Studies on boundary values of eigenfunctions on spaces of constant negative curvature

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(1)UPPSALA DISSERTATIONS IN MATHEMATICS 57. Studies on boundary values of eigenfunctions on spaces of constant negative curvature Pierre Bäcklund. Department of Mathematics Uppsala University UPPSALA 2008.

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(147) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Bäcklund, P. (2008) Automorphic distributions and Selberg zeta functions. Manuscript. II Bäcklund, P. (2008) Families of equivariant differential operators and anti de Sitter spaces. Manuscript.. iii.

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(149) Contents. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Hyperbolic spaces, anti de Sitter spaces and their boundaries 1.2 Selberg zeta functions and Patterson’s conjecture . . . . . . . . 1.3 Paper I: Automorphic distributions and Selberg zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 One-parameter families of intertwinors and conformal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Paper II: Equivariant differential operators and anti de Sitter spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Summary in Swedish: Studier angående randvärden för egenfunktioner på rum med konstant negativ krökning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1 2 6 10 11. 17 19 21. v.

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(151) 1. Introduction. In this thesis we study two problems in the spectral geometry of locally symmetric spaces concerning relations between analysis on the symmetric space and its boundary. Our work is motivated by a conjecture of Patterson on Selberg zeta functions of Kleinian groups, and by conformal geometry.. 1.1 Hyperbolic spaces, anti de Sitter spaces and their boundaries In any dimension n, hyperbolic space and the universal cover of anti de Sitter space are the Riemannian and Lorentzian simply connected spaces of constant negative sectional curvature, respectively. Hn and AdSn can be realized as the upper sheet (t > 0) of the hyperboloid t2 − x21 − x22 − · · · − x2n = 1 and the one-sheeted hyperboloid −t21 − t22 + x21 + · · · + x2n−1 = −1 in Rn+1 , respectively. The pseudo-Riemannian metrics ds2 = dt2 − dx21 − dx22 − · · · − dx2n and. ds2 = −dt21 − dt22 + dx21 + · · · + dx2n−1. on Rn+1 induce Riemannian and Lorentzian metrics on the hyperboloids. Convenient alternative models are the upper half-space realizations Hn = {(x1 , . . . , xn ) ∈ Rn | xn > 0} with the Riemannian metric x−2 n. n . dx2i. i=1. and Un = {(x1 , . . . , xn ) ∈ Rn | xn > 0} 1.

(152) with the Lorentzian metric. . x−2 n. −dx21 +. n . dx2i. i=2. (being a chart of anti de Sitter space). The set of lines through the origin in the light cone t2 − x21 − x22 − · · · − x2n = 0 can be regarded as a boundary of Hn . It can be identified with the sphere S n−1 (at infinity); [Rat94]. Similarly, the set of lines through the origin in the cone −t21 − t22 + x21 + · · · + x2n−1 = 0 defines a boundary of AdSn which can be identified with the quotient of S 1 × S n−2 by the product of antipodal maps; [BCD+ 07], [Fra02]. The group O(1, n) acts by isometries on Hn and by Möbius transformations on the sphere S n−1 . The action on the latter space is by conformal diffeomorphisms with respect to the round metric gS n−1 . Similarly, O(2, n) acts isometrically on AdSn and by conformal diffeomorphisms on S 1 × S n−2 with respect to the Lorentzian metric (−gS 1 ) ⊕ gS n−2 . Discrete subgroups Γ of O(1, n) and O(2, n) give rise to quotient spaces Γ\Hn and Γ\AdSn , which inherit constant curvature metrics. There is a Selberg type zeta function associated to any hyperbolic manifold Γ\Hn . These motivate the results of the first paper.. 1.2 Selberg zeta functions and Patterson’s conjecture For a hyperbolic surface X = Γ\H2 , the Selberg zeta function of X (or Γ) is defined by (1 − e−(s+N )l(c) ), Re(s) > 1, ZX (s) = c N ≥0. where c runs over all oriented closed primitive geodesics in X and l(c) is the length of the geodesic c. The equivalent definition ZX (s) = (1 − N (γ)−(s+N ) ) {γ}Γ N ≥0. for finite area surfaces is due to Selberg [Sel56]. In the latter definition, the outer product runs over the non-trivial hyperbolic conjugacy classes of Γ and N (·) denotes the norm of such classes ([Sel56], [Hej76]). Selberg proved that for a finite area surface, the function ZX is holomorphic for Re(s) > 1, and admits a meromorphic continuation to the 2.

(153) complex plane. Moreover, he related the zeros and poles of ZX to the spectral theory of the Laplace-Beltrami operator ΔX and the topology of the surface X. For a finite-area surface X = Γ\H2 , the spectrum of the unbounded operator ΔX on L2 (X) consists of a discrete and, if X is non-compact, a continuous part. More precisely, let L2 (X) be the space of square integrable functions on X with respect to the inner product f, g = f (z)g(z)dμ(z), Γ\H2. where dμ is the area measure of X. Then L2 (X) = L2d (X) ⊕ L2c (X),. (1.1). and any f ∈ L2d (X) admits a decomposition f (z) =. ∞ . f, uj uj (z),. j=0. in terms of orthonormal L2 -eigenfunctions uj of ΔX , i.e., Δuj + λuj = 0, 0 = λ0 < λ1 ≤ λ2 ≤ . . . .. (1.2). The elements of L2c (X) can be represented as superpositions of Eisenstein series. Eisenstein series are generalized Laplace-Beltrami eigenfunctions associated to the cusps of the surface. If X has a single cusp at ∞, then any f ∈ L2c (X) can be written ∞ 1 1 1 f, E(·, + ir)E(z, + ir)dr, f (z) = 4π −∞ 2 2 where E(·, s) is the Eisenstein series associated to the cusp at ∞. For Re(s) > 1, it is given by the absolutely convergent series Im(γz)s , (1.3) γ∈Γ∞ \Γ. where Γ∞ is the stabilizer subgroup of ∞ ∈ ∂H2 . Since Eisenstein series are not square-integrable, the inner product f, E(·, 12 + ir) has to be interpreted appropriately. For more details see [Iwa02], [Hej76], [Hej83]. The spectrum of ΔX generates zeros of ZX as follows. The zeros outside 12 − 12 N0 are those points s ∈ C such that, either s(1−s) is an L2 eigenvalue for the Laplacian, or s is a pole of the determinant det(ϕ(s)) of the scattering matrix ϕ(s) of X. ϕ is a meromorphic function on C with values in matrices of size (m, m), where m is the number of cusps of X. 3.

(154) Later authors introduced Selberg type zeta functions for finite volume quotients of non-compact symmetric spaces of rank one by discrete subgroups ([Gan77], [Wak85], [GW80]). In addition, original work of Patterson and Elstrodt stimulated the interest in Selberg type zeta functions of more general classes of hyperbolic manifolds. In particular, it is a challenge to understand the zeta functions ZX of geometrically finite hyperbolic spaces X = Γ\Hn . In order to elucidate the relations between the geometry and the spectral theory of X, which are encoded in the zeta function ZX , it is first of all crucial to prove that ZX admits a meromorphic continuation to the complex plane. For general geometrically cofinite Γ, the meromorphic continuation of the corresponding zeta function has not yet been established. That problem, in particular, requires understanding the appropriate analogs of the scattering operators and studying their meromorphic continuation; for partial results in that direction see [FHP91], [BO99b]. However, in recent years the particular case of convex-cocompact discrete groups Γ has been studied intensively, and the meromorphy of the scattering operator and the zeta function are well-known. For a review of the spectral geometry of such spaces we refer to [Per07]. Here Γ is called convex-cocompact, if it acts with a compact quotient on the geodesic convex hull of its limit set Λ(Γ). We recall that the limit set of Γ is the set of accumulation points of orbits of Γ acting on Hn . Λ(Γ) is a closed subset of the boundary ∂Hn = S n−1 . Γ acts properly discontinuously on the complement Ω(Γ) of Λ(Γ) in S n−1 . The set Ω(Γ) is called the proper set. For convex-cocompact Γ, the quotient Γ\Ω(Γ) is compact. For a convex-cocompact Γ, Patterson and Perry ([PP01]) described the divisor of ZX in terms of L2 -eigenvalues of the Laplacian of X, poles of the associated scattering operator, and topological data. For such Γ, the discrete spectrum of the Laplacian is finite, and the continuous spectrum can be described in terms of so-called Eisenstein integrals ([Man89], [Pat75]). An alternative description of divisors of Selberg zeta functions for geometrically cofinite groups was conjectured by Patterson in [Pat]. It characterizes the divisor of the zeta function ZX of the hyperbolic manifold X = Γ\Hn in terms of the action of Γ on the limit set Λ(Γ) ⊆ S n−1 . The original version of Patterson’s conjecture claimed that ords (ZX ) = −. ∞ . −∞ p(−1)p dimH p (Γ, Vs−(n−1) (Λ(Γ))).. (1.4). p=0. (1.4) calculates the multiplicities ords (ZX ) of zeros and poles of ZX in terms of group cohomology with values in distributions supported on 4.

(155) the limit set Λ(Γ). Here we use the convention that zeros (poles) have positive (negative) multiplicity. One of the main features of (1.4) is that it provides a uniform description of all zeros and poles. In addition, it is motivated by the following observation. The zeta function ZX is completely defined by the action of Γ at its fixed points on the sphere S n−1 at infinity: these points are the end points of the lifts of closed geodesics to the universal cover. Thus, from the point of view of dynamics, it is natural to expect a description of the divisor of ZX in terms of analysis in a neighborhood of the limit set. The characterizations of the divisor of ZX in terms of spectral theory on X take a different perspective. The relations between both perspectives can be made visible by using the extension operator of Bunke and Olbrich ([BO97], [BO99a], [BO00]). A refined version of Patterson’s conjecture has been established for cocompact and convex-cocompact Γ in arbitrary dimension (see [BO95], [BO99a] and [Juh01]). However, for geometrically cofinite groups, Patterson’s conjecture is widely open, and certainly requires modifications already for cofinite Γ ([BO98]). In (1.4), the notation Vs−∞ (Λ(Γ)) indicates the space of distributions on S n−1 with support in Λ(Γ), regarded as a Γ-module with respect to the spherical principal series representation πs . More precisely, let Vs (S n−1 ) denote the space of smooth functions on S n−1 with the SO(1, n)-module structure given by the spherical principal series representation (πs (g)u) (b) = e−sg·O,b u(g −1 b).. (1.5). Here x, b denotes the hyperbolic distance from the origin O ∈ Hn of hyperbolic space in the ball model, to the horocycle through x ∈ Hn and b ∈ ∂Hn (the distance has negative sign if O lies inside the corresponding horoball and positive otherwise). Moreover, Vs−∞ (S n−1 ) = C −∞ (S n−1 ) is the space of distributions defined on S n−1 , with the action −∞ πs (g)ω, ϕ = ω, π−s−(n−1) (g −1 )ϕ , ϕ ∈ C ∞ (S n−1 ) and finally Vs−∞ (Λ(Γ)) = {ω ∈ Vs−∞ (S n−1 ) | supp (ω) ⊂ Λ(Γ)}.. (1.6). For generic s, the spaces H p (Γ, Vs−∞ (Λ(Γ)) vanish for p ≥ 2, and there are isomorphisms H 1 (Γ, Vs−∞ (Λ(Γ)) H 0 (Γ, Vs−∞ (Λ(Γ)) ([BO99a], [Juh01]). Thus, for generic s, (1.4) states that −∞ ords (Z) = dim H 1 (Γ, Vs−(n−1) (Λ(Γ)) −∞ = dim H 0 (Γ, Vs−(n−1) (Λ(Γ)).. (1.7) 5.

(156) The latter observation relates the divisor of the Selberg zeta function to spaces of Γ-invariant distributions H 0 (Γ, Vs−∞ (Λ(Γ)) Vs−∞ (Λ(Γ))Γ . Following [Sch00], such distributions will be called automorphic distributions supported on the limit set. Higher Γ-cohomology contributes only to multiplicities at non-generic real s. Finally, equivariant Poisson transforms −∞ Ps−(n−1) : Vs−(n−1) (S n−1 ) → Esn. map automorphic distributions to automorphic eigenfunctions on Hn . Here Esn := {f ∈ C ∞ (Hn ) | − Δf = s(n − 1 − s)f }. This establishes the connection between (1.4) and the more traditional type characterization of the divisor in terms of spectral theory as given in [PP01].. 1.3 Paper I: Automorphic distributions and Selberg zeta functions Paper I is motivated by the attempt to extend Patterson’s conjecture (1.4) to geometrically finite hyperbolic manifolds. Here we consider three-dimensional hyperbolic manifolds which are cylinders with a finite area surface as cross-section. In comparison with the convex-cocompact case, new difficulties arise from the fact that the fundamental domains of the corresponding discrete groups Γ touch the limit set. The cylindrical hyperbolic manifolds are constructed as follows. Any discrete subgroup Γ of SL(2, R) can be viewed as a subgroup of SL(2, C). For such a Γ, we shall write Γ2 and Γ3 to indicate the ambient group. Γ3 acts on H3 with a quotient given by a cylinder X 3 diffeomorphic to (0, π) × Γ2 \H2 . Such cylinders with compact cross-sections correspond to convex-cocompact discrete groups Γ3 , and have been studied in e.g. [GZ95] and [PP01]. In Paper I we analyze this situation for a cofinite Γ2 with one cusp. The resulting cylinder Γ3 \H3 is geometrically finite, but not convexcocompact. Now the Selberg zeta function Z 3 of the cylinder X 3 satisfies the factorization formula ([PP01]) Z 3 (s) = Z 2 (s + N ). (1.8) N ≥0. Using (1.8), the meromorphy of Z 2 ([Sel56]) implies the meromorphy of Z 3 . Moreover, it follows that the divisor of the zeta function Z 3 of the 6.

(157) cylinder is completely determined by the divisor of the zeta function Z 2 of the cross-section. Any zero (pole) s of Z 2 induces a ladder s − N0 of zeros (poles) of Z 3 and ords (Z 3 ) = ords+N (Z 2 ). (1.9) N ≥0. Now the relation (1.7) raises the question of how the ladder structure of the divisor of Z 3 is reflected in the structure of the spaces of Γ3 automorphic distributions supported on the limit set. Assuming that (1.7) is correct in the geometrically finite case, a zero of Z 2 at s would 2 −∞ correspond to an automorphic distribution in Vs−1 (Λ(Γ2 ))Γ . Moreover, by (1.9), Z 3 has a ladder of zeros in s − N0 . Therefore, (1.7) would imply that to any zero of Z 3 at s − N , N ≥ 0, there corresponds an automor3 −∞ (Λ(Γ3 ))Γ . This suggests the existence of a phic distribution in Vs−2−N sequence of operators 2. 3. −∞ −∞ Vs−1 (Λ(Γ2 ))Γ → Vs−2−N (Λ(Γ3 ))Γ , N ≥ 0,. which generate the spaces of automorphic distributions. We shall see how such maps actually appear in connection with the extension operator exts : Vs (Ω(Γ))Γ → Vs−∞ (S n−1 )Γ ,. (1.10). which was introduced and analyzed by Bunke and Olbrich for convexcocompact groups Γ acting on Hn ([BO97], [BO99a], [BO00]). Here Vs (Ω(Γ)) denotes the space C ∞ (Ω(Γ)), regarded as a Γ-module for the principal series representation πs . For convex-cocompact Γ, exts is a meromorphic family of operators which satisfies the relation resΩ(Γ) ◦ exts = id, where resΩ(Γ) is the map that restricts distributions on S n−1 to the open set Ω(Γ). This relation justifies saying that exts extends Γ-invariant functions on the proper set Ω(Γ) through the limit set Λ(Γ) to Γ-invariant distributions on S n−1 . The ranges of the residues of exts form spaces of automorphic distributions which are supported on Λ(Γ). For convex-cocompact Γ, generic zeros of the corresponding Selberg zeta function correspond to poles of the scattering operators of Γ\Hn . In turn, generic poles of the scattering operator correspond to poles of the extension operator, and the ranges of the residues of these poles generate spaces of automorphic distributions. This yields a relation between automorphic distributions supported on Λ(Γ) and zeros of the Selberg zeta function for convex-cocompact Γ, extending the corresponding relation in the cocompact case. For the details we refer to the last chapter of [Juh01]. 7.

(158) The problem of finding a clear correspondence between zeros of ZX and automorphic distributions is open for geometrically finite Γ, even for the above cylinders. This motivates the detailed investigation of the extension operator in Paper I. Now let Γ2 be a cofinite group with one cusp at infinity and without non-trivial elliptic elements. Then the scattering matrix ϕ∞ (s) of the finite area surface Γ2 \H2 is a scalar meromorphic function on C. If we view Γ2 as a subgroup Γ3 of SL(2, C) which acts on H3 , its limit set and proper set in ∂H3 = R2 ∪ {∞} are given by Λ(Γ3 ) = R ∪ {∞}. and. Ω(Γ3 ) = R2+ ∪ R2− .. We identify H2 with R2+ = {(x, y) ∈ R2 | y > 0}, and use the maps 3. T (s) : C ∞ (Γ2 \H2 ) → Vs (Ω(Γ3 ))Γ , which are defined by. y s f (x, y) if y > 0 , (T (s)f )(x, y) = 0 if y < 0. (1.11). to identify automorphic functions on H2 with functions on Ω(Γ) which are invariant under πs . 2 Now for f ∈ C ∞ (Γ2 \H2 ) = C ∞ (H2 )Γ , we define y s f (x, y)ϕ(x, y)dxdy, ϕ ∈ C0∞ (R2 ). (1.12) exts (T (s)f ), ϕ = R2+. If Γ2 is cocompact, then f is bounded and the integral in (1.12) converges if Re(s) > −1. If Γ2 is only cofinite, then (1.12) is well-defined for Re(s) > −1 only if f is subject to suitable growth conditions. In Paper I we analyze extλ (T (λ)f ) for typical functions f which are associated to cusps: packets of Eisenstein series (incomplete theta series) and Eisenstein series. We establish the meromorphic continuation to the complex plane and read off the residues. This yields automorphic distributions which are supported on the limit set Λ(Γ3 ). The main results of the first paper are the following. For ψ ∈ C0∞ (R+ ), the incomplete theta series, associated to the cusp at ∞, is defined as θψ (z) = ψ(Im(γz)). γ∈Γ∞ \Γ2. θψ is a bounded Γ2 -automorphic function satisfying σ+i∞ 1 ˜ ψ(s)E(z, s)ds θψ (z) = 2πi σ−i∞ for σ > 1, where ψ˜ is the Mellin transform of ψ ([Iwa02, p. 57]). 8.

(159) Theorem 1. Let ψ ∈ C0∞ (R+ ). Let S be the set of poles of the scattering matrix ϕ∞ of Γ2 . Assume that ϕ∞ has only simple poles. Let 1 P := S ∪ ( − N0 ). 2 Then for any ϕ ∈ C0∞ (R2 ), the function y λ θψ (x + iy)ϕ(x, y)dxdy G(λ) = extλ ((T (λ)θψ ), ϕ = R2+. is holomorphic for sufficiently large Re(λ), and has a meromorphic continuation to C with simple poles in the set P − 2 − N0 , and no other poles. The residue of G at λ0 ∈ P − 2 − N0 is given by −. ψ(p) ˜ ∗ DN (−p + N )(ω− (p)), ϕ , N!. (1.13). (p,N ). where the sum runs over all pairs (p, N ) ∈ C × N0 such that p ∈ P and 2 −∞ (R)Γ λ0 = p − 2 − N . Here the automorphic distributions ω− (p) ∈ Vp−1 are related to the Eisenstein series according to ⎧ −1 ⎪ E(z, s)) if p ∈ 12 − N0 , ⎨c(p)(Pp−1 ) (Res s=p (1.14) ω− (p) = ⎪ ⎩ Res c(s) (Pp−1 )−1 (E(z, p)) if p ∈ 12 − N0 , s=p. where Pp denotes the Poisson transform and −∞ −∞ ∗ 2 DN (−λ + N ) : Vλ−1 (R) → Vλ−N −2 (R ) √. is the adjoint of the family DN (λ), cf (1.17). Finally, c(s) =. πΓ(s− 12 ) . Γ(s). Moreover, we prove Theorem 2. Suppose ϕ∞ is holomorphic at s ∈ 12 Z. Let ϕ ∈ C0∞ (R2 ). Then the function 1 extλ (T (λ)E(·, s0 )), ϕ 2πi 1 = y λ E(x + iy, s0 )ϕ(x, y)dxdy 2πi R2+. G(λ) =. is holomorphic for sufficiently large Re(λ), and has a meromorphic continuation to C. It has poles in (1−s)−2−N0 and s−2−N0 , all of which 9.

(160) are simple, and no other poles. The residues of G at λ = (1 − s) − 2 − N and λ = s − 2 − N are given by the respective formulas 1 ∗ DN (s − 1 + N )(c(1 − s)(P−s )−1 (E(z, s))), ϕ N!. (1.15). and. 1 ∗ DN (−s + N )(c(s)(Ps−1 )−1 (E(z, s))), ϕ . N! In Theorem 1 and Theorem 2 the operators DN (λ) : Vλ (S 2 ) → Vλ−N (S 1 ), λ ∈ C. (1.16). (1.17). play an important role. These are polynomial families of differential operators which intertwine spherical principal series representations on S 1 and S 2 (see [Juh08]). The adjoint families −∞ −∞ ∗ 1 2 (λ) : V−λ+N DN −1 (S ) → V−λ−2 (S ). are defined by ∗ (λ)ω, ϕ = ω, DN (λ)ϕ, ϕ ∈ C ∞ (S 2 ). DN. If Γ is convex-cocompact, the set of complex numbers s for which the space of πs -automorphic distributions are supported on Λ(Γ) is discrete. In contrast, Theorem 2 shows that, for geometrically cofinite groups Γ, it is natural to expect one-parameter families of automorphic distributions which are supported on the limit set. In particular, the correspondence between zeros of Selberg zeta functions and automorphic distributions on the limit set is more subtle, and a characterization of (most of) the zeros in terms of automorphic distributions is not known. On the other hand, Theorem 1 indicates that such a characterization could be related to the range of the singular part of the extension operator. One of the problems here is to specify a suitable functional analytic framework so that the extension operator becomes a meromorphic family of operators. For an attempt in this direction we refer to [BO99b].. 1.4 One-parameter conformal geometry. families. of. intertwinors. and. The equivariant families DN (λ) : Vλ (S 2 ) → Vλ−N (S 1 ), which appear in Theorem 1 (see (1.17)), are special cases of SO(1, n)equivariant one-parameter families DN (λ) : Vλ (S n ) → Vλ−N (S n−1 ) 10. (1.18).

(161) of differential operators. The families (1.18) are the conformally flat special cases of conformally covariant families DN (X, M ; g; λ) : C ∞ (X) → C ∞ (M ) of differential operators. Here M → X is a hypersurface in a Riemannian manifold (X, g), and conformal covariance of DN (X, M ; g; λ) means that DN (X, M ; e2ϕ g; λ)eλϕ = e(λ−N )ϕ DN (X, M ; g; λ) for all ϕ ∈ C ∞ (X). Such families are studied in [Juh08] in connection with Branson’s Q-curvature ([Bra95]) for Riemannian metrics. Specifically, the so-called residue families res DN (h; λ) : C ∞ ([0, ε) × M ) → C ∞ (M ). contain information on Q-curvature of a Riemannian metric h on M . In [GJ07] it is shown how this leads to a formula for Q-curvature in terms of holographic data associated to the Fefferman-Graham Poincaré metric of h on [0, ε) × M . The concept of Q-curvature ([Bra95]) is defined in the broader setting of pseudo-Riemannian metrics. We recall that its definition rests on the conformally covariant powers of the Laplacian of pseudo-Riemannian metrics constructed in [GJMS92]. In order to extend the methods of [Juh08] to the case of general signature, it is necessary to establish analogs of (1.18) on pseudo-spheres S (p,q) . Here, the pseudo-sphere is S (p,q) = S p × S q with the metric (−gS p ) ⊕ gS q . Paper II deals with these problems in the case of signature (1, n) (Lorentzian signature).. 1.5 Paper II: Equivariant differential operators and anti de Sitter spaces n be the compact Lorentzian manifold S 1 × S n−1 with the metric Let Ein (−gS 1 ) ⊕ gS n−1 , and let n−1 → Ein n i : Ein be the isometric embedding induced by the equatorial embedding ˆ c (λ), N ≥ 0, of S n−2 → S n−1 of spheres. We construct a sequence D N polynomial families c n ) → C ∞ (Ein n−1 ) ˆN (λ) : C ∞ (Ein D. of O(2, n − 1)-equivariant differential operators. Here equivariance is understood with respect to spherical principal series representations πλ n ) and C ∞ (Ein n−1 ). on C ∞ (Ein 11.

(162) ˆ c (λ) descends to a polynomial family For even N , the family D N c (λ) : C ∞ (Einn ) → C ∞ (Einn−1 ) DN. of O(2, n − 1)-equivariant differential operators, where Einn = (S 1 × S n−1 )/Z2 , and the Z2 -action comes from the involution S 1 × S n−1 → S 1 × S n−1 given by (x, y) → (−x, −y). n−1 can alternatively be described as the set of rays The space Ein through the origin in the cone Cn = −t21 − t22 + x21 + · · · + x2n−1 = 0 . n−1 , and we let Pˆ n ⊂ Gn be Gn = O(2, n − 1) acts transitively on Ein the isotropy group of the ray generated by (1, 0, 0, 1, 0, . . . , 0) ∈ Rn+1 . In a similar way, Einn−1 is given by the set of lines in Cn through the origin, and we let P n ⊂ Gn denote the isotropy group of the line through (1, 0, 0, 1, 0, . . . , 0) ∈ Rn+1 . For n ≥ 4 let gn = o(2, n − 1) be the Lie algebra of Gn = O(2, n − 1). The isotropy group P n is a maximal parabolic subgroup with Langlands decomposition P n = M n A(N + )n . The corresponding decomposition of the Lie algebra of P n is pn = mn ⊕ a ⊕ n+ n with a one-dimensional space a. Here a is spanned by H and the Lie subalgebras n± n are the respective ±-eigenspaces of ad(H). Let (N − )n = exp(n− ). n For λ ∈ C we define the character ξλ : pn → C = Cλ by ξλ (tH) = tλ (ξλ acts by 0 on mn ⊕ n+ n ). Let U(gn ) be the universal enveloping algebra of gn . For λ ∈ C the character ξλ of pn gives rise to the generalized Verma module Mλ (gn ) = U(gn ) ⊗U(pn ) Cλ . Similarly, for the corresponding character ξλ of P we consider the induced representation IndG P (ξλ ) of G. Let i : U(gn ) → U(gn+1 ) be the map induced by the inclusion gn ⊂ gn+1 . Theorem 3. For any non-negative integer N , there exists a polynomial 0 (λ) ∈ U(n− ) such that the map family DN n+1 0 (λ) ⊗ 1 ∈ U(gn+1 ) ⊗ Cλ U(gn ) ⊗ Cλ−N T ⊗ 1 → i(T )DN. induces a homomorphism Mλ−N (gn ) → Mλ (gn+1 ) 0 (λ) spans the of U(gn )-modules for all λ. Furthermore, for any λ ∈ C, DN space of all homomorphisms Mλ−N (gn ) → Mλ (gn+1 ) of U(gn )-modules.. 12.

(163) Theorem 3 admits the following interpretation in terms of intertwining operators between principal series representations. Theorem 4. (a) For any even integer N ≥ 0, the polynomial family 0 (λ) defines a family of Gn -equivariant operators DN c DN (λ) : C ∞ (Einn )λ → C ∞ (Einn−1 )λ−N . 0 (λ) defines (b) For any non-negative integer N , the polynomial family DN n a family of G -equivariant operators c ˆN n )λ → C ∞ (Ein n−1 )λ−N . D (λ) : C ∞ (Ein. Here the representation spaces of spherical principal series are m−1 ), C ∞ (Einm−1 ) and C ∞ (Ein m )λ emphasizes the respectively. The notation C ∞ (Einm )λ and C ∞ (Ein respective module structure. Now let Mn be the Minkowski space with the Lorentzian metric ds2 = −dx21 + dx22 + · · · + dx2n . c (λ) in Theorem 4 give rise to families The families DN nc DN (λ) : C ∞ (Mn ) → C ∞ (Mn−1 ). (1.19). as follows. Einm is conformally flat. More precisely, the composition of the inclusion (N − )m → Gm with the projection Gm → Gm /P m defines a conformal embedding j : Mm−1 → Einm−1 . Here we identify Mm−1 (N − )m and Einm−1 Gm /P m . The embedding j yields a well-known conformal compactification of Minkowski space; see also [BCD+ 07]. Now nc (λ) are defined by restriction to the open sets j(Mm ) ⊂ the families DN Einm . nc (λ) is a polynomial in the Laplacian Δ The family D2N Mn−1 and 2 2 ∂ /∂xn with polynomial coefficients in λ. In particular, n n−1 nc ∗ N nc ∗ +N = ΔN D2N − +N = i ΔMn and D2N − Mn−1 i , 2 2 where i is the inclusion map Mn−1 → Mn . The powers (ΔMm )k of the Laplacian ΔMm on flat Minkowski space are very special cases of the conformally covariant powers of the Laplacian on pseudo-Riemannian manifolds constructed in [GJMS92]. These are the famous GJMS-operators. From the general perspective, the O(2, m)equivariance of (ΔMm )k is a consequence of the conformal covariance of the GJMS-operators. For a discussion of this equivariance within representation theory see also [Kos75] and [JV77]. 13.

(164) nc (λ) in Now we turn to an alternative construction of the families DN terms of the asymptotics of eigenfunctions of the Laplacian on anti de Sitter space AdSn . In the language of [Juh08], this is the residue family of the Lorentzian metric on Minkowski space. These families are conformally covariant families of differential operators which are associated to arbitrary pseudo-Riemannian metrics. In [Juh08], the theory of these families was restricted to the case of Riemannian signature. The terminology is motivated by the fact that the families arise as residues of meromorphic functions which constitute curved analogs of the extension operator of Paper I. On the n-dimensional Lorentzian upper half space with the metric n −2 2 2 xn −dx1 + dxi , i=2. we consider formal approximate eigenfunctions of the Laplacian with eigenvalue λ(n − 1 − λ). The ansatz u(x) ∼ xnλ+2j c2j (x1 , . . . , xn−1 ) j≥0. yields recursive relations for the coefficients c2j by which all coefficients are determined by the leading one. More precisely, there are differential operators T2j (λ) : C ∞ (Rn−1 ) → C ∞ (Rn−1 ) of order 2j, parametrized by λ, such that T2j (λ)c0 = c2j . It is easily seen that T2j (λ) = A2j (λ)(ΔMn−1 )j , (1.20) where the coefficients A2j satisfy the recursive relations A2j−2 (λ) + 2j(2j + 2λ + 1 − n)A2j (λ) = 0, A0 (λ) = 1.. (1.21). Here the d’Alembertian ∂2 ∂2 =− 2 + ∂x1 i=2 ∂x2i n−1. ΔMn−1. is the Laplacian of Minkowski space. Let S2N (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ) be defined by S2N (λ) =. N j=0. 1 T2j (λ)i∗ (2N −2j)!. . ∂ ∂xn. 2N −2j .. (1.22). The following theorem shows that both constructions yield the same results (up to normalization). 14.

(165) nc (λ) coincide, Theorem 5. The families S2N (λ + n − 1 − 2N ) and D2N up to a rational function in λ.. By Theorem 5, the residue families, which are defined in terms of the asymptotics of eigenfunctions of the Laplacian for the Poincaré-Einstein metric on anti de Sitter space, coincide with the families induced by homomorphisms of Verma modules. This is a pseudo-Riemannian version of the holographic duality of [Juh08]. Theorem 5 is expected to have natural consequences for Q-curvature along the lines of [Juh08].. 15.

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(167) 2. Summary in Swedish: Studier angående randvärden för egenfunktioner på rum med konstant negativ krökning. Olika frågeställningar inom spektralgeometri behandlas i denna avhandling genom att överföra problemen från det symmetriska rummet till dess rand. Här ges randen till det hyperboliska rummet Hn av sfären S n−1 , och randen till anti de Sitter-rummet AdSn kan identifieras med kvoten av pseudo-sfären S 1 × S n−2 med produkten av antipodavbildningar. I detta sammanhang är det användbart med representationsteori för isometrigrupperna till de symmetriska rummen. Liegruppen O(1, n) verkar isometriskt på Hn samt konformt på sfären S n−1 med avseende på den runda metriken gS n−1 . På samma sätt verkar O(2, n) isometriskt på AdSn och genom konforma diffeomorfismer på S 1 × S n−2 med avseende på Lorentzmetriken (−gS 1 ) ⊕ gS n−2 . Till varje hyperbolisk mångfald Γ\Hn finns en associerad zetafunktion ZΓ\Hn av Selbergtyp, som bestäms av de slutna geodeterna på mångfalden. För hyperboliska ytor Γ\H2 har vi Z(s) =. . (1 − e−(s+N )l(c) ),. Re(s) > 1,. c N ≥0. där den yttre produkten tas över alla slutna primitiva geodeter c på ytan Γ\H2 , och l(c) är längden på geodeten c. Trots att definitionen av Selbergs zetafunktion är geometrisk, så ges dess divisor (mängden av poler och nollställen) av spektraldata för mångfalden; för kompakta hyperboliska mångfalder bestäms divisorn av L2 egenvärdena till Laplace-Beltrami-operatorn. På icke-kompakta hyperboliska mångfalder har Laplace-Beltrami-operatorn ett såväl kontinuerligt som diskret spektrum, och divisorn hos Z bestäms av L2 -egenvärden samt så kallade spridningsresonanser. En alternativ beskrivning av divisorn för Selbergs zetafunktion för geometriskt ändliga grupper har formulerats som en förmodan av Patterson: ords (ZΓ\Hn ) = −. ∞ . −∞ p(−1)p dimH p (Γ, Vs−(n−1) (Λ(Γ))).. (2.1). p=0. 17.

(168) Multipliciteten av nollställen och poler hos Z ges i (2.1) av gruppkohomologi med värden i distributioner med stöd i hopningsmängden Λ(Γ). En av formelns fördelar är att den ger en likformig beskrivning av nollställen och poler. Pattersons förmodan har bevisats (i något modifierad form) för kokompakta och konvex-kokompakta grupper Γ. Det är fortfarande okänt vad som är den korrekta omformuleringen av Pattersons förmodan för det allmänna geometriskt ändliga fallet. Detta är en motivering till den första artikeln. I själva verket gäller generiskt att högerledet i (2.1) kan beskrivas som dimensionen för vissa rum av automorfa distributioner på randen till det hyperboliska rummet med stöd i hopningsmängden Λ(Γ). Vi inriktar oss i artikeln på studiet av BunkeOlbrichs utvidgningsoperator, eftersom residyer till denna operator ger upphov till sådana automorfa distributioner. I artikeln behandlas geometriskt ändliga hyperboliska cylindrar med icke-kompakta Riemannytor som tvärsnitt samt Selbergs zetafunktion för dessa cylindrar. Vi ger en detaljerad analys av Bunke-Olbrichs utvidgningsoperator för dessa cylindrar, under antagandet att tvärsnittet för cylindern har en spets. Vi bevisar den meromorfa fortsättningen för utvidgningen av Eisenstein-serier och ofullständiga theta-serier. Residyerna beskrivs explicit som randvärden till automorfa funktioner med hjälp av en familj DN (λ) av differentialoperatorer. Den ekvivarianta familjen DN (λ) är ett specialfall av de SO(1, N )ekvivarianta enparameterfamiljer som behandlas i monografin [Juh08], där även kopplingar ges till beräkningen av Bransons Q-krökning för Riemannska mångfalder. Begreppet Q-krökning har definierats även för pseudo-Riemannska mångfalder. För att utvidga metoderna i [Juh08] till fallet med allmän signatur, så är det nödvändigt att hitta analoga familjer DN (λ) för pseudo-sfären S p × S q med metriken (−gS p ) ⊕ gS q . Den andra artikeln behandlar detta problem för Lorentz-signaturen (1, n). Vi bevisar existens och entydighet för en följd av differentialoperatorer som sammanflätar sfäriska huvudserierepresentationer. Dessa huvudserierepresentationer realiseras geometriskt på randen till anti de Sitter-rummet. Vi beskriver också familjerna DN (λ) som asymptoter till egenfunktioner på anti de Sitter-rummet.. 18.

(169) Acknowledgements. First of all I would like to express my deep gratitude towards my advisor professor Andreas Juhl for all his help and support, for providing me with interesting research problems and for giving invaluable suggestions during my work on this thesis. I am grateful to my second advisor, docent Andreas Strömbergsson, for mathematical discussions, which have been very helpful. My thanks to Johan Öhman for his substantial effort involving a detailed reading of the preliminary manuscript and for suggesting many improvements. The constructive criticism of Leo Larsson has also been very helpful. I would also like to take this opportunity to thank all friends, inside and outside the Department of Mathematics. Thanks to Keizo Matsubara and Mårten Stenmark for all discussions about mathematics and physics. Finally, I would like to extend my thanks to my always supporting parents.. This work was funded by the Graduate School in Mathematics and Computing.. 19.

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