Hermitian theta series and Maa{\ss} spaces under the action of the maximal discrete extension of the Hermitian modular group

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary quadratic number field $\mathbb{K}$ and $\Delta_{n,\mathbb{K}}^*$ its maximal discrete extension in the special unitary group $SU(n,n;\mathbb{C})$. In this paper we study the action of $\Delta_{n,\mathbb{K}}^*$ on Hermitian theta series and Maass spaces. For $n=2$ we will find theta lattices such that the corresponding theta series are modular forms with respect to $\Delta_{2,\mathbb{K}}^*$ as well as examples where this is not the case. Our second focus lies on studying two different Maass spaces. We will see that the new found group $\Delta_{2,\mathbb{K}}^*$ consolidates the different definitions of the spaces.

1. Introduction. In 1950, Hel Braun introduced the Hermitian modular group Γ n (O K ) as a generalization of the elliptic modular group [1]. Its maximal discrete extension ∆ * n,K in SU (n, n; C), called the extended Hermitian modular group, was determined in [7]. We study the action of this extended group on theta series Θ(Z, Λ) (n) where Z is an element of the Hermitian half space H n and Λ is a theta lattice. We will find a sufficient condition such that Θ(Z, Λ) (n) is a modular form with respect to ∆ * n,K for all n ∈ N and a less strict condition such that it is a modular form for a fixed n ∈ N. For n = 2 we consider specific theta lattices for which the corresponding theta series is a modular form with respect to ∆ * 2,K as well as examples where this is not the case. Furthermore, we study Maaß spaces introduced by Sugano in 1985 [9] and Krieg in 1991 [6]. It is known that Krieg's Maaß space is always a subspace of Suganos' Maaß space. Moreover, there are known cases for which they coincide as well as for which they differ. By studying the extended Hermitian modular group once more, we will prove that they coincide if and only if all elements in Sugano's Maaß space are modular forms with respect to ∆ * n,K .
where J = 0 −I I 0 and I denotes the n × n-identity matrix. Then SU (n, n; C) acts on the Hermitian half space as in (1). For m ∈ N, m squarefree, we consider the imaginary quadratic number field K : Its ring of integers is given by By O * K we denote its unit group. The Hermitian modular group of degree n is where Γ 1 (O K ) = SL 2 (Z). By [7] it is known that its maximal discrete extension in SU (n, n; C) is given by 3. Theta series. In the following we study theta series. Let V ⊆ C r be an r-dimensional K-vector space with the standard Hermitian form We call a set Λ ⊆ V an O K -lattice of rank r if Λ is an O K -module and a Z-lattice of rank 2r. We consider .
is even and unimodular, then (Λ, h) is called a theta lattice of rank r. Let (Λ, h) be a theta lattice. Then by [3] it is known that the associated theta series with Fourier coefficients is a Hermitian modular form of degree n and weight r. Clearly the theta series associated to theta lattices Λ and Λ ′ coincide whenever Λ and Λ ′ are isometric. Considering where Λ denotes the complex conjugate of Λ, we see that applying an automorphism of the Hermitian half space can yield a theta series with respect to an altered theta lattice.
In the following, we study Λ under the action of ∆ * n,K . Theorem 1. Let (Λ, h) be a theta lattice of rank r and M ∈ ∆ * n,K of the form (2). Then Particularly, the theta series coincide whenever Λ and Λ ′ are isometric.
Proof. As Λ ′ is an O K -module by definition and we have with arbitrary λ ′ = 1 u j l j λ j with l j ∈ I(L) and λ j ∈ Λ and using l j l k ∈ N (I(L))O K . Denote by Λ ′# the dual lattice of Λ ′ , then Λ ′# = Λ ′ follows directly from the definition and the fact that Λ is unimodular. Hence, Λ ′ is a theta lattice.
Let Λ ′ 1 × . . . × Λ ′ n := Λ n A. Then Λ ′ j does not depend on the representative A as a multiplication by a unimodular matrix does not change the lattice. If i = j ∈ {1, . . . , n}, let U denote the permutation matrix of i and j. Then AU is a representative of the same coset as A because the modular group ist normal in ∆ * n,K due to [7]. We obtain Λ ′ j = Λ ′ i and hence Λ ′ 1 = . . . = Λ ′ n . By definition of I(L) we immediately obtain (2). Now let Z ∈ H n . We consider Theorem 2. If a theta lattice Λ of rank r is strongly modular, the corresponding theta series Θ(Z, Λ) (n) is a modular form of weight r with respect to the maximal discrete extension ∆ * n,K for all n ∈ N.
By applying Theorem 1 from [7], we obtain the following corollary for fixed n ∈ N.
IΛ ∼ = Λ for all integral ideals I whose order in the class group is a divisor of n.
Strong modularity is a restrictive requirement but allows us to make a statement for arbitrary degree. As seen in Theorem 1, strong modularity is not a necessary condition for Θ(Z, Λ) (n) to be a modular form with respect to ∆ * n,K . Isometry is at most necessary for all integral ideals which can be generated by matrices L of the form (2). For fixed n ∈ N we know from [7] that these are precisely those ideals in O K whose n-th power is a principal ideal, as was formulated in Corollary 1. These can be constructed explicitely for n = 2. In this case we have [10], [7]. Because of V d SL 2 (O K ) = SL 2 (O K )V d and the form of ∆ * 2,K , V d does not depend on the choice of α, β, γ, δ in our setting and thus is well defined. Considering the special structure of ∆ * 2,K , we see that in order to understand the behavior of theta lattices under the action of ∆ * 2,K , we only need to study finitely many lattices A d corresponding to V d .

Hence, it is sufficient to study Λ
We consider a specific lattice given by [5].
In particular, we see that whenever a theta series is a modular form with respect to ∆ * 2,K for squarefree m 100, we immediately have Λ ∼ = 1 √ d A d Λ for all squarefree divisors d of d K .

The Maaß Spaces. Let
By [2], [6] it is known that any Hermitian modular form of degree 2 and weight r ∈ Z has a Fourier expansion with Fourier coefficients α f (T ) and For T = 0 let ǫ(T ) := max q ∈ N, 1 q T ∈ Λ(2; O K ) . The Maaß space by Sugano [9], denoted by S(r, O K ), consists of all f ∈ [Γ 2 (O K ), r] whose Fourier coefficients satisfy for all T = k t t l ∈ Λ(2, O K ), T 0, T = 0. In [6], Krieg defines another Maaß space, denoted by M (O r,K ), of all f ∈ [Γ 2 (O K ), r] for which there is an α * f : N 0 → C such that the Fourier coefficients of f satisfy for all T ∈ Λ(2, O K ), T 0, T = 0. One immediately sees that Krieg's Maaß space is contained in Sugano's Maaß space. Furthermore, the Maaß spaces coincide whenever d K is a prime discriminant.
Proof. The proof is very intuitive when applying the isomorphism in [10] and [7] and considering the orthogonal instead of the Hermitian setting. In this setting, the matrix T becomes the vector ( k v w l ) tr ∈ R 4 with v + ω K w = t and the action of V d becomes a multiplication by a matrix in GL 4 (Z). Alternatively, Lemma 1 can be verified by simple computation.
For T = k t t l ∈ Λ(2, O K ) we write α f (k, t, l) := α f (T ). We use this notation in the following Lemma.
Lemma 2. Let r ∈ Z, f ∈ S(r, O K ) and d|m. Then the following are equivalent: Proof. The proof is based on the proof of Theorem 2 in [4]. The equivalence of (i) and (ii) follows from the uniqueness of the Fourier expansion. In order to prove the equivalence of (ii) and (iii) we first acknowledge the form of α f (T ) in (3), i.e. it suffices to consider k = 1, as well as α f (1, t + u, l ′ ) = α f (1, t, l) for u ∈ O K and suitable l ′ . Now assume that (iii) holds. We consider which is not necessarily true for m ≡ 1 (mod 4) and 2|d. Using this and a suitable shift about u ∈ O K one finds a which satisfies (5). Applying (iii) yields the result. Now assume that (ii) holds. For T, T ′ ∈ Λ(2, O K ) which satisfy (5) we find T ′′ as before, i.e.
which satisfies (5) and apply the Chinese remainder Theorem to obtain (iii).
With this Lemma we are now able to prove the connection between Sugano's and Krieg's Maaß spaces in the following Theorem. Proof. As for Lemma 2, this proof is based on a proof for the paramodular group in [4]. Let m ≡ 3 (mod 4) and consider T, T ′ with det(T ) = det(T ′ ). As we can shift t, t ′ about O K without affecting the Fourier coefficient, we can assume ℜ(t) = ℜ(t ′ ) = 0. Using this and t, t ′ ∈ O ⋆ K , there are t 0 , t ′ 0 ∈ Z such that det(T ) = det(T ′ ) then yields t 0 ≡ t ′ 0 (mod 2). We define d := p∈P p|m p|(t 0 +t ′ 0 ) p.