Nearly holomorphic modular forms and differential operators

Nearly holomorphic modular forms and differential operators

Atsuo YAMAUCHI

2004 Oct. 25th and 26th in Hakuba

1 The case of elliptic modular forms

This lecture is a review of [3], [7], [8] and [9] in case of Siegel modular forms.

As is well known, the elliptic Eisenstein series of weight 2 is non-holomorphic.

In [1], Hecke showed that

E₂(z; Γ) =









∑



 a b c d



∈(P∩Γ)\Γ

(cz+d)|cz+d|^−2s









s=0

on z ∈H={z ∈C|Im(z)>0}is as follows.

E₂(z; Γ) = c₀ Im(z)+

∑∞ n=0

a_n·exp(2π√

−1nz/N),

where c₀, a_n ∈ C and a positive integer N. Using this, we can consider a non-holomorphic modular form f as

f(z) =

∑p i=0

f_i(z)

Im(z)ⁱ (1.1)

with holomorphic functions f_i onH.

We can define differential operatorsD_k and E (with k ∈Z) as D_kf = 2√

−1·∂f

∂z + kf Im(z), Ef = 2√

−1·Im(z)²∂f

∂z.

Then for any C^∞-function f on H and any α∈SL(2,R), we have (D_kf)|k+2α =D_k(f|kα),

(Ef)|k−2α =E(f|kα).

For f as in (1.1), we obtain D_kf =

∑p i=0

((k−i)f_i

Im(z)ⁱ⁺¹ + 2√

−1 Im(z)ⁱ · ∂f_i

∂z )

,

Ef =

∑p i=1

if_i Im(z)ⁱ⁻¹.

It can be proved that f ∈ C^∞(H,C) can be written as (1.1) if and only if E^p+1f = 0.

Now we can define the space of nearly holomorphic (elliptic) modular forms as

N_k^p(Γ) =



f ∈C^∞(H,C)

f|kγ =f for any γ ∈Γ, E^p+1f = 0,

f is finite at every cusp,





for any congruence subgroup Γ of SL(2,Z). Then anyf ∈ N_k^p(Γ) is written as (1.1).

2 Nearly holomorphic Siegel modular forms

For a positive integer m, put Sp(m,R) =

{

γ ∈GL(2m,R) ^tγ

( 0 −1_m

1_m 0

) γ =

( 0 −1_m 1_m 0

)}

, H_m ={Z ∈C^mm|^tZ =Z and Im(Z) is positive definite}.

Then H_m is an ^m(m+1)₂ -dimensional complex manifold. As is well known, Sp(m,R) acts on H_m by

α(Z) = (A_αZ +B_α)(C_αZ+D_α)⁻¹, where α =

( A_α B_α C_α D_α

)

∈ Sp(m,R) with A_α, B_α, C_α, D_α ∈ R^m_m, and Z ∈ H_m. Set

µ(α, Z) =C_αZ+D_α, η(Z) = 2√

−1(Z−Z) = 2·Im(Z).

For any rational representation (ρ, X) of GL(m,C) (i.e. ρ: GL(m,C)→ GL_C(X)), any α∈Sp(m,R), and a function f :H_m →X, we define another X-valued function f|ρα onHm by

(f|ρα)(Z) =ρ(µ(α, Z))⁻¹f(α(Z)).

In case ρ(a) = det(a)^k, f|ρα is written asf|kα.

Put

T =Tm ={

u∈C^mm^tu=u} .

For f ∈ C^∞(H_m, X), define Hom(T, X)-valued functions Df, Df, D_ρf and Ef by

(Df)(u) =

∑m i=1

∑m j=1

(1

2(1 +δ_ij) ∂f

∂z_ij )

·u_ij,

(Df)(u) =

∑m i=1

∑m j=1

(1

2(1 +δ_ij) ∂f

∂z_ij )

·u_ij, (Ef)(u) = (Df)(ηu^tη),

D_ρf =ρ(η)⁻¹D(ρ(η)f), where u= (u_ij)_1≤i,j≤m ∈T.

Let us define representationsρ⊗τ and ρ⊗π of GL(m,C) on Hom(T, X) by {(ρ⊗τ)(a)h}(u) =ρ(a)h(^taua) for any u∈T,

{(ρ⊗π)(a)h}(u) =ρ(a)h(a⁻¹u^ta⁻¹) for any u∈T, for h∈Hom(T, X).

Then by a formal computation, we obtain D_ρ(f|ρα) = (D_ρf)|ρ⊗τα,

E(f|ρα) = (Ef)|ρ⊗πα,

for any α ∈Sp(m,R) and any f ∈C^∞(H_m, X).

The k-th iterates of D_ρ and E take values in

Hom(T,Hom(T, . . . ,Hom(T, X)· · ·)).

This vector space can be identified with the space of all multilinear maps of T^k intoX, which we denote byM↕_k(T, X). We define representationsρ⊗τ^k and ρ⊗π^k of GL(m,C) on M↕_k(T, X) by

{(ρ⊗τ^k)(a)h}

(u₁, . . . , u_k) =ρ(a)h(^tau₁a, . . . ,^tau_ka), {(ρ⊗π^k)(a)h}

(u₁, . . . , u_k) =ρ(a)h(a⁻¹u₁^ta⁻¹, . . . , a⁻¹u_k^ta⁻¹), for h∈ M↕_k(T, X) and (u₁, . . . , u_k)∈T^k. Then clearly

(D_ρ⊗τ^k−1· · ·Dρ⊗τDρ)(f|ρα) = (D_ρ⊗τ^k−1· · ·Dρ⊗τDρf)|ρ⊗τ^kα, E^k(f|ρα) = (E^kf)|ρ⊗π^kα,

for any α ∈Sp(m,R).

We can view that τ (resp. π) is a representation of GL(m,C) on T by τ(a)u = ^taua (resp. π(a) = a⁻¹u^ta⁻¹), and ρ⊗τ^k (resp. ρ⊗π^k) can be identified with ρ⊗ (τ^⊗k) (resp. ρ ⊗ (π^⊗k)). Then we identify the space M↕_k(T, X) with X⊗T^⊗k or Hom(T^⊗k, X).

But actually, the images of D_ρ⊗τ^k−1· · ·D_ρ⊗τD_ρ and E^k take values in Sk(T, X) =

{

h ∈ M↕_k(T, X)

h(u_ε(1), . . . , h_ε(k)) =h(u₁, . . . , u_k) for any ε ∈S_k

} ,

which is identified with X⊗Sym^kT or Hom(Sym^kT, X).

Put

r(Z) = (r_ij(Z))_1≤i,j≤m = (Z −Z)⁻¹. Then we have the following lemma.

Lemma 2.1. Assume that f ∈ C^∞(H_m, X) and E^p+1f = 0 for some non- negative integer p. Then f can be written as a polynomial of {r_ij}_1≤i,j≤m of degree at most p, with coefficients in holomorphic functions on Hm.

Note that, iff is written as a polynomial of {r_ij}_1≤i,j≤m of degree pwith coefficients in holomorphic functions on H_m, D_ρf (resp. Ef) is written as that of degree p+ 1 (resp. p−1).

For a non-negative integer p and a congruence subgroup Γ of Sp(m,Z), we denote byNρ^p(Γ), the space of allf ∈C^∞(H_m, X) satisfying the following properties (1)–(3).

(1) f|ργ =f for any γ ∈Γ.

(2) E^p+1f = 0 .

(3) f is finite at every cusp ifm= 1.

We denote by Nρ^p the union of Nρ^p(Γ) for all congruence subgroups Γ of Sp(m,Z). Ifm is odd, the Eisenstein series of weight (m+ 3)/2 is contained in N_(m+3)/2^m . (See, Proposition 4.1 of [8].)

Consider a subspace Y of Sym^kT which is stable under the action of GL(m,C) by Sym^kτ (resp. Sym^kπ) and denote by D_ρ^Yf (resp. E^Yf), the restriction of (D_ρ⊗τk−1◦ · · · ◦Dρ)f (resp. E^kf) toY for any f ∈C^∞(Hm, X).

In this case D_ρ^Yf and E^Yf are Hom(Y, X)-valued.

Since det²is a subrepresentation of Sym^mτ, there exists a one-dimensional subspace Y of Sym^mT such that

(Sym^mτ)(a)v_Y = det(a)²v_Y,

for any a ∈GL(m,C), where 0 ̸=v_Y ∈Y. Hence we can define the differen- tial operator ∆_ρ:C^∞(H_m, X)→C^∞(H_m, X) by

(∆_ρf) = (D_ρ⊗τ^k−1 ◦ · · · ◦D_ρ◦f)(v_Y).

Then we have

det(µ(α, Z))⁻²(∆_ρf)|ρα= ∆_ρ(f|ρα),

for anyα∈Sp(m,R). Ifρ(a) = det(a)^k, then ∆_ρis essentially same as Maass operator M_k, which is concretely written in [2]. (Strictly, ∆_ρ = const× det(Im(Z))⁻¹M_k.)

3 Holomorphic projection

Given a rational representation (ρ, X) of GL(m,C), we define a contraction map θ_X :M↕_k(T,M↕_k(T, X))→X by

θ_Xφ=∑

φ(c₁, . . . , c_k;c₁, . . . , c_k),

where c₁, . . . , c_k run independently over the standard basis of T. Then we have

θ_X ◦(ρ⊗τ^k⊗π^k)(a) =θ_X ◦(ρ⊗π^k⊗τ^k)(a)

=ρ(a)◦θ_X,

for any a∈GL(m,C).

Theorem 3.1. (Proposition 3.4 of [7], Proposition 3.3 of [8])

Let ρ(a) = det(a)^νρ₀(a) for a ∈ GL(m,C) with ν ∈ Z and a rational repre- sentation (ρ₀, X) of GL(m,C). Let f ∈ Nρ^p(Γ) with a congruence subgroup Γ of Sp(m,Z). If ν is larger than an integer N(ρ₀, p) depending omly on ρ₀ and p, then

f =

∑p l=0

(θ_X ◦D_ρ⊗πl⊗τ^l−1 ◦ · · · ◦D_ρ⊗πl⊗τ ◦D_ρ⊗πl)(g_l), (3.1) with holomorphic modular forms g_l with respect to Γ and the representations ρ⊗π^l.

For example, elliptic Eisenstein series of weight 2 can not be written as (3.1). But any nearly holomorphic (scalar-valued) elliptic modular form can be made from holomorphic ones andE₂ (non-holomorphic Eisenstein series of weight 2) by using differential operatorsDk. It is because nearly holomorphic elliptic modular form of weight k is contained in N_k^k/2 orN_k^(k−1)/2 according to the parity of k. In case of f ∈ N_k^p(Γ) (k >2p), it can be written as

f =

∑p l=0

D_k−2◦ · · · ◦D_k−2l◦g_l

with holomorphic modular forms g_l of weightsk−2l with respect to Γ. On the other hand, f ∈ N2p^p(Γ) can be written as

f =c·D_2p−2◦ · · · ◦D₂◦E₂+

p−1

∑

l=0

D_2p−2◦ · · · ◦D_2p−2l◦g_l

with a constant c, (non-holomorphic) Eisenstein series E₂ ∈ N2¹(Γ), and holomorphic modular forms g_l of weights 2p−2l with respect to Γ.

Define an operatorL_ρ:C^∞(H_m, X)→C^∞(H_m, X) by L_ρ=−θ_X ◦D_ρ⊗π◦E.

Then clearly we have

(L_ρf)|ρα=L_ρ(f|ρα),

for any α ∈ Sp(m,R). Hence we have L_ρ(Nρ^p(Γ)) ⊂ Nρ^p(Γ). Moreover, the image of L_ρ is orthogonal to any cusp forms with respect to ρ.

In caseρ(a) = det(a)^k, we writeLρbyLk. Using thisLk, we can make an orthogonal projection of nearly holomorphic modular forms to holomorphic ones as follows.

Theorem 3.2. (Theorem 3.3 of [9], Proposition 15.3 of [10]) Let p, k be non-negative integers such that

k > m+p or k < m+3−p 2 .

For any irreducible subspace Y of SymⁱT with respect to Symⁱτ, put α_Y =i(k−(m+ 1) + (1−i)c_Y),

where −1/2≤ c_Y ≤1 denotes a rational number depending only on Y (not on k). Put A_i = ∏

Y(1−α⁻_Y¹L_k) for 0 < i ≤ p, where Y runs over all the irreducible subspaces of SymⁱT (with respect to Symⁱτ), and A = ∏_p

i=1Ai. Take any f ∈ Nk^p. Then Af is a holomorphic modular form of weightk and f =Af +L_kt with t∈ N_k^p.

Note that< f, g >=<Af, g >for any holomorphic cusp formgof weight k.

This theorem essentially uses the fact that the space SymⁱT is a direct sum of irreducible (rational) representations of GL(m,C), and each irreducible constituent has multiplicity one. (See, §12 of [10] and [6].) The author doesn’t know how to generalize this theorem to the case of vector-valued modular forms. In fact, letρbe a rational representation of GL(m,C) on aC- vector spaceX. Then the irreducible decomposition of the spaceX⊗SymⁱT does not satisfy the property of multiplicity one in general.

Professor B¨oecherer conjectures that both holomorphic projections of Theorems 3.1 and 3.2 are equal. It means thatAf in Theorem 3.2 coincides with g0 in Theorem 3.1 for any scalar valued modular form f (of sufficiently high weight.)

References

[1] E. Hecke, Theorie der Eisensteinschen Reihen h¨oherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem.

Hamburg 5(1927), 199-224.

[2] H. Maass, Siegel’s Modular Forms and Dirichlet Series, Lecture Notes in Math. 216(1971), Springer-Verlag.

[3] G. Shimura, Arithmetic of differential operators on symmetric domains, Duke Math. J. 48(1981), 813-843.

[4] , Confluent hypergeometric functions on tube domains, Math. An- nalen 260(1982), 269-302.

[5] , On Eisenstein series, Duke Math. J. 50(1983), 417-476.

[6] , On differential operators attached to certain representations of classical groups, Invent. Math. 77(1984), 463-488.

[7] , On a class of nearly holomorphic automorphic forms, Ann. of Math. 123(1986), 347-406.

[8] , Nearly holomorphic functions on hermitian symmetric spaces, Math. Annalen 278(1987), 1-28.

[9] , Differential operators, holomorphic projection, and singular forms.

Duke Math. J. 76(1994), 141-173.

[10] , Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs vol.82(2000), AMS.