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A Note on Maass-Jacobi Forms

In this paper, we introduce the notion of Maass-Jacobi forms and investigate some properties of these new automorphic forms. We also characterize these automorphic forms in several ways.

The aim of this paper is to define the notion of Maass-Jacobi forms generalizing that of Maass wave forms and study some properties of these new automorphic forms. For the convenience of the reader, we review Maass wave forms. For s ∈ C, we denote by W s (Γ 1 ) the vector space of all smooth bounded functions f : SL(2, R) −→ C satisfying the following conditions (a) and (b) : (a) f (γgk) = f (g) for all γ ∈ Γ 1 , g ∈ SL(2, R) and k ∈ K.
The paper is organized as follows. In Section 2, we calculate the algebra of all invariant differential operators under the action (1.2) of G on H × C completely. In addition, we provide a G-invariant Riemannian metric on H × C and compute its Laplace-Beltrami operator. In Section 3, using the above Laplace-Beltrami operator, we introduce a concept of Maass-Jacobi forms generalizing that of Maass wave forms. We characterize Maass-Jacobi forms as smooth functions on G or SP 2 ×R (1,2) satisfying a certain invariance property, where SP 2 denotes the symmetric space consisting of all 2 × 2 positive symmetric real matrices Y with det Y = 1. In Section 4, we find the unitary dual of G and present some properties of G. In Section 5, we describe the decomposition of the Hilbert space L 2 (Γ\G). In the final section, we make some comments on the Fourier expansion of Maass-Jacobi forms.
Notations. We denote by Z, R and C the ring of integers, the field of real numbers and the field of complex numbers respectively. Z + denotes the set of all positive integers. F (k,l) denotes the set of all k × l matrices with entries in a commutative ring F . For a square matrix A, σ(A) denotes the trace of A. For any M ∈ F (k,l) , t M denotes the transpose of M . For A ∈ F (k,l) and B ∈ F (k,k) , we set B[A] = t ABA. We denote the identity matrix of degree n by E n . H denotes the Poincaré upperhalf plane.

Invariant Differential Operators on H × C
We recall that SP 2 is the symmetric space consisting of all 2 × 2 positive symmetric real matrices Y with det Y = 1. Then G acts on SP 2 × R (1,2) transitively by where g ∈ SL(2, R), α ∈ R (1,2) , Y ∈ SP 2 and V ∈ R (1,2) . It is easy to see that K is a maximal compact subgroup of G stabilizing the origin (E 2 , 0). Thus SP n × R (m,n) may be identified with the homogeneous space G/K as follows : where g ∈ SL(2, R) and α ∈ R (1,2) .
We know that SL(2, R) acts on H transitively by Now we observe that the action (1.2) of G on H × C may be rewritten as (1,2) , and (τ, z) ∈ H × C. Since the action (1.2) is transitive and K is the stabilizer of this action at the origin (i, 0), H × C can be identified with the homogeneous space G/K as follows : We see that we can express an element Y of SP 2 uniquely as Lemma 2.1. We define the mapping T : where Y is of the form (2.4) and V = (v 1 , v 2 ) ∈ R (1,2) . Then the mapping T is a bijection which is compatible with the above two actions (1.2) and (2.1).
For any Y ∈ SP 2 of the form (2.4), we put Proof. It is easy to prove the lemma. So we leave the proof to the reader. Now we give a complete description of the algebra D(H × C) of all differential operators on H × C invariant under the action (1.2) of G. First we note that the Lie algebra g of G is given by g = (X, Z) | X ∈ R (2,2) , σ(X) = 0, Z ∈ R (1,2) equipped with the following Lie bracket where [X 1 , X 2 ] 0 = X 1 X 2 −X 2 X 1 denotes the usual matrix bracket and (X 1 , Z 1 ), (X 2 , Z 2 ) ∈ g. And g has the following decomposition We observe that k is the Lie algebra of K and that we have the following relations Thus the coset space G/K ∼ = H × C is a reductive homogeneous space in the sense of [12], p. 284. It is easy to see that the adjoint action Ad of K on p is given by where k ∈ K and (X, Z) ∈ p with X = t X, σ(X) = 0. The action (2.9) extends uniquely to the action ρ of K on the polynomial algebra Pol (p) of p given by (2.10) ρ : K −→ Aut ( Pol(p )).
Let Pol (p) K be the subalgebra of Pol (p) consisting of all invariants of the action ρ of K. Then according to [12], Theorem 4.9, p. 287, there exists a canonical linear bijection is any basis of p and P ∈ Pol (p) K , then whereg ∈ G and f ∈ C ∞ (H × C).
We put Then e 1 , e 2 , f 1 , f 2 form a basis of p. We write for coordinates (X, Z) by with real variables x, y, z 1 and z 2 .
Lemma 2.2. The following polynomials are algebraically independent generators of Pol (p) K .
Proof. We leave the proof of the above lemma to the reader. Now we are ready to compute the G-invariant differential operators D, Ψ, D 1 and D 2 corresponding to the K-invariants P, ξ, P 1 and P 2 respectively under the canonical linear bijection (2.11). For real variables t = (t 1 , t 2 ) and s = (s 1 , s 2 ), we have For brevity, we write a j , b k for a j (t, s), b k (t, s) ( j = 1, 2, 3, k = 1, 2 ) respectively. We now fix an element (g, α) ∈ G and write Here x(t, s), y(t, s), u(t, s) and v(t, s) are real. By an easy calculation, we obtain By an easy calculation, at t = s = 0, we have We note thatãd −bc = 1, a 1 a 2 − a 2 3 = 1 and g 1 g 2 − g 12 g 21 = 1. Using the above facts and applying the chain rule, we can easily compute the differential operators D, Ψ, D 1 and D 2 . It is known that the images of generators P, ξ, P 1 and P 2 under λ are generators of D(H × C) ( cf. [11]).
Summarizing, we have the following.
Theorem 2.3. The algebra D(H × C) is generated by the following differential operators (2.14) where τ = x + iy and z = u + iv with real variables x, y, u, v. Moreover, we have In particular, the algebra D(H × C) is not commutative. Thus the homogeneous space H × C is not weakly symmetric in the sense of A. Selberg ([19]).
Now we provide a natural G-invariant Kähler metric on H × C.
Proposition 2.4. The Riemannian metric ds 2 on H × C defined by is invariant under the action (1.2) of G and is a Kähler metric on H × C. The Laplace-Beltrami operator ∆ of the Riemannian space ( H × C, ds 2 ) is given by That is, ∆ = D + Ψ.
Proof. For Y ∈ SP 2 of the form (2.4) and (v 1 , v 2 ) ∈ R (1,2) , it is easy to see that and dV = (dv 1 , dv 2 ). Then we can show that the following metric ds 2 on SP 2 ×R (1,2) defined by is invariant under the action (2.1) of G. Indeed, since we can easily show that . For an element ( g, α ) ∈ G with g ∈ SL(2, R) and α ∈ R (1,2) , we put Since Y * = gY t g and V * = ( V + α ) t g, we get dY * = g dY t g and V * = ( V + α ) t g. Therefore by a simple calculation, we can show that Hence the metric ds 2 is invariant under the action (2.1) of G.
Using this fact and Lemma 2.1, we can prove that the metric ds 2 in the above theorem is invariant under the action (1.2). Since the matrix form ( g ij ) of the metric ds 2 is given by and det ( g ij ) = y −6 , the inverse matrix ( g ij ) of ( g ij ) is easily obtained by Now it is easily shown that D + Ψ is the Laplace-Beltrami operator of ( H×C, ds 2 ).
Remark 2.5. We can show that for any two positive real numbers α and β, the following metric is also a Riemannian metric on H × C which is invariant under the action (1.2) of G.
In fact, we can see that the two-parameter family of ds 2 α,β ( α > 0, β > 0 ) provides a complete family of Riemannian metrics on H × C invariant under the action of (1.2) of G. It can be easily seen that the Laplace-Beltrami operator ∆ α,β of ds 2 α,β is given by Remark 2.6. By a tedious computation, we see that the scalar curvature of (H × C, ds 2 ) is −3.
(3) y s v, y s uv, y s xv with eigenvalue s(s + 1).
(MJ2) f is an eigenfunction of the Laplace-Beltrami operator ∆.
(MJ3) f has a polynomial growth, that is, f fulfills a boundedness condition.
For a complex number λ ∈ C, we denote by M J(Γ, λ) the vector space of all Maass-Jacobi forms f such that ∆f = λf. We note that, since ∆f = λf is an elliptic partial differential equation, Maass-Jacobi forms are real analytic (see [8]). Professor Berndt kindly informed me that he also considered such automorphic forms in ( [1]) (also see [4], p.82).
Let f ∈ M J(Γ, λ) be a Maass-Jacobi form with eigenvalue λ. Then it is easy to see that the function φ f : G −→ C defined by (MJ2) 0 φ f is an eigenfunction of the Laplace-Beltrami operator ∆ 0 of (G, ds 2 0 ), where ds 2 0 is a G-invariant Riemannian metric on G induced by (H × C, ds 2 ). (MJ3) 0 φ f has a suitable polynomial growth (cf. [5]).
For any right K-invariant function φ : G −→ C on G, we define the function α) is an element of G such that (g, α) • (i, 0) = (τ, z). Obviously it is well defined because (3.2) is independent of the choice of (g, α) ∈ G such that (g, α) • (i, 0) = (τ, z). It is easy to see that if φ is a smooth bounded function on G satisfying the conditions (MJ1) 0 -(MJ3) 0 , then the function f φ defined by (3.2) is a Maass-Jacobi form.
Now we characterize Maass-Jacobi forms as smooth eigenfunctions on SP n × R (m,n) satisfying a certain invariance property.
(MJ3) * h f has a suitable polynomial growth.
Proof. First of all, we note that h f is well defined because (3.3) is independent of the choice of g with Y = g t g. If (γ, δ) ∈ Γ with γ ∈ Γ 1 , δ ∈ Z (1,2) and (Y, V ) ∈ SP 2 × R (1,2) with Y = g t g for some g ∈ SL(2, R), then the element g γ := γg satisfies γY t γ = γg t (γg). Thus according to the definition of h f , for all (γ, δ) ∈ Γ and (Y, V ) ∈ SP n × R (m,n) , we have Therefore this proves the condition (MJ1) * . ds 2 and∆ are obtained from Lemma 2.1 and Proposition 2.3. Hence h f is an eigenfunction of∆. Clearly h f satisfies the condition (MJ3) * .

On the group SL 2,1 (R)
For brevity, we set H = R (1,2) . Then we have the split exact sequence We see that the unitary dualĤ of H is isomorphic to R 2 . The unitary character χ (λ,µ) of H corresponding to (λ, µ) ∈ R 2 is given by G acts on H by conjugation and hence this action induces the action of G onĤ as follows.
where the character χ g is defined by χ g (a) = χ(gag −1 ), a ∈ H. If g = (g 0 , α) ∈ G with g 0 ∈ SL(2, R) and α inH, it is easy to check that for each (λ, µ) ∈ R 2 , We see easily from (4.2) that the G-orbits inĤ ∼ = R 2 consist of two orbits Ω 0 , Ω 1 given by We observe that Ω 0 is the G-orbit of (0, 0) and Ω 1 is the G-orbit of any element (λ, µ) = 0. Now we choose the element δ = χ (1,0) ofĤ. That is, δ(x, y) = e 2πix for all (x, y) ∈ R 2 . It is easy to check that the stabilizer of χ (0,0) is G and the stabilizer G δ of δ is given by We see that H is regularly embedded. This means that for every G-orbit Ω inĤ and for every σ ∈ Ω with stabilizer G σ of σ, the canonical bijection G σ \G −→ Ω is a homeomorphism.
According to G. Mackey ([18]), we obtain (a) The irreducible unitary representations π, where the restriction of π to H is trivial and the restriction of π to SL(2, R) is an irreducible unitary representation of SL(2, R). For the unitary dual of SL(2, R), we refer to [7] or [15], p. 123.
(b) The representations π (r) = Ind G G δ σ r (r ∈ R) induced from the unitary character σ r of G δ defined by Proof. The proof of the above theorem can be found in [22], p. 850.
We put and Clearly W 1 , · · · , W 5 form a basis of g.
Lemma 4.2. We have the following relations.
Proof. The proof follows from an easy computation.
Let g C = g ⊗ R C be the complexfication of g. We put Then we have We note that k C is the complexification of the Lie algebra k of K.
We set a = R W 3 . By Lemma 4.2, the roots of g relative to a are given by ±e, ±2e, where e is the linear functional e : a −→ C defined by e(W 3 ) = 1. The set Σ + = {e, 2e} is the set of positive roots of g relative to a. We recall that for a root α, the root space g α is defined by Then we see easily that Proposition 4.3. The Killing form B of g is given by where (X 1 , Z 1 ), (X 2 , Z 2 ) ∈ g with X 1 , X 2 ∈ sl(2, R) and Z 1 , Z 2 ∈ R (1,2) . Hence the Killing form is highly nondegenerate. The adjoint representation Ad of G is given by where (g, α) ∈ G with g ∈ SL(2, R), α ∈ R (1,2) and (X, Z) ∈ g with X ∈ sl(2, R), Z ∈ R (1,2) .
Proof. The proof follows immediately from a direct computation.
An Iwasawa decomposition of the group G is given by (1,2) and A = a 0 0 a −1 , 0 ∈ G a > 0 .
An Iwasawa decomposition of the Lie algebra g of G is given by In fact, a is the Lie algebra of A and n is the Lie algebra of N . Now we compute the Lie derivatives for functions on G explicitly. We define the differential operators L k , R k ( 1 ≤ k ≤ 5 ) on G by and where f ∈ C ∞ (G) andg ∈ G.
In fact, the calculation for L 3 and R 5 can be found in [22], p. 837-839.
We define the differential operators L j ( 1 ≤ j ≤ 5 ) on H × C by where f ∈ C ∞ (H × C). Using the coordinates τ = x + iy and z = u + iv with x, y, u, v real and y > 0, we can easily compute the explicit formulas for L j 's. They are given by 5. The decomposition of L 2 (Γ\G) Let R be the right regular representation of G on the Hilbert space L 2 (Γ\G). We set G 1 = SL(2, R). Then the decomposition of R is given by where L 2 disc (Γ 1 \G 1 ) (resp. L 2 cont (Γ 1 \G 1 )) is the discrete (resp. continuous) part of L 2 (Γ 1 \G 1 ) (cf. [14], [15]) and H (r) is the representation space of π (r) (cf. Theorem 4.1. (b)).