For a given fixed positive integer n, we let

be the Siegel upper half plane of degree n and let

be the symplectic group of degree n, where F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F for two positive integers k and l, ^tM denotes the transposed matrix of a matrix M and

Then Sp(n,R) acts on Hn transitively by

where M=ABCD∈Sp(n,R) and Ω∈Hn. Let

be the Siegel modular group of degree n. This group acts on Hn properly discontinuously. C. L. Siegel investigated the geometry of Hn and automorphic forms on Hn systematically. Siegel[16] found a fundamental domain Fn for Γn\Hn and described it explicitly. Moreover he calculated the volume of Fn. We also refer to [13, 16] for some details on Fn.

For two positive integers m and n, we consider the Heisenberg group

endowed with the following multiplication law

with (λ,μ κ),(λ',μ' κ')∈HR(n,m). We define the Jacobi group GJ of degree n and index m that is the semidirect product of Sp(n,R) and HR(n,m)

endowed with the following multiplication law

with M,M'∈Sp(n,R),(λ,μ κ),(λ',μ' κ')∈HR(n,m) and (λ˜,μ˜)=(λ,μ)M'. Then GJ acts on Hn×C(m,n) transitively by

where M=ABCD∈Sp(n,R),(λ,μ κ)∈HR(n,m) and (Ω,Z)∈Hn×C(m,n). We note that the Jacobi group GJ is not a reductive Lie group and the homogeneous space Hn×C(m,n) is not a symmetric space. From now on, for brevity we write Hn,m=Hn×C(m,n). The homogeneous space Hn,m is called the Siegel-Jacobi space of degree n and index m.

In this short article, we propose the basic and natural problems in the geometry of the Siegel-Jacobi space.

Notations: We denote by

Q,R and C the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by Z the ring of integers. The symbol “:=” means that the expression on the right is the definition of that on the left. For two positive integers k and l, F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F. For a square matrix A∈F(k,k) of degree k, σ(A) denotes the trace of A. For any M∈F(k,l)tM denotes the transpose of a matrix M. In denotes the identity matrix of degree n. For a complex matrix A, A¯ denotes the complex conjugate of A. For a number field F, we denote by AF the ring of adeles of F. If F=Q, the subscript will be omitted.

We let G:=Sp(n,R) and K=U(n). The stabilizer of the action (1.1) at iIn is

Thus we get the biholomorphic map

Hn is a Hermitian symmetric manifold.

For Ω=(ωij)∈Hn, we write Ω=X+iY with X=(xij),Y=(yij) real. We put dΩ=(dωij) and dΩ¯=(dω¯ij). We also put

C. L. Siegel [16] introduced the symplectic metric dsn A2 on Hn invariant under the action (1.1) of Sp(n,R) that is given by

It is known that the metric dsn A2 is a Kähler-Einstein metric. H. Maass [12] proved that its Laplace operator Δn A is given by

And

is a Sp(n,R)-invariant volume element on Hn(cf.[17, p.130]).

Siegel proved the following theorem for the Siegel space (Hn,dsn 12).

Theorem 2.1. (Siegel[16]). (1) There exists exactly one geodesic joining two arbitrary points Ω0,Ω1 in Hn. Let R(Ω0,Ω1) be the cross-ratio defined by

For brevity, we put R*=R(Ω0,Ω1). Then the symplectic length ρ(Ω0,Ω1) of the geodesic joining Ω0 and Ω1 is given by

where

(2) For M∈Sp(n,R), we set

Then R(Ω1,Ω0) and R(Ω˜1,Ω˜0) have the same eigenvalues.

(3) All geodesics are symplectic images of the special geodesics

where a1,a2,⋯,an are arbitrary positive real numbers satisfying the condition

The proof of the above theorem can be found in [16, pp.289-293].

Let D(Hn) be the algebra of all differential operators on Hn invariant under the action (1.1). Then according to Harish-Chandra [5, 6],

where D1,⋯,Dn are algebraically independent invariant differential operators on Hn. That is, D(Hn) is a commutative algebra that is finitely generated by n algebraically independent invariant differential operators on Hn. Maass [13] found the explicit D1,⋯,Dn. Let gC be the complexification of the Lie algebra of G. It is known that D(Hn) is isomorphic to the center of the universal enveloping algebra of gC(cf.[7]).

Example 2.2. We consider the simplest case n=1 and A=1. Let H be the Poincaré upper half plane. Let ω=x+iy∈H with x,y∈R and y>0. Then the Poincaré metric

is a SL(2,R)-invariant Kähler-Einstein metric on H. The geodesics of (H,ds2) are either straight vertical lines perpendicular to the x-axis or circular arcs perpendicular to the x-axis (half-circles whose origin is on the x-axis). The Laplace operator Δ of (H,ds2) is given by

and

is a SL(2,R)-invariant volume element. The scalar curvature, i.e., the Gaussian curvature is -1. The algebra D(H) of all SL(2,R)-invariant differential operators on H is given by

The distance between two points ω1=x1+iy1 and ω2=x2+iy2 in (H,ds2) is given by

For a coordinate (Ω,Z)∈Hn,m with Ω=(ωμν) and Z=(zkl), we put dΩ,dΩ¯,∂∂Ω,∂∂Ω¯ as before and set

The author proved the following theorems in [18].

Theorem 3.1. For any two positive real numbers A and B,

is a Riemannian metric on Hn,m which is invariant under the action (1.2) of GJ.

Proof. See [18, Theorem 1.1].

Theorem 3.2. The Laplace operator Δm,m A,B of the GJ-invariant metric dsn,m A,B2 is given by

where

and

Furthermore M1 and M2 are differential operators on Hn,m invariant under the action (1.2) of GJ.

Proof. See [18, Theorem 1.2].

Remark 3.3. Erik Balslev [2] developed the spectral theory of Δ1,1 1,1 on H1,1 for certain arithmetic subgroups of the Jacobi modular group to prove that the set of all eigenvalues of Δ1,1 1,1 satisfies the Weyl law.

Remark 3.4. The scalar curvature of (H1,1,ds1,1 A,B2) is - and hence is independent of the parameter B. We refer to [21] for more detail.

Remark 3.5. Yang and Yin [22] showed that dsn,m A,B2 is a Kähler metric. For some applications of the invariant metric dsn,m A,B2 we refer to [22].

Now we propose the basic and natural problems.

Problem 1. Find all the geodesics of (Hn,m,dsn,m A,B2) explicitly.

Problem 2. Compute the distance between two points (Ω1,Z1) and (Ω2,Z2) of Hn,m explicitly.

Problem 3. Compute the Ricci curvature tensor and the scalar curvature of (Hn,dsn,m A,B2).

Problem 4. Find all the eigenfunctions of the Laplace operator Δn,m A,B.

Problem 5. Develop the spectral theory of Δn,m A,B.

Problem 6. Describe the algebra of all GJ-invariant differential operators on Hn,m explicitly. We refer to [19, 20, 22] for some details.

Problem 7. Find the trace formula for the Jacobi group GJ(A).

Problem 8. Discuss the behaviour of the analytic torsion of the Siegel-Jacobi space Hn,m or the arithmetic quotients of Hn,m.

We make some remarks on the above problems.

Remark 3.6. Problem 1 reduces to trying to solve a system of ordinary differential equations explicitly. If Problem 2 is solved, the distance formula would be a very beautiful one that generalizes the distance formula ρ(Ω0,Ω1) given by Theorem 2.1 (the Siegel space case).

Remark 3.7. Problem 3 was recently solved in the case that n=1 and m is arbitrary. Precisely the scalar and Ricci curvatures of the Siegel-Jacobi space (H1,m,ds1,m A,B2)(m≥1) were completely computed by G. Khan and J. Zhang [8, Proposition 8, pp.825–826]. Furthermore Khan and Zhang proved that (H1,m,ds1,m A,B2)(m≥1) has non-negative orthogonal anti-bisectional curvature (cf.[8, Proposition 9, p.826]).

Remark 3.8. Concerning Problem 4 and Problem 5, computing eigenfunctions explicitly is a tall order, but if this can be done it will shed a lot of light onto the geometry of this space. And understanding the spectral geometry seems to be a central question which will likely have applications in number theory and other areas.

Remark 3.9. The algebra D(Hn,m) of all GJ-invariant differential operators on Hn,m is not commutative. Concerning Problem 6, the case n=m=1 was completely solved by M. Itoh, H. Ochiai and J.-H. Yang in 2013. They proved that the noncommutative algebra D(H1,1) is generated by four explicit generators D1,D2,D3,D4, and found the relations among those Di(1≤i≤4). For more precise statements, we refer to [19, pp.56–58] and [20, pp.285–290]. We note that the above four generators Di(1≤i≤4) are not algebraically independent.

Remark 3.10 The solution of Problem 7 will provide lots of arithmetic properties of the Siegel-Jacobi space.

Let Γn(N) be the principal congruence subgroup of the Siegel modular group Γn. Let Xn(N):=Γn(N)\Hn be the moduli of n-dimensional principally polarized abelian varieties with level N-structure. The Mumford school [1] found toroidal compactifications of Xn(N) which are usefully applied in the study of the geometry and arithmetic of Xn(N). D. Mumford [14] proved the Hirzebruch's Proportionality Theorem in the non-compact case introducing a good singular Hermitian metric on an automorphic vector bundle on a smooth toroidal compactification of Xn(N) with N≥3.

We set

where

Let

be the universal abelian variety. An arithmetic toroidal compactification of Xn,m(N) was intensively investigated by R. Pink [15]. D. Mumford described very nicely a toroidal compactification of the universal elliptic curve X1,1(N) (cf.[1, pp.14–25]). The geometry of Xn,m(N) is closely related to the theory of Jacobi forms (cf. [3, 9-11]). Jacobi forms play an important role in the study of the geometric and arithmetic of Xn,m(N). We refer to [4, 23] for the theory of Jacobi forms.