Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space $X$ over $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$, we show that the resolvent $R(z)=(\Box-z\operatorname{Id})^{-1}$ of the Laplace-Beltrami operator $-\Box$ on $X$ can be extended meromorphically across the spectrum of $\Box$ as a family of operators $C_c^\infty(X)\to \mathcal{D}'(X)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $\mathcal{D}'(X)$ forms a representation of the isometry group of $X$, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello-Will and Hilgert-Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which $R(z)$ extends, and the residue representations can be infinite-dimensional.

To describe the resonances of , i.e. the poles of the meromorphic extension of R(ζ), we divide the complex plane into three regions: the upper half plane {Im ζ > 0}, the real line {Im ζ = 0} and the lower half plane {Im ζ < 0}.

On the real line
In particular, all singularities of R(ζ) are on the imaginary axis ζ = is, s ∈ R. The residue of R(ζ) at ζ = is is a convolution operator of the form for a so-called H-spherical distribution ϕ ∈ D ′ (X), i.e. ϕ is an H-invariant eigendistribution: ϕ = −(s 2 − ρ 2 )ϕ. Since is G-invariant, the resolvent R(ζ) and its residues are G-equivariant, so their images form subrepresentations of D ′ (X) called residue representations. We identify the residue representations in all cases. The statement of the results requires a certain amount of notation, and we refer to Section 4 for details. At this point we remark that for F = R with pq even or F = C, H, O the residue representation at ζ = −is, s = ρ + 2k, k ∈ N, is finite-dimensional, while all other residue representations are infinitedimensional. This is one of the main differences to the Riemannian situation, i.e. the case p = 1, where all residue representations are finite-dimensional (see [7]). Moreover, in the Riemannian case all poles are of order one while we also encounter poles of order two in the case F = R with p even and q odd.
Our method of proof is very similar to the one employed by Hilgert-Pasquale [7]. We use the explicit spectral decomposition of on L 2 (X) which is equivalent to the decomposition of the left-regular representation of G on L 2 (X) into irreducible unitary representations, also referred to as Plancherel formula. This formula is due to Faraut [4] for the cases F = R, C, H and due to Kosters [13] for the exceptional case F = O. The key point is the explicit description of the Plancherel measure for the continuous part which has to be extended meromorphically using a shift of contour of integration. In this way, we pick up residues coming from poles of the Plancherel density. In contrast to the Riemannian case, some of these residues actually cancel with the residues arising from the discrete spectrum of , leading for instance to the fact that for F = R with p and q odd the resolvent R(ζ) is holomorphic in the lower half plane.
Let us also mention some related work for the case of Riemannian symmetric spaces. In addition to the work of Hilgert-Pasquale [7] who treated all Riemannian symmetric spaces of rank one, there have been some attempts at Riemannian symmetric spaces of higher rank by Hilgert-Pasquale-Przebinda [8,9,10]. They were able to treat most cases of rank two. Working in a different direction, Will [22] and Roby [18] studied resonances of the Laplacian acting on sections of vector bundles over Riemannian symmetric spaces of rank one.
One might wonder about an extension of our results to homogeneous vector bundles over the pseudo-Riemannian symmetric space X. In fact, the only homogeneous vector bundles which carry an invariant Hermitian metric are the ones associated with an irreducible representation of the compact factor U(1; F) of H. The corresponding Plancherel formula for such vector bundles was obtained by Shimeno [20]. We expect that using his work, similar methods as used in this paper are applicable. This might be particularly interesting for the case can be identified with even functions on the hyperboloid O(p, q)/ O(p− 1, q). To also treat odd functions, one has to consider the line bundle associated with the non-trivial character of O(1). For q = 1 this would describe resonances on de Sitter space dS n = O(n, 1)/ O(n − 1, 1) and for p = 2 this would treat Anti-de Sitter space AdS n = O(2, n − 1)/ O(1, n − 1). We plan to study these cases in a subsequent paper.
Acknowledgements. Both authors were partly supported by a research grant from the Villum Foundation (Grant No. 00025373), and, in addition to that, the second author was partly supported by the DFG Research Training Group 2491 "Fourier Analysis and Spectral Theory".

Pseudo-Riemannian hyperbolic spaces
We introduce pseudo-Riemannian hyperbolic spaces over F = R, C, H, O, their isometry groups, polar coordinates and the Poisson kernel. For more details, we refer the reader to the work of Faraut [4] and Shimeno [20] for F = R, C, H and of Kosters [13] for F = O.

Hyperbolic spaces and isometry groups
Let F = R, C, H and write d = dim R F = 1, 2, 4. We further let p ≥ 1 and q ≥ 0, write n = p + q and consider on F n the standard sesquilinear form of signature (dp, dq) given by On the open subset {y ∈ F n : [y, y] > 0}, the pseudo-Riemannian metric In this way, X becomes a semisimple symmetric space (although the group G itself is not semisimple for F = C, but can be replaced by the semisimple SU(p, q)).
The symmetric spaces constructed in this way are all isotropic, i.e. the isotropy group H at x 0 acts transitively on each level set in T x0 X of the quadratic form induced by the metric g. Actually, the above construction gives all but three isotropic symmetric spaces (see [23, page 382]). The missing ones can be constructed in a similar fashion using the octonions F = O. More precisely, we let G = U(p, q; O) for (p, q) ∈ {(3, 0), (2, 1), (1, 2)}, meaning U(3; O) = F 4 , the compact simple Lie group of type F 4 , or U(2, 1; O) = U(1, 2; O) = F 4(−20) , the non-compact simple Lie group of type F 4 and real rank one. Further, let H = U(p − 1, q; O), using the interpretation U(2; O) = Spin (9) and U(1, 1; F) = Spin 0 (1,8). Then X = G/H is an isotropic symmetric space called octonionic hyperbolic space (see [13,Chapter 3.4] for details). For p = 3, it is compact and Riemannian, for p = 1 it is non-compact and Riemannian, and for p = 2 it is non-compact and pseudo-Riemannian.
In this paper, we will only be concerned with non-compact pseudo-Riemannian hyperbolic spaces, so from now on we assume that p ≥ 2 and q ≥ 1.

Polar coordinates
Realizing G as n × n matrices with entries in F, we choose the maximal compact subgroup K = U(p; F) × U(q; F) of G (resp. K = Spin(9) for G = F 4(−20) ) and consider the one-parameter group A = {a t : t ∈ R} given by The map K ×A → X, (k, a) → ka·x 0 is surjective and induces a diffeomorphism onto an open dense subset of X (see [4, page 403] and [13, Lemma 3.9.1]): where A(t) = (cosh t) dp−1 (sinh t) dq−1 , db is the normalized K-invariant measure on K/M 0 and the constant only depends on the normalization of the measures.

The isotropic cone and the Poisson kernel
The isotropic cone {y ∈ F n \ {0} : [y, y] = 0} is invariant under multiplication by λ ∈ U(1; F) from the right and we denote by Ξ the corresponding quotient: G acts transitively on Ξ and the stabilizer of with z * = −z ⊤ 1 p−1,q−1 and Im F = {w ∈ F : Re(w) = 0} (see [4, page 391] and [13, Chapter 3.5]). We therefore identify Ξ ≃ G/M N . We introduce the Poisson kernel P : For fixed ξ ∈ Ξ, its complex powers (as functions of x ∈ X where P (x, ξ) = 0) are eigenfunctions of the Laplace-Beltrami operator − on X (see [4, page 394] and note that in our notation corresponds to − in Faraut's notation):

Harmonic analysis on X
We recall from [4,20] the Fourier transform, the Poisson transform, the corresponding spherical distributions and the inversion formula for the hyperbolic spaces X. For this whole section we assume p ≥ 2 and q ≥ 1.

Degenerate principal series representations
We briefly discuss the degenerate principal series representations that occur in the Plancherel formula for X. Recall the subgroups M , A and N of G. The product P = M AN is a maximal parabolic subgroup of G. For s ∈ C we define a character of P = M AN by and consider the induced representation π s = Ind G P (χ −s ), realized as the leftregular representation on the space then Ξ = R + B and therefore every function f ∈ E s (Ξ) is uniquely determined by its restriction to B. Note that K acts transitively on B and the stabilizer of b 0 = [(1, 0, . . . , 0, 1)] equals M 0 , so we can identify B ≃ K/M 0 . In particular, there is a unique normalized K-invariant measure db on B. The bilinear form [4,Proposition 5.1] and [13,Chapter 3.6]).
Under the action of K, the space L 2 (B) decomposes into a direct sum of invariant subspaces if q > 1.

The Fourier transform
For f ∈ C ∞ c (X) the following integral converges whenever Re(s) > ρ − d: By [4, Proposition 7.1] and [13,Chapter 3.9], the function (s, ξ) c (R) even or odd (depending on whether m is even or odd, see [4, page 403]). For such functions we have the constant being positive and independent of ℓ and m (see [

The Poisson transform
We further define the Poisson transform P s g of a function g ∈ E s (Ξ): Then P s g ∈ C ∞ (X) is an eigenfunction of : and P s g depends holomorphically on s ∈ C (see [20,Remark 4.5]). For g = Y ∈ Y ℓ,m we have (see [20,Lemma 4.3]):

Spherical distributions
Following [4, page 403], we define a distribution ϕ s on X by They have the following properties: (1) ϕ s is entire in s ∈ C.
In the case where ϕ s = 0, i.e. s = ±(ρ + 2k), the space of spherical distributions is two-dimensional and spanned by two distributions η k and θ k (see [

The inversion formula
.

Theorem 2.2.
For f ∈ C ∞ c (X) the following inversion formula holds: (1) For F = R and q odd: (2) For F = R and q even, or F = C, H: Here, c −2 [h(s); s 0 ] denotes the coefficient of (s − s 0 ) −2 in the Laurent expansion of a meromorphic function h(s) around s 0 .
Proof. For K-finite f ∈ C ∞ c (X) the inversion formula can either be deduced from the Plancherel formula in [4, Théorème 10] and [13, Theorem 3.13.1], or can be found in [20,Theorem 5.2]. Using the estimates in Lemma 3.1 and 3.2 it can be extended to all f ∈ C ∞ c (X). To simplify notation, we put ; ρ + 2k .

Images of Poisson transforms
In order to determine the residue representations in Section 4, we need to identify the representation of G on the image of the map for some values of s. We start with the values s for which s 2 −ρ 2 is an eigenvalue of − on L 2 (X). (2) Assume that F = R with q even or F = C, H, O and s = ρ + 2k ∈ ρ + 2N.  We also need to study the image of f → f * ϕ s for some other values of s.

Proposition 2.4.
(1) Assume that either F = R and p − q ∈ 4Z + 2 or that F = C and p − q ∈ 2Z + 1. For q > 1 and s = 0, the unitary degenerate principal series E 0 (Ξ) decomposes into the direct sum E + 0 (Ξ) ⊕ E − 0 (Ξ) of two irreducible subrepresentations consisting of the K-modules The image of the map f → f * ϕ s is equivalent to E + 0 (Ξ). For q = 1, the unitary degenerate principal series E 0 (Ξ) is irreducible and the image of the map f → f * ϕ s is equivalent to E 0 (Ξ).
(2) Assume that F = R with p odd and q even and s ∈ ρ + 2Z + 1, This subrepresentation is not unitarizable.
(3) Assume that F = R with p even and q odd and s ∈ ρ + 2Z, s > 0. The image of the map f → f * ϕ s is for 0 < s < ρ equivalent to the subrepresentation of E s (Ξ) with K-types Y ℓ,m for m − ℓ < s − ρ + p occurring with multiplicity one, and for s ≥ ρ equivalent to the finite-dimensional subrepresentation of E s (Ξ) with K-types Y ℓ,m for ℓ + m ≤ s − ρ occurring with multiplicity one. Apart from the trivial representation which occurs for s = ρ, these subrepresentations are not unitarizable.
Proof. The statements about the irreducibility and unitarizability of the subrepresentations with given K-types can for instance be deduced from the statements in [11]. To show that the image of f → f * ϕ s is the claimed representation in all cases, we apply (2.3) together with (2.1) and (2.2). This reduces the proof to the study of the zeros of (ℓ, m) → β ℓ,m (s) and β ℓ,m (−s).

Meromorphic continuation
We now prove the meromorphic continuation of the resolvent operator Applying the resolvent R(ζ) to the inversion formula in Theorem 2.2 yields is the contribution of the continuous spectrum and ; ρ + 2k the contribution of the discrete part. While D(ζ) clearly is meromorphic in ζ ∈ C, the integral expression (3.2) only shows that I(ζ) is holomorphic in the upper half plane {Im ζ > 0} and we have to shift the contour to extend it to all ζ ∈ C. Contour shifts are possible by Cauchy's Integral Theorem since (f * ϕ s )(x) is holomorphic in s ∈ C by Lemma 2.1 (1) and (c(s)c(−s)) −1 is meromorphic as a quotient of gamma functions (see (2.5)). Further, we can show that the boundary terms at infinity vanish using the following statements on the growth/decay of (f * ϕ s )(x) and (c(s)c(−s)) −1 as | Im s| → ∞: . For every compact subset Ω ⊆ X, R > 0 and M ∈ N: We give a proof of this statement in Appendix B.

Lemma 3.2. For every
Proof. This follows from Lemma A.1.

Resonances and residue representations
We now study the location of the poles of the meromorphic extension of R(ζ) and the corresponding residue representations. In view of (3.1), (3.3) and (3.5) the possible poles of R(ζ) can be divided into three families which we treat separately.

Remark 4.2.
At first glance, it seems that (3.5) suggests that I(ζ) also has poles in the upper half plane. However, the poles of the second and the third term in (3.5) cancel at all values ζ, Im ζ > 0, where (c(iζ)c(−iζ)) −1 (f * ϕ −iζ )(x) has a pole.

Poles on the real line
The only term in (3.5) that could produce a pole at ζ ∈ R is the term Proof. We use the expression (4.1). First note that the gamma factors Γ(±s) have a single pole at s = 0 while the gamma factors Γ( ±s+ρ 2 ) are regular at s = 0. Further, for s = 0 the two sine terms become sin( dp−ρ 2 π) and sin( dq−ρ 2 π), and since dq − ρ = 2 − (dp − ρ) they are either both non-zero or both zero, the latter being the case iff dp − ρ = d(p−q) 2 + 1 ∈ 2Z. This show the claim.
Theorem 4.4. The only possible pole of the resolvent R(ζ) for ζ ∈ R is at ζ = 0, and this is indeed a pole if and only if either F = C and p − q ∈ 2Z or if F = R and p − q ∈ 4Z + 2. In this case the corresponding residue representation is equivalent to the representation E + 0 (Ξ) for q > 1, and E 0 (Ξ) for q = 1 (see Proposition 2.4 (1)).
Proof. The first part of the statement follows from Lemma 4.3. For the second part, we note that the residue representation is the image of f → f * ϕ 0 , so the statement follows from Proposition 2.4 (1).
We now use Remark 3.3 to compute the residues of (4.2).

4.3.1
The case F = R with p and q even and F = C, H, O We first note that by Corollary 4.6 the first two terms in (4.2) have poles at the same places ζ = −is, s ∈ E. The corresponding residue is Moreover E = D = D 1 ∪ D 2 , so the third and the fourth term in (4.2) have poles at the same places. We look at s ∈ D 1 and s ∈ D 2 separately: • For s ∈ D 1 , the third term in (4.2) has a single pole at ζ = −is with residue .
Since in this case, ϕ s = 0, this term cancels with (4.4), so R(ζ) has no pole at ζ = −is for s ∈ D 1 .
To compare this term with (4.4), we rewrite the latter using (2.4): Res ; s .
It follows that the terms with (f * θ k )(x) cancel and the residue of (4.2) ; s .

The case F = R with p odd and q even
In view of Corollary 4.6, all computations in Section 4.3.1 remain valid. In addition, there are simple poles of (c(iζ)c(−iζ)) −1 ϕ iζ at ζ = −is with s > 0 and s ∈ ρ + 2Z + 1. Here, the same computation as in (4.4) shows that the corresponding residue of (4.2) is where we have used that ϕ s = 0 by Lemma 2.

The case F = R with p and q odd
By Corollary 4.6 the singularities of ζ → (c(iζ)c(−iζ)) −1 ϕ iζ in the lower half plane are single poles at ζ = −is, s > 0 with s ∈ ρ + 2Z + 1, so the same computation as in (4.4) shows that the residue of the first two terms in (4.2) at ζ = −is equals i 2s Res While the fourth term in (4.2) vanishes in this case (here D 2 = ∅), the third term also has a single pole at ζ = −is, s > 0 with s ∈ ρ + 2Z + 1, with residue equal to .
Since ϕ s = 0 by Lemma 2.1, these two residues cancel, so we have: Theorem 4.9. For F = R with p and q odd, the resolvent R(ζ) is holomorphic in the lower half plane {Im ζ < 0}.

The case F = R with p even and q odd
As in Section 4.3.3, we have D 2 = ∅ and hence (4.2) only contains three terms. In view of Corollary 4.6, the potential poles of the expression (4.2) are at ζ = −is with either s ∈ D 1 = (ρ + 2Z + 1) ∩ (0, ∞) or s ∈ (ρ + 2Z) ∩ (0, ∞). For s ∈ ρ + 2Z + 1, s > 0, the same computations as in Section 4.3.3, show that all three terms in (4.2) have a simple pole at ζ = −is, but the residues cancel, so R(ζ) is regular at these points.
For s ∈ (ρ + 2Z), 0 < s < ρ, one can argue as in Section 4.3.1 that the first two terms in (4.2) have simple poles at ζ = −is and the same computation as in (4.4) produces the following residue of R(ζ) at ζ = −is: Res Finally, for s ∈ ρ + 2N, the function (c(σ)c(−σ)) −1 has a double pole at σ = s by Corollary 4.6. In view of Remark 3.3 we can rewrite the first two terms of (4.2) as ; σ = s .
The first and the last term in this expression have a pole of order two at ζ = −is, and adding the c −2 -terms of their Laurent expansion around ζ = −is gives ; σ = s .
(We remark at this point that in the same expression the singularities of the three terms around ζ = +is cancel, so there is no contradiction with the observation from Section 4.

B Proof of Lemma 3.1
Some of the arguments are inspired by the proof of the Paley-Wiener Theorem for the case F = R and p, q ∈ 2Z in [1]. We first prove estimates for H ∩ Kinvariant functions in a specific K-isotypic component and then combine these estimates to show the claim for all compactly supported smooth functions. For T > 0 we let C ∞ T (X) ⊆ C ∞ c (X) denote the space of all f ∈ C ∞ (X) such that f (ka t · x 0 ) = 0 for t > T . Note that C ∞ T (X) is invariant under the action of K. Further, A(t) ∼ e 2ρt as t → ∞, so for F ∈ C ∞ c (R) with supp F ⊆ [−T, T ] we have for all x = g · x 0 , g ∈ Ω and all s ∈ C with | Re s| ≤ R. Since the map g → (1 − ∆ B ) N N L g −1 f ∞ is continuous, it is bounded on Ω and the proof is complete.