero Fourier coefficients af (T ) with T occur unless r is zero (i.e. t1 is positive-definite with rank n). Suppose that we have a Fourier coefficient af (T ) with r > 0. This can only occur if F itself has a non-zero Fourier coefficient aF (T) such that Re(T) = T . Then the positive semidefinite matrix T must itself be of the form ! t 0 (6.1) T= 0 0 with t being positive Hermitian of rank n − r < n. It is now sufficient to show that for all such Hermitian matrices Dkν,0 FT = 0,

with FT (Z) = exp(trT · Z).

We observe that Dkν,0 FT = P (T) exp(trT · Z) with a function P in the entries of T satisfying P (g t Tg) = det(g)ν P (T) for all g ∈ GL(n, R). The vanishing of FT follows from acting by g(λ) = diag(1, . . . , 1, λ) ∈ GL(n, R) on P (T) if T is of the form (6.1) and ν ≥ 1.  7. Explicit Combinatorics 7.1. The explicit form of Dkν,0 for n = 2. From now on, we assume that n = 2. In this section, we take an alternative and more direct approach for the construction of holomorphic differential operators on holomorphic functions on H2 having an automorphy under Sp(2, R) before restriction to the Siegel half-space. Hence they can be iterated, serving our purpose of mapping Hermitian

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

14

modular forms to Siegel modular forms, thereby increasing the weight by ν. First we recall some notation and preliminaries. τ z Let F be a holomorphic function on H2 . Writing Z = ( w τ ′ ) for the variable on H2 , for convenience, let us define the operators ! ∂ ∂ ∂ ∂ ∂ τ ∂ z − and D := det ∂ . ∆ := ∂ ∂z ∂w ∂w ∂ τ′

We find in this case that φt = (z − w)t (cf. (5.1)). Recall from Corollary 3.10 that the determinant operator D has the following automorphy property : D (F |1 γ) = D(F ) |3 γ, for all γ ∈ SU (2, 2). Hence using the automorphy property of φt under the embedded Sp(2, R) inside SU (2, 2) from (5.2), we can construct a differential operator (which is however not holomorphic) as: Definition 7.1. For a function F : H2 → C,

 (Dk′ F ) (Z) := (z − w)−(k−2) D F · (z − w)k−1   ∂F ∂F + (k − 1)(k − 2)(z − w)−1 F. − = (z − w) D(F ) + (k − 1) ∂z ∂w

For a holomorphic function F : H2 → C we therefore have the following: (7.1)

Dk′ (F |k γ) = Dk′ F |k+1 γ, for all γ ∈ Sp(2, R).

Again using the fact that φ1 has an automorphy under Sp(2, R), we can define a holomorphic differential operator in the following way: ˜ k := Dk′ − (k − 1)(k − 2)(z − w)−1 , D which has the required automorphy as in (7.1) and increases the weight by 1. After iterating ν times we have: ˜ ν := D ˜ k+ν−1 ◦ · · · D ˜k ◦ D ˜k , D ˜ 0 = Id.. D k k

˜ ν,0 = D ˜ ν |z=w , we have obtained the required differential operator intrinsically, Therefore defining D k k i.e., without the use of Rankin–Cohen brackets. ˜ ν,0 is a polynomial in D and ∆; but the combinatorics It can be proved by induction that D k evolving in the quest for an explicit formula seem to be difficult. In the next section, we will use the Rankin–Cohen bracket to define the differential operator Dkν,0 having the requested properties; ˜ ν,0 by Proposition 9.2. which must be a scalar multiple of D k Here we list the first few iteration of the differential operators constructed as above: Example 7.2. ˜ k1 = (z − w)D2 + (k − 1)∆, D (7.2)

˜ k2 = (z − w)2 D22 + 2k(z − w)∆D2 + 2kD2 + k(k − 1)∆2 , D

˜ 3 = (z − w)3 D3 + 3(k + 1)(z − w)2 ∆D2 + 3k(k + 1)(z − w)∆2 D2 D k 2 2 + 6(k + 1)(z − w)D22 + 6k(k + 1)∆D2 + (k − 1)k(k + 1)∆3 .

ON HOLOMORPHIC DIFFERENTIAL OPERATORS

15

7.2. Holomorphic operator via the Rankin–Cohen bracket. In this section, we elaborate and explicitly compute the combinatorial description of the differential operators mapping holomorphic functions on H2 to those on H2S (after restriction of variables) having an automorphy under SU (2, 2) (before restriction) thereby increasing the weight by ν, using the ν-th Rankin– Cohen bracket. An explicit formula for the ν-th Rankin–Cohen bracket for Hermitian modular group of degree 2 has been obtained by H. Kim in [K]. However his arguments actually hold for SU (2, 2), and we state the more general result as the next theorem. Theorem 7.3. For any two holomorphic functions F, G on H2 and nonnegative integers k, k ′ , ν, define X (−1)p (k + k ′ + 2ν − p − 3)! (7.3) Dp (Dr (F )Ds (G)) . [F, G]ν = ′ + s − 2)! r!s!p!(k + r − 2)!(k r+s+p=ν Then

[F, G]ν |k+k′ +2ν γ = [F |k γ, G |k′ γ]ν , ∀ γ ∈ SU (n, n).

(7.4)

We normalize the above definition to suit our needs, and define Dkν := (−1)ν (k + ν − 3)!(k ′ + ν − 1)! [F, (z − w)ν ]ν .

(7.5)

By (7.4) and the automorphy property of φν = (z−w)ν , we see that Dkν has the desired automorphy property under the group Sp(2, R). From the previous subsection, we know that the desired operator is Dkν,0 := Dkν |z=w . Looking at (7.3) and (7.5), we see that for any m ≥ 0, therefore we need to compute the coefficients qαp : Dp ((z − w)m X) =

(7.6) qαp (D, δ)

m X

α=0

qαp (D, δ)(X) · (z − w)α ,

are polynomials in D and ∆ and X is a holomorphic function on H2 . In the above where and henceforth we will continue to write the differential operators on the left of (z − w)n for convenience and suppress X; but keep in mind that it has no effect on (z − w)n . We denote by qαp := qαp (x, y) the corresponding polynomial in the variables x and y. We wish to compute q0p . To this end, we will compute the generating function for the qαp ’s for each α. We p = xp and that qαp = 0 for all α > m. Further, note that qα0 = δα,m . The following note that qm recurrence relation holds: p p qαp+1 = xqαp + (α + 1)yqα+1 + (α + 1)(α + 2)qα+2 , (α ≥ 0).

Let Zα be the generating function for the functions qαp (p = 0, 1, 2 · · · ): Zα =

∞ X

qαp W p .

p=0

Clearly, Zm

 = 1 (1 − W x), Zα = 0 for α > m and the following recurrence relation holds:

(1 − W x)Zα = δα,m + (α + 1)yW Zα+1 + (α + 1)(α + 2)W Zα+2 , (α ≥ 0).

It is only possible to compute Zα backwards, i.e., starting from m. So we define the polynomials Qα := Qα (W ) ∈ C[x, y][W ] by the formula (which is readily checked by induction): Zm−α =

m(α) Qα , m(α) = m(m − 1) · · · (m − α + 1). (1 − W x)α+1

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For convenience of notation, let θ := 1 − W x. We note that Qα = 0 for α > m, and the Qα ’s obey the following recursive rule: Qα = W (yQα−1 + θQα−2 ), (α ≥ 1); Q0 = 1, Q1 = W y. We now compute the generating function for the Qα ’s. Define Ω=

∞ X

Qα T α .

α=0

Using the fact that Q0 = 1, we find that ∞ X 1 W s T s (y + θT )s = 1 − W T (y + θT ) s=0 s   ∞ X X s W s θt y s−t T s+t = t s=0 t=0 ∞ X  X s t s−t s+t = θy T t t=0 s≥t  ∞  ∞ X X t + u t u+t u u+2t θW y T ; = t t=0 u=0

Ω=

after putting s = u + t. This clearly gives Qα for all α. However, we will only need Qm , as we will put z = w in the resulting operator. Hence X t + u  Qm = θt y u W u+t . t u+2t=m We have Z0 = m! θ

−(m+1)

Qm

X t + um − t + v  xv y u W t+u+v . = t v v=0 u+2t=m ∞ X

Hence we have the following formula for q0p : X t + um − t + v  p xv y u . q0 = m! t v u+2t=m t+u+v=p

From the formula of q0p above, we see that q0p = 0 unless p ≥ m/2 and we can rewrite:    X p−v p v u q0p = m! x y . u v u+2v=2p−m From (7.5), we get after simplication and restriction: X Dkν,0 F = cν (α, β)Dα ∆β |z=w ; 2α+β=ν

where cν (α, β) =

α α X X (−1)r (k + r + s − 3)! ν! (ν − s + 1)! r=0 s=0

r! s! (α − r)! (α − s)! β! (k + r − 2)!

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17

7.3. Preservation of the Maaß space. In this subsection we prove that the differential operators Dkν,0 map the Hermitian Maaß space to the Siegel Maaß space of increased weight. The case ν = 0, i.e., where we just restrict Hermitian modular forms to Siegel modular forms, has been proved by A. Krieg in [Kr2, Kr3]. Let us define the Hermitian and Siegel Maaß spaces of weight k and degree 2. Let L denote r either Q or an imaginary quadratic field K of discriminant −DL . For T = ( m r¯ n ) ∈ PSym(2, OL ) define  ǫ(T ) := max q ∈ N | q −1 T ∈ PSym(2, OL ) . The Maaß space Mk (L) consists of all modular forms F in [Γ2 (OL ), k] with the Fourier expansion (2.1) such that a function c∗F : N → C exists satisfying X cF (T ) = dk−1 c∗F (DL det T /d2 ). d|ǫ(T )

We note that these spaces are nonzero only when k is even. Our next theorem shows that the differential operators are quite natural, in the sense that they preserve the Maaß spaces. Theorem 7.4. Let k ≡ 0 (mod wK ). Then, Dkν,0 (Mk (K)) ⊆ Mk+ν (Q). Proof. To prove the theorem, we use the fact that Dkν,0 is a polynomial in ∆ and D. Let G ∈ Mk (Γ2 ) be the image of F ∈ Mk (Γ2 (O)) under Dkν,0 . Further let us denote by bG (S) (resp. aF (T )), the Fourier coefficients of G (resp. F ). Recall that for a matrix A, φ1 (A) denotes the Pffafian of A − A′ . Then one obtains X X bG (S) = (2πi)ν cν (α, β) (7.7) (detT )α φβ1 (T )aF (T ). 2α+β=ν

T ∈PSym(2,OK ) T ≥0,Re(T )=S

We define b∗G (n) = 0, if n ≡ 1, 2 (mod 4) and for s = 0, 1 X X (7.8) b∗G (4n − s) = (2πi)ν (n − N (t))α φβ1 cν (α, β) 2α+β=ν

# t∈OK ,Re(t)=s N (t)≤n

Then it follows that for S ∈ PSym(2, Z), X b∗G (4 det S) = cν (α, β) 2α+β=ν

X

0 t t¯ 0



a∗F (DK (n − N (t))).

(detT )α φβ1 (T )a∗F (DK detT ).

T ∈PSym(2,OK ) T ≥0,Re(T )=S

The rest of the proof is then completed from the expression (7.7) of bG (S) and the definition (7.8) b∗G of along the lines of [Kr2, Kr3].  × Remark 7.5. We note that Mk (K) = {0} unless k ≡ 0 (mod wK ), where wK = #OK . In ν,0 particular, we must have that Dk (Mk (K)) = {0} unless k ≡ ν (mod 2). But this is indeed the case, as is evident from (7.7) noting that in the sum above, β is odd if k 6≡ ν (mod 2).

7.4. An application : Holomorphic operators on Hermitian Jacobi forms. For holomorphic functions φ = φ(τ, z, w) defined on H × C2 , we can consider the function φ∗ : H2 → C defined τ z ′ by φ∗ (( w τ ′ )) = φ(τ, z, w)e(mτ ). Then one can check easily (see [H] for example) that φ∗ |k M (Z) = φ∗ (Z) for all M ∈ Γ2,1 ⊂ Sp(2, R),

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where Γ2,1 is the group generated by the image of the group SL(2, R) ⋉ R2 inside Sp(2, R). See [EZ, H] for the details. Then using the automorphy property of our differential operator Dkν (see (7.5)) applied to φ∗ J,ν we can deduce the corresponding properties of the differential operator Dk,m defined via (7.9)

J,ν Dkν φ∗ = Dk,m φ e(mτ ′ ).

J,ν Namely, Dk,m then has the following automorphy property : J,ν J,ν Dk,m (φ |k,m γ) = (Dk,m φ) |k+ν,m γ,

(γ ∈ Γ2,1 )

where |k,m denotes the action of Γ2,1 on H×C2 which is induced from the action of U (1, 1)⋉C2 on H × C2 ; see [H] for the details. Now, we can restrict the variables z = w and consider equivariance for the Jacobi group SL(2, Z) ⋉ Z2 ⊂ Γ2,1 . Thus, if Φ(τ, z, w) ∈ Jk,m (OK ), the space of Hermitian Jacobi forms of weight k and index m for the Hermitian Jacobi group defined over the ring of integers OK of an imaginary quadratic field K (see [H]), the above procedure produces a classical J,ν Jacobi form Dk,m Φ |z=w ∈ Jk+ν,m of increased weight. J,ν Example 7.6. We find the first few operators Dk,m after a normalisation. For this, we just use the definition (7.9) and the formulas in Example 7.2. Let ϕ = ϕ(τ, z, w) ∈ Jk,m (OK ),

(7.10)

J,1 Dk,m ϕ = ∆ϕ.    2  ∂2 ∂2 ∂ ∂ J,2 ϕ. − 2k + (k − 1) + Dk,m ϕ = 4πim ∂τ ∂z 2 ∂w2 ∂z ∂w     2    ∂ ∂ ∂2 ∂2 J,3 Dk,m ϕ = 12πim ∆ + (k − 1) ϕ. − 3(k + 1) ∆ − ∂τ ∂z 3 ∂w3 ∂z ∂w

8. The explicit form of Dk For convenience, we switch in this section from n to 2n (to avoid the condition “n even”). 8.1. Differential operators on spaces of skew-symmetric matrices. In this subsection we consider differential operators acting on functions defined on the space Skew2n (C) of skewsymmetric matrices V = (vi,j ) of size 2n. As before, we collect the partial differential operators ∂ ∂vi,j in a (skew-symmetric) matrix ∂V ; as before, we can define minors.  the set of all subsets of 2N with r We put 2N := {1, 2, . . . , 2n} and we denote by 2N r [r] elements. The aim of this subsection is to determine ∂V Pf(V ) explicitly for 0 ≤ r ≤ n (for r > n  it is obviously equal to zero). For that purpose, we choose a, b ∈ 2N and study | ∂V |ab . r  Lemma 8.1. Let r ≤ n and a, b ∈ 2N with a = {a1 , . . . , ar }, b = {b1 , . . . , br }. Then r | ∂ |ab Pf(V ) = 0

if a ∩ b 6= ∅.

Proof. First we recall an expansion principle for Pfaffians. We denote by Π2n the set of all partions of 2N into non-ordered pairs. We write an element α of Π2n in the form α = {(i1 , j1 ), . . . , (in , jn )}

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with i1 <2 < · · · < in and im < jm for all 1 ≤ m ≤ n. We attach to such an α ∈ Π2n a permutation ! 1 2 3 4 . . . 2n πα := i1 j 1 i2 j 2 . . . j n and for a skew-symmetric matrix V = (vi,j ), we put Aα := sign(πα )vi1 ,j1 . . . vin ,jn . Then we have the following expression of Pf(V ) in terms of monomials: X Aα . Pf(V ) = α∈Π2n

From this formula and using Leibnitz’s formula for the determinant | ∂V |ab , we see that we have to consider, for σ ∈ Sr and α ∈ Π2n an expression of type (8.1)

∂a1 ,bσ(1) . . . ∂ar ,bσ(r) (vi1 ,j1 . . . vin ,jn )

Assume that there are indices s, t ∈ {1, . . . , r} with as = bt . The expression (8.1) can be nonzero only if the (possibly equal) pairs (as , bσ(s) ) and (aσ−1 (t) , bt ) appear among the pairs (im , jm ). This means as is equal to some ik and bt is equal to some jℓ ; this is impossible in view of the assumption a s = bt .  We first treat a special case of a ∩ b = ∅. The general case follows. Lemma 8.2. For r ≤ n we put a = {1, . . . , r} b = {r + 1, . . . , 2r}. Then   2N \a∪b | ∂V |ab Pf(V ) = (−1)r(r−1)/2 r! Pf V2N \a∪b . Here, for any subset c of 2N we denote by Vcc the submatrix of V , whose rows and columns are indicated by the set c. Proof. Using the same notation as in the previous lemma, we see that a typical summand of | ∂V |ab is of the form sign(σ)sign(πα ) ∂1,σ(r+1) , . . . , ∂rσ(2r) (vi1 ,j1 , . . . , vin ,jn ), where we view σ ∈ Sr as group of permutations of {r + 1, . . . , 2r}. A nonzero contribution occurs only if the following conditions are satisfied: ik = k for 1 ≤ k ≤ r

and

jk = σ(k + r)

for

1 ≤ k ≤ r.

Therefore πα must be of the special form πα =

1 1

2 σ(r + 1)

... ...

2r − 1 2r r σ(2r)

2r + 1 2r + 2 . . . i2r+1 j2r+1 . . .

2n j2n

!

.

We observe that sign(πα ) = (−1)r(r−1)/2 sign(σ) · sign(πβ ),

where β = {(i2r+1 , j2r+1 ), . . . , (i2n , j2n )} denotes a partition of 2N \ (a ∪ b) into unordered pairs. We get then for these special sets a and b

  2N \(a∪b) | ∂V |ab Pf(V ) = (−1)r(r−1)/2 r! Pf V2N \(a∪b) .

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Note that the factor r! comes from summing up over σ ∈ Sr .



The general case is then obtained by applying appropriate permutations of rows and columns; which we record as a proposition: Proposition 8.3. For a, b ∈

2N r



with r ≤ n we have

  2N \(a∪b) | ∂V |ab Pf(V ) = (−1)r(r−1)/2 r! ǫ(a, 2N \ a)ǫ(b, 2N \ b) Pf V2N \(a∪b) ,

where supposing that a = {a1 < a2 < · · · < ar }, 2N \ a = {c1 < c2 < · · · < c2n−r }, ǫ(a, 2N \ a) denotes the sign of the permutation which transfers the tuple (a1 , a2 , . . . ar , c1 , . . . c2n−r ) into the naturally ordered 2n-tuple. 8.2. A simple expression for Dk . Now we switch back to the Hermitian half-space H2n . Clearly the formulas above imply the same type of formula for | ∂ |ab Pf(Z − Z ′ ). Here ∂ is the 2n × 2n matrix of differentials ( ∂z∂i,j ), Z ∈ H2n . The formula for ∂ [α] F ⊓ ∂ [β] Pf(Z − Z ′ ) with α + β = 2n simplifies considerably due to the results in the previous section. By definition, it equals  −1 X ′ ′′ 2n ǫ(a′ , a′′ )ǫ(b′ , b′′ )(| ∂ |ab′ F )· | ∂ |ab′′ Pf(Z − Z ′ ). β ′ ′′ ′ ′′ a ∪a =2N,b ∪b =2N

′′

′′

Only the cases a ∩ b = ∅ will contribute. We end up with ∆(β)F := X

a′′ ∩b′′ =∅

 −1   β(β−1) 2n 2N \a′′ 2N \(a′′ ∪b′′ ) ǫ(2N \ a′′ , a′′ )ǫ(2N \ b′′ , b′′ )(−1) 2 β! · Pf (Z − Z ′ )2N \(a′′ ∪b′′ ) (| ∂ |2N \b′′ F ) β

where a′′ and b′′ run over disjoint subsets of 2N with β many elements. Then we get Dk =

X

A(α, β)∆(β).

α+β=2n,β≤n

8.3. On the nonvanishing of Dkν,0 . If we apply the differential operator ∆(β) from the previous section to a function f which depends only on z2 , where ! ∗ z2 Z= ∗ ∗ is the decomposition of Z in n by n block matrices. Then only 2n−β ≥ n can produce a contribution different from zero, i.e. only β = n has to be considered. This implies {1,...,n}

Dk f ∼ Dn (−n)Dn (k − n + 1)∂{n+1,...,2n} . In particular, if we use fs (Z) := det(z2 )s , we get Dk fs ∼ Dn (−n)Dn (k − n + 1)Dn (s)fs−1 .

ON HOLOMORPHIC DIFFERENTIAL OPERATORS

21

We used the symbol ∼ to indicate “equality up to elementary nonzero factors”. The formula above can be iterated: We get Dkν,0 fν = Dn (−n)ν Dn (ν) . . . Dn (1)Dn (k − n + 1) . . . Dn (k − n + ν). Proposition 8.4. The differential operator Dkν,0 is different from zero as long as Dn (ν) . . . Dn (1)Dn (k − n + 1) . . . Dn (k − n + ν) is different from zero; this holds in particular for k ≥ n. (h),0

8.4. On the vector-valued operators Dk (h),0 Dk (F )

. We can of course define the operators

:= [F, φ1 ]k,−1,h |Z=Z ′

(0 ≤ h ≤ 2n)

S . In the sum over α and β which map functions on H2n to M(2n) , C)- valued functions on H2n h defining the operator [∗, φ1 ]k,−1,h , only the summand with ∂ [n] Pf(Z − Z ′ ) does not vanish under restriction to the Siegel upper half-space, in particular, this operator is zero, if h < n. For h ≥ n it is a multiple of ∂ [h−n] ⊓ ∂ [n] Pf(Z − Z ′ ).

Note that in this case ∂ [n] Pf(Z − Z ′ ) is a constant matrix, whose value was determined explictly in the preceeding sections.

We pick out the particular case h = n = 1 to explain some amusing feature of these operators: (1),0 We get in this case that Dk (F ) is a multiple of ! 0 1 F· . −1 0 The transformation law (h),0

Dk

(h),0

(F |k M ) = det(CZ + D)−k+1 (CZ + D)−1 Dk

holds (as it should), because of ! 0 1 0 A A′ = det(A) −1 0 −1

! 1 0

(F )(CZ + D)′−1

(A ∈ GL(2, C)).

A similar computation can be made for arbitrary n, using an expansion of determinants of 2n– rowed matrices in terms of it’s minors of order n. The reason for this is that the representation of GL(2n, C) on M(2n) (C) given by (A, X) 7→ n A · X · A′[n] is not irreducible, in particular, it contains the representation det if h = n. [n]

9. On harmonic polynomials and theta series 9.1. Differential operators and harmonic polynomials with invariance under U (k, C). Our differential operators are closely related to Harmonic polynomials. It should be possible to exhibit this relation along the lines of Ibukiyama’s work [I]. Here we follow [B1]. For X, Y ∈ Rk,n we define a polynomial P = Pk,ν of variables in R2k,n by ! ! !   X X ν,0 Dk f (Z) = Pk,ν ( ) exp tr(12k [ ] · Z) (Z ∈ HnS ) Y Y

22

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

with f being defined on Hn by f (Z) = exp (tr(1k {X + iY } · Z)) . Proposition 9.1. Pk,ν is a pluriharmonic polynomial satisfying Pk,ν (XA) = det(A)ν Pk,ν (X)

(A ∈ GL(n, R), X ∈ R2k,n ).

Proof. One can use the same method of proof as in [B1, Satz 15] where the corresponding statement was given for the Siegel case: One shows that the polynomial P equals its Gauss transform. To do this, one uses theta functions and computes the Gauss transform in two ways, using the reciprocity law for the theta function. Then one uses the characterisation of pluriharmonic forms by their Gauss transform [Fr]. We omit details and refer to [B1, Satz 15].  In the situation of [B1], the harmonic polynomials were generalized Gegenbauer polynomials, carrying an invariance property for the action of an orthogonal group. Here we get an invariance property with respect to the unitary group U (k, C): We put z := X + iY ∈ Ck,n . Then in fact, the polynomial Pk,ν (X) depends only on the entries of the Hermitian Gram matrix associated to z. When we write Pk,ν as a polynomial Qk,ν (z), this means that (9.1)

Qk,ν (U · z) = Qk,ν (z)

∀ U ∈ U (k, C).

Now we recall that the space Hν,C (2k, n) of harmonic polynomials on C2k,n satisfying P (XA) = det(A)ν P (X) is an irreducible space for the action of the orthogonal group O(2k, C), see [KV]. We may view U (k, C) as a subgroup of O(2k, R) using the identification of z ∈ Ck,n with X ∈ R2k,n as above. Then (9.1) above says (after passing from U (k, C) to its Zariski-closure GL(k, C)): Proposition 9.2. Up to scalars, the polynomial Qk,ν is the unique non-zero element of Hν,C (2k, n) which is fixed by the action of GL(k, C) ֒→ O(2k, C). In particular, the differential operators Dkν,0 are unique up to scalars. Proof. By Representation theory, this space of fixed vectors is one-dimensional, see e.g. [GW, section 12.3.3.]. In the context of compact real Lie groups, one could also use the theorem of Cartan-Helgason [He, Ch.V,§4].  Our work provides a method to write down these operators completely explicitly in the case n = 2, ν ≥ 0. For n > 2 the combinatorics of the coefficients of the differential operators is not as explicit as desired, but we get them after iteration of an explicitly given differential operator with nonconstant coefficients.

9.2. Hermitian theta series and Siegel theta series with harmonic polynomials. We describe here the precise action of our differential operators on Hermitian theta series, generalizing well known statements about restriction of Hermitian theta series. We use the setting of lattices (follow closely the exposition in [SSS]): We fix an imaginary quadratic number filed K and consider a free OK -module of OK -rank m, together with a (positive definite) Hermitian form h : M ×M −→ OK . At the same time we can view M as a Z-module of rank 2m together with the bilinear form

ON HOLOMORPHIC DIFFERENTIAL OPERATORS

23

b : M × M −→ Z, defined by b(x, y) := trK/Q h(x, y). The Hermitian theta series of degree n is defined by X exp(πi tr(h(x) · Z) (Z ∈ Hn ), θn (M, h, Z) := x∈M n

where h(x) denotes the matrix h ((xi , xj )) for x := (x1 , . . . , xn ) ∈ M n . For the automorphy properties of such theta series we refer to [SSS]. We apply now our differential operators to this series:   X Qk,ν (h(x)) exp(πi tr(b(x) · Z) (Z ∈ HSn ), Dkν,0 θn (M, h, ∗) (Z) = x∈M n

in other words, the application of the differential operator maps Hermitian theta series to Siegel theta series with the (essentially unique) harmonic polynomials Qk,ν .

References [B1]

¨ S. B¨ ocherer: Uber die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen II. Math. Z. 189, 81-

[B2]

110(1985) S. B¨ ocherer: Bilinear differential operators for the Jacobi group. Comment. Math. Univ. St. Pauli 47, 135154(1998)

[B3] [CE]

S. B¨ ocherer: Holomorphic differential operators with iteration. Proceedings of the 7-th Number Theory workshop, Hakuba, 2004. Y. Choie, W. Eholzer: Rankin-Cohen Operators for Jacobi and Siegel forms. J. Number Theory Vol. 68

[Co]

(1998), 160–177. H. Cohen: Sums involving the Values at negative Integers of L-Functions of Quadratic Characters. Math.

[EI] [EZ] [Fr]

Ann. 217, (1975), 271–285. W. Eholzer, T. Ibukiyama: Rankin-Cohen type differential operators for Siegel modular forms. International Journal of Mathematics Vol. 9, (1998), No. 4, 443–463. M. Eichler and D. Zagier: The Theory of Jacobi Forms. Progress in Mathematics, Vol. 55. Boston-BaselStuttgart: Birkh¨ auser, 1985. E. Freitag: Siegelsche Modulfunktionen. Grundl. Math. Wiss. 254, Springer–Verlag (1983).

[GW] R. Goodman, N. Wallach: Symmetry, Representations and Invariants. Grauate Texts in Mathematics 255. Springer 2009. [H] K. Haverkamp: Hermitsche Jacobiformen. Schriftenreihe des Mathematischen Instituts der Universitt Mn[He] [I]

ster. 3. Serie, Vol.15, 105 pp., 1995. S. Helgason: Groups and Geometric Analysis. Academic Press 1984. T. Ibukiyama: On differential operators on automorphic forms and invariant pluriharmonic polynomials.

Comment. Math. Univ. St. Pauli 48, (1999), 103–118. [K] H. Kim: Differential operators on Hermitian Jacobi forms. Arch. Math. (Basel) 79, (2002), 208–215. [Kr1] A. Krieg: Modular Forms on Half-Spaces of Quaternions. Lecture Notes in Mathematics 1143, Springer– Verlag, 1985. [Kr2] A. Krieg: The Maaß- Space on the Half-Space of Quaternions of Degree 2. Math. Ann. 276, (1987), 675–686. [Kr3] A. Krieg: The Maaß spaces on the Hermitian half-space of degree 2. Math. Ann. 289, (1991), 663–681. [KV] [Ma]

M. Kashiwara, M. Vergne: On the Segal-Shale-Weil representation and harmonic polynomials. Invent. Math. 44, (1978), 1–47. H. Maaß: Siegel’s Modular Forms and Dirichlet Series. Lecture notes in Mathematics, 216 Springer-Verlag

[Mi]

(1971). M. Miyawaki: Explict Construction of Rankin-Cohen-Type Differential Operators for vector-valued Siegel

[Mu] [Re]

Modular Forms. Kyushu J. Math. 55, (2001), 369–385. T. Muir: A Treatise on the Theory of Determinants. Macmillan and Co., (1882). H. L. Resnikoff: On a Class of Linear Differential Equations for Automorphic Forms in Several Complex

[Sh]

Variables. Amer. J. Math. 95 (1973), No. 2, 321–332. G. Shimura: Arithmetic of Differential operators on symmetric domains. Duke Math. J. 48, (1981), 813–843.

¨ SIEGFRIED BOCHERER AND SOUMYA DAS

24

[SSS] R.Scharlau, A. Schiemann, R.Schulze-Pillot: Theta series of modular, extremal, and Hermitian lattices In: Integral Quadratic Forms and Lattices, Contemporary Mathematics 249, (1999), 221–233.

Kunzenhof 4B, 79117 Freiburg, Germany. E-mail address: [email protected]

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai – 400005, India. E-mail address: [email protected],[email protected]

ON HOLOMORPHIC DIFFERENTIAL OPERATORS ...

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