Journal of Lie Theory Volume 10 (2000) 311–322 C ◯2000 Heldermann Verlag

Harmonic analysis on SU(n,n)/SL(n,C) x R_{+}^{∗}

Nils Byrial Andersen and Jérémie M. Unterberger^{∗}

Communicated by J. Hilgert

Abstract. We find an explicit expression for the spherical functions on the ordered symmetric space M=SU(n,n)/SL(n,ℂ)×ℝ_{+}^{∗}, we formulate and prove a Paley-Wiener theorem for the spherical Laplace transform on M and we find an inversion formula for the Abel transform on M.

0. Introduction

Let M = SU(n, n)/SL(n, ℂ) × ℝ_{+}^{∗}, let 𝔞^{−} be the negative Weyl chamber of a certain Cartan subspace 𝔞 for M, let λ ∈ 𝔞_{ℂ}^{∗}, the complex dual of 𝔞, and let A^{−} = exp 𝔞^{−}. Let Φ_{λ} denote the Harish-Chandra series on the Riemannian dual M^{d} = SU(n, n)/S(U(n) × U(n)) of M. G. Ólafsson proved in [9], §5 an expansion formula (for general ordered symmetric spaces):

φ_{λ}(a) = ∑_{w∈W_{0}} c(wλ)Φ_{wλ}(a), a ∈ A^{−},

for the spherical functions φ_{λ} on M (see §3 for a precise definition and construction of φ_{λ}), where c(λ) is the c-function for M and W_{0} is some Weyl group.

The Berezin-Karpelevič formula for the spherical functions ψ_{λ}^{d} on M^{d} was proved by B. Hoogenboom, see [6], using the Harish-Chandra expansion of ψ_{λ}^{d} and an explicit expression for Φ_{λ}. We use the expansion formula above to prove a similar (explicit) formula for the spherical functions φ_{λ} on M.

The spherical Laplace transform L on M is defined in terms of integrating against the spherical functions. We use the explicit formulae for the spherical functions on M and M^{d} to prove a Paley-Wiener Theorem for the spherical Laplace transform, generalizing results in the rank 1 case obtained by G. Ólafsson and the first author, see [1].

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag

∗ The authors were supported by postdoc fellowships from the European Commission within the European TMR Network ”Harmonic Analysis” 1998-2001 (Contract ERBFMRX-CT97-0159). The first author was also supported by The Carlsberg Foundation.

The Abel transform on M is related to the spherical Laplace transform L by the classical Laplace transform on the cone c_{max} ⊂ 𝔞. We find an inversion formula for the Abel transform, using an approach similar to the method used by C. Meaney for the inversion formula for the Abel transform on M^{d}, see [8].

We have tried to keep notations and proofs to a minimum in order to make the presentation as clear as possible, we refer to [3], [5] and [9] for more details on spherical functions and the spherical Laplace and Abel transforms defined on ordered symmetric spaces.

We would like to thank J. Faraut, G. Ólafsson and H. Schlichtkrull for helpful discussions and comments.

1. Notation and preliminaries

Let n be a positive integer and let G^{c} = SU(n, n) denote the connected group of matrices with determinant 1 preserving the hermitian form

(x, y) = x_{1}ȳ_{1} + ⋯ + x_{n}ȳ_{n} − x_{n+1}ȳ_{n+1} − ⋯ − x_{2n}ȳ_{2n}, x, y ∈ ℂ^{2n}.

The Lie algebra 𝔤^{c} = 𝔰𝔲(n, n) is given by 2n × 2n-matrices of the form

𝔤^{c} = \{( \begin{matrix}a & c \\ c^{∗} & b \end{matrix} )| a = −a^{∗}, b = −b^{∗}, tr(a + b) = 0\} ,

where a, b and c are n × n-matrices. It is isomorphic (by c-duality) to

𝔤 = \{( \begin{matrix}α & β \\ γ & −α^{∗} \end{matrix} )| β = β^{∗}, γ = γ^{∗}, ℑ tr α = 0\} .

We embed 𝔥 = 𝔰𝔩(n, ℂ) ⊕ ℝ ≅ {α ∈ 𝔤𝔩(n, ℂ) | ℑ tr α = 0} in the diagonal as follows:

α ↦ ( \begin{matrix}α \\ & −α^{∗} \end{matrix} ) .

Let G and H denote the analytic subgroups of GL(2n, ℂ) with Lie algebras 𝔤 and 𝔥 respectively. The involution σ on 𝔤 given by

σ ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = ( \begin{matrix}α & −β \\ −γ & δ \end{matrix} ) ,

fixes 𝔥. The −1 eigenspace 𝔮 of σ is given by:

𝔮 = \{( \begin{matrix}& β \\ γ \end{matrix} )| β = β^{∗}, γ = γ^{∗}\} .

Let M = G/H ≅ SU(n, n)/SL(n, ℂ) × ℝ_{+}^{∗}, then G/H is an ordered symmetric space of Cayley type, see [5] or [9], §1.

Let θ be the classical Cartan involution on 𝔤, i.e. θ(X) = −X^{∗}, X ∈ 𝔤, and let 𝔨 and 𝔭 denote the ±1-eigenspaces of θ. Let K ≅ S(U(n) × U(n)) denote the maximal compact subgroup of G with Lie algebra 𝔨. Then G/K is isometric to the Riemannian dual M^{d} of M, see [5] and [9], §1 for details.

We choose a Cartan subspace 𝔞 ⊂ 𝔭 ∩ 𝔮 for M as follows:

𝔞 = \{X_{T} = ( \begin{matrix}0 & T \\ T & 0 \end{matrix} )| T = diag(t_{1}/2, …, t_{n}/2), t_{1}, …, t_{n} ∈ ℝ\} .

We note that 𝔞 also is a Cartan subspace of 𝔭. We identify 𝔞 and ℝ^{n} via the map ℝ^{n} ∋ t = (t_{1}, …, t_{n}) ↦ T = diag(t_{1}/2, …, t_{n}/2). Let γ_{i} ∈ 𝔞^{∗} be defined by: γ_{i}(t) = −t_{i} for i = 1, …, n. We identify the complexified dual 𝔞_{ℂ}^{∗} and ℂ^{n} by the map:

ℂ^{n} ∋ λ = (λ_{1}, …, λ_{n}) ↦ − ∑_{j} λ_{j}γ_{j}.

The root system Δ = Δ(𝔤, 𝔞) is given by Δ = \{±γ_{i}\}∪\{\frac{γ_{j}±γ_{i}}{2} \}, with multiplicity

m_{α} = 2 for the short roots α = \frac{γ_{j}±γ_{i}}{2} and m_{α} = 1 for the long roots α = ±γ_{i}.

Let Δ^{+} = \{γ_{i}\} ∪ \{\frac{γ_{j}±γ_{i}}{2} , i < j\} be a set of positive roots. Let furthermore Δ_{0}

denote the root system Δ_{0} = \{\frac{γ_{j}−γ_{i}}{2} \} with positive roots Δ_{0}^{+} = \{\frac{γ_{j}−γ_{i}}{2} , i < j\}. The negative Weyl chamber 𝔞^{−} is given by:

𝔞^{−} = {t ∈ ℝ^{n}| 0 < t_{1} < t_{2} < ⋯ < t_{n−1} < t_{n}}.

Let W ≅ {±1}^{n} × 𝔖_{n} and W_{0} ≅ 𝔖_{n} (the permutation group of n elements) denote the Weyl groups of the root systems Δ and Δ_{0} respectively. Let finally 𝔫 = ∑_{α∈Δ^{+}} 𝔤_{α}, 𝔫̄ = ∑_{α∈Δ^{+}} 𝔤_{−α}, A = exp 𝔞, A^{−} = exp 𝔞^{−}, N = exp 𝔫 and N̄ = exp 𝔫̄, where exp is the exponential mapping from 𝔤 to G.

Let x ∈ ℝ^{n} and r ∈ ℝ. We will use the notation x ≥ r (x > r) if x_{j} ≥ r (x_{j} > r) for all j. Let C_{max} be the (unique) closed H -invariant cone in 𝔮 defined by C_{max} ∩ 𝔞 := c_{max} = {t ∈ ℝ^{n}|t ≥ 0}. Let S = exp(C_{max})H be the associated semigroup in G, and let S^{o} denote the interior of S. Let finally S_{A}^{o} := S^{o} ∩ A = exp c_{max}^{o}.

Let η : 𝔻(M) → 𝔻(M^{d}) denote the Flensted-Jensen isomorphism between the commutative algebras of invariant differential operators on M and M^{d} respectively (mapping the Laplace-Beltrami operator Δ on M onto the Laplace-Beltrami operator Δ^{d} = η(Δ) on M^{d}). Let Π(D) and Π^{d}(D^{d}) denote the radial part (on A^{−}) of D ∈ 𝔻(M) and D^{d} ∈ 𝔻(M^{d}) respectively. There exists a unique map C_{c}^{∞}(H\S^{o}/H) ∋ f ↦ f^{d} ∈ C_{c}^{∞}(K\G/K) such that f_{|A^{−}} = f_{|A^{−}}^{d} and Π(D)f_{|A^{−}} = Π^{d}(η(D))f_{|A^{−}}^{d}, see [5] or [9], §4 for more details.

Let P_{λ} and Q_{λ} denote Legendre functions of the first and second kind. We note that

P_{λ− \frac{1}{2}} (cosh t) = φ_{iλ}^{(0,− \frac{1}{2} )}(t) = φ_{2iλ}^{(0,0)}(t/2),

and

\frac{Γ (λ + 1)}{Γ (\frac{1}{2}) Γ (λ + \frac{1}{2})}Q_{λ− \frac{1}{2}} (cosh t) = Φ_{−iλ}^{(0,− \frac{1}{2} )}(t) = Φ_{−2iλ}^{(0,0)}(t),

where φ_{λ}^{(α,β)} and Φ_{λ}^{(α,β)} denote Jacobi functions of the first and second kind. We can furthermore view P_{λ− \frac{1}{2}} (cosh t) and Q_{λ− \frac{1}{2}} (cosh t) as spherical functions on the Riemannian symmetric space SO_{o}(1, 2)/SO(2), respectively on the ordered symmetric space SO_{o}(1, 2)/SO_{o}(1, 1), of rank 1. From e.g. [7], §2, we get the following estimates on P_{λ− \frac{1}{2}} (cosh t) and Q_{λ− \frac{1}{2}} (cosh t):

|P_{λ− \frac{1}{2}} (cosh t)| ≤ ce^{(|ℜλ|− \frac{1}{2} )|t|},

for all t ∈ ℝ and λ ∈ ℂ, for some constant c; and, for any r > 0:

|\frac{Γ(λ + 1)}{Γ(λ + \frac{1}{2})}| |Q_{λ− \frac{1}{2}} (cosh t)| ≤ c_{r}e^{−(ℜλ+ \frac{1}{2} )t},

for ℜλ ≥ 0 and t ≥ r > 0, where c_{r} is a constant only depending on r.

2. The spherical Fourier transform on M^{d} = SU(n, n)/S(U(n) × U(n))

In this section we recall some well-known definitions and results for the spherical Fourier transform on M^{d}, see e.g. [4], Chapter 4.

Let λ ∈ ℂ^{n}. The Poisson kernel for M^{d} is defined by:

NAK ∋ nak = x ↦ a^{λ+ρ} =: p_{λ}^{d}(x),

where ρ = ∑_{α∈Δ^{+}} m_{α}α. The spherical functions on M^{d} can be written as:

ψ_{λ}^{d}(x) = ∫_{K} p_{λ}^{d}(kx)dk,

for x ∈ G. The spherical functions are bi-K -invariant, ψ_{λ}^{d}(exp 0) = 1 and Dψ_{λ}^{d} = γ(D)(λ)ψ_{λ}^{d} for all D ∈ D(M^{d}) and all λ ∈ ℂ^{n}, where γ is the Harish-Chandra isomorphism. They are furthermore invariant under the action of the Weyl group W, i.e. ψ_{wλ}^{d} = ψ_{λ}^{d} for all w ∈ W.

Let Λ denote the simple roots in Δ^{+} . The Harish-Chandra series:

Φ_{λ}(a) = a^{ρ−λ} ∑_{μ∈(ℕ∪{0})Λ} a^{μ}Γ_{μ}(λ), a ∈ A^{−},

is a solution of the differential equation Δ^{d}Φ_{λ}(a) = (λ^{2} − ρ^{2})Φ_{λ}(a) for a ∈ A^{−}, where Γ_{0}(λ) ≡ 1 and Γ_{μ}(λ), μ ∈ ℕΛ is determined by recursion. The Harish-Chandra expansion formula states that:

ψ_{−λ}^{d}(a) = ψ_{λ}^{d}(a^{−1}) = ∑_{w∈W} c^{d}(wλ)Φ_{wλ}(a), a ∈ A^{−},

where the Harish-Chandra c-function c^{d} for M^{d} is given by (modulo constants):

c^{d}(λ) := ∫N̄ p_{λ}^{d}(n̄)dn̄ = ∏_{j} \frac{Γ(−λ_{j})}{Γ(−λ_{j} + \frac{1}{2})} ∏_{i<j}(λ_{i}^{2} − λ_{j}^{2})^{−1}.

The Harish-Chandra series on M^{d} is given by:

(1)  Φ_{λ}(exp t) = π^{−n/2} ∏_{i} \frac{Γ(λ_{i} + 1)}{Γ(λ_{i} + \frac{1}{2})} \frac{∏_{i} Q_{λ_{i}− \frac{1}{2}} (cosh t_{i})}{δ_{1}(t)},

for t > 0, where

δ_{1}(t) = ∏_{α= \frac{γ_{j} ±γ_{i}}{2} ,i<j} sinh ⟨−α, t⟩ = 2^{n(n−1)/2} ∏_{i<j}(cosh t_{j} − cosh t_{i}),

see [6], Theorem 2. Using the Harish-Chandra expansion formula, this yields the Berezin-Karpelevič formula for the spherical functions on M^{d}:

ψ_{λ}^{d}(exp t) = \frac{c}{∏_{i<j}(λ_{j}^{2} − λ_{i}^{2})} \frac{det (P_{λ_{i}− \frac{1}{2}} (cosh t_{j}))}{δ_{1}(t)},

for all t ∈ ℝ^{n}, where c is a constant, see [6] for more details.

The spherical Fourier transform F on M^{d} is defined for any function f ∈ C_{c}^{∞}(K\G/K) as:

F(f)(λ) = ∫_{G} f(x)ψ_{−λ}^{d}(x)dx = ∫_{A^{−}} f(a)ψ_{−λ}^{d}(a)δ(a)da,

where δ(exp t) = ∏_{α∈Δ^{+}} sinh^{m_{α}}⟨−α, t⟩ = δ_{1}(t)^{2} ∏_{j} sinh t_{j}. The inversion formula for F reads (after normalizing dλ suitably):

f(x) = ∫_{iℝ^{n}} F(f)(λ)ψ_{λ}^{d}(x)|c^{d}(λ)|^{−2}dλ,

for all f ∈ C_{c}^{∞}(K\G/K) and x ∈ G.

Let R > 0. Let C_{R}^{∞}(K\G/K) := {f ∈ C_{c}^{∞}(K\G/K)|suppf ⊂ exp B_{R}}, where B_{R} := {t ∈ ℝ^{n}| |t| ≤ R}. Define the Paley-Wiener space H_{R}(ℂ^{n}) as the space of W -invariant holomorphic functions g on ℂ^{n} of exponential type R, i.e. satisfying the estimate:

sup_{λ∈ℂ^{n}} e^{−R|ℜλ|}(1 + |λ|)^{N}|g(λ)| < ∞,

for all N ∈ ℕ. Furthermore denote by H(ℂ^{n}) the union of the spaces H_{R}(ℂ^{n}) for all R > 0.

Theorem 1 (The Paley-Wiener Theorem). The Fourier transform is a bijection of C_{c}^{∞}(K\G/K) onto H(ℂ^{n}). More precisely it is a bijection of C_{R}^{∞}(K\G/K) onto H_{R}(ℂ^{n}) for all R > 0.

3. Spherical functions on M = SU(n, n)/SL(n, ℂ) × ℝ_{+}^{∗}

We define spherical functions on M according to [9], Definition 4.1:

Definition 2. An H -biinvariant continuous function φ : S^{o} → ℂ is called a spherical function if there exists a character χ of 𝔻(M) such that (in the sense of distributions) Dφ = χ(D)φ for all D ∈ 𝔻(M).

Define the Poisson kernel for M (and the open orbit NAH) by:

NAH ∋ nah = x ↦ a^{ρ−λ} =: p_{λ}(x),

and p_{λ} ≡ 0 on G\NAH. We note that hx ∈ S ⊂ NAH for all h ∈ H and x ∈ S, see [3], Theorem 4.2. We can construct spherical functions φ_{λ} as follows:

φ_{λ}(x) := ∫_{H} p_{λ}(hx)dh,

for x ∈ S^{o}, and Dφ_{λ} = γ(D)(λ)φ_{λ} for all D ∈ 𝔻(M), whenever the integral exists, see [3], §5 and [9], Theorem 4.10.

The asymptotic behavior of φ_{λ} as t → ∞, t ∈ 𝔞^{−} is given by:

lim_{t→∞} e^{(λ−ρ)t}φ_{λ}(exp t) = c(λ) = c_{0}(λ)c_{Ω}(λ),

see [3], §6 for details, where c is the c-function for M given by:

c(λ) := ∫_{N̄∩NAH} p_{−λ}(n̄)dn̄,

the function c_{Ω} is given by (modulo constants):

c_{Ω}(λ) := ∫_{K∩NAH} p_{−λ}(k)dk = ∏_{j} \frac{Γ(λ_{j} + \frac{1}{2})}{Γ(λ_{j} + 1)} ∏_{i<j}(λ_{i} + λ_{j})^{−1},

see [2], Corollaire 5.2, and c_{0} is the c-function for a Riemannian symmetric space with root system Δ_{0} , given by (modulo constants):

c_{0}(λ) = ∏_{i<j}(λ_{j} − λ_{i})^{−1}.

We note that c_{Ω} is W_{0} -invariant, i.e. c_{Ω}(wλ) = c_{Ω}(λ) for w ∈ W_{0} .

Considering asymptotics of the spherical functions and the correspondence between (the radial parts of) invariant differential operators on M, respectively on M^{d}, we obtain the following expansion formula for φ_{λ}:

(2)  φ_{λ}(a) = c_{Ω}(λ) ∑_{w∈W_{0}} c_{0}(wλ)Φ_{wλ}(a), a ∈ A^{−},

for λ in a dense open subset of ℂ^{n}, see [9], Theorem 5.7. We use this expansion formula to find an explicit expression for φ_{λ}:

Theorem 3. The spherical functions on M are given by:

φ_{λ}(exp t) = \frac{c}{∏_{i<j}(λ_{j}^{2} − λ_{i}^{2})} \frac{det (Q_{λ_{i}− \frac{1}{2}} (cosh t_{j}))}{δ_{1}(t)},

for λ ≥ 0 and t > 0, where c is a constant. The map λ → φ_{λ}(exp t) extends (for fixed t > 0) to a meromorphic function with simple poles for λ_{i} ∈ −ℕ + \frac{1}{2} , (i = 1, … n) and λ_{i} = −λ_{j} (i ≠ j).

Proof. The expansion formula (2) yields:

φ_{λ}(exp t) = c_{Ω}(λ)c_{0}(λ) ∑_{w∈W_{0}} ε(w)Φ_{wλ}(exp t) = c(λ) ∑_{w∈W_{0}} ε(w)Φ_{wλ}(exp t),

since c_{0}(wλ) = ε(w)c_{0}(λ) for all w ∈ W_{0} = 𝔖_{n}, where ε(w) denotes the sign of the permutation w ∈ 𝔖_{n}. Inserting the explicit expression (1) of the Harish-Chandra series Φ_{λ} gives the result by definition of the determinant.

We easily get the following estimates of the spherical functions on M:

Lemma 4. Let r > 0. There exists a constant c_{r} such that

|δ_{1}(t)φ_{λ}(exp t)/c(λ)| ≤ c_{r}e^{− min_{w∈W_{0}} ⟨wℜλ,t⟩} ≤ c_{r}e^{−⟨ℜλ,rt_{o}⟩},

for ℜλ ≥ 0 and t ≥ r, where t_{o} = (1, …, 1).

Proof. Let r > 0, then:

|δ_{1}(t)φ_{λ}(exp t)/c(λ)| = c |\frac{Γ(λ_{j} + 1)}{Γ(λ_{j} + \frac{1}{2})} det (Q_{λ_{i}− \frac{1}{2}} (cosh t_{j}))|
≤ ce^{− min_{w∈W_{0}} ⟨wℜλ,t⟩},

for λ ≥ 0 and t ≥ r, for some constants c.

From the two expansion formulae for the spherical functions we finally obtain the following correspondence between the spherical functions on M^{d} and M:

ψ_{λ}^{d}(a^{−1}) = ψ_{−λ}^{d}(a) = ∑_{w∈W_{0}\W} \frac{c^{d}(wλ)}{c(wλ)}φ_{wλ}(a), a ∈ A^{−},

see also [9], Theorem 5.9. We note that the fraction \frac{c^{d}(λ)}{c(λ)} is W_{0} -invariant.

4. The spherical Laplace transform on M

We define the normalized spherical Laplace transform L^{o} on M as (cf. [3], §8):

L^{o}(f)(λ) = c_{Ω}(λ)^{−1} ∫_{A^{−}} f(a)φ_{λ}(a)δ(a)da,

for any f ∈ C_{c}^{∞}(H\S^{o}/H) ≅ C_{c}^{∞}(S_{A}^{o})^{W_{0}} (the left-W_{0} -invariant functions in C_{c}^{∞}(S_{A}^{o})), whenever the integral converges. From the explicit expression for φ_{λ}, we see that the function λ ↦ L^{o}(f)(λ) extends to a meromorphic function on ℂ^{n} with at most simple poles for λ_{i} ∈ −ℕ (i = 1, …, n).

Let f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} . We see that L^{o}f satisfies the following functional equation:

(3)  F(f^{d})(λ) = ∑_{w∈W_{o}\W} c_{1}(wλ)L^{o}(f)(wλ),

almost everywhere (and the right hand side extends to an analytic function), where

c_{1}(λ) := c^{d}(λ)/c_{0}(λ) = ∏_{j} \frac{Γ(−λ_{j})}{Γ(−λ_{j} + \frac{1}{2})} ∏_{i<j}(−λ_{i} − λ_{j})^{−1}.

The inversion formula for the normalized spherical Laplace transform is an easy consequence of (3) and the inversion formula for the spherical Fourier transform, see also [9], Theorem 6.13:

Theorem 5 (The Inversion Formula). Let f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} . Then

f(a) = \frac{|W |}{|W_{0}|} ∫_{iℝ^{n}} L^{o}(f)(λ)ψ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)},

for all a ∈ S_{A}^{o}.

Let R > r > 0 and define C_{r,R}^{∞}(S_{A}^{o})^{W_{0}} := {f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}}|suppf ⊂ exp(C_{r} ∩ B_{R})}, where C_{r} := {t ∈ ℝ^{n}|t ≥ r}. Lemma 4 and (3) suggest the following definition of the Paley-Wiener space, the supposed image space of the normalized spherical Laplace transform acting on C_{c}^{∞}(S_{A}^{o})^{W_{0}} (or on the subspaces C_{r,R}^{∞}(S_{A}^{o})^{W_{0}}):

Definition 6. Let R > r > 0. We define the Paley-Wiener space P W_{r,R}(ℂ^{n}) as the space of W_{0} -invariant meromorphic functions g on ℂ^{n}, with at most simple poles for λ_{i} ∈ −ℕ (i = 1, …, n), such that (i)

sup_{ℜλ≥0} e^{ℜ⟨λ,rt_{o}⟩}(1 + |λ|)^{N}|g(λ)/c_{0}(λ)| < ∞,

for all N ∈ ℕ, and (ii) the c_{1} -weighted average

P^{av}g(λ) = ∑_{w∈W_{0}\W} c_{1}(wλ)g(wλ)

extends to a function in H_{R}(ℂ^{n}). Furthermore denote by P W (ℂ^{n}) the union of the spaces P W_{r,R}(ℂ^{n}) over all R > r > 0.

It is easily seen that L^{o} maps C_{r,R}^{∞}(S_{A}^{o})^{W_{0}} into P W_{r,R}(ℂ^{n}) for all R > r > 0 (since L^{o}(Δf)(λ) = (λ^{2} − ρ^{2})L^{o}f(λ) for all f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} ). We remark that P^{av}L^{o} acts injectively on C_{c}^{∞}(S_{A}^{o})^{W_{0}} , since P^{av}L^{o}(f) = F(f^{d}) = 0 implies f = f^{d} = 0 on A^{−} for any f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} by injectivity of the spherical Fourier transform. The following lemma, due to H. Schlichtkrull in the rank 1 case, see [1], Lemma 7, shows that P^{av} is injective on P W (ℂ^{n}):

Lemma 7. Let g be meromorphic function on ℂ^{n} that satisfies item (i) of Definition 6 (for some r > 0). Assume that P^{av}g = 0. Then g = 0.

Proof. Let g_{1}(λ) = g(λ)/c^{d}(−λ)c_{0}(λ) and let W_{1} := {±1}^{n} ≅ W_{0}\W. Then P^{av}g(λ) = |W_{1}|c^{d}(λ)c^{d}(−λ)avg_{1}(λ), where

avg_{1}(λ) := \frac{1}{|W_{1}|} ∑_{w∈W_{1}} g_{1}(wλ)

is the average of g_{1} over W_{1} . It follows from the assumption P^{av}g = 0 that avg_{1} = 0. The function g_{1} also satisfies item (i) of Definition 6, in particular, g_{1}(i ·) ∈ L^{1}(ℝ^{n}). Let

γ(s) = ∫_{ℝ^{n}} g_{1}(iλ)e^{i⟨s,λ⟩} dλ, s ∈ ℝ^{n},

denote the Euclidean Fourier transform of g_{1}(i ·). The condition (i) implies that g_{1} is holomorphic in an open set containing {z ∈ ℂ^{n}|ℜz ≥ 0}, and the standard argument with Cauchy’s theorem gives that γ is supported on C_{r}. On the other hand, the average avγ of γ is the Fourier transform of avg_{1}(i·), which vanishes, hence avγ vanishes as well. Hence γ = 0 by the support condition. Since the Euclidean Fourier transform is injective on L^{1}(ℝ^{n}), we conclude that g_{1} , and hence also g, vanishes.

Theorem 8 (The Paley-Wiener Theorem). The normalized spherical Laplace transform L^{o} is a bijection of C_{c}^{∞}(S_{A}^{o})^{W_{0}} onto P W (ℂ^{n}). More precisely it is a bijection of C_{r,R}^{∞}(S_{A}^{o})^{W_{0}} onto P W_{r,R}(ℂ^{n}) for all R > r > 0.

Proof. It only remains to show that the normalized spherical Laplace transform maps C_{r,R}^{∞}(S_{A}^{o})^{W_{0}} onto P W_{r,R}(ℂ) for all R > r > 0.

We define an auxiliary function Ξ_{λ}^{d} by:

Ξ_{λ}^{d}(exp t) = ∑_{w∈W_{1}} c^{d}(wλ)Φ_{wλ}(exp t) = \frac{c}{∏_{i<j}(λ_{j}^{2} − λ_{i}^{2})} \frac{∏_{j} P_{λ_{j}− \frac{1}{2}} (cosh t_{j})}{δ_{1}(t)}

= c^{d}(−λ)c ∏_{j} \frac{Γ(λ_{j} + \frac{1}{2})}{Γ(λ_{j})} \frac{∏_{j} P_{λ_{j}− \frac{1}{2}} (cosh t_{j})}{δ_{1}(t)},

for λ_{i} ≠ ±λ_{j} (i ≠ j) and t_{i} ≠ t_{j} (i ≠ j). Hence ψ_{λ}^{d} = ∑_{w∈W_{0}} Ξ_{wλ}^{d}, and we can rewrite the inversion formula as:

\begin{aligned}f(a) & = \frac{|W |}{|W_{0}|} ∫_{iℝ^{n}} L^{o}f(λ)ψ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)} \\ & = |W | ∫_{iℝ^{n}} L^{o}f(λ)Ξ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)}, \end{aligned}

for all a ∈ A^{−}, by W_{0} -invariance of the measure dλ.

Consider the wave packet Ig ∈ C^{∞}(S_{A}^{o})^{W_{0}} of g ∈ P W_{r,R}(𝔞_{ℂ}^{∗}) defined by the inversion formula(e) (for a ∈ A^{−}):

\begin{aligned}Ig(a) & = \frac{|W |}{|W_{0}|} ∫_{iℝ^{n}} g(λ)ψ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)} \\ & = |W | ∫_{iℝ^{n}} g(λ)Ξ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)}. \end{aligned}

Fix r > 0 and assume that t ∉ C_{r}. There exists λ_{o} > 0 such that ⟨λ_{o}, t−rt_{o}⟩ = −ε < 0 (t_{o} = (1, …, 1)). This yields the following estimate:

|Ξ_{λ+μλ_{o}}^{d}(exp t)/c^{d}(−λ − μλ_{o})| ≤ c(1 + |λ + μλ_{0}|)^{n/2}e^{μ⟨λ_{o},rt_{o}⟩}e^{−με},

for μ ≥ 0 and λ ∈ iℝ^{n}, for some constants c not depending on λ. By Cauchy’s theorem and a contour shift we get:

Ig(exp t) = |W | ∫_{iℝ^{n}} \frac{g(λ)}{c_{0}(λ)} \frac{Ξ_{λ}^{d}(exp t)}{c^{d}(−λ)}dλ

= |W | ∫_{iℝ^{n}} \frac{g(λ + μλ_{o})}{c_{0}(λ + μλ_{o})} \frac{Ξ_{λ+μλ_{o}}^{d}(exp t)}{c^{d}(−λ − μλ_{o})}dλ

→ 0 for μ → ∞.

By continuity and W_{0} -invariance this shows that Ig is identically zero on S_{A}^{o}\ exp C_{r}.

An easy calculation shows that (for a ∈ A^{−}):

\begin{aligned}Ig(a) & = \frac{|W |}{|W_{o}|} ∫_{iℝ^{n}} g(λ)ψ_{λ}^{d}(a)\frac{dλ}{c_{0}(λ)c^{d}(−λ)} \\ & = ∫_{iℝ^{n}} P^{av}g(λ)ψ_{λ}^{d}(a) |c^{d}(λ)|^{−2} dλ, \end{aligned}

which we recognize as the inverse Fourier transform of P^{av}g ∈ H_{R}(ℂ), whence Ig(a) = 0 for a ∈ S_{A}^{o}\ exp B_{R} by the Paley-Wiener theorem for the spherical Fourier transform on M^{d}.

Since P^{av}L^{o}f = Ff^{d} for all f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} , the above also yields:

P^{av}L^{o}Ig = F(Ig)^{d} = P^{av}g,

for all g ∈ P W (ℂ^{n}), hence Lemma 7 implies that L^{o}Ig = g for all g ∈ P W (ℂ^{n}) and we conclude that L^{o} maps C_{r,R}^{∞}(S_{A}^{o})^{W_{0}} onto P W_{r,R}(ℂ^{n}) for all R > r > 0.

5. The Abel transform on M = SU(n, n)/SL(n, ℂ) × ℝ_{+}^{∗}

The Abel transform A of an H -invariant function f on the semigroup S is defined as (cf. [3], §8):

Af(a) = a^{−ρ} ∫_{N} f(na)dn,

for a ∈ A, whenever this integral exists (we put f(x) ≡ 0 for x ∈ NAH\S). It has the following connection to the spherical Laplace transform (for λ ≫ 0 and otherwise by analytic continuation):

Lf(λ) = ∫_{exp c_{max}} a^{−λ}Af(a)da = L_{A}(Af)(λ),

where L_{A} is the Euclidean Laplace transform on A with respect to the cone c_{max} , see [3], Proposition 8.5.

Using the explicit expression of the spherical functions from Theorem 3, we get (modulo constants):

∏_{i<j}(λ_{j}^{2} − λ_{i}^{2})L(f)(λ) = ∫_{t_{n}>t_{n−1}>...t_{2}>t_{1}>0} f(exp t) det (Q_{λ_{i}− \frac{1}{2}} (cosh t_{j})) \frac{δ(t)}{δ_{1}(t)}dt

= ∫_{t_{n}>t_{n−1}>...t_{2}>t_{1}>0} f(exp t) det (Q_{λ_{i}− \frac{1}{2}} (cosh t_{j})) δ_{1}(t) ∏_{j} sinh t_{j}dt

= ∑_{w∈W_{0}} ∫_{t_{n}>t_{n−1}>...t_{2}>t_{1}>0} f(exp t)ϵ(w) ∏_{j} Q_{λ_{j}− \frac{1}{2}} (cosh wt_{j})δ_{1}(t) ∏_{j} sinh t_{j}dt

= ∑_{w∈W_{0}} ∫_{t_{n}>t_{n−1}>...t_{2}>t_{1}>0} f(exp t) ∏_{j} Q_{λ_{j}− \frac{1}{2}} (cosh wt_{j})δ_{1}(wt) ∏_{j} sinh t_{j}dt

= ∫_{c_{max}} f(exp t)δ_{1}(t) \{∏_{j} Q_{λ_{j}− \frac{1}{2}} (cosh t_{j}) sinh t_{j}\} dt

= L_{1}^{⊗}(f(exp ·) · δ_{1})(λ) = L_{A}A_{1}^{⊗}(f(exp ·) · δ_{1})(λ),

where L_{1}^{⊗} is the n-fold tensor product of the Laplace transform L_{1} on the ordered symmetric space SO_{o}(1, 2)/SO_{o}(1, 1) of rank 1:

L_{1}f(λ) = ∫_{0}^{∞} f(t)Q_{λ− \frac{1}{2}} (cosh t) sinh tdt,

for f ∈ C_{c}^{∞}(ℝ_{+}), and A_{1}^{⊗} is the n-fold tensor product of the Abel transform A_{1} on SO_{o}(1, 2)/SO_{o}(1, 1):

A_{1}f(t) = ∫_{0}^{t} f(τ)(2 cosh t − 2 cosh τ)^{−1/2} sinh τdτ,

for f ∈ C_{c}^{∞}(ℝ_{+}), see [3], §10 for details (we have identified A^{−} in the rank 1 case with ℝ_{+} via the map a_{t} ↦ t).

We furthermore have:

∏_{i<j} (λ_{j}^{2} − λ_{i}^{2}) L(f) = L_{A} (∏_{i<j}(∂_{j}^{2} − ∂_{i}^{2})A(f)) (λ),

which implies that:

(∏_{i<j} (∂_{j}^{2} − ∂_{i}^{2}) A(f)) = A_{1}^{⊗}(f(exp ·) · δ_{1}),

by injectivity of the Laplace transform L_{A}. Finally, inverting one coordinate at a time, we get by [3], §10:

Theorem 9. Let f ∈ C_{c}^{∞}(S_{A}^{o})^{W_{0}} . Then:

f(exp t) = cδ_{1}(t)^{−1} ∏_{j} (\frac{1}{sinh t_{j}} \frac{d}{dt_{j}} ) ∫_{0}^{t_{n}} ⋯ ∫_{0}^{t_{1}} (∏_{k<l}(∂_{l}^{2} − ∂_{k}^{2})Af) (exp τ)

× ∏_{j} ((cosh t_{j} − cosh τ_{j})^{−1/2} sinh τ_{j}) dτ_{1} … dτ_{n},

for t ∈ 𝔞^{−}, for some constant c.

References

[1] Andersen, N. B., and G. Ólafsson, Paley-Wiener Theorems for the Spherical Laplace transform on Causal Symmetric Spaces of rank 1, Proc. Amer. Math. Soc, to appear.

[2] Faraut, J., Fonctions Sphériques sur un Espace Symétrique Ordonné de type Cayley, Cont. Math. 191 (1995), 41–55.

[3] Faraut, J., J. Hilgert, and G. Ólafsson, Spherical functions on ordered symmetric spaces, Ann. Inst. Fourier 44 (1994), 927–966.

[4] Helgason, S., “Groups and Geometric Analysis,” Academic Press, Orlando, 1984.

[5] Hilgert, J., and G. Ólafsson, “Causal Symmetric Spaces, Geometry and Harmonic Analysis,” Perspectives in Mathematics 18, Academic Press, San Diego, 1997.

[6] Hoogenboom, B., Spherical functions and invariant differential operators on complex Grassmann manifolds, Ark. Mat. 20 (1982), 69–85.

[7] Koornwinder, T. H., A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat. 13 (1975), 145–159.

[8] Meaney, C., The inverse Abel transform for SU(p, q), Ark. Mat. 24 (1986), 131–140.

[9] Ólafsson, G., Spherical Functions and Spherical Laplace Transform on Ordered Symmetric Spaces, Preprint, 1997.

Nils Byrial Andersen Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA E-mail: byrial@math.lsu.edu Jérémie M. Unterberger Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 TORINO ITALY

Received March 29, 1999 and in final form December 8, 1999