A Characterization of Planar Mixed Automorphic Forms

Abstract

We characterize the functional space of the planar mixed automorphic forms with respect to an equivariant pair and given lattice as the image of the Landau automorphic forms (involving special multiplier) by an appropriate isomorphic transform.

1. Introduction

Mixed automorphic forms of type (𝜈,𝜇) arise naturally as holomorphic forms on elliptic varieties [1] and appear essentially in the context of number theory and algebraic geometry. Roughly speaking, they are a class of functions defined on a given (Hermitian symmetric) space 𝑋 and satisfying a functional equation of type𝐹𝛾⋅𝑥=𝑗^𝜈𝛾,𝑥𝑗^𝜇𝜌𝛾,𝜏(𝑥)𝐹(𝑥), (1.1) for every 𝑥∈𝑋 and 𝑔∈Γ. Here, 𝑗^𝛼(𝛾,𝑥); 𝛼∈ℝ is an automorphic factor associated to an appropriate action of a group Γ on 𝑋, and (𝜌,𝜏) is an equivariant pair for the data (Γ,𝑋). Such notion was introduced by Stiller [2] and extensively studied by Lee in the case of 𝑋 being the upper half-plane. They include the classical ones as a special case. Nontrivial examples of them have been constructed in [3, 4]. We refer to [5] for an exhaustive list of references.

In this paper, we are interested in the space of mixed automorphic forms ℳ^𝜈,𝜇_Γ,𝜏(ℂ) defined on the complex plane 𝑋=ℂ with respect to a given lattice Γ in ℂ and an equivariant pair (𝜌,𝜏). We find that ℳ^𝜈,𝜇_Γ,𝜏(ℂ) is isomorphic to the space of Landau automorphic forms [6],𝐹𝑧+𝛾=𝜒_𝜏𝛾𝑗^{𝐵𝜏𝜈,𝜇}𝛾,𝑧𝐹(𝑧);𝑧∈ℂ,𝛾∈Γ, (1.2) of “weight” 𝐵_𝜏^𝜈,𝜇(𝑧)=𝜈+𝜇𝜕𝜏𝜕𝑧²−𝜕𝜏𝜕𝑧²=𝐵_𝜏^𝜈,𝜇, (1.3) with respect to a special pseudocharacter 𝜒_𝜏 defined on Γ and given explicitly through (5.3) below. The crucial point in the proof is to observe that the quantity 𝐵_𝜏^𝜈,𝜇 is in fact a real constant independent of the complex variable 𝑧.

The exact statement of our main result (Theorem 5.1) is given and proved in Section 5. In Sections 2 and 3, we establish some useful facts that we need to introduce the space of planar mixed automorphic forms ℳ^𝜈,𝜇_Γ,𝜏(ℂ). We have to give necessary and sufficient condition to ensure the nontriviality of such functional space. In Section 4, we introduce properly the function 𝜑_𝜏^𝜈,𝜇 that serves to define the pseudocharacter 𝜒_𝜏.

2. Group Action

Let𝐺=𝐓⋊ℂ=𝑔∶=[𝑎,𝑏]≡𝑎𝑏01;𝑎∈𝐓,𝑏∈ℂ (2.1) be the semidirect product group of the unitary group 𝐓={𝜆∈ℂ;|𝜆|=1} and the additive group (ℂ,+). 𝐺 acts on the complex plane ℂ by the holomorphic mappings 𝑔⋅𝑧=𝑎𝑧+𝑏; 𝑔=[𝑎,𝑏], 𝑧∈ℂ, so that ℂ can be realized as Hermitian symmetric space ℂ=𝐺/𝐓.

By a 𝐺-equivariant pair (𝜌,𝜏), we mean that 𝜌 is a 𝐺-endomorphism and 𝜏∶ℂ→ℂ is a compatible mapping, that is,𝜏𝑔⋅𝑧=𝜌𝑔⋅𝜏(𝑧);𝑔∈𝐺,𝑧∈ℂ. (2.2) Now, for given real numbers 𝜈,𝜇, and an equivariant pair (𝜌,𝜏), we define 𝐽^𝜈,𝜇_𝜌,𝜏 to be the complex-valued mapping𝐽^𝜈,𝜇_𝜌,𝜏𝑔,𝑧∶=𝑗^𝜈𝑔,𝑧𝑗^𝜇𝜌𝑔,𝜏(𝑧);𝑔,𝑧∈𝐺×ℂ, (2.3) where 𝑗^𝛼; 𝛼∈ℝ is the “automorphic factor” given by𝑗^𝛼𝑔,𝑧=𝑒^{2𝑖𝛼ℑ⟨𝑧,𝑔−1⋅0⟩}. (2.4) Here and elsewhere, ℑ𝑧 denotes the imaginary part of the complex number 𝑧 and ⟨𝑧,𝑤⟩=𝑧𝑤 the usual Hermitian scalar product on ℂ. Thus, one can check the following.

Proposition 2.1. The mapping 𝐽^𝜈,𝜇_𝜌,𝜏 satisfies the chain rule 𝐽^𝜈,𝜇_𝜌,𝜏𝑔𝑔′,𝑧=𝑒^{2𝑖𝜙𝜌𝜈,𝜇(𝑔,𝑔′)}𝐽^𝜈,𝜇_𝜌,𝜏𝑔,𝑔′⋅𝑧𝐽^𝜈,𝜇_𝜌,𝜏𝑔′,𝑧, (2.5) where 𝜙_𝜌^𝜈,𝜇(𝑔,𝑔′) is the real-valued function defined on 𝐺×𝐺 by 𝜙_𝜌^𝜈,𝜇𝑔,𝑔′∶=ℑ𝜈𝑔⁻¹⋅0,𝑔′⋅0+𝜇𝜌𝑔⁻¹⋅0,𝜌𝑔′⋅0. (2.6)

Proof. For every 𝑔,𝑔′∈𝐺 and 𝑧∈ℂ, we have 𝐽^𝜈,𝜇_𝜌,𝜏𝑔𝑔′,𝑧=𝑗^𝜈𝑔𝑔′,𝑧𝑗^𝜇𝜌𝑔𝑔′,𝜏(𝑧)=𝑗^𝜈𝑔𝑔′,𝑧𝑗^𝜇𝜌𝑔𝜌𝑔′,𝜏(𝑧). (2.7) Next, one can see that the automorphic factor 𝑗^𝛼(⋅,⋅) satisfies 𝑗^𝛼ℎℎ′,𝑤=𝑒^{2𝑖𝛼ℑ⟨ℎ−1⋅0,ℎ′⋅0⟩}𝑗^𝛼ℎ,ℎ′⋅𝑤𝑗^𝛼ℎ′,𝑤, (2.8) for every ℎ,ℎ′∈𝐺 and 𝑤∈ℂ. This gives rise to 𝐽^𝜈,𝜇_𝜌,𝜏𝑔𝑔′,𝑧=𝑒^{2𝑖𝜙𝜌𝜈,𝜇(𝑔,𝑔′)}𝑗^𝜈𝑔,𝑔′⋅𝑧𝑗^𝜇𝜌𝑔,𝜌𝑔′⋅𝜏(𝑧)𝐽^𝜈,𝜇_𝜌,𝜏𝑔′,𝑧. (2.9) Finally, (2.5) follows by making use of the equivariant condition 𝜌(𝑔′)⋅𝜏(𝑧)=𝜏(𝑔′⋅𝑧).

Remark 2.2. According to Proposition 2.1 above, the unitary transformations 𝒯_𝑔^𝜈,𝜇𝑓(𝑧)∶=𝐽^𝜈,𝜇_𝜌,𝜏𝑔,𝑧𝑓𝑔⋅𝑧 (2.10) for varying 𝑔∈𝐺 define then a projective representation of the group 𝐺 on the space of 𝒞^∞ functions on ℂ.

3. The Space of Planar Mixed Automorphic Forms

Let Γ be a uniform lattice of the additive group (ℂ,+) that can be seen as a discrete subgroup of 𝐺 by the identification 𝛾∈Γ⟼1,𝛾=1𝛾01, (3.1) so that the action of Γ on ℂ is the one induced from this of 𝐺, that is,𝛾⋅𝑧=𝑧+𝛾. (3.2) Associated to such Γ and given fixed data of 𝜈,𝜇>0 and (𝜌,𝜏) as above, we perform ℳ^𝜈,𝜇_Γ,𝜏(ℂ) the vector space of smooth complex-valued functions 𝐹 on ℂ satisfying the functional equation𝐹𝛾⋅𝑧=𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧𝐹(𝑧)=𝑗^𝜈𝛾,𝑧𝑗^𝜇𝜌𝛾,𝜏(𝑧)𝐹(𝑧), (3.3) for every 𝛾∈Γ and 𝑧∈ℂ.

Definition 3.1. The space ℳ^𝜈,𝜇_Γ,𝜏(ℂ) is called the space of planar mixed automorphic forms of biweight (𝜈,𝜇) with respect to the equivariant pair (𝜌,𝜏) and the lattice Γ.

We assert the following.

Proposition 3.2. The functional space ℳ^𝜈,𝜇_Γ,𝜏(ℂ) is nontrivial if and only if the real-valued function (1/𝜋)𝜙_𝜌^𝜈,𝜇 in (2.6) takes integral values on Γ×Γ.

Proof. The proof can be handled in a similar way as in [6] making use of (2.5) combined with the equivariant condition (2.2). Indeed, assume that ℳ^𝜈,𝜇_Γ,𝜏(ℂ) is nontrivial, and let 𝐹 be a nonzero function belonging to ℳ^𝜈,𝜇_Γ,𝜏(ℂ). According to (2.5), we get 𝐹𝛾+𝛾′⋅𝑧=^()3.2𝐹𝛾′⋅𝛾⋅𝑧=^()3.3𝐽^𝜈,𝜇_𝜌,𝜏𝛾′⋅𝛾,𝑧_()2.5𝐹(𝑧)=𝑒^{2𝑖𝜙𝜌𝜈,𝜇(𝛾′,𝛾)}𝐽^𝜈,𝜇_𝜌,𝜏𝛾′,𝛾⋅𝑧𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧𝐹(𝑧), (3.4) for every 𝛾,𝛾′∈Γ. On the other hand, we can write 𝐹𝛾+𝛾^′⋅𝑧=𝐹𝑧+𝛾+𝛾^′=𝐹𝛾^′⋅𝑧+𝛾=𝐹𝛾^′⋅𝛾⋅𝑧_()3.3=𝐽^𝜈,𝜇_𝜌,𝜏𝛾^′,𝛾⋅𝑧𝐹𝛾⋅𝑧_()3.3=𝐽^𝜈,𝜇_𝜌,𝜏𝛾^′,𝛾⋅𝑧𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧𝐹(𝑧). (3.5) Now, by equating the right hand sides of (3.4) and (3.5), keeping in mind that 𝐹 is not identically zero, we get necessarily 𝑒^{2𝑖𝜙𝜌𝜈,𝜇(𝛾′,𝛾)}=1,∀𝛾,𝛾^′∈Γ. (3.6)
Conversely, by classical analysis, we pick an arbitrary nonzero 𝒞^∞ and compactly supported function 𝜓 with support contained in a fundamental domain Λ(Γ) of the lattice Γ. Next, we consider the associated Poincaré series 𝒫_Γ^𝜈,𝜇𝜓 given by 𝒫_Γ^𝜈,𝜇𝜓(𝑧)=_𝛾∈Γ𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧𝜓𝛾⋅𝑧. (3.7) Then, it can be shown that the function 𝒫_Γ^𝜈,𝜇𝜓 is 𝒞^∞ and a nonzero function on ℂ for Γ being discrete and Supp𝜓⊂Λ(Γ). Indeed, for every 𝑧∈Supp𝜓, we have 𝒫_Γ^𝜈,𝜇𝜓(𝑧)=𝜓(𝑧). (3.8) Furthermore, under the condition that (1/𝜋)𝜙_𝜌^𝜈,𝜇 takes integral values on Γ×Γ, we see that 𝒫_Γ^𝜈,𝜇𝜓 belongs to ℳ^𝜈,𝜇_Γ,𝜏(ℂ). In fact, for every 𝛾∈Γ and 𝑧∈ℂ^𝑛, we have 𝒫_Γ^𝜈,𝜇𝜓𝛾⋅𝑧=_𝑘∈Γ𝐽^𝜈,𝜇_𝜌,𝜏𝜓=𝑘,𝛾⋅𝑧𝑘⋅𝛾⋅𝑧_𝑘∈Γ𝐽^𝜈,𝜇_𝜌,𝜏𝑘,𝛾⋅𝑧𝜓𝑘+𝛾⋅𝑧_{ℎ=𝛾+𝑘}=_ℎ∈Γ𝐽^𝜈,𝜇_𝜌,𝜏ℎ⋅𝛾⁻¹,𝛾⋅𝑧𝜓(ℎ⋅𝑧_()2.5)=_ℎ∈Γ𝑒^{−2𝑖𝜙𝜌𝜈,𝜇(ℎ,𝛾−1)}𝐽^𝜈,𝜇_𝜌,𝜏ℎ,𝛾⁻¹⋅𝛾⋅𝑧𝐽^𝜈,𝜇_𝜌,𝜏𝛾⁻¹,𝛾⋅𝑧𝜓(ℎ⋅𝑧). (3.9) Finally, since 𝑒^{2𝑖𝜙𝜌𝜈,𝜇(𝛾,ℎ)}=1 (by hypothesis on 𝜙_𝜌^𝜈,𝜇), it follows that 𝒫_Γ^𝜈,𝜇𝜓𝛾⋅𝑧=_ℎ∈Γ𝐽^𝜈,𝜇_𝜌,𝜏(ℎ,𝑧)𝐽^𝜈,𝜇_𝜌,𝜏𝛾⁻¹𝜓()=,𝛾⋅𝑧ℎ⋅𝑧𝐽^𝜈,𝜇_𝜌,𝜏𝛾⁻¹,𝛾⋅𝑧_ℎ∈Γ𝐽^𝜈,𝜇_𝜌,𝜏(ℎ,𝑧)𝜓(ℎ⋅𝑧)=𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧𝒫_Γ^𝜈,𝜇𝜓(𝑧). (3.10) The last equality, that is, 𝐽^𝜈,𝜇_𝜌,𝜏𝛾⁻¹,𝛾⋅𝑧=𝐽^𝜈,𝜇_𝜌,𝜏𝛾,𝑧 (3.11) holds for every 𝛾∈Γ using the chain rule (2.5) and taking into account the assumption made on 𝜙_𝜌^𝜈,𝜇. This completes the proof.

Remark 3.3. The condition involved in Proposition 3.2 ensures that ℳ^𝜈,𝜇_Γ,𝜏(ℂ) can be realized as the space of cross-sections on a line bundle over the complex torus ℂ/Γ.

4. On the Function 𝜑_𝜏^𝜈,𝜇

In order to prove the main result of this paper, we need to introduce the function 𝜑_𝜏^𝜈,𝜇.

Proposition 4.1. The first-order differential equation 𝜕𝜑_𝜏^𝜈,𝜇𝜕𝑧𝜏𝜕=−𝑖𝜇𝜏𝜕𝑧−𝜏𝜕𝜏𝜕𝑧−𝜕𝜏𝜕𝑧²−𝜕𝜏𝜕𝑧²𝑧 (4.1) admits a solution 𝜑_𝜏^𝜈,𝜇∶ℂ→ℂ such that ℑ𝜑_𝜏^𝜈,𝜇 is constant.

Proof. By writing the 𝐺-endomorphism 𝜌∶𝐺→𝐺=𝐓⋊ℂ as 𝜌(𝑔)=[𝜒(𝑔),𝜓(𝑔)], and differentiating the equivariant condition 𝜏𝑔⋅𝑧=𝜌𝑔⋅𝜏(𝑧)=𝜒𝑔𝜏(𝑧)+𝜓𝑔, (4.2) it follows that 𝜕𝑔⋅𝑧𝜕𝑧𝜕𝜏𝑔𝜕𝑧𝑔⋅𝑧=𝜒𝜕𝜏(𝑧),𝜕𝜕𝑧𝑔⋅𝑧𝜕𝑧𝜕𝜏𝜕𝑧𝑔⋅𝑧=𝜒𝑔𝜕𝜏𝜕𝑧(𝑧). (4.3) Hence, for (𝜕𝑔⋅𝑧)/(𝜕𝑧) and 𝜒(𝑔) being in 𝐓, we deduce that 𝐵_𝜏^𝜈,𝜇𝑔⋅𝑧=𝜈+𝜇𝜕𝜏𝜕𝑧𝑔⋅𝑧²−𝜕𝜏𝜕𝑧𝑔⋅𝑧²=𝜈+𝜇𝜕𝜏(𝑧𝜕𝑧)²−𝜕𝜏𝜕𝑧(𝑧)²=𝐵_𝜏^𝜈,𝜇(𝑧). (4.4) Therefore, 𝑧↦𝐵_𝜏^𝜈,𝜇(𝑧) is a real-valued constant function (since the only 𝐺-invariant functions on ℂ are the constants). Now, by considering the differential 1-form 𝜃_𝜏^𝜈,𝜇(𝑧)∶=𝑖𝜈(𝑧𝑑𝑧−𝑧𝑑𝑧)+𝜇(𝜏𝑑𝜏−𝜏𝑑𝜏), (4.5) one checks that 𝑑𝜃_𝜏^𝜈,𝜇=𝑑𝜃^{𝐵𝜏𝜈,𝜇}, where 𝜃^{𝐵𝜏𝜈,𝜇}∶=𝑖𝐵_𝜏^𝜈,𝜇(𝑧𝑑𝑧−𝑧𝑑𝑧). Therefore, there exists a function 𝜑_𝜏^𝜈,𝜇∶ℂ→ℂ such that ℑ𝜑_𝜏^𝜈,𝜇=𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and satisfying the first-order partial differential equation 𝜕𝜑_𝜏^𝜈,𝜇𝜕𝑧=−𝑖𝜈−𝐵_𝜏^𝜈,𝜇𝜏𝜕𝑧+𝜇𝜏𝜕𝑧−𝜏𝜕𝜏𝜕𝑧𝜏𝜕=−𝑖𝜇𝜏𝜕𝑧−𝜏𝜕𝜏𝜕𝑧−𝜕𝜏𝜕𝑧²−𝜕𝜏𝜕𝑧²𝑧. (4.6) This completes the proof.

Remark 4.2. The partial differential equation (4.6) satisfied by 𝜑_𝜏^𝜈,𝜇 can be reduced further to the following: 𝜕𝜓_𝜏^𝜈,𝜇𝜕𝑧=𝜏𝜕𝜏𝜕𝑧,𝜕𝜓_𝜏^𝜈,𝜇=1𝜕𝑧𝜇𝜈−𝐵_𝜏^𝜈,𝜇𝑧+𝜏𝜕𝜏,𝜕𝑧 (4.7) with 𝜑_𝜏^𝜈,𝜇(𝑧)=𝑖([𝐵_𝜏^𝜈,𝜇−𝜈]|𝑧|²−𝜇|𝜏(𝑧)|²)+2𝑖𝜇𝜓_𝜏^𝜈,𝜇(𝑧).

5. Main Result

Let 𝜑_𝜏^𝜈,𝜇 be the real part of 𝜑_𝜏^𝜈,𝜇−𝜑_𝜏^𝜈,𝜇(0), where 𝜑_𝜏^𝜈,𝜇 is a complex-valued function on ℂ as in Proposition 4.1. Define 𝒲_𝜏^𝜈,𝜇 to be the special transformation given by𝒲_𝜏^𝜈,𝜇𝑓(𝑧)∶=𝑒^{𝑖𝜑𝜏𝜈,𝜇(𝑧)}𝑓(𝑧). (5.1) We have the following.

Theorem 5.1. The image of ℳ^𝜈,𝜇_Γ,𝜏(ℂ) by the transform (5.1) is the space of Landau (Γ,𝜒_𝜏)-automorphic functions. More exactly, one has 𝒲_𝜏^𝜈,𝜇ℳ^𝜈,𝜇_Γ,𝜏(ℂ)=𝐹;𝒞^∞,𝐹𝑧+𝛾=𝜒_𝜏𝛾𝑗^{𝐵𝜏𝜈,𝜇}𝛾,𝑧𝐹(𝑧), (5.2) with 𝐵_𝜏^𝜈,𝜇=𝜈+𝜇(|𝜕𝜏/𝜕𝑧|²−|𝜕𝜏/𝜕𝑧|²)∈ℝ and 𝜒_𝜏 is the pseudocharacter defined on Γ by 𝜒_𝜏𝛾=exp2𝑖𝜑_𝜏^𝜈,𝜇𝛾−2𝑖𝜇ℑ𝜏(0),𝜌𝛾⁻¹⋅0. (5.3)

For the proof, we begin with the following.

Lemma 5.2. The function 𝜒_𝜏 defined on ℂ×Γ by 𝜒_𝜏𝑧;𝛾∶=𝑒^{𝑖(𝜑𝜏𝜈,𝜇(𝑧+𝛾)−𝜑𝜏𝜈,𝜇(𝑧))}𝑒^{−2𝑖([𝐵𝜏𝜈,𝜇−𝜈]ℑ⟨𝑧,𝛾⟩+𝜇ℑ⟨𝜏(𝑧),𝜌(𝛾)−1⋅0⟩)} (5.4) is independent of the variable 𝑧.

Proof. Differentiation of 𝜒_𝜏(𝑧;𝛾) with respect to the variable 𝑧 gives 𝜕𝜒_𝜏𝜕𝑧=𝑖𝜕𝜑_𝜏^𝜈,𝜇−𝜕𝑧𝑧+𝛾𝜕𝜑_𝜏^𝜈,𝜇(𝑧)𝜒𝜕𝑧^𝜏−𝐵_𝜏^𝜈,𝜇−𝜈𝛾+𝜇𝜌𝛾⁻¹⋅0𝜕𝜏(𝑧)−𝜌𝛾𝜕𝑧⁻¹𝜕⋅0𝜏(𝑧𝜒𝜕𝑧)_𝜏. (5.5) On the other hand, using the equivariant condition 𝜏(𝑧+𝛾)=𝜌(𝛾)⋅𝜏(𝑧) and (4.6), one gets 𝑖𝜕𝜑_𝜏^𝜈,𝜇−𝜕𝑧𝑧+𝛾𝜕𝜑_𝜏^𝜈,𝜇(𝑧)𝜕𝑧=𝐵_𝜏^𝜈,𝜇𝛾+𝑆_𝜏^𝜈,𝜇(𝑧)−𝑆_𝜏^𝜈,𝜇=𝐵𝑧+𝛾_{𝜏𝜈,𝜇}−𝜈𝛾+𝜇𝑎_𝛾𝜕𝜏(𝑧)𝜕𝑧−𝑎_𝛾𝜕𝜏(𝑧,𝜕𝑧) (5.6) where we have set 𝑎_𝛾=𝜌(𝛾⁻¹)⋅0. Thus, from (5.5) and (5.6), we conclude that 𝜕𝜒_𝜏/𝜕𝑧=0. Similarly, one gets also 𝜕𝜒_𝜏/𝜕𝑧=0. This ends the proof of Lemma 5.2.

Proof of Theorem 5.1. We have to prove that 𝒲_𝜏^𝜈,𝜇𝐹 belongs to ℱ^{𝐵𝜏𝜈,𝜇}_Γ,𝜒𝜏∶=𝐹∶ℂ⟶^𝒞∞ℂ;𝐹𝑧+𝛾=𝜒_𝜏𝛾𝑗^{𝐵𝜏𝜈,𝜇}𝛾,𝑧𝐹(𝑧), (5.7) whenever 𝐹∈ℳ^𝜈,𝜇_Γ,𝜏(ℂ), where 𝜒_𝜏𝛾∶=exp2𝑖𝜑_𝜏^𝜈,𝜇𝛾−2𝑖𝜇ℑ𝜏(0),𝜌𝛾⁻¹⋅0. (5.8) Indeed, we have 𝒲_𝜏^𝜈,𝜇𝐹𝑧+𝛾∶=𝑒^{𝑖𝜑𝜏𝜈,𝜇(𝑧+𝛾)}𝐹𝑧+𝛾=𝑒^{𝑖𝜑𝜏𝜈,𝜇(𝑧+𝛾)}𝑗^𝜈𝛾,𝑧𝑗^𝜇𝜌𝛾,𝜏(𝑧)𝐹(𝑧)=𝑒^{𝑖(𝜑𝜏𝜈,𝜇(𝑧+𝛾)−𝜑𝜏𝜈,𝜇(𝑧))}𝑗^𝜈𝛾,𝑧𝑗^𝜇𝜌𝛾,𝜏(𝑧)𝒲_𝜏^𝜈,𝜇𝐹𝑧)=𝜒_𝜏(𝑧;𝛾𝑗^{−𝐵𝜏𝜈,𝜇}𝛾,𝑧𝒲_𝜏^𝜈,𝜇𝐹(𝑧). (5.9) Whence by Lemma 5.2, we see that 𝜒_𝜏(𝑧;𝛾)=𝜒_𝜏(0;𝛾)=∶𝜒_𝜏(𝛾), and therefore 𝒲_𝜏^𝜈,𝜇𝐹𝑧+𝛾=𝜒_𝜏𝛾𝑗^{−𝐵𝜏𝜈,𝜇}𝛾,𝑧𝒲_𝜏^𝜈,𝜇𝐹(𝑧). (5.10) The proof is complete.

Corollary 5.3. The function 𝜒_𝜏(𝛾)=exp(2𝑖𝜑_𝜏^𝜈,𝜇(𝛾)−2𝑖𝜇ℑ⟨𝜏(0),𝜌(𝛾)⁻¹⋅0⟩) satisfies the following pseudocharacter property: 𝜒_𝜏𝛾+𝛾′=𝑒^{2𝑖𝐵𝜏𝜈,𝜇ℑ⟨𝛾,𝛾′⟩}𝜒_𝜏𝛾𝜒_𝜏𝛾′ (5.11) if and only if 𝜙_𝜌^𝜈,𝜇 in (2.6) takes its values in 𝜋ℤ on Γ×Γ.

Acknowledgment

The author is indebted to Professor A. Intissar for valuable discussions and encouragement.