MAASS-JACOBI FORMS

Jae-Hyun Yang

Abstract. In this paper, we introduce the notion of new automorphic forms on GL(n, ℝ)⋉ℝ^{(m,n)} generalizing that of automorphic forms on the general linear group GL(n, ℝ).

1. Introduction

Let n and m be two fixed positive integers. Let

Sp(n, ℝ) = { M ∈ ℝ^{(2n,2n)} | ^{t}MJ_{n}M = J_{n} }

be the real symplectic group of degree n, where

J_{n} := ( \begin{matrix}0 & E_{n} \\ −E_{n} & 0 \end{matrix} ) .

Here we denote by ^{t}A the transpose of any matrix A and by F^{(k,l)} the set of all k × l matrices with entries in a commutative ring F. Let

H_{n} := { Z ∈ ℂ^{(n,n)} | Z = ^{t}Z, Im Z > 0 }

be the Siegel upper half plane of degree n. Then Sp(n, ℝ) acts on H_{n} transitively by

(1.1)  M < Z >:= (AZ + B)(CZ + D)^{−1},

wher M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ) and Z ∈ H_{n}. We let

P_{n} := { Y ∈ ℝ^{(n,n)} | Y = ^{t}Y > 0 }

be the open convex cone of positive definite matrices of degree n in the Euclidean space ℝ^{n(n+1)/2}. Then the action (1.1) gives a natural action of the general linear group GL(n, ℝ) on P_{n} defined by

(1.2)  (g, Y) ⟼ g < Y >:= gY ^{t}g, g ∈ GL(n, ℝ), Y ∈ P_{n}.

Typeset by AMS-TEX

This work was supported financially by TGRC-KOSEF and Max-Planck Institut für Mathematik.

1991 Mathematics Subject Classification. Primary 11F55, 32C55, 32N10, 43A85, 43A90.

In his famous paper [S1], Selberg proved that the algebra 𝔻(P_{n}) of all differential operators on P_{n} which are invariant under the action (1.2) of GL(n, ℝ) has a commutative ring structure and is generated by n algebraically independent ones ( see Remark 4.13 ). Thus 𝔻(P_{n}) may be identified with the commutative ring ℂ[Z_{1}, ⋯, Z_{n}] of polynomials in n indeterminates. Using the action (1.2) and the GL(n, ℝ)-invariant differential operators on P_{n}, the concept of automorphic forms for GL(n, ℤ) is defined together with a certain suitable growth condition. These automorphic forms on GL(n, ℝ) have been investigated intensively by Langlands school ( cf. [B-J], [G], [J-L] ) adelically and by some people ( cf. [Gr], [I-T] ) analytically.

The aim of this paper is to introduce a new notion of automorphic forms on the new group GL(n, ℝ) ⋉ ℝ^{(m,n)} generalizing the concept of automorphic forms on GL(n, ℝ) and also the notion of the so-called Maass-Jacobi forms. Now I will explain the procedure in some detail. We consider the Heisenberg group

H_{ℝ}^{(n,m)} := { (λ, μ; κ) | λ, μ ∈ ℝ^{(m,n)}, κ ∈ ℝ^{(m,m)}, κ + μ^{t}λ symmetric }

endowed with the following multiplication law (λ, μ; κ) ∘ (λ′, μ′; κ′) := (λ + λ′, μ + μ′, κ + κ′ + λ^{t}μ′ − μ^{t}λ′).

We define the semidirect product of Sp(n, ℝ) and H_{ℝ}^{(n,m)}

(1.3)  G_{n,m}^{J}(ℝ) := Sp(n, ℝ) ⋉ H_{ℝ}^{(n,m)}

endowed with the following multiplication law (M, (λ, μ; κ)) · (M′, (λ′, μ′; κ′))

:= (MM′, (λ̃ + λ′, μ̃ + μ′,; κ + κ′ + λ̃^{t}μ′ − μ̃^{t}λ′)),

where M, M′ ∈ Sp(n, ℝ) and (λ̃, μ̃) := (λ, μ)M′. The group G_{n,m}^{J}(ℝ) is called the Jacobi group of degree (n, m). It is easy to see that the Jacobi group G_{n,m}^{J}(ℝ) acts on the homogeneous space H_{n} × ℂ^{(m,n)} transitively by

(1.4)  (M, (λ, μ; κ)) · (Z, W) := (M < Z >, (W + λZ + μ)(CZ + D)^{−1}),

where M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ), (λ, μ; κ) ∈ H_{ℝ}^{(n,m)} and (Z, W) ∈ H_{n} × ℂ^{(m,n)}.

We let

(1.5)  G_{n,m}(ℝ) := GL(n, ℝ) ⋉ ℝ^{(m,n)}

be the semidirect product of GL(n, ℝ) and the commutative additive group ℝ^{(m,n)} equipped with the following multiplication law

(1.6)  (g, a) ∗ (h, b) := (gh, a^{t}h^{−1} + b),

where g, h ∈ GL(n, ℝ) and a, b ∈ ℝ^{(m,n)}. Then the action (1.4) of G_{n,m}^{J}(ℝ) on H_{n} × ℂ^{(m,n)} gives a canonical action of G_{n,m}(ℝ) on the reductive homogeneous space P_{n} × ℝ^{(m,n)} given by

(1.7)  (g, a) · (Y, V) := (gY ^{t}g, (V + a)^{t}g),

where g ∈ GL(n, ℝ), a ∈ ℝ^{(m,n)}, Y ∈ P_{n} and V ∈ ℝ^{(m,n)}. At this stage we naturally consider the discrete subgroup

(1.8)  G_{n,m}(ℤ) := GL(n, ℤ) ⋉ ℤ^{(m,n)}

of G_{n,m}(ℝ) and investigate the algebra 𝔻(P_{n} × ℝ^{(m,n)}) of all differential operators on P_{n} × ℝ^{(m,n)} which is invariant under the action (1.7) of G_{n,m}(ℝ).

Now it is natural for us to introduce the new concept of automorphic forms for GL(n, ℤ) ⋉ ℤ^{(m,n)} generalizing that of well-known automorphic forms for GL(n, ℤ) and the new notion of Maass-Jacobi forms generalizing that of Maass wave forms on SL(2, ℝ). With some modifications, we may define the notion of Eisenstein series, L-functions and other related materials on G_{n,m}(ℝ) analytically or adelically.

The main result of this paper is the following theorem.

Main Theorem. The algebra 𝔻(P_{n} ×ℝ^{(m,n)}) is generated by the following invariant differential operators

D_{j} : = σ (( 2Y \frac{∂}{∂Y} )^{j}) , 1 ≤ j ≤ n

and

Ψ_{kp} : = \{ \frac{∂}{∂V} Y ^{t} (\frac{∂}{∂V} )\}_{kp} , 1 ≤ k ≤ p ≤ m.

These differential operators satisfies the following relations :

D_{i}D_{j} = D_{j}D_{i},
Ψ_{kp}Ψ_{lq} − Ψ_{lq}Ψ_{kp} = 0,
D_{j}Ψ_{kp} − Ψ_{kp}D_{j} ≠ 0

for all 1 ≤ i, j ≤ n, 1 ≤ k ≤ p ≤ m and 1 ≤ l ≤ q ≤ m. The operators D_{j}’s are algebraically independent and the operators Ψ_{kp}’s are also algebraically independent.

Here σ(A) denotes the trace of a square matrix A. For the precise definitions of D_{j} and Ψ_{kp}, we refer to (4.1), (4.17) and (4.18).

This paper is organized as follows. In Section 2, we obtain some basic properties of the new group G_{n,m}(ℝ) which are to be used in the subsequent sections. In Section 3, we give some geometric properties of P_{n} × ℝ^{(m,n)}. In Section 4, we investigate the algebra 𝔻(P_{n}×ℝ^{(m,n)}) of all G_{n,m}(ℝ)-invariant differential operators on P_{n} × ℝ^{(m,n)}. According to Helgason [He2], Theorem 4.9, p. 287, the problem of finding the complete description of the algebra 𝔻(P_{n} × ℝ^{(m,n)}) is reduced to that of finding all the invariants of the action of the orthogonal group O(n) on the symmetric algebra S(𝔭) of 𝔭 induced from the adjoint action Ad of O(n) on 𝔭 given by

(1.9)  Ad(k)((X, Z)) := (kX^{t}k, Z^{t}k), k ∈ O(n), (X, Z) ∈ 𝔭.

Here 𝔭 denotes a certain subspace of the Lie algebra of G_{n,m}(ℝ) which can be identified with the tangent space of P_{n}×ℝ^{(m,n)} at the origin (E_{n}, 0). For more detail, we refer the reader to (4.5) and (4.6) in Section 4. We give a complete description of algebraically independent generators of the subalgebra S(𝔭)^{K} of K-invariants and generators of the algebra 𝔻(P_{n}×ℝ^{(m,n)}). We also provide many invariant differential operators on P_{n} × ℝ^{(m,n)}. And we show that for any positive integers n and m, the algebra 𝔻(P_{n} × ℝ^{(m,n)}) is not commutative and hence P_{n} × ℝ^{(m,n)} is not a weakly symmetric Riemannian manifold in the sense of Selberg [S1], p. 51. It is well known that if G is a noncompact semisimple real Lie group and K is a maximal compact subgroup of G, then the algebra 𝔻(G/K) of all G-invariant differential operators on G/K is a commutative ring and that actually it is a polynomial ring with rank (G) algebraically independent generators. But in our case G_{n,m}(ℝ) is not semisimple and the algebra 𝔻(G_{n,m}(ℝ)/O(n)) = 𝔻(P_{n} × ℝ^{(m,n)}) is not commutative. So we need a new approach to the study of subjects related to 𝔻(P_{n} × ℝ^{(m,n)}). In Section 5, we define the spherical functions on G_{n,m}(ℝ) and characterize these functions following the general theory of spherical functions on a Lie group. We describe the spherical functions on G_{n,m}(ℝ) explicitly. In Section 6, using the results of Section 2 and 4, we define the concept of new automorphic forms on the new group G_{n,m}(ℝ). In Section 7, we define the notion of Maass-Jacobi forms on the group SL(n, ℝ) ⋉ ℝ^{(m,n)}. This notion generalizes that of Maass wave forms on SL(2, ℝ). The reason why I name “Maass-Jacobi” to such new automorphic forms is that the new automorphic forms are implicitly related to the concepts of Maass wave forms ( cf. [M1] ) and Jacobi forms ( cf. [E-Z], [K], [Y1-6], [Z] ). We deal with the simple case n = 2 and m = 1. In this case, we characterize Maass-Jacobi forms as smooth eigenfunctions on ℍ × ℂ satisfying a certain invariance property, where ℍ denotes the Poincaré upper half plane. Also we discuss the actions of the group SL_{2,1}(ℝ) := SL(2, ℝ) ⋉ ℝ^{(1,2)} on SP_{2} × ℝ^{(1,2)} and ℍ × ℂ ( cf. see (7.4) and (7.5) ). We provides the algebra of all differential operators on SP_{2} × ℝ^{(1,2)} or ℍ × ℂ invariant under the actions (7.4) and (7.5) of SL_{2,1}(ℝ) without proof. And we present SL_{2,1}(ℝ)-invariant Riemannian metrics on ℍ × ℂ and SP_{2} × ℝ^{(1,2)} together with their Laplace-Beltrami operators. The detail can be found in [Y7-8]. In the final section, we briefly remark a geometrical aspect of the quotient space G_{n,m}(ℤ)\P_{n} × ℝ^{(m,n)} and provide some problems related to our subjects.

Acknowledgement. I would like to thank Professor R. Bott, W. Schmid, R. Howe, S. Helgason and B. Kostant for some useful discussions and their kind advices about the invariant theory. This work was mostly done during my stay in Cambridge. I am very grateful to Department of Mathematics, Harvard University for its wonderful hospitality during my stay in Cambridge. The final writing was completed in a way during my stay at Max-Planck-Institut für Mathematik in Bonn. I would like to give my hearty thanks to the institute for its financial support and its wonderful hospitality during my stay in Bonn.

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. The symbol “:=” means that the expression on the right is the definition of that on the left. 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. E_{n} denotes the identity matrix of degree n.

2. The Group GL(n, ℝ) ⋉ ℝ^{(m,n)}

In this section, we obtain some basic properties of the group

G_{n,m}(ℝ) := GL(n, ℝ) ⋉ ℝ^{(m,n)}

endowed with the multiplication law (1.6). Clearly the stabilizer K of G_{n,m}(ℝ) at (E_{n}, 0) is given by

K := { (k, 0) ∈ G_{n,m}(ℝ) | k ∈ O(n) }

which is isomorphic to the orthogonal group O(n). Since the action (1.7) of G_{n,m}(ℝ) on P_{n} ×ℝ^{(m,n)} is transitive, P_{n} ×ℝ^{(m,n)} is diffeomorphic to the homogeneous space G_{n,m}(ℝ)/O(n). We note that G_{n,m}(ℝ) acts on P_{n} × ℝ^{(m,n)} effectively. It is easy to see that the Lie algebra 𝔤 of G_{n,m}(ℝ) is given by

(2.1)  𝔤 = { (X, Z) | X ∈ ℝ^{(n,n)}, Z ∈ ℝ^{(m,n)} }

equipped with the following Lie bracket

(2.2)  [(X_{1}, Z_{1}), (X_{2}, Z_{2})] := ([X_{1}, X_{2}]_{0}, Z_{2} ^{t}X_{1} − Z_{1} ^{t}X_{2}),

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. The adjoint representation Ad of G_{n,m}(ℝ) is given by

(2.3)  Ad ((g, a))(X, Z) = (gXg^{−1}, (Z − a^{t}X)^{t}g),

where (g, a) ∈ G_{n,m}(ℝ) and (X, Z) ∈ 𝔤. And the adjoint representation ad of 𝔤 on End (𝔤) is given by

(2.4)  ad ((X, Z))((X_{1}, Z_{1})) = [(X, Z), (X_{1}, Z_{1})].

We easily see that the Killing form B of 𝔤 is given by

(2.5)  B((X_{1}, Z_{1}), (X_{2}, Z_{2})) = 4(n + m + 1) σ(X_{1}X_{2}).

The Lie algebra 𝔨 of K is

(2.6)  𝔨 = { (X, 0) ∈ 𝔤 | X + ^{t}X = 0 } .

We let 𝔭 be the subspace of 𝔤 defined by

(2.7)  𝔭 := { (X, Z) ∈ 𝔤 | X = ^{t}X ∈ ℝ^{(n,n)}, Z ∈ ℝ^{(m,n)} } .

Then we have the following relation

(2.8)  [𝔨, 𝔨] ⊂ 𝔨 and [𝔨, 𝔭] ⊂ 𝔭.

In addition, we have

(2.9)  𝔤 = 𝔨 ⊕ 𝔭 ( direct sum ).

Thus the coset space G_{n,m}(ℝ)/O(n) ≅ P_{n} × ℝ^{(m,n)} is a reductive homogeneous space in the sense of [He2], p. 284. We note that the restriction of the Killing form B to 𝔨 is negative definite and the restriction of B to the abelian subalgebra 𝔯 := { (0, Y) ∈ 𝔤 } is identically zero. Since 𝔯 is the radical of B, B is degenerate. The subspace 𝔭 of 𝔤 may be identified with the tangent space of the reductive homogeneous space P_{n} × ℝ^{(m,n)} = G_{n,m}(ℝ)/O(n) at the origin (E_{n}, 0).

An Iwasawa decomposition of the group G_{n,m}(ℝ) is given by

(2.10)  G_{n,m}(ℝ) = NAK, N = N_{0} ⋉ ℝ^{(m,n)},

where A is an abelian subgroup of G_{n,m}(ℝ) consisting of all n × n real diagonal matrices with positive real diagonals and N_{0} is the subgroup of GL(n, ℝ) consisting of all n × n upper triangular real matrices whose diagonal entries are all equal to one. The Iwasawa decomposition of the Lie algebra 𝔤 of G_{n,m}(ℝ) is given by

(2.11)  𝔤 = 𝔫 + 𝔞 + 𝔨, 𝔫 = 𝔫_{0} ⊕ ℝ^{(m,n)},

where 𝔞 is the subalgebra of 𝔤 consisting of all n × n real diagonal matrices and 𝔫_{0} denotes the subalgebra of 𝔤 consisting of all n × n upper triangular real matrices whose diagonal entries are all equal to zero. In fact, 𝔞 is the Lie algebra of A and 𝔫 is the Lie algebra of N.

Now we will determine the roots of 𝔤. Let E_{ij} ( 1 ≤ i, j ≤ n) be the n × n matrix with entry 1 where the i-throw and the j-th column meet, all other entries being 0. We let F_{kl} ( 1 ≤ k ≤ m, 1 ≤ l ≤ n) be the m × n matrix with 1 in the (k, l)-entry and zero elsewhere. We put

H_{i} : = (E_{ii}, 0), 1 ≤ i ≤ n,
E_{ij}^{0} : = (E_{ij}, 0), 1 ≤ i, j ≤ n, i ≠ j,
F_{kl}^{0} : = (0, F_{kl}), 1 ≤ k ≤ m, 1 ≤ l ≤ n.

Then the set { H_{i}, E_{ij}^{0}, F_{kl}^{0} } forms a basis for 𝔤. Clearly 𝔞 = ∑_{i=1}^{n} ℝ H_{i} and 𝔫 = ∑_{i<j} ℂ E_{ij}^{0} + ∑_{k,l} ℝ F_{kl}^{0}. Using the rules

E_{ij}E_{kl} = δ_{jk}E_{il}, [E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{li}E_{kl},

we obtain the following

[H, E_{ij}^{0}] = ( e_{i}(H) − e_{j}(H)) E_{ij}^{0}, i ≠ j,
[H, F_{kl}^{0}] = e_{l}(H)F_{kl}^{0}, 1 ≤ k ≤ m, 1 ≤ l ≤ n

for all H ∈ 𝔞, where e_{i} ( 1 ≤ i ≤ n) is the linear functional e_{i} : 𝔞 ⟶ ℂ on 𝔞 defined by

e_{i}(H_{j}) = δ_{ij}, 1 ≤ j ≤ n.

Thus the roots of 𝔤 relative to 𝔞 are given by

(2.12)  ±e_{l}, ±(e_{i} − e_{j}), 1 ≤ l ≤ n, 1 ≤ i < j ≤ n.

We denote by Σ := Σ(𝔤, 𝔞) the set of all roots of 𝔤 relative to 𝔞. The set Σ^{+} of positive roots of 𝔤 relative to 𝔞 is given by

(2.13)  Σ^{+} = { e_{l}, e_{i} − e_{j} | 1 ≤ l ≤ n, 1 ≤ i < j ≤ n } .

The roots

e_{1} − e_{2}, e_{2} − e_{3}, ⋯, e_{n−1} − e_{n}, e_{n}

form the set of simple roots. We observe that the root subspaces of 𝔤 are given by

𝔤_{−e_{l}} = 𝔤_{e_{l}} = ∑_{k=1}^{m} ℝ F_{kl}^{0}, 1 ≤ l ≤ n,
𝔤_{e_{i}−e_{j}} = ℝ E_{ij}^{0}, 1 ≤ i, j ≤ n, i ≠ j.

So we obtain

(2.14)  dim 𝔤_{e_{l}} = dim 𝔤_{−e_{l}} = m, dim 𝔤_{e_{i}−e_{j}} = 1, 1 ≤ i, j, l ≤ n, i ≠ j.

We define

(2.15)  ρ := \frac{1}{2} ∑_{α∈Σ^{+}} ( dim 𝔤_{α}) · α.

By an easy computation, we have

(2.16)  ρ = \frac{1}{2} ∑_{h=1}^{n}(n + m + 1 − 2h) e_{h}.

3. Some geometry on P_{n} × ℝ^{(m,n)}

For a variable (Y, V) ∈ P_{n} × ℝ^{(m,n)} with Y ∈ P_{n} and V ∈ ℝ^{(m,n)}, we put

Y = (y_{μν}), V = (v_{kl}), dY = (dy_{μν}), dV = (dv_{kl})

and

[dY] := ∏_{μ≤ν} dy_{μν}, [dV] := ∏_{k,l} dv_{kl},

where 1 ≤ μ, ν, l ≤ n and 1 ≤ k ≤ m.

Lemma 3.1. The metric ds^{2} on P_{n} × ℝ^{(m,n)} defined by

(3.1)  ds^{2} := σ(Y^{−1}dY Y^{−1}dY) + σ(Y^{−1} ^{t}(dV) dV)

is invariant under the action (1.7) of G_{n,m}(ℝ) on P_{n} × ℝ^{(m,n)}. So G_{n,m}(ℝ) is a subgroup of the isometry group of the Riemannian manifold (P_{n} × ℝ^{(m,n)}, ds^{2}). The Laplace-Beltrami operator of (P_{n} × ℝ^{(m,n)}, ds^{2}) is a differential operator of order 2 which is invariant under the action (1.7) of G_{n,m}(ℝ).

Proof. For a fixed element (g, a) ∈ G_{n,m}(ℝ), we set

(Y^{∗}, V^{∗}) := (g, a) · (Y, V).

The first statement follows immediately from the fact that

dY^{∗} = gdY g^{−1} and dV^{∗} = dV ^{t}g.

The last statement follows from the fact that a diffeomorphism Φ on a Riemannian manifold leaves the Laplace-Beltrami operator invariant if and only if it is an isometry ( see [He1], proposition 2, p. 245 ). □

Lemma 3.2. The following volume element dv on P_{n} × ℝ^{(m,n)} defined by

(3.2)  dv := ( det Y )^{− \frac{n+m+1}{2}} [dY][dV]

is invariant under the action (1.7) of G_{n,m}(ℝ).

Proof. For a fixed element (g, a) ∈ G_{n,m}(ℝ), we set

(Y^{∗}, V^{∗}) := (g, a) · (Y, V) = ( gY ^{t}g, (V + a)^{t}g).

Let \frac{∂(Y^{∗},V^{∗})}{∂(Y,V)} be the Jacobian determinant of the action (1.7) of G_{n,m}(ℝ) on P_{n} × ℝ^{(m,n)}. It is known that the Jacobian determinant of the action Y ⟼ Y^{∗} is given by

\frac{∂(Y^{∗})}{∂(Y)} = ( det g)^{n+1}.

Take the diagonal matrix g = (d_{1}, ⋯, d_{n}) with distinct real numbers d_{i}. Obviously if a = (a_{kl}), V = (v_{kl}) and V^{∗} = (v_{kl}^{∗}), then v_{kl}^{∗} = (v_{kl} + a_{kl})d_{l} for all k, l. Thus we have

(3.3)  \frac{∂(V^{∗})}{∂(V)} = (d_{1} ⋯ d_{n})^{m} = ( det g)^{m}.

Since the set of all n×n real matices whose eigenvalues are all distinct is everywhere dense in GL(n, ℝ) , and \frac{∂(V^{∗})}{∂(V)} is a rational function, the relation (3.3) holds for any g ∈ GL(n, ℝ). It is easy to see that

\frac{∂(Y^{∗}, V^{∗})}{∂(Y, V)} = \frac{∂(Y^{∗})}{∂(Y)} · \frac{∂(V^{∗})}{∂(V)}.

Thus we obtain

[dY^{∗}][dV^{∗}] = ( det g)^{m+n+1}[dY][dV].

Since det Y^{∗} = ( det g)^{2} det Y, we have

( det Y^{∗})^{− \frac{m+n+1}{2}} [dY^{∗}][dV^{∗}] = ( det Y )^{− \frac{m+n+1}{2}} [dY][dV].

Hence the volume element (3.2) is invariant under the action (1.7). □

Theorem 3.3. Any geodesic through the origin (E_{n}, 0) is of the form

γ(t) = ( λ(2t)[k], Z (∫_{0}^{t} λ(t − s)ds) [k]) ,

where k is a fixed element of O(n), t is a real variable, λ_{1}, λ_{2}, ⋯, λ_{n} are fixed real numbers but not all zero and

λ(t) := diag (e^{λ_{1}t}, ⋯, e^{λ_{n}t}).

Furthermore, the tangent vector γ′(0) of the geodesic γ(t) at (E_{n}, 0) is (D[k], Z), where D = diag (2λ_{1}, ⋯, 2λ_{n}).

Proof. Let W := (X, Z) be an element of 𝔭 with X ≠ 0. Then the curve

α(t) := exp tW = ( e^{tX}, Z (∫_{0}^{t} e^{−sX}ds)) , t ∈ ℝ

is a geodesic in G_{n,m}(ℝ) with α′(0) = W passing through the identity of G_{n,m}(ℝ). Thus the curve

γ(t) := α(t) · (E_{n}, 0) = ( e^{2tX}, Z (∫_{0}^{t} e^{−sX}ds) e^{tX} )

is a geodesic in P_{n} × ℝ^{(m,n)} passing through the origin (E_{n}, 0). Since X is a symmetric real matrix, there is a diagonal matrix Λ = diag (λ_{1}, ⋯, λ_{n}) such that

X = ^{t}kΛk for some k ∈ O(n),

where λ_{1}, ⋯, λ_{n} are real numbers and not all zero. Thus we may write

γ(t) = ( (δ_{kl}e^{2λ_{k}t})[k], Z (∫_{0}^{t} e^{(t−s)Λ}ds) [k] ) .

Hence this completes the proof. □

4. Proof of Main Theorem

In this section, we investigate the differential operators on P_{n} × ℝ^{(m,n)} which are invariant under the action (1.7) of G_{n,m}(ℝ).

We set

(4.1)  \frac{∂}{∂Y} := (\frac{1 + δ_{μν}}{2} \frac{∂}{∂y_{μν}} ) , \frac{∂}{∂V} := (\frac{∂}{∂v_{kl}} ) ,

where 1 ≤ μ, ν, l ≤ n, 1 ≤ k ≤ m and δ_{μν} denotes the Kronecker delta symbol. For a fixed element (g, a) ∈ G_{n,m}(ℝ), we set

(Y^{∗}, V^{∗}) := (g, a) · (Y, V) = (gY ^{t}g, (V + a)^{t}g),

where (Y, V) ∈ P_{n} × ℝ^{(m,n)}. Then we have

(4.2)  Y^{∗} = gY ^{t}g, V^{∗} = (V + a)^{t}g

and

(4.3)  \frac{∂}{∂Y^{∗}} = ^{t}g^{−1} \frac{∂}{∂Y} g^{−1}, \frac{∂}{∂V^{∗}} = \frac{∂}{∂V} · g^{−1}.

In [Y4], I introduced the following differential operator M_{n,m,M} on P_{n} × ℝ^{(m,n)} defined by

(4.4)  M_{n,m,M} := det (Y) · det (\frac{∂}{∂Y} + \frac{1}{8π} ^{t} (\frac{∂}{∂V} ) M^{−1} (\frac{∂}{∂V} )) ,

where M is a positive definite, symmetric half-integral matrix of degree m. This differential operator characterized singular Jacobi forms. For more detail, we refer the reader to [Y4].

Lemma 4.1. The differential operator M_{n,m,M} is invariant under the action (1.7) of G_{n,m}(ℝ).

Proof. It follows immediately from (4.2) and (4.3). □

We recall that the adjoint action of K := O(n) on 𝔭 is given by

(4.5)  Ad (k)((X, Z)) = (kX^{t}k, Z^{t}k),

where k ∈ K and (X, Z) ∈ 𝔭 with X = ^{t}X ∈ ℝ^{(n,n)}. The action (4.5) induces uniquely the action ρ of K on the symmetric algebra S(𝔭)

(4.6)  ρ : K ⟶ Aut (S(𝔭)).

Now we have the following natural questions :

Problem A. Find the algebra 𝔻(P_{n} × ℝ^{(m,n)}) of all differential operators on P_{n} × ℝ^{(m,n)} which are invariant under the action (1.7) of G_{n,m}(ℝ) on P_{n} × ℝ^{(m,n)}. In other words, this problem is to find a set of algebraically independent differential operators which generate the algebra 𝔻(P_{n} × ℝ^{(m,n)}).

Problem B. Compute all the joint eigenfunctions of 𝔻(P_{n} × ℝ^{(m,n)}).

According to [He2], Problem A is reduced to the following problem :

Problem A’. Find all the invariants of the action (4.6) of K on S(𝔭).

We define the nondegerate symmetric bilinear form B̃ on 𝔭 by

(4.7) B̃((X_{1}, Z_{1}), (X_{2}, Z_{2})) := σ(X_{1}X_{2}) + σ(Z_{1} ^{t}Z_{2}), (X_{i}, Z_{i}) ∈ 𝔭, i = 1, 2.

We identify the dual space 𝔭^{∗} of 𝔭 with 𝔭 via B̃ by letting v ↦ v^{∗} ( v ∈ 𝔭), where v^{∗}(w) = B̃(w, v) for all w ∈ 𝔭. Let Ad^{∗} be the contragredient of the adjoint representation Ad of K defined by

(4.8) Ad^{∗}(k)v^{∗}(w) := v^{∗}(Ad (k^{−1})w) = B̃(Ad (k^{−1})w, v), k ∈ K, v, w ∈ 𝔭.

We also identify 𝔭^{∗∗} with 𝔭 via B̃ by letting v ↦ v^{∗∗} ( v ∈ 𝔭), where v^{∗∗}(w^{∗}) = w^{∗}(v) = B̃(v, w), v, w ∈ 𝔭. Let Pol (𝔭^{∗}) = S(𝔭^{∗∗}) be the polynomial algebra on 𝔭^{∗}, i.e., the algebra of all polynomial functions on 𝔭^{∗}. Then K acts on Pol (𝔭^{∗}) by

(4.9)  Ad^{0}(k)P (v) := P (Ad(k^{−1})v), k ∈ K, P ∈ Pol (𝔭^{∗}), v ∈ 𝔭.

Then it is easy to see that S(𝔭) is isomorphic to Pol (𝔭^{∗}) and S(𝔭)^{K} is isomorphic to Pol (𝔭^{∗})^{K}. Here S(𝔭)^{K} denotes the subalgebra of all K-invariants under the action (4.6) of K and Pol (𝔭^{∗})^{K} denotes the subalgebra of all K-invariants under the action (4.9) of K.

By the above argument, the problem of finding the K-invariants of the action (4.6) of K is equivalent to that of finding the polynomials on 𝔭 which are invariant under the action (4.5) of K.

Before we give a complete set of algebraically independent generators of the algebra 𝔻(P_{n} × ℝ^{(m,n)}), we give some examples of K-invariants under the action (4.6) and invariant differential operators on P_{n} × ℝ^{(m,n)}. Also we describe the relation between the two invariants.

For variables (X, Z) in 𝔭, we define the polynomials P_{j} and Q_{k} on 𝔭 by

(4.10)  P_{j}(X, Z) := σ(X^{j}), 1 ≤ j ≤ n,
(4.11)  ξ_{kp}(X, Z) := (Z^{t}Z)_{kp}, 1 ≤ k ≤ p ≤ m

and

(4.12)  Q_{k}(X, Z) := σ((^{t}ZZ)^{k})), 1 ≤ k ≤ m,

where (Z^{t}Z)_{kp} denotes the (k, p)-entry of the matrix function Z^{t}Z.

And we define the polynomials R_{jk} on 𝔭 by

(4.13)  R_{jk}(X, Z) := σ(X^{j}(^{t}ZZ)^{k}), 1 ≤ j ≤ n, 1 ≤ k ≤ m.

For any real matrix S of degree m, we define the polynomials M_{j,S}, Q_{k,S} and R_{j,k,l,S} on 𝔭 by

(4.14)  M_{j,S}(X, Z) := σ ((X + ^{t}ZSZ)^{j}) , 1 ≤ j ≤ n,

(4.15)  Q_{k,S}(X, Z) := σ ((^{t}ZSZ)^{k}) , 1 ≤ k ≤ m

and

(4.16)  R_{j,k,l,S}(X, Z) := σ(X^{j}(^{t}ZSZ)^{k}(X + ^{t}ZSZ)^{l}), 1 ≤ l ≤ n.

Proposition 4.2. The set { P_{j}, ξ_{kp} | 1 ≤ j ≤ n, 1 ≤ k ≤ p ≤ m } is a set of algebraically independent K-invariants. Furthermore the polynomials Q_{k}, R_{jk}, M_{j,S}, Q_{k,S} and R_{j,k,l,S} are also K-invariants, where S is any real matrix of degree m.

Proof. The proof follows immediately from (4.5) and (4.6). □

We define the differential operators D_{j} and L_{k} on P_{n} × ℝ^{(m,n)} by

(4.17)  D_{j} := σ (( 2Y \frac{∂}{∂Y} )^{j}) , 1 ≤ j ≤ n,

(4.18)  Ψ_{kp} := \{ \frac{∂}{∂V} Y ^{t} (\frac{∂}{∂V} )\}_{kp} , 1 ≤ k ≤ p ≤ m

and

(4.19)  L_{k} := σ (\{Y ^{t} (\frac{∂}{∂V} ) \frac{∂}{∂V} \}^{k}) , 1 ≤ k ≤ m.

Also we define the differential operators S_{jk} by

(4.20)  S_{jk} := σ (( 2Y \frac{∂}{∂Y} )^{j} \{Y ^{t} (\frac{∂}{∂V} ) \frac{∂}{∂V} \}^{k}) ,

where 1 ≤ j ≤ n and 1 ≤ k ≤ m.

For any real matrix S of degree m, we define the differential operators Φ_{j,S} and L_{k,S} by

(4.21)  Φ_{j,S} := σ ( \{ Y ( 2\frac{∂}{∂Y} + ^{t} (\frac{∂}{∂V} ) S (\frac{∂}{∂V} ))\}^{j}) , 1 ≤ j ≤ n,

(4.22)  L_{k,S} := σ ( \{ Y ^{t} (\frac{∂}{∂V} ) S (\frac{∂}{∂V} )\}^{k}) , 1 ≤ k ≤ m

and

(4.23)  Φ_{j,k,l,S}(X, Z) :=

σ ( ( 2Y \frac{∂}{∂Y} )^{j} (Y ^{t} (\frac{∂}{∂V} ) S (\frac{∂}{∂V} ))^{k} \{Y ( 2\frac{∂}{∂Y} + ^{t} (\frac{∂}{∂V} ) S (\frac{∂}{∂V} ))\}^{l}) .

Proposition 4.3. The set { D_{j}, Ψ_{kp} | 1 ≤ j ≤ n, 1 ≤ k ≤ p ≤ m } is a set of algebraically independent G_{n,m}(ℝ)-invariant differential operators. Furthermore, L_{k}, S_{jk}, Φ_{j,S}, L_{k,S} and Φ_{j,k,l,S} ( 1 ≤ j ≤ n, 1 ≤ k ≤ m) are invariant differential operators under the action (1.7) of G_{n,m}(ℝ).

Proof. The proof follows immediately from (4.2) and (4.3). □

We recall that S(𝔭)^{K} is the algebra of all invariants of the action (4.6) of K. According to [He2], Theorem 4.9, p. 287, there exists a canonical linear bijection λ of S(𝔭)^{K} onto 𝔻(P_{n} × ℝ^{(m,n)}). We recall that if (ξ_{α}) is any basis of 𝔭 and P ∈ S(𝔭)^{K}, then

(4.24)  ( D_{λ(P)}f ) (gK) = [P (\frac{∂}{∂t_{α}} ) f (g exp (∑_{α=1}^{N} t_{α}ξ_{α}) K)]_{(t_{α})=0} ,

where f ∈ C^{∞}(P_{n} × ℝ^{(m,n)}) and N = n(n + 1)/2 + mn is the dimension of 𝔭.

Proposition 4.4. D_{j}, Ψ_{kp}, L_{k}, S_{jk}, Φ_{j,S}, L_{k,S} and Φ_{j,k,l,S} are the G_{n,m}(ℝ)-invariant differential operators on P_{n}×ℝ^{(m,n)} corresponding to the K-invariants P_{j}, ξ_{kp}, Q_{k}, R_{jk}, M_{j,S}, Q_{k,S} and R_{j,k,l,S} respectively under a canonical linear bijection λ : S(𝔭)^{K} ⟶ 𝔻(P_{n} × ℝ^{(m,n)}).

Proof. Using the bijection λ in (4.24), it follows from a direct computation. □

Now we prove the main theorem mentioned in the introduction. The proof will be divided into several steps.

First we let G be a reductive real group and H a closed subgroup of G with their Lie algebras 𝔤_{0} and 𝔥 respectively. We assume that 𝔤_{0} has a decomposition 𝔤_{0} = 𝔥 + 𝔪 ( the direct sum ) satisfying the following conditions

[𝔥, 𝔥] ⊂ 𝔥, [𝔥, 𝔪] ⊂ 𝔪, [𝔪, 𝔪] ⊂ 𝔥.

The adjoint representation Ad_{G} of H on 𝔪 induces the action Ad_{G} of H on the polynomial algebra Pol(𝔪) on 𝔪. Let Pol(𝔪)^{H} be the subalgebra of all Ad_{G}(H)- invariants in Pol(𝔪) and 𝔻(G/H) the algebra of all G-invariant differential operators on the homogeneous space G/H. According to [He2], there is a canonical linear bijection Pol(𝔪)^{H} ⟶ 𝔻(G/H)

(*)  P ⟼ D_{P} , P ∈ Pol(𝔪)^{H}

defined in a similiar way as (4.24).

Lemma 4.5. Assume that 𝔻(G/H) is commutative. Let R_{1}, R_{2}, ⋯, R_{q} be algebraically independent generators of Pol(𝔪)^{H} and R̂_{i} := D_{R_{i}} ( 1 ≤ i ≤ q) the elements of 𝔻(G/H) given by (*). Then R̂_{1}, ⋯, R̂_{q} are algebraically independent generators of 𝔻(G/H).

Proof. We can prove the above lemma following the line of the argument in [G-H], Lemma 4.2 and [He1], pp.269-270. For the convenience of the reader, we will give a proof of the above lemma. First we show that the operators R̂_{i} := D_{R_{i}} ( 1 ≤ i ≤ q) are algebraically independent. Suppose that they are not algebraically independent. Then there exists a nonzero polynomial

P = ∑ a_{n_{1},··· ,n_{q}}x_{1}^{n_{1}} ⋯ x_{q}^{n_{q}}

such that P (R̂_{1}, ⋯, R̂_{q}) = 0. Let r_{i} ( 1 ≤ i ≤ q) be the degree of the polynomial R_{i}. We put N := max(∑ r_{i}n_{i}), where the maximum is taken over the set of all q-tuples (n_{1}, ⋯, n_{q}) with a_{n_{1},··· ,n_{q}} ≠ 0. We consider the following two polynomials S and Q defined by

S := ∑ a_{n_{1},··· ,n_{q}}R_{1}^{n_{1}} ⋯ R_{q}^{n_{q}}

and

Q := ∑ ; ∑ d_{i}n_{i}=N a_{n_{1},··· ,n_{q}}R_{1}^{n_{1}} ⋯ R_{q}^{n_{q}}.

Obviously Q ≠ 0 by assumption and the polynomial R := S − Q has degree less than N. It is easy to see that the operator

U := ∑ a_{n_{1},··· ,n_{q}} R̂_{1}^{n_{1}} ⋯ R̂_{q}^{n_{q}} − D_{S}

has order less than N. And we note that since P (R̂_{1}, ⋯, R̂_{q}) = 0, we have

U = 0 − D_{S} = −D_{Q} − D_{R}.

Therefore the operator U has order N by the linear bijection (*). This contradicts our assumption. It remains to prove that the operators R̂_{i} ( 1 ≤ i ≤ q) are generators. Suppose D is an element of 𝔻(G/H). Then there exists a polynomial F ∈ Pol(𝔪)^{H} such that D = D_{F} via the linear bijection (*). We write the polynomial F as

F = ∑ β_{m_{1},··· ,m_{q}}R_{1}^{m_{1}} ⋯ R_{q}^{m_{q}}, β_{m_{1},··· ,m_{q}} ∈ ℂ.

We put M := max(∑ r_{i}m_{i}), where the maximum is taken over the set of all qtuple (m_{1}, ⋯, m_{q}) with β_{m_{1},··· ,m_{q}} ≠ 0. Then it is easy to see that the differential operator

D − ∑ ; ∑ r_{i}m_{i}=M βR̂_{1}^{N_{1}} ⋯ R̂_{q}^{N_{q}}

is of lower order than D. Hence it follows from the induction on the order that the R̂_{i} ( 1 ≤ i ≤ q) are generators of 𝔻(G/H). □

Secondly, we review the well-known classical results. Let V_{1} := Symm (ℝ^{n}) be the vector space of all n × n symmetric real matrices. The polynomial algebra Pol (V_{1}) on V_{1} has a standard grading :

Pol (V_{1}) ≅ ∑_{d=0}^{∞} Pol^{d}(V_{1}).

The elements of Pol^{d} (V_{1}) ( d ≥ 0 ) is called homogeneous of degree d. We consider the action of K on V_{1} given by

(4.25)  (k, X) ⟼ kX^{t}k, k ∈ K, X ∈ V_{1}.

Then this action extends uniquely to the action of K on Pol (V_{1})

(4.26)  ρ_{1} : K ⟶ Aut (Pol (V_{1}))

by algebra automorphisms preserving the grading. For a typical element X = (x_{ij}) ∈ V_{1} with x_{ij} = x_{ji} ( 1 ≤ i, j ≤ n), we define the polynomials P_{j} on V_{1} by

(4.27)  P_{j}(X) := σ(X^{j}), 1 ≤ j ≤ n.

Theorem 4.6. Let Pol (V_{1})^{K} be the subalgebra of Pol (V_{1}) consisting of Kinvariants under the action (4.26) of K. Then the polynomials P_{j}(X) ( 1 ≤ j ≤ n) are algebraically independent generators of Pol (V_{1})^{K}.

Proof. It is obvious that each polynomial P_{j}(X) is an element of Pol (V_{1})^{K}. The GL(n, ℝ)-invariant differential operators D_{j}^{0} for the action (1.2) of GL(n, ℝ) on P_{n} corresponding to the K-invariant polynomials P_{j}(X) under a canonical linear bijection

λ_{1} : Pol (V_{1})^{K} ⟶ 𝔻(P_{n}), P_{n} = GL(n, ℝ)/K

are given by

D_{j}^{0} = σ ((2Y \frac{∂}{∂Y} )^{j}) , 1 ≤ j ≤ n.

According to Maass [M2], pp. 64-66, D_{j}^{0} ( 1 ≤ j ≤ n) are algebraically independent generators of the algebra 𝔻(P_{n}) consisting of all GL(n, ℝ)-invariant differential operators on P_{n} for the action (1.2). According to Lemma 4.5 and the commutativity of the algebra 𝔻(P_{n}) ( cf. [M2] and [S1] ), the polynomials P_{j}(X) ( 1 ≤ j ≤ n) must be algebraically independent generators of Pol (V_{1}). □

Another proof of Theorem 4.6. We first observe that K := O(n) is a maximal compact subgroup of GL(n, ℝ). Then the Lie algebra 𝔤𝔩(n, ℝ) of GL(n, ℝ) has the Cartan decomposition

𝔤𝔩(n, ℝ) = 𝔨 + V_{1}, V_{1} := Symm(ℝ^{n}),

where 𝔨 is the subalgebra of 𝔤𝔩(n, ℝ) consisting of all skew-symmetric real matrices of degree n. Let 𝔞 be the maximal abelian subspace of V_{1} consisting of all n × n diagonal real matrices. It is well known that the Weyl group W of GL(n, ℝ) is isomorphic to the symmetric group S_{n} of degree n. A classical result on invariant theory tells us that the following symmetric polynomials

C_{1}(λ) := λ_{1} + ⋯ + λ_{n}, C_{2}(λ) := λ_{1}^{2} + ⋯ λ_{n}^{2}, ⋯, C_{n}(λ) := λ_{1}^{n} + ⋯ + λ_{n}^{n}

are algebraically independent generators of the algebra Pol(𝔞)^{W} of all W -invariant polynomials on 𝔞, where λ_{1}, ⋯, λ_{n} are n variables on 𝔞. It is also known that Pol(V_{1})^{K} is isomorphic to Pol(𝔞)^{W} via the restriction to 𝔞 ( cf. [He2], corollary 5.12, p. 299 ). It is easily seen that the polynomials on V_{1} corresponding to W -invariant polynomials C_{j}(λ), 1 ≤ j ≤ n are given by P_{j}(X) := σ(X^{j}), 1 ≤ j ≤ n respectively. Hence the polynomials P_{1}, ⋯, P_{n} are algebraically independent generators of Pol(V_{1})^{K}. □

Thirdly we let V_{2} := ℝ^{(m,n)} be the vector space of all m × n real matrices. The polynomial algebra Pol (V_{2}) on V_{2} has a natural grading

Pol (V_{2}) ≅ ∑_{d=0}^{∞} Pol^{d}(V_{2}).

We consider the action of K on V_{2} given by

(4.28)  (k, Z) ⟼ Z^{t}k, k ∈ K, Z ∈ V_{2}.

Then the above action extends uniquely to the action of K on Pol (V_{2})

(4.29)  ρ_{2} : K ⟶ Aut(Pol (V_{2}))

by algebra automorphisms preserving the grading. For a typical element Z = (z_{kl}) ∈ V_{2} with 1 ≤ k ≤ m, 1 ≤ l ≤ n, we define the polynomials ξ_{kp} on V_{2} by

(4.30)  ξ_{kp}(Z) := (Z^{t}Z)_{kp}, 1 ≤ k, p ≤ m,

where (Z^{t}Z)_{kp} denotes the (k, p)-entry of the matrix Z^{t}Z. Obviously ξ_{kp} = ξ_{pk} for all 1 ≤ k, p ≤ m.

Theorem 4.7. Let Pol (V_{2})^{K} be the subalgebra of Pol (V_{2}) consisting of Kinvariants under the action (4.29) of K. Then the polynomials ξ_{kp} ( 1 ≤ k ≤ p ≤ m) are algebraically independent generators of Pol (V_{2})^{K}.

Proof. The proof of the above theorem can be found in [Ho], Theorem 3.4.4., pp. 34-35. □

Fourthly we prove the following theorem.

Theorem 4.8. The following polynomials

P_{j}(X, Z) := σ(X^{j}), 1 ≤ j ≤ n ( see (4.10) )

and

ξ_{kp}(X, Z) := (Z^{t}Z)_{kp}, 1 ≤ k ≤ p ≤ m ( see (4.11) )

are algebraically independent generators of Pol (𝔭)^{K}.

Proof. Since 𝔭 = V_{1} ⊕ V_{2}, we have the following natural isomorphism

(4.31)  Pol (𝔭) ≅ Pol (V_{1}) ⊗ Pol (V_{2}).

Using the gradings of Pol (V_{1})^{K} and Pol (V_{2})^{K}, we can show that

(4.32)  Pol (𝔭)^{K} = Pol (V_{1})^{K} ⊗ Pol (V_{2})^{K}.

According to Theorem 4.6, Theorem 4.7 and (4.32), we see that the elements P_{j}(X) ⊗ 1 ( 1 ≤ j ≤ n) and 1 ⊗ ξ_{kp}(Z) ( 1 ≤ k ≤ p ≤ m) generate Pol (𝔭)^{K}. It remains that the elements P_{j}(X, Z) = P_{j}(X) ⊗ 1 ( 1 ≤ j ≤ n) and ξ_{kp}(X, Z) := 1 ⊗ ξ_{kp}(Z) ( 1 ≤ k ≤ p ≤ m) are algebraically independent. Assume that there exists a nonconstant polynomial Q such that

(4.33)  Q(P_{1}(X, Z), ⋯, ξ_{11}(X, Z), ⋯, ξ_{mm}(X, Z)) = 0.

Taking a bigrading of the left hand side of (4.33), we have the following grading

Q(P_{1}, P_{2}, ⋯, ξ_{11}, ⋯, ξ_{mm}) = ∑_{a,b} Q_{a,b},

where

Q_{a,b} = ∑_{α} P_{a,α} ⊗ ξ_{b,α}, P_{a,α} = P_{a,α}(X), ξ_{b,α} = ξ_{b,α}(Z)

is a homogeneous component of bidegree (a, b). Since Q(P_{1}, P_{2}, ⋯, ξ_{mm}) = 0, the following conditions have to be satisfied

Q_{a,b} = 0 for all the bidegree (a, b),

that is,

P_{a,α} = 0 or ξ_{b,α} = 0 for all a, b, α.

Therefore Q is identically zero. This contradicts our assumption. Our theorem is completely proved. □

Finally we prove the main theorem.

Proof of Main Theorem. By Proposition 4.4, D_{j} and Ψ_{kp} ( 1 ≤ j ≤ n, 1 ≤ k ≤ p ≤ m) are invariant differential operators on P_{n}×ℝ^{(m,n)} corresponding to the elements P_{j} and ξ_{kp} of Pol(𝔭)^{K} via (4.24). By the argument similiar to that of the latter part in the proof of Lemma 4.5, we can prove that D_{j} and Ψ_{kp} ( 1 ≤ j ≤ n, 1 ≤ k ≤ p ≤ m) generate the algebra 𝔻(P_{n} × ℝ^{(m,n)}). It follows from [M2] and [S1] that the operators D_{j}’s commute with each other and that they are algebraically independent. It is easy to see that the operators Ψ_{kp} ( 1 ≤ k ≤ p ≤ m) commute with each other and also that D_{j}Ψ_{kp} ≠ Ψ_{kp}D_{j} for all j, k, p. We also see easily that the Ψ_{kp}’s are algebraically independent. □

Example 4.9. We consider the case such that n = 1 and m is an arbitrary positive integer. We let G^{0} := ℝ^{+} ⋉ ℝ^{(m,1)} be the connected Lie group equipped with the following multiplication

(g_{1}, a_{1}) ∗ (g_{2}, a_{2}) := (g_{1}g_{2}, g_{2}^{−1}a_{1} + a_{2}), g_{1}, g_{2} ∈ ℝ^{+}, a_{1}, a_{2} ∈ ℝ^{(m,1)},

where ℝ^{+} denotes the multiplicative group of all positive real numbers. G^{0} acts on P_{1} × ℝ^{(m,1)} transitively by

(g, a) · (y, v) := (g^{2}y, g(v + a)), y ∈ ℝ^{+}, v = (v_{k}) ∈ ℝ^{(m,1)}.

The stabilizer K^{0} of G^{0} at (1, 0) is just the identity element {1}. Thus 𝔨^{0} = (0) and 𝔭^{0} = 𝔤^{0} ( cf. (2.6) and (2.7) ). Here 𝔨^{0} and 𝔤^{0} denotes the Lie algebra of K^{0} and G^{0} respectively. Since the adjoint representation of K^{0} on 𝔭^{0} is the trivial one, the symmetric algebra S(𝔤^{0}) is none other than the algebra of all invariants of the action (4.6) of K^{0} = {1} on S(𝔤^{0}). We denote by e_{k} ( 1 ≤ k ≤ m) the vector t(0, ⋯, 1, ⋯, 0) ∈ ℝ^{(m,1)} with 1 in the k-th position. We let

(4.34)  Y_{0} := (1, 0), X_{1} := (0, e_{1}), ⋯, X_{m} := (0, e_{m})

be the standard basis of 𝔤^{0}. We denote by 𝔻^{0}(P_{1} × ℝ^{(m,1)}) the algebra of all G^{0}-invariant differential operators on P_{1} × ℝ^{(m,1)}. By a simple computation, the G^{0}-invariant differential operators L_{Y_{0}}^{0}, L_{X_{1}}^{0}, ⋯, L_{X_{m}}^{0} on P_{1}×ℝ^{(m,1)} corresponding to Y_{0}, X_{1}, ⋯, X_{m} under a canonical linear bijection λ : S(𝔤^{0}) ⟶ 𝔻^{0}(P_{1} ×ℝ^{(m,1)}) are given by

(4.35)  L_{Y_{0}}^{0} = 2y \frac{∂}{∂y} , L_{X_{1}}^{0} = y^{1/2} \frac{∂}{∂v_{1}} , ⋯, L_{X_{m}}^{0} = y^{1/2} \frac{∂}{∂v_{m}}

respectively. Obviously the above (n + 1) G^{0}-invariant differential operators (4.35) generate 𝔻^{0}(P_{1} × ℝ^{(m,1)}). Since

L_{Y_{0}}^{0}L_{X_{j}}^{0} − L_{X_{j}}^{0}L_{Y_{0}}^{0} = L_{X_{j}}^{0}, 1 ≤ j ≤ m,

the algebra 𝔻^{0}(P_{1} × ℝ^{(m,1)}) is not commutative.

On the other hand, for the group G_{1,m}(ℝ) = GL(1, ℝ) ⋉ ℝ^{(m,1)} with GL(1, ℝ) = ℝ^{∗} − {0}, the stabilizer K := O(1) of G_{1,m}(ℝ) at (1, 0) is {±1}. Thus 𝔨 = 0 and 𝔭 = 𝔤 ( cf. (2.6) and (2.7) ). We let Y_{0}, X_{1}, ⋯, X_{m} as (4.34). Then Y_{0}, X_{1}, ⋯, X_{m} is the standard basis of 𝔭. The algebra S(𝔭)^{K} of K-invariants under the action (4.6) is generated by algebraically independent elements Y_{0}, X_{k}X_{l} ( 1 ≤ k ≤ l ≤ m). By a simple computation, the G_{1,m}(ℝ)-invariant differential operators L_{Y_{0}}, L_{X_{k}X_{l}} (1 ≤ k ≤ l ≤ m) corresponding to the the K-invariants Y_{0}, X_{k}X_{l} under a canonical linear bijection λ : S(𝔭)^{K} ⟶ 𝔻(P_{1} × ℝ^{(m,1)}) are given by

(4.36)  L_{Y_{0}} = 2y \frac{∂}{∂y} , L_{X_{k}X_{l}} = y \frac{∂^{2}}{∂v_{k}∂v_{l}} , 1 ≤ k ≤ l ≤ m.

It is easy to see that L_{Y_{0}}, L_{X_{k}X_{l}} ( 1 ≤ k ≤ l ≤ m) generate 𝔻(P_{1} × ℝ^{(m,1)}). Although L_{X_{k}X_{l}} ( 1 ≤ k ≤ l ≤ m) commute with each other, L_{Y_{0}} does not commute with any L_{X_{k}X_{l}}. Indeed, we have the noncommutation relation

L_{Y_{0}}L_{X_{k}X_{l}} − L_{X_{k}X_{l}}L_{Y_{0}} = 2L_{X_{k}X_{l}}.

Hence the algebra 𝔻(P_{1} × ℝ^{(m,1)}) is not commutative.

18 JAE-HYUN YANG

Example 4.10. We consider the case n = 2 and m = 1. In this case,

G_{2,1}(ℝ) := GL(2, ℝ) ⋉ ℝ^{(1,2)}, K := O(2) and G_{2,1}(ℝ)/K = P_{2} × ℝ^{(1,2)}. According to (2.7),

𝔭 = { (X, Z) | X = ^{t}X ∈ ℝ^{(2,2)}, Z ∈ ℝ^{(1,2)} } .

We put

e_{1} := ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , e_{12} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , e_{2} := ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )

and

f_{1} := (1, 0), f_{2} := (0, 1).

Then { e_{1}, e_{12}, e_{2}, f_{1}, f_{2} } forms a basis for 𝔭. We write for variables (X, Z) ∈ 𝔭 by

X = ( \begin{matrix}x_{1} & x_{3} \\ x_{3} & x_{2} \end{matrix} ) and Z = (z_{1}, z_{2}).

The following polynomials

P_{1}(X, Z) := σ(X) = x_{1} + x_{2}, P_{2}(X, Z) := σ(X^{2}) = x_{1}^{2} + x_{2}^{2} + 2x_{3}^{2} and

ξ_{11}(X, Z) := z_{1}^{2} + z_{2}^{2}

are invariant under the action (4.6) of K.

Now we will compute the G_{2,1}(ℝ)-invariant differential operators D_{1}, D_{2}, Ψ_{11} on P_{2} × ℝ^{(1,2)} corresponding to the K-invariants P_{1}, P_{2}, ξ_{11} respectively under a canonical linear bijection

λ : S(𝔭)^{K} ⟶ 𝔻(P_{2} × ℝ^{(1,2)}).

For real variables t = (t_{1}, t_{2}, t_{3}) and s = (s_{1}, s_{2}), we have exp (t_{1}e_{1} + t_{2}e_{2} + t_{3}e_{3} + s_{1}f_{1} + s_{2}f_{2})

= (( \begin{matrix}a_{1}(t, s) & a_{3}(t, s) \\ a_{3}(t, s) & a_{2}(t, s) \end{matrix} ) , ( b_{1}(t, s), b_{2}(t, s) ) ) ,

where

a_{1}(t, s) = 1 + t_{1} + \frac{1}{2!}(t_{1}^{2} + t_{3}^{2}) + \frac{1}{3!}(t_{1}^{3} + 2t_{1}t_{3}^{2} + t_{2}t_{3}^{2}) + ⋯,
a_{2}(t, s) = 1 + t_{2} + \frac{1}{2!}(t_{2}^{2} + t_{3}^{2}) + \frac{1}{3!}(t_{1}t_{3}^{2} + 2t_{2}t_{3}^{2} + t_{2}^{3}) + ⋯,
a_{3}(t, s) = t_{3} + \frac{1}{2!}(t_{1}t_{3} + t_{2}t_{3}) + \frac{1}{3!}(t_{1}t_{2}t_{3} + t_{1}^{2}t_{3} + t_{2}^{2}t_{3} + t_{3}^{3}) + ⋯,
b_{1}(t, s) = s_{1} − \frac{1}{2!}(s_{1}t_{1} + s_{2}t_{3}) + \frac{1}{3!} \{ s_{1}(t_{1}^{2} + t_{3}^{2}) + s_{2}(t_{1}t_{3} + t_{2}t_{3})\} − ⋯,
b_{2}(t, s) = s_{2} − \frac{1}{2!}(s_{1}t_{3} + s_{2}t_{2}) + \frac{1}{3!} \{ s_{1}(t_{1}t_{3} + t_{2}t_{3}) + s_{2}(t_{2}^{2} + t_{3}^{2})\} − · · · .

For brevity, we write a_{i}, b_{k} for a_{i}(t, s), b_{k}(t, s) ( i = 1, 2, 3, k = 1, 2 ) respectively. We now fix an element (g, c) ∈ G_{2,1}(ℝ) and write

g = ( \begin{matrix}g_{1} & g_{12} \\ g_{21} & g_{2} \end{matrix} ) and c = (c_{1}, c_{2}).

We put

(Y (t, s), V (t, s)) := ( (g, c) · exp (∑_{i=1}^{3} t_{i}e_{i} + ∑_{k=1}^{2} s_{k}f_{k}) ) · (E_{2}, 0)

with

Y (t, s) := ( \begin{matrix}y_{1}(t, s) & y_{3}(t, s) \\ y_{3}(t, s) & y_{2}(t, s) \end{matrix} ) and V (t, s) := (v_{1}(t, s), v_{2}(t, s)).

By an easy computation, we obtain

y_{1} = (g_{1}a_{1} + g_{12}a_{3})^{2} + (g_{1}a_{3} + g_{12}a_{2})^{2},
y_{2} = (g_{21}a_{1} + g_{2}a_{3})^{2} + (g_{21}a_{3} + g_{2}a_{2})^{2},
y_{3} = (g_{1}a_{1} + g_{12}a_{3})(g_{21}a_{1} + g_{2}a_{3}) + (g_{1}a_{3} + g_{12}a_{2})(g_{21}a_{3} + g_{2}a_{2}),
v_{1} = (c_{1} + b_{1}a_{1} + b_{2}a_{3})g_{1} + (c_{2} + b_{1}a_{3} + b_{2}a_{2})g_{12},
v_{2} = (c_{1} + b_{1}a_{1} + b_{2}a_{3})g_{21} + (c_{2} + b_{1}a_{3} + b_{2}a_{2})g_{2}.

Using the chain rule, we can easily compute the G_{2,1}(ℝ)-invariant differential operators D_{1} := D_{λ(P_{1})}, D_{2} := D_{λ(P_{2})} and Ψ_{11} := D_{λ(ξ_{11})}. They are given by

D_{1} = 2 σ ( Y \frac{∂}{∂Y} ) = 2 ( y_{1} \frac{∂}{∂y_{1}} + y_{2} \frac{∂}{∂y_{2}} + y_{3} \frac{∂}{∂y_{3}} ) ,

D_{2} = σ (( 2Y \frac{∂}{∂Y} )^{2})

= 3 D_{1} + 8 ( y_{3}^{2} \frac{∂^{2}}{∂y_{1}∂y_{2}} + y_{1}y_{3} \frac{∂^{2}}{∂y_{1}∂y_{3}} + y_{2}y_{3} \frac{∂^{2}}{∂y_{2}∂y_{3}} )

+ 4 \{ y_{1}^{2} \frac{∂^{2}}{∂y_{1}^{2}} + y_{2}^{2} \frac{∂^{2}}{∂y_{2}^{2}} + \frac{1}{2}(y_{1}y_{2} + y_{3}^{2}) \frac{∂^{2}}{∂y_{3}^{2}} \}

and

Ψ_{11} = σ ( Y ^{t} (\frac{∂}{∂V} ) (\frac{∂}{∂V} ))

= y_{1} \frac{∂^{2}}{∂v_{1}^{2}} + 2y_{3} \frac{∂^{2}}{∂v_{1}∂v_{2}} + y_{2} \frac{∂^{2}}{∂v_{2}^{2}} .

Clearly D_{1} commutes with D_{2} but Ψ_{11} does not commute with D_{1} and D_{2}. Indeed, we have the following noncommutation relations

[D_{1}, Ψ_{11}] : = D_{1}Ψ_{11} − Ψ_{11}D_{1}
= 2 Ψ_{11}

and

[D_{2}, Ψ_{11}] : = D_{2}Ψ_{11} − Ψ_{11}D_{2}

= 2 ( 2 D_{1} − 1 ) Ψ_{11}

− 8 det (Y) · det (\frac{∂}{∂Y} + ^{t}(\frac{∂}{∂V} ) \frac{∂}{∂V} )

+ 8 det (Y) · det (\frac{∂}{∂Y} ) − 4 (y_{1}y_{2} + y_{3}^{2}) \frac{∂^{3}}{∂y_{3}∂v_{1}∂v_{2}} .

Hence the algebra 𝔻(P_{2} × ℝ^{(1,2)}) is not commutative. According to Theorem 4.8, D_{1}, D_{2} and Ψ_{11} generate the algebra 𝔻(P_{n} × ℝ^{(m,n)}).

Remark 4.11. It is easily seen that for all 1 ≤ k ≤ p ≤ m,

[D_{1}, Ψ_{kp}] := D_{1}Ψ_{kp} − Ψ_{kp}D_{1} = 2 Ψ_{kp}.

However in general for j ≥ 2, it is not easy to express the following operators

[D_{j}, Ψ_{kp}] := D_{j}Ψ_{kp} − Ψ_{kp}D_{j}

in terms of the generators D_{j}’s and Ψ_{kp}’s.

Definition 4.12. Let G be a locally compact group of isometries of a Riemannian manifold M which acts transitively on M. We assume that there is a fixed isometry ϕ of M ( possibly not in G). We say that M is a weakly symmetric Riemannian manifold with respect to G and ϕ if the following two conditions (WS1) and (WS2) are satisfied :

(WS1) ϕ commutes with G and ϕ^{2} belongs to G.

(WS2) For any pair of points x and y of M, there exists an element λ ∈ G such that λ(x) = ϕ(y) and λ(y) = ϕ(x).

Remark 4.13. Selberg [S1] proved that the algebra 𝔻(M) of all G-invariant differential operators on a weakly symmetric Riemannian manifold M is commutative. It is well known [S1] that the algebra 𝔻(P_{n}) of all differential operators on P_{n} which are invariant under the action (1.2) of GL(n, ℝ) are generated by the following n algebraically independent differential operators

(4.37)  σ ((Y \frac{∂}{∂Y} )^{j}) , 1 ≤ j ≤ n.

Furthermore, 𝔻(P_{n}) has a commutative ring structure which is isomorphic to the ring ℂ[Z_{1}, ⋯, Z_{n}] of polynomials in n indeterminates Z_{1}, ⋯, Z_{n}. In fact, P_{n} is a weakly symmetric Riemannian manifold. For more detail, we refer to [M2] and [S1].

Proposition 4.14. For any positive integers n and m, the algebra 𝔻(P_{n} × ℝ^{(m,n)}) is not commutative. Consequently the homogeneous space P_{n} × ℝ^{(m,n)} is not a weakly symmetric Riemannian manifold in the sense of the above definition.

Proof. We already know that the operators D_{j} and L_{k} ( cf. (4.17) and (4.19) ) do not commute. Since the ring 𝔻(P_{n} × ℝ^{(m,n)}) is not commutative, according to Selberg’s theorem, P_{n} × ℝ^{(m,n)} is not weakly symmetric. We will leave the detail to the reader. □

5. Spherical Functions

In this section, we define the spherical functions on the group G_{n,m}(ℝ) and characterize these functions.

Let M be a smooth manifold and let ϕ : M ⟶ M be a diffeomorphism of M onto itself. We put

(5.1)  f^{ϕ} = f ∘ ϕ^{−1}, f ∈ C^{∞}(M),

where C^{∞}(M) denotes the space of all smooth functions on M. If D is a differential operator on M, we define D^{ϕ} by

(5.2)  D^{ϕ}(f) := (D(f ∘ ϕ)) ∘ ϕ^{−1}, f ∈ C^{∞}(M).

We note that if D is invariant under ϕ, then D^{ϕ} = D, i.e., D(f ∘ ϕ) = (Df) ∘ ϕ for all f ∈ C^{∞}(M).

Now we return to our stage. For an element g of G_{n,m}(ℝ), we denote by L_{g} and R_{g} the left translation and the right translation of G_{n,m}(ℝ) by g. Let 𝔻(G_{n,m}(ℝ)) be the algebra of all differential operators on G_{n,m}(ℝ) invariant under all the left translations L_{g} of G_{n,m}(ℝ). We note that 𝔻(G_{n,m}(ℝ)) is generated by all the left invariant vector fields on G_{n,m}(ℝ). We define

(5.3)  𝔻_{K}(G_{n,m}(ℝ)) := { D ∈ 𝔻(G_{n,m}(ℝ)) | D^{R_{k}} = D for all k ∈ K } .

Let π : G_{n,m}(ℝ) ⟶ G_{n,m}(ℝ)/K be the natural projection. We put for any function f on G_{n,m}(ℝ)/K

(5.4)  f̃:= f ∘ π.

So f̃ is a function on G_{n,m}(ℝ). Since G_{n,m}(ℝ)/K is reductive, we have a natural homomorphism μ : u ↦ D_{u} of 𝔻_{K}(G_{n,m}(ℝ)) onto 𝔻(G_{n,m}(ℝ)/K) defined by

(5.5)  (μ(u)f)^{}˜ = (D_{u}f)^{}˜ = uf̃, f ∈ C^{∞}(G_{n,m}(ℝ)/K).

The kernel of μ is given by

(5.6)  ker μ = 𝔻_{K}(G_{n,m}(ℝ)) ∩ 𝔻(G_{n,m}(ℝ))𝔨.

So we have the isomorphism

(5.7)  𝔻(P_{n} × ℝ^{(m,n)}) ≅ 𝔻(G_{n,m}(ℝ)/K) ≅ 𝔻_{K}(G_{n,m}(ℝ))/ker μ.

For the proof of (5.5), (5.6) and (5.7), we refer the reader to [He2], pp. 285-286.

Now we are ready to define the concept of spherical functions on G_{n,m}(ℝ).

Definition 5.1. A function

ϕ : G_{n,m}(ℝ) ⟶ ℂ

is called a spherical function if ϕ satisfies the following properties (S1)-(S3) :

(S1) ϕ(k_{1}gk_{2}) = ϕ(g) for all k_{1}, k_{2} ∈ K and g ∈ G_{n,m}(ℝ), i.e., ϕ is bi-invariant under K.

(S2) ϕ is an eigenfunction for all the differential operators in 𝔻_{K}(G_{n,m}(ℝ)). (S3) ϕ(E_{n}, 0) = 1.

We can say that a spherical function is a rotation-invariant eigenfunction for the algebra 𝔻_{K}(G_{n,m}(ℝ)) which is normalized to have the value 1 at the identity (E_{n}, 0).

Let us descend a spherical function ϕ on G_{n,m}(ℝ) to the coset space P_{n}×ℝ^{(m,n)} ≅ G_{n,m}(ℝ)/K. We define the function f_{ϕ} : P_{n} × ℝ^{(m,n)} ⟶ ℂ by

(5.8)  f_{ϕ}(Y, V) := ϕ(g, V ^{t}g^{−1}), (Y, V) ∈ P_{n} × ℝ^{(m,n)},

where Y = g^{t}g with g ∈ GL(g, ℝ). We note that g is unique up to K. Then f_{ϕ} satisfies the following properties :

(S1)^{0} f_{ϕ}(kY ^{t}k, V ^{t}k) = f_{ϕ}(Y, V) for all (Y, V) ∈ P_{n} × ℝ^{(m,n)} and k ∈ K.

(S2)^{0} f_{ϕ} is an eigenfunction for all the differential operators in 𝔻(P_{n} × ℝ^{(m,n)}).

(S3)^{0} f_{ϕ}(E_{n}, 0) = 1.

We also call a function on P_{n} × ℝ^{(m,n)} satisfying the above three properties spherical.

First we recall the Iwasawa decompositions (2.10) and (2.11) of G_{n,m}(ℝ) and 𝔤 respectively. For g ∈ G_{n,m}(ℝ), we let A(g) denote the unique element of 𝔞 such that

g ∈ N exp A(g) K.

Since the algebra 𝔻(A) of all invariant differential operators on A is a commutative ring, each linear function ν : 𝔞 ⟶ ℂ extends uniquely to a homomorphism of 𝔻(A) into ℂ, also denoted by ν : 𝔻(A) ⟶ ℂ. Let 𝔞^{∗} be the dual space of 𝔞 and let 𝔞_{c}^{∗} be the set of all linear functionals of 𝔞 into ℂ.

Lemma 5.2. For each ν ∈ 𝔞_{c}^{∗}, the function p_{ν} on G_{n,m}(ℝ) defined by

(5.9)  p_{ν}(g) := e^{ν(A(g))}, g ∈ G_{n,m}(ℝ)

is an eigenfunction of each D ∈ 𝔻_{K}(G_{n,m}(ℝ)) wirh eigenvalue ν(D_{𝔞}), where D_{𝔞} denotes the radial part of D. The function p_{ν} is called the power function on G_{n,m}(ℝ) associated with ν ∈ 𝔞_{c}^{∗}. Moreover, the function φ_{ν} on G_{n,m}(ℝ) defined by

(5.10)  φ_{ν}(g) := ∫_{K} e^{ν(A(kg))} dk, g ∈ G_{n,m}(ℝ)

is an eigenfunction of each D ∈ 𝔻_{K}(G_{n,m}(ℝ)) with eigenvalue ν(D_{𝔞}), where dk is the Haar measure on K normalized to be ∫_{K} dk = 1.

Proof. The proof of the above lemma can be found in [He2], p. 303. □

Theorem 5.3. For each ν ∈ 𝔞_{c}^{∗}, we define the function ϕ_{ν} on G_{n,m}(ℝ) by

(5.11)  ϕ_{ν}(g) := ∫_{K} e^{(iν+ρ)(A(kg))}dk, g ∈ G_{n,m}(ℝ),

where ρ is defined in (2.15). Then

ϕ_{sν} = ϕ_{ν} for all s ∈ W,

where W denotes the Weyl group and (sν)(H) := ν(s^{−1}H), H ∈ 𝔞. The functions ϕ_{ν} ( ν ∈ 𝔞_{c}^{∗}) define the class of spherical functions on G_{n,m}(ℝ). Moreover, two such functions ϕ_{ν} and ϕ_{λ} are identical if and only if ν = sλ for some s ∈ W.

Proof. We can prove the above theorem following the line of the argument in [He2], pp. 303-304 and p. 418. □

We describe the functions mentioned above in more detail. Let ν ∈ 𝔞_{c}^{∗} be the linear functional ν : 𝔞 ⟶ ℂ given by

ν(D(a_{1}, ⋯, a_{n})) := ν_{1}a_{1} + ⋯ + ν_{n}a_{n}, ν_{1}, ⋯, ν_{n} are fixed complex numbers,

where D(a_{1}, ⋯, a_{n}) denotes the n×n diagonal real matrix with entries a_{1}, ⋯, a_{n}. Now we define the function p̃_{ν} : P_{n} × ℝ^{(m,n)} ⟶ ℂ by

(5.12)  p̃_{ν}(Y, V) := p_{ν}(g_{0}, V ^{t}g_{0}), (Y, V) ∈ P_{n} × ℝ^{(m,n)},

where Y = g_{0} ^{t}g_{0} with g_{0} ∈ GL(n, ℝ). We observe that p̃_{ν}(Y, V) is independent of the choice of g_{0} ∈ GL(n, ℝ) with Y = g_{0} ^{t}g_{0} and hence p̃_{ν} is well defined. We may choose an upper triangular matrix g_{0} with diagonal entries t_{1}, ⋯, t_{n} > 0 such that Y = g_{0} ^{t}g_{0}. Then we have

p_{ν}(g_{0}, V ^{t}g_{0}) = e^{ν(A(g_{0}, V ^{t}g_{0}^{−1}))} = e^{ν(D(log t_{1},··· ,log t_{n}))}
= e^{ν_{1}log t_{1}+···+ν_{n}log t_{n}}
= t_{1}^{ν_{1}} ⋯ t_{n}^{ν_{n}}.

We define the n-tuple s = (s_{1}, ⋯, s_{n}) ∈ ℂ^{n} of complex numbers by

(5.13)  ν_{h} := 2(s_{n−h+1} + ⋯ + s_{n}), h = 1, 2, ⋯, n.

Then we have

(5.14)  s_{h} := \frac{1}{2}(ν_{n−h+1} − ν_{n−h}), h = 1, ⋯, n − 1 and s_{n} = \frac{1}{2}ν_{1}.

For each h ∈ ℤ^{+} with 1 ≤ h ≤ n, we denote by Y_{h} the lower right corner h × h matrix of Y so that

Y = ( \begin{matrix}∗ & ∗ \\ ∗ & Y_{h} \end{matrix} ) .

From the fact that

det Y_{h} = (t_{n−h+1}t_{n−h+2} ⋯ t_{n})^{2},

it follows immediately that

(5.15)  p̃_{ν}(Y, V) = p_{ν}(g_{0}, V ^{t}g_{0}) = ∏_{h=1}^{n} ( det Y_{h})^{s_{h}} .

For ν = (ν_{1}, ⋯, ν_{n}) ∈ ℂ^{n} we define the function Φ_{ν} : P_{n} × ℝ^{(m,n)} ⟶ ℂ by

(5.16)  Φ_{ν}(Y, V) := ∫_{K} e^{(iν̃+ρ)(A(kg))}dk, (Y, V) ∈ P_{n} × ℝ^{(m,n)},

where g = (g_{0}, V ^{t}g_{0}^{−1}) with Y = g_{0} ^{t}g_{0}, g_{0} ∈ GL(n, ℝ) and ν̃ : 𝔞 ⟶ ℂ is the linear function on 𝔞 defined by

ν̃(D(a_{1}, ⋯, a_{n})) = −i(ν_{1}a_{1} + ⋯ + ν_{n}a_{n}), D(a_{1}, ⋯, a_{n}) ∈ 𝔞.

Then it is easily seen that Φ_{ν} is a joint eigenfunction for all D ∈ 𝔻(P_{n} × ℝ^{(m,n)}). If Y = g_{0} ^{t}g_{0} such that g_{0} is a upper triangular n × n matrix with diagonals t_{1}, ⋯, t_{n} > 0, then according to (2.16) we have

(5.17)  Φ_{ν}(Y, V) = ∏_{h=1}^{n} t_{h}^{\frac{1}{2} (n+m+1−2h)+ν_{h}}.

Summarizing the above, we have Theorem 5.4. For any ν ∈ 𝔞_{c}^{∗}, the function

(5.18)  Φ_{ν}(Y, V) = ( det Y )^{\frac{1}{4} (n+m+1)} ∏_{h=1}^{n} ( det Y_{h})^{s_{h}− \frac{1}{2}}

is a joint eigenfunction for all D ∈ 𝔻(P_{n}×ℝ^{(m,n)}), where the n-tuple s = (s_{1}, ⋯, s_{n}) of complex numbers is given by (5.14).

Proof. The proof follows from (5.17) and the fact that

det Y_{h} = (t_{n−h+1}t_{n−h+2} ⋯ t_{n})^{2}, h = 1, ⋯, n.
□

6. Automorphic Forms on G_{n,m}(ℝ)

In this section, we define the concept of automorphic forms on G_{n,m}(ℝ).

First of all, following the article of Borel and Jacquet [B-J] we define

Definition 6.1. A norm || || on G_{n,m}(ℝ) is a function of the form

||x|| = \{ tr (ρ(x)^{∗}ρ(x)) \}^{\frac{1}{2}} , x ∈ G_{n,m}(ℝ),

where ρ : G_{n,m}(ℝ) ⟶ GL(V_{ρ}) is a finite dimensional complex representation with finite kernel whose image is closed in End (V_{ρ}) and ρ(x)^{∗} denotes the adjoint of ρ(x) with respect to a Hilbert space structure on V_{ρ} invariant under K. We say that a function ϕ on G_{n,m}(ℝ) is slowly increasing if there exists a norm || || on G_{n,m}(ℝ), a positive constant C and a positive integer ℓ such that

(6.2)  |ϕ(x)| ≤ C · ||x||^{ℓ} for all x ∈ G_{n,m}(ℝ).

Borel and Jacquet [B-J] introduced the notion of automorphic forms on a real reductive Lie group. We will define the notion of automorphic forms on the group G_{n,m}(ℝ) slightly different from theirs.

Definition 6.2. A smooth function

f : G_{n,m}(ℝ) ⟶ ℂ

is called an automorphic form with respect to G_{n,m}(ℤ) if f satisfies the following conditions :

(a) f(γx) = f(x) for all γ ∈ G_{n,m}(ℤ) and x ∈ G_{n,m}(ℝ).

(b) f is K-finite on the right.

(c) f is an eigenfunction for all the differential operators in 𝔻_{K}(G_{n,m}(ℝ)).

(d) f is slowly increasing.

For a system λ = (λ_{α}) of eigenvalues for 𝔻_{K}(G_{n,m}(ℝ)), we denote by A(G_{n,m}(ℤ), λ) the vector space of all automorphic forms f with respect to G_{n,m}(ℤ) such L_{α}f = λ_{α}f for all L_{α} ∈ 𝔻_{K}(G_{n,m}(ℝ)). And an automorphic form ϕ is said to be cuspidal if

∫_{N_{ℤ}\N(ℝ)} f(nx) dn = 0, N_{ℤ} := N(ℝ) ∩ G_{n,m}(ℤ)

for all x ∈ G_{n,m}(ℝ), where N is the unipotent radical of any proper ℚ-subgroup of G_{n,m}(ℝ). In fact, it suffices to require this for any proper maximal parabolic ℚ-subgroup ( cf. [HC], Lemma 3, p. 8 ). We denote by A(G_{n,m}(ℤ), λ)_{0} the subspace of all cuspidal forms in A(G_{n,m}(ℤ), λ).

Remark 6.3. The condition (b) means that by definition the right translates of f by elements of K span a finite dimensional vector space of functions. The condition (c) in the above definition is slightly different from the condition 1.3 (c) in [B-J].

Theorem 6.4. The space A(G_{n,m}(ℤ), λ) is finite dimensional.

Proof. The proof follows from [HC], Theorem 1, p. 8 and [B-J], p. 191. □

7. Maass-Jacobi Forms

We let

(7.1)  SL_{n,m}(ℝ) := SL(n, ℝ) ⋉ ℝ^{(m,n)}

be the semidirect product of the special linear group SL(n, ℝ) and the commutative additive group ℝ^{(m,n)} with the multiplication law (1.6). Obviously SL_{n,m}(ℝ) is a subgroup of G_{n,m}(ℝ). We let

(7.2)  SL_{n,m}(ℤ) := SL(n, ℤ) ⋉ ℤ^{(m,n)}

be the discrete subgroup of SL_{n,m}(ℝ) and K_{0} := SO(n, ℝ).

Now we introduce the notion of Maass-Jacobi forms.

Definition 7.1. A smooth bounded function

ϕ : SL_{n,m}(ℝ) ⟶ ℂ

is called a Maass -Jacobi form if ϕ satisfies

(MJ1) ϕ(γxk) = ϕ(x) for all γ ∈ SL_{n,m}(ℤ), x ∈ SL_{n,m}(ℝ) and k ∈ K_{0}.

(MJ2) ϕ is an eigenfunction for all the differential operators in 𝔻_{K_{0}}(SL_{n,m}(ℝ)).

(MJ3) ϕ is slowly increasing.

For a fixed system λ = (λ_{α}) of eigenvalues for 𝔻_{K_{0}}(SL_{n,m}(ℝ)), we denote by MJ(SL_{n,m}(ℤ), λ) the vector space of all Maass-Jacobi forms ϕ such that L_{α}ϕ = λ_{α}ϕ for all Ł_{α} ∈ 𝔻_{K_{0}}(SL_{n,m}(ℝ)).

We set

SP_{n} := { Y ∈ P_{n} | det Y = 1 } .

Let us descend a Maass-Jacobi form ϕ to the coset space SP_{n} × ℝ^{(m,n)} ≅ SL_{n,m}(ℝ)/K_{0}. We define the function f_{ϕ} : SP_{n} × ℝ^{(m,n)} ⟶ ℂ by

(7.3)  f_{ϕ}(Y, V) := ϕ(g, V ^{t}g^{−1}), (Y, V) ∈ SP_{n} × ℝ^{(m,n)},

where Y = g^{t}g for some g ∈ SL(n, ℝ). (7.3) is well-defined. Then f_{ϕ} satisfies the following conditions : (MJ1)^{0} f_{ϕ}(γY ^{t}γ, (V + α)^{t}γ) = f_{ϕ}(Y, V) for all (γ, α) ∈ SL_{n,m}(ℤ). (MJ2)^{0} f_{ϕ} is an eigenfunction for all the differential operators in 𝔻(SP_{n} ×ℝ^{(m,n)}). (MJ3)^{0} f_{ϕ} has a suitable polynomial growth condition.

For a system s = (s_{α}) of eigenvalues for 𝔻(SP_{n}×ℝ^{(m,n)}), we denote by MJ_{s}(SL_{n,m}(ℤ)) the vector space of all Maass-Jacobi forms f on SP_{n} × ℝ^{(m,n)} such that L_{α}f = s_{α}f for all L_{α} ∈ 𝔻(SP_{n} × ℝ^{(m,n)}).

Theorem 7.2. For any fixed system λ = (λ_{α}), the space MJ(SL_{n,m}(ℤ), λ) is finite dimensional.

The proof of the above theorem is due to Harish-Chandra [HC], Theorem 1, p. 8 and [B-J], p 191.

Now we consider the simple case n = 2 and m = 1. We recall that SL_{2,1}(ℝ) acts on SP_{2} × ℝ^{(1,2)} transitively by

(7.4)  (g, α) · (Y, V) := ( gY ^{t}g, (V + α)^{t}g),

where g ∈ SL(2, ℝ), α ∈ ℝ^{(1,2)}, Y ∈ SP_{2} and V ∈ ℝ^{(1,2)}. We recall that SL(2, ℝ) acts on ℍ transitively by

g < τ >:= (aτ + b)(cτ + d)^{−1}, ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ), τ ∈ ℍ,

where ℍ denotes the Poincaré upper half plane.

Now we define the action of SL_{2,1}(ℝ) on ℍ × ℂ by

(7.5)  (g, α) ∘ (τ, z) := (^{t}g^{−1} < τ >, (z + α_{1}τ + α_{2})(−bτ + a)^{−1} ),

where g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ) and α = (α_{1}, α_{2}) ∈ ℝ^{(1,2)}, and (τ, z) ∈ ℍ × ℂ.

Since the action (7.5) is transitive and the special orthogonal real group SO(2) is the stabilizer of this action at the origin (i, 0), ℍ × ℂ can be identified with the homogeneous space SL_{2,1}(ℝ)/SO(2) as follows :

(7.6)  SL_{2,1}(ℝ)/SO(2) ∋ (g, α) SO(2) ⟼ (g, α) ∘ (i, 0).

We observe that we can express an element Y of SP_{2} uniquely as

(7.7)  Y = ( \begin{matrix}y^{−1} & 0 \\ 0 & y \end{matrix} ) [( \begin{matrix}1 & −x \\ 0 & 1 \end{matrix} )] = ( \begin{matrix}y^{−1} & −xy^{−1} \\ −xy^{−1} & x^{2}y^{−1} + y \end{matrix} )

with x, y ∈ ℝ and y > 0.

Proposition 7.3. We define the mapping T : SP_{2} × ℝ^{(1,2)} ⟶ ℍ × ℂ by

(7.8)  T (Y, V) := (x + iy, v_{1}(x + iy) + v_{2} ),

where Y is of the form (7.7) and V = (v_{1}, v_{2}) ∈ ℝ^{(1,2)}. Then the mapping T is a bijection which is compatible with the above two actions (7.4) and (7.5).

For any Y ∈ SP_{2} of the form (7.7), we put

(7.9)  g_{Y} := ( \begin{matrix}1 & 0 \\ −x & 1 \end{matrix} ) ( \begin{matrix}y^{−1/2} & 0 \\ 0 & y^{1/2} \end{matrix} ) = ( \begin{matrix}y^{−1/2} & 0 \\ −xy^{−1/2} & y^{1/2} \end{matrix} ) .

and

(7.10)  α_{Y,V} := V ^{t}g_{Y}^{−1}.

Then we have

(7.11)  T (Y, V) = ( g_{Y} , α_{Y,V} ) ∘ (i, 0).

Proof. It is obvious that T is a bijection. Let T_{0} : SP_{2} ⟶ ℍ be the mapping defined by

T_{0}(Y) := x + i y,

where Y is of the form (7.7). We observe that

Y = g_{Y} ^{t}g_{Y} and T_{0}(Y) = ^{t}g_{Y}^{−1} < i >,

where g_{Y} is an element of SL(2, ℝ) defined by (7.9).

For g ∈ SL(2, ℝ) and Y ∈ SP_{2},

Y_{∗} := gY ^{t}g = g_{∗} ^{t}g_{∗} for some g_{∗} ∈ SL(2, ℝ).

Since Y_{∗} = g_{Y_{∗}} ^{t}g_{Y_{∗}}, there is an element k ∈ K such that g_{Y_{∗}} = g_{∗}k. So we have

T_{0}(Y_{∗}) = ^{t}g_{Y_{∗}}^{−1} < i > = ^{t}(g_{∗}k)^{−1} < i >
= ^{t}g_{∗}^{−1} ^{t}k^{−1} < i > = ^{t}g_{∗}^{−1} < i > .

So far we showed that if Y_{0} ∈ SP_{2} such that Y_{0} = g_{0} ^{t}g_{0} with g_{0} ∈ SL(2, ℝ), then T_{0}(Y_{0}) = g_{0} < i > . If Y is of the form (7.7), we take

g_{∗} := g · g_{Y} .

Then g_{∗} ^{t}g_{∗} = gY ^{t}g. Therefore we have

T_{0}(gY ^{t}g) = ^{t}g_{∗}^{−1} < i > = ^{t}g^{−1} ·^{t} g_{Y}^{−1} < i > = ^{t}g^{−1} < τ >,

where we put τ := x + i y.

Let (g, α) ∈ SL_{2,1}(ℝ) with g := ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ), α := (α_{1}, α_{2}) ∈ ℝ^{(1,2)}

and (Y, V) ∈ SP_{2} × ℝ^{(1,2)} such that Y ∈ SP_{2} is of the form (7.7) and V := (v_{1}, v_{2}) ∈ ℝ^{(1,2)} with v_{1}, v_{2} ∈ ℝ. Now we have, if τ = x + i y ∈ ℍ,

(g, α) ∘ T (Y, V)
= (g, α) ∘ (τ, v_{1}τ + v_{2})
= (^{t}g^{−1} < τ >, { ( v_{1} + α_{1} ) τ + ( v_{2} + α_{2} ) } ( − b τ + a)^{−1} ).

On the other hand.

T ( (g, α) · (Y, V) ) = T ( gY ^{t}g, ( V + α)^{t}g)
= ( T_{0}(gY ^{t}g), ṽ_{1} · T_{0}(gY ^{t}g) + ṽ_{2} )
= (^{t}g^{−1} < τ >, ṽ_{1} ·^{t} g^{−1} < τ > + ṽ_{2} ),

where ( ṽ_{1}, ṽ_{2} ) := (V + α)^{t}g.

Since

ṽ_{1} : = (v_{1} + α_{1}) a + (v_{2} + α_{2}) b,
ṽ_{2} : = (v_{1} + α_{1}) c + (v_{2} + α_{2}) d,

by a simple calculation, we get

ṽ_{1} ^{t}g^{−1} < τ > + ṽ_{2} = { (v_{1} + α_{1}) τ + (v_{2} + α_{2}) } (−bτ + a)^{−1}.

Hence we obtain

T ((g, α) · (Y, V)) = (g, α) ∘ T (Y, V).

Consequently T is compatible with the actions (7.4) and (7.5). It is easily checked that (7.11) holds. □

Theorem 7.4. Let 𝔻(ℍ × ℂ) be the algebra of all differential operators on ℍ × ℂ invariant under the action (7.5) of SL_{2,1}(ℝ). Then the algebra 𝔻(ℍ×ℂ) is generated by the following differential operators

(7.12)  D := y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + v^{2} ( \frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} )

+ 2 y v ( \frac{∂^{2}}{∂x∂u} + \frac{∂^{2}}{∂y∂v} )

and

(7.13)  Ψ := y ( \frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} ) ,

where τ = x + iy and z = u + iv with real variables x, y, u, v. Moreover, we have

[D, Ψ] := DΨ − ΨD = 2 y^{2} \frac{∂}{∂y} ( \frac{∂^{2}}{∂u^{2}} − \frac{∂^{2}}{∂v^{2}} ) − 4 y^{2} \frac{∂^{3}}{∂x∂u∂v}

− 2 ( v \frac{∂}{∂v} Ψ + Ψ ) .

In particular, the algebra 𝔻(ℍ × ℂ) is not commutative.

Proof. The proof can be found in [Y7]. □

For any right K-invariant function ϕ : SL_{2,1}(ℝ) ⟶ ℂ on SL_{2,1}(ℝ), we define the function f_{ϕ} : ℍ × ℂ ⟶ ℂ by

(7.14)  f_{ϕ}(τ, z) := ϕ(g, α), τ ∈ ℍ, z ∈ ℂ,

where (g, α) is an element of SL_{2,1}(ℝ) with (g, α) ∘ (i, 0) = (τ, z). Clearly it is well defined because (7.14) is independent of the choice of (g, α) ∈ SL_{2,1}(ℝ) such that (g, α) ∘ (i, 0) = (τ, z).

Theorem 7.5. Let ϕ : SL_{2,1}(ℝ) ⟶ ℂ be a nonzero Maass-Jacobi form. Then the function f_{ϕ} defined by (7.14) satisfies the following conditions : (MJ1)^{0} f_{ϕ}(γ̃ ∘ (τ, z)) = f_{ϕ}(τ, z) for all γ̃ ∈ SL_{2,1}(ℤ) := SL(2, ℤ) ⋉ ℤ^{(1,2)}. (MJ2)^{0} f_{ϕ} is a joint eigenfunction for two invariant differential operators D

and Ψ. (MJ3)^{0} f_{ϕ} has a polynomial growth.

Conversely, if a smooth function f : ℍ × ℂ ⟶ ℂ satisfies the above conditions (MJ1)^{0} − (MJ3)^{0}, then the function ϕ_{f} : SL_{2,1}(ℝ) ⟶ ℂ defined by

(7.15)  ϕ_{f}(g, α) := f((g, α) ∘ (i, 0)), (g, α) ∈ SL_{2,1}(ℝ)

is a Maass-Jacobi form.

Proof. Take g̃ ∈ SL_{2,1}(ℝ) such that g̃ ∘ (i, 0) = (τ, z). Then for any γ̃ ∈ SL_{2,1}(ℤ), we have

f_{ϕ}(γ̃ ∘ (τ, z)) = f_{ϕ}((γ̃ ∗ g̃) ∘ (i, 0))
= ϕ(γ̃ ∗ g̃) = ϕ(g̃) = f_{ϕ}(τ, z).

So (MJ1)^{0} is proved.

Let π : SL_{2,1}(ℝ) ⟶ SL_{2,1}(ℝ)/SO(2) be the natural projection. For any function f on SL_{2,1}(ℝ)/SO(2), we put

f̃:= f ∘ π.

Thus f̃ is a function on SL_{2,1}(ℝ). Since SL_{2,1}(ℝ)/SO(2) is a reductive homogeneous space, we have a natural homomorphism μ : u ⟼ D_{u} of 𝔻_{SO(2)}(SL_{2,1}(ℝ)) onto 𝔻(SL_{2,1}(ℝ)/SO(2)) defined by

(7.16)  (μ(u)f)^{}˜ = (D_{u}f)^{}˜ := uf̃, f ∈ C^{∞}(SL_{2,1}(ℝ)/SO(2)).

The kernel of μ is given by

(7.17)  ker μ = 𝔻_{SO(2)}(SL_{2,1}(ℝ)) ∩ 𝔻(SL_{2,1}(ℝ)) 𝔰𝔬(2),

where 𝔰𝔬(2) denotes the Lie algebra of SO(2). Therefore we have the isomorphism

(7.18)  𝔻(SL_{2,1}(ℝ)/SO(2)) = 𝔻(ℍ × ℂ) ≅ 𝔻_{SO(2)}(SL_{2,1}(ℝ))/ker μ.

For the proof of the above facts (7.16), (7.17) and (7.18), we refer to [He2], pp. 285-

286. It follows from (7.18) that the function f_{ϕ} is a joint eigenfunction for 𝔻(SL_{2,1}(ℝ)/SO(2)) = 𝔻(ℍ × ℂ).

It is easy to check (MJ3)^{0}. We leave the detail to the reader. If we note ϕ = f_{ϕ}∘π, the proof of the converse can be done in a similiar way. □

For a right K-invariant function ϕ : SL_{2,1}(ℝ) ⟶ ℂ, we define another function h_{ϕ} : SP_{2} ⟶ ℂ by

(7.19)  h_{ϕ}(Y, V) := ϕ(g, V ^{t}g^{−1} ), Y ∈ SP_{2}, V ∈ ℝ^{(1,2)},

where Y = g^{t}g with some g ∈ SL(2, ℝ). The definition (7.19) is well defined because it is independent of the choice of g with Y = g^{t}g. Indeed, such g is unique up to SO(2).

Theorem 7.6. Let ϕ : SL_{2,1}(ℝ) ⟶ ℂ be a nonzero Maass-Jacobi form. Then the function h_{ϕ} defined by (7.19) satisfies the following conditions : (MJ1)^{∗} h_{ϕ}(γY ^{t}γ, (V + δ)^{t}γ) = h_{ϕ}(Y, V) for all (γ, δ) ∈ SL_{2,1}(ℤ) with γ ∈

SL(2, ℤ) and δ ∈ ℤ^{(1,2)}. (MJ2)^{∗} h_{ϕ} is a joint eigenfunction for all the differential operators in

𝔻(SP_{2} × ℝ^{(1,2)}). (MJ3)^{∗} h_{ϕ} has a suitable polynomial growth.

Conversely, if a smooth function h : SP_{2} × ℝ^{(1,2)} ⟶ ℂ satisfies the conditions (MJ1)^{∗} − (MJ3)^{∗}, the function ϕ_{h} : SL_{2,1}(ℝ) ⟶ ℂ defined by

(7.20)  ϕ_{h}(g, α) := h(g^{t}g, α^{t}g), g ∈ SL(2, ℝ), α ∈ ℝ^{(1,2)}

is a Maass-Jacobi form.

Proof. Let (γ, δ) ∈ SL_{2,1}(ℤ) and (Y, V) ∈ SP_{2} × ℝ^{(1,2)}. We take g ∈ SL(2, ℝ) such that g^{t}g = Y. Then we have

h_{ϕ}(γY ^{t}γ, (V + δ)^{t}γ) = h_{ϕ}(γg^{t}(γg), (V + δ)^{t}γ)
= ϕ(γg, (V + δ)^{t}g^{−1}) ( by (7.19) )
= ϕ((γ, δ) ∗ (g, V ^{t}g^{−1}))
= ϕ(g, V ^{t}g^{−1}) = h_{ϕ}(Y, V).

So (MJ1)^{∗} is proved. Following the line of the proof of (MJ2)^{0} in Theorem 7.5, we can prove (MJ2)^{∗}. It is easy to check (MJ3)^{∗}. The converse may be proved in a similiar way. We leave the detail of the rest to the reader. □

Definition 7.7. A smooth function on ℍ × ℂ is also called a Maass-Jacobi form if it satisfies the conditions (MJ1)^{0}−(MJ3)^{0}. For a system λ = (λ_{α}) of eigenvalues for 𝔻(ℍ × ℂ), we also denote by MJ(SL_{2,1}(ℤ), λ) the vector space of all Maass-Jacobi forms f such that L_{α}f = λ_{α}f for all L_{α} ∈ 𝔻(ℍ × ℂ).

Remark 7.8. We note that Maass wave forms are special ones of Maass-Jacobi forms. Thus the number of λ’s with MJ(SL_{2,1}(ℤ), λ) ≠ 0 is infinite.

Theorem 7.9. The Riemannian metric ds^{2} on ℍ × ℂ defined by

ds^{2} = \frac{y + v^{2}}{y^{3}} ( dx^{2} + dy^{2} ) + \frac{1}{y} ( du^{2} + dv^{2} )

− \frac{2v}{y^{2}} ( dx du + dy dv)

is invariant under the action (7.5) of SL_{2,1}(ℝ). The Laplace-Beltrami operator Δ of the Riemannian space ( ℍ × ℂ, ds^{2} ) is given by

Δ = y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + ( y + v^{2} ) ( \frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} )

+ 2 y v ( \frac{∂^{2}}{∂x∂u} + \frac{∂^{2}}{∂y∂v} ) .

That is, Δ = D + Ψ. In particular, SL_{2,1}(ℝ) is a subgroup of the isometry group of ( ℍ × ℂ, ds^{2} ).

Proof. The proof can be found in [Y7]. □

Now we introduce the notion of weak Maass -Jacobi forms.

Definition 7.10. A smooth bounded function ϕ : SL_{2,1}(ℝ) ⟶ ℂ is called a weak Maass -Jacobi form if it satisfies the conditions (MJ1), (MJ3) and the following condition (WMJ2) : (WMJ2) ϕ is an eigenfunction for the Laplace-Beltrami operator in 𝔻_{SO(2)}(SL_{2,1}(ℝ)).

In other words, the function f_{ϕ} : ℍ × ℂ ⟶ ℂ defined by (7.14) from a weak Maass-Jacobi form ϕ satisfies the conditions (MJ1)^{0}, (MJ3)^{0} and the following condition (WMJ2)^{0} : (WMJ2)^{0} f_{ϕ} is an eigenfunction for the Laplace-Beltrami operator Δ := D + Ψ.

Obviously Maass-Jacobi forms are weak Maass-Jacobi forms but in general the converse does not hold. We observe that Maass wave forms are the special ones of weak Maass-Jacobi forms.

Theorem 7.11. Let 𝔻(SP_{2} × ℝ^{(1,2)}) be the algebra of all differential operators on SP_{2} × ℝ^{(1,2)} invariant under the action (7.4) of SL_{2,1}(ℝ). Then the algebra 𝔻(SP_{2} × ℝ^{(1,2)}) is generated by the following differential operators

(7.21)  D̃ := y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) .

and

(7.22)  Ψ̃ := y^{−1} \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2 x \frac{∂^{2}}{∂v_{1}∂v_{2}} + ( x^{2} + y^{2} ) \frac{∂^{2}}{∂v_{2}^{2}} \} ,

where Y is of the form (7.7) and V = ( v_{1}, v_{2} ) with real variables x, y, v_{1}, v_{2} and y > 0. Moreover, we have

[D̃, Ψ̃] : = D̃ Ψ̃ − Ψ̃ D̃

= − 4 y \frac{∂^{3}}{∂x∂v_{1}∂v_{2}} + 4 x y \frac{∂^{3}}{∂x∂v_{2}^{2}} − 2 \frac{∂^{3}}{∂y∂v_{1}^{2}} + 4 x \frac{∂^{3}}{∂y∂v_{1}∂v_{2}}

+ 2 ( y^{2} − x^{2} ) \frac{∂^{3}}{∂y∂v_{2}^{2}} + \frac{2 x}{y} \frac{∂^{2}}{∂v_{1}∂v_{2}} + 2 Ψ.

In particular, the algebra 𝔻(SP_{2} × ℝ^{(1,2)}) is not commutative.

Proof. The proof can be found in [Y8]. □

Now we present a SL_{2,1}(ℝ)-invariant metric on SP_{2} × ℝ^{(1,2)}.

Theorem 7.12. The following Riemannian metric ds_{0}^{2} on SP_{2} × ℝ^{(1,2)} defined by

ds_{0}^{2} := \frac{1}{y^{2}} ( dx^{2} + dy^{2} ) + \frac{1}{y} \{ ( x^{2} + y^{2} ) dv_{1}^{2} + 2 x dv_{1} dv_{2} + dv_{2}^{2} \}

is invariant under the action (7.4) of SL_{2,1}(ℝ), where we used the coordinates ( Y, V ) such that Y is of the form (7.7) and V = ( v_{1}, v_{2} ) with v_{1}, v_{2} real variables. The Laplace-Beltrami operator Δ_{0} of the Riemannian space ( SP_{2} × ℝ^{(1,2)}, ds_{0}^{2} ) is given by

Δ_{0} = y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + \frac{1}{y} \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2 x \frac{∂^{2}}{∂v_{1}∂v_{2}} + ( x^{2} + y^{2} ) \frac{∂^{2}}{∂v_{2}^{2}} \} .

That is, Δ_{0} = D̃ + Ψ̃. In particular, SL_{2,1}(ℝ) is a subgroup of the isometry group of ( SP_{2} × ℝ^{(1,2)}, ds_{0}^{2} ).

Proof. The proof can be found in [Y8]. □

8. Final Remarks

In this final section, we make some comments on a fundamental domain for G_{n,m}(ℤ)\P_{n} × ℝ^{(m,n)} and provide some problems related to the previous subjects.

We define the domain ℜ_{n,m} in P_{n} × ℝ^{(m,n)} by

ℜ_{n,m} = { (Y, λ) | Y ∈ ℜ_{n}, λ = (λ_{kl}) ∈ ℝ^{(m,n)}, 0 ≤ λ_{kl} ≤ 1 for all k, l } ,

where ℜ_{n} denotes the Minkowski fundamental domain for GL(n, ℤ)\P_{n}. For more detail on ℜ_{n}, we refer the reader to [M2] or [Si1]. It is easy to see that ℜ_{n,m} is a fundamental domain for G_{n,m}(ℤ)\P_{n} × ℝ^{(m,n)}. We see that ℜ_{n,m} may be regarded as the fibre bundle over the Minkowski fundamental domain ℜ_{n} with fibres as tori of dimension mn. Thus we call the fundamental domain ℜ_{n,m} the universal torus or the universal family of tori. The concept of the universal torus is analogous to that of the universal abelian variety ( cf. [F-C] and [Y6] ).

We propose the following problem:

Problem 8.1. Describe the boundary of the universal torus in terms of automorphic forms on G_{n,m}(ℝ).

We review Maass wave forms. Let s ∈ ℂ and Γ := Γ_{0}(N) for some positive integer N > 0. We denote by W_{s}(Γ) the vector space of all smooth bounded functions ϕ : SL(2, ℝ) ⟶ ℂ such that

(a) ϕ(γgk) = ϕ(g) for all γ ∈ Γ, g ∈ SL(2, ℝ) and k ∈ SO(2).

(b) Δϕ = \frac{1−s^{2}}{4} ϕ, where

Δ = −y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − y \frac{∂^{2}}{∂x∂θ}

is the Laplace-Beltrami operator on SL(2, ℝ) whose coordinates x, y, θ ( x ∈ ℝ, y > 0, 0 ≤ θ < 2π) are given by

g = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}y^{1/2} & 0 \\ 0 & y^{−1/2} \end{matrix} ) ( \begin{matrix}cos θ & −sin θ \\ sin θ & cos θ \end{matrix} ) , g ∈ SL(2, ℝ)

by means of the Iwasawa decomposition of SL(2, ℝ). The functions in W_{s}(Γ) are called Maass wave forms. It is well known that W_{s}(Γ) is nontrivial for infinitely many values of s.

Let π_{s}^{+} and π_{s}^{c} be the irreducible unitary representations of SL(2, ℝ) induced by the characters

( \begin{matrix}a & u \\ 0 & a^{−1} \end{matrix} ) ⟼ |a|^{s}, Re (s) = 0 ; a unitary character of B

and

( \begin{matrix}a & u \\ 0 & a^{−1} \end{matrix} ) ⟼ sgn (a) |a|^{s}, Re (s) = 0 ; a non-unitary character of B

respectively, where B := \{ ( \begin{matrix}a & u \\ 0 & a^{−1} \end{matrix} ) | a > 0, u ∈ ℝ \} is a Borel subgroup of

SL(2, ℝ). Let L_{0}^{2} (Γ\SL(2, ℝ)) be the subspace of all cuspidal forms in L^{2} (Γ\SL(2, ℝ)) . It is well known that the multiplicity of π_{s}^{+} or π_{s}^{c} in the right regular representation of SL(2, ℝ) on L_{0}^{2} (Γ\SL(2, ℝ)) is equal to the dimension of W_{s}(Γ). For more detail, we refer to [G] and [I].

Problem 8.2. Decompose the Hilbert space L^{2} (SL_{n,m}(ℤ)\SL_{n,m}(ℝ)) into irreducibles. Let L_{0}^{2} (SL_{n,m}(ℤ)\SL_{n,m}(ℝ)) be the subspace of all cuspidal automorphic forms in L^{2} (SL_{n,m}(ℤ)\SL_{n,m}(ℝ)) . Investigate the role which Maass-Jacobi forms play in the decomposition of the right regular representation of SL_{n,m}(ℝ) on L_{0}^{2}(SL_{n,m}(ℤ)\SL_{n,m}(ℝ)).

Let μ : 𝔻 (G_{n,m}(ℝ)/K) ⟶ ℂ be a homomorphism. Let

E_{μ} := { f ∈ C^{∞} (G_{n,m}(ℝ)/K) | Df = μ(D) for all D ∈ 𝔻 (G_{n,m}(ℝ)/K) }

be the joint eigenspace associated with μ. Then we have the natural representation of G_{n,m}(ℝ) on E_{μ} defined by

( T_{μ}(g)f ) (xK) := f(g^{−1}xK), g, x ∈ G_{n,m}(ℝ), f ∈ E_{μ}.

We call T_{μ} the eigenspace representation of G_{n,m}(ℝ) on E_{μ}.

Problem 8.3. For which μ is T_{μ} irreducible ? Decompose T_{μ} into irreducibles.

Problem 8.4. Write down M_{n,m,M} ( cf. 4.1 ) in terms of D_{j}’s and Ψ_{kp}’s.

Problem 8.5. Construct automorphic forms G_{n,m}(ℝ) and Maass-Jacobi forms. Develope the Fourier expansion of automorphic forms on G_{n,m}(ℝ) and Maass-Jacobi forms. And develope the theory of these automorphic forms adelically.

Selberg [S2] discussed the invariant differential operators and the integral operators on bounded symmetric domains in the sense of Elie Cartan ( cf. [Si2] ). We propose the following problem.

Problem 8.6. Discuss the G_{n,m}^{J}(ℝ)-invariant differential operators on H_{n} ×ℂ^{(m,n)} under the action of (1.4) of G_{n,m}^{J}(ℝ) and the integral operators on H_{n} × ℂ^{(m,n)}. We observe that H_{n} × ℂ^{(m,n)} is not a symmetric space and not a bounded domain in ℂ^{d}, d = n(n + 1)/2 + mn.

Finally we review the results of G. Shimura [Sh]. We let G/K be a hermitian symmetric space of classical and noncompact type. Here G is classical, usually a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. Let ρ be a continuous representation of K on a finite dimensional complex vector space V_{ρ}. We denote by C^{∞}(ρ) the set of all V_{ρ}-valued smooth functions f : G ⟶ V_{ρ} such that

f(gk^{−1}) = ρ(k)f(g) for all k ∈ K and g ∈ G.

We let 𝔻(ρ) the ring of all left-invariant differential operators on G which map C^{∞}(ρ) into itself.

G. Shimura proved that if ρ is one-dimensional, then 𝔻(ρ) is commutative and there exists a canonically defined set of generators L_{1}, L_{2}, ⋯, L_{r} for 𝔻(ρ) which are algebraically independent. Thus we have 𝔻(ρ) = ℂ[L_{1}, ⋯, L_{r}]. Here r denotes the rank of the hermitian symmetric space G/K.

Returning to our case, since G_{n,m}^{J}(ℝ) acts on H_{n} × ℂ^{(m,n)} transitively, H_{n} × ℂ^{(m,n)} may be regarded as a homogeneous space G_{n,m}^{J}(ℝ)/K_{n,m}^{J} via the identification

G_{n,m}^{J}(ℝ)/K_{n,m}^{J} ⟶ H_{n} × ℂ^{(m,n)}, g · K_{n,m}^{J} ⟼ g · (iE_{n}, 0), g ∈ G_{n,m}^{J}(ℝ),

where K_{n,m}^{J} := U(n)×Symm(ℝ^{m}) is the stabilizer of G_{n,m}^{J}(ℝ) at (iE_{n}, 0). Let τ be a continuous representation of K_{n,m}^{J} on a finite dimensional complex vector space V_{τ} . We denote by C^{∞}(τ) the set of all V_{τ} -valued smooth functions on G_{n,m}^{J}(ℝ) such that

f(gk^{−1}) = τ(k)f(g), g ∈ G_{n,m}^{J}(ℝ), k ∈ K_{n,m}^{J}.

We let 𝔻(τ) the ring of all left-invariant differential operators on G_{n,m}^{J}(ℝ) which map C^{∞}(τ) into itself.

Problem 8.7. Discuss the 1st and 2nd main theorems for 𝔻(τ) in the sense of invariant theory. Also discuss the existence of a canonically defined set of algebraically independent generators for 𝔻(τ).

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Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse 26 53225 Bonn Germany

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