Hyperbolic Eisenstein series on n -dimensional hyperbolic spaces

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Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces Yosuke Irie Graduate School of Mathematics, Kyushu University, Motooka, Fukuoka 819-0395, Japan Received 11 December 2018; received in revised form 18 July 2019; accepted 19 July 2019 Communicated by G. Cornelissen

Abstract We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion. c 2019 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝ Keywords: Hyperbolic Eisenstein series; Hyperbolic spaces; Spectral expansion

1. Introduction The hyperbolic Eisenstein series is the Eisenstein series associated to hyperbolic fixed points, or equivalently a primitive hyperbolic element of Fuchsian groups of the first kind. It was first introduced by S.S. Kudla and J.J. Millson [11] in 1979 as an analogue of the ordinary Eisenstein series associated to a parabolic fixed point. They established an explicit construction of the harmonic 1-form dual to an oriented closed geodesic on an oriented Riemann surface M of genus greater than 1. Furthermore, they proved the meromorphic continuation of the hyperbolic Eisenstein series to all of C and gave the location of the possible poles when M is compact. After that, they generalized the results of [11] to compact n-dimensional hyperbolic manifold and its totally geodesic hyperbolic (n−k)-manifolds. In [11,12], they constructed the hyperbolic Eisenstein series by averaging certain smooth closed k-form. Following Kudla and Millson’s point of view, the scalar-valued analogue of the hyperbolic Eisenstein series is defined in [3,4], and [10]. It is defined as follows. Let H2 := {z = x + i y ∈ E-mail address: [email protected]. https://doi.org/10.1016/j.indag.2019.07.005 c 2019 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝

Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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C | x, y ∈ R, y > 0} be the upper-half plane and Γ ⊂ PSL(2, R) a Fuchsian group of the first kind acting on H2 by the fractional linear transformations. Then the quotient Γ \H2 is a hyperbolic Riemann surface of finite volume. Let γ ∈ Γ be a primitive hyperbolic element and Γγ = ⟨γ ⟩ be its centralizer group in Γ . Consider the coordinates x = eρ cos θ and y = eρ sin θ . In this setting, the hyperbolic Eisenstein series associated to γ is defined by the series ∑ E hyp,γ (z, s) := (1) (sin θ (Aηz))s , η∈Γγ \Γ

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where s ∈( C with sufficiently large Re(s) and A is an element in PSL(2, R) such that ) a(γ ) 0 −1 Aγ A = 0 a(γ )−1 for some a(γ ) ∈ R with |a(γ )| > 1. The hyperbolic Eisenstein series (1) converges for any z ∈ H2 and s ∈ C with Re(s) > 1 and defines a Γ -invariant function where it converges. Furthermore, it is known that the hyperbolic Eisenstein series E hyp,γ (z, s) satisfies the following differential equation (−∆ + s(s − 1)) E hyp,γ (z, s) = s 2 E hyp,γ (z, s + 2), where ∆ is the hyperbolic Laplace–Beltrami operator. There are many researches on this hyperbolic Eisenstein series. M.S. Risager [16] studied the hyperbolic Eisenstein series twisted by modular symbols. T. Falliero [2] and D. Garbin, J. Jorgenson and M. Munn [3] studied the asymptotic behavior of the hyperbolic Eisenstein series for a degenerating family of finite volume hyperbolic Riemann surfaces. They proved that the limit of the hyperbolic Eisenstein series associated to a pinching geodesic for degenerating family is equal to the parabolic Eisenstein series associated to the newly formed cusp on the limit surface. D. Garbin and A.-M. v. Pippich [4] studied the asymptotic behavior of parabolic, hyperbolic and elliptic Eisenstein series for an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. Furthermore, J. Jorgenson, J. Kramer and A.-M. v. Pippich [10], in 2010, proved that the hyperbolic Eisenstein series is a square integrable function on Γ \H2 and obtained the spectral expansion associated to the hyperbolic Laplace–Beltrami operator −∆ precisely. They also derived the meromorphic continuation and the location of the possible poles and residues from the spectral expansion. We are interested in the generalization of the scalar-valued hyperbolic Eisenstein series. In our previous paper [5], we defined the hyperbolic Eisenstein series for a loxodromic element of the cofinite Kleinian groups acting on the 3-dimensional hyperbolic space and proved the results analogous to [10]. We also in [6] considered the asymptotic behavior of the hyperbolic Eisenstein series for the degeneration of 3-dimensional hyperbolic manifolds and obtained the results corresponding to [2,3]. Our purpose in this article is to obtain a generalization of the hyperbolic Eisenstein series (1) for the n-dimensional hyperbolic spaces and study its fundamental properties. Let Hn := {x = (x1 , x2 , . . . , xn ) ∈ Rn | xn > 0} , Hn is the be the n-dimensional upper-half space of Rn . With the hyperbolic metric |dx| xn n-dimensional hyperbolic space. Let G := O(n, 1) be the orthogonal group of signature (n, 1) and G 0 the connected component of G containing the unit element. Then G 0 acts on Hn transitively and any element of G 0 determines an element of the group of orientation preserving isometries of Hn . Let Γ ⊂ G 0 be a torsion-free cofinite discrete subgroup. Then the quotient Γ \Hn is a n-dimensional hyperbolic manifold. Let σ ∈ O(n + 1) be the involution which is invariant on a hyperbolic (n − k)-plane Dσ ⊂ Hn . We denote by G σ and Γσ the centralizer group of σ in G 0 and the intersection Γ ∩G σ respectively. We assume the following assumption σΓσ = Γ. Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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Then the quotient Γσ \Dσ is naturally identified (n − k)-submanifold of Γ \Hn . In addition, we assume Γσ \Dσ is compact. Without loss of generality, we may identify Dσ with the subset {x = (x1 , . . . , xn ) ∈ Hn | xi = 0, 1 ≤ i ≤ k} ⊂ Hn .

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Then, for x ∈ Hn and s ∈ C with sufficiently large Re(s), the hyperbolic Eisenstein series associated to the involution σ is defined as follows. ∑ E hyp,σ (x, s) := (2) (sin ϕ0 (ηx))s ,

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η∈Γσ \Γ

where ϕ0 is given by sin ϕ0 (ηx) = cosh(dhyp (ηx, Dσ ))−1 and dhyp (ηx, Dσ ) denotes the hyperbolic distance from ηx to Dσ . The hyperbolic Eisenstein series E hyp,σ (x, s) converges absolutely and locally uniformly for any x ∈ Hn and s ∈ C with Re(s) > n − 1 and satisfies the following differential shift equation (−∆ + s(s − n + 1))E hyp,σ (x, s) = s(s − n + k + 1)E hyp,σ (x, s + 2),

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where −∆ denotes the Laplace–Beltrami operator associated to the hyperbolic metric. In this article, about this hyperbolic Eisenstein series E hyp,σ (x, s), we obtain the following results:

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Main results (See Theorems 4.3 and 4.4). Let 0 = λ0 < λ1 ≤ λ2 · · · be the eigenvalues of −∆ and em the eigenfunction corresponding to λm . Let D ⊂ N be an index set for a complete orthogonal system of eigenfunctions {em }m∈D . Then, for any s ∈ C with Re(s) > n − 1, the Eisenstein series E hyp,σ (x, s) admits the following spectral expansion. ∑ E hyp,σ (x, s) = am,σ (s)em (x)

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m∈D

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) ( ∫ ∞ n−1 1 ∑ a n−1 +iµ,σ (s)E par,ν x, + iµ dµ, 4π ν:cusps −∞ 2 2

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where E par,ν is the ordinary Eisenstein series associated to the cusp ν. Then this series converges absolutely and locally uniformly. The coefficients am,σ (s) and a n−1 +iµ,σ (s) are given by 2 ( ) 1 k am,σ = vol(S k−1 ) · Γ 2 2 (( ) ) (( ) ) n−1 Γ s − 2 + µm /2 Γ s − n−1 − µm /2 2 × Γ (s/2) Γ ((s − n + k + 1) /2) ∫ × em dv2 (5)

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Γσ \Dσ

and

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a n−1 +iµ,σ 2

( ) k 1 k−1 = vol(S ) · Γ 2 2 (( ) ) (( ) ) Γ s − n−1 + iµ /2 Γ s − n−1 − iµ /2 2 2 × Γ (s/2) Γ ((s − n + k + 1) /2) ( ) ∫ n−1 × E par,ν x, + iµ dv2 , 2 Γσ \Dσ

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)2 − λm and dv2 is the hyperbolic volume element restricted on Γσ \Dσ . In where µ2m = ( n−1 2 addition, vol(S k−1 ) denotes the Euclidean volume of the unit (k − 1)-dimensional sphere S k−1 := {x = (x1 , . . . , xk ) ∈ Rk | |x|2 = x12 + · · · + xk2 = 1}. Besides, we derive the meromorphic continuation of E hyp,σ (x, s) to all complex plane C and the possible poles and residues from the spectral expansion. (See Theorem 4.4.) We finally introduce the outline of this article. In Section 2, we establish basic notations and recall known results needed later in this article. Section 3 is devoted to the definition of the hyperbolic Eisenstein series on n-dimensional hyperbolic spaces and its fundamental properties. In Section 4, we state details of main results and their proofs.

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2. Preliminaries

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2.1. The hyperboloid model of the hyperbolic n-space

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Let Rn+1 be the (n + 1)-dimensional real vector space and ei (1 ≤ i ≤ n + 1) be the standard basis of Rn+1 . For any vector x ∈ Rn+1 , we write the coordinate representation in standard basis of Rn+1 as x = (x1 , x2 , . . . , xn+1 ). We consider the Lorentzian inner product ( , ) on Rn+1 . It is defined for any two vectors x and y in Rn+1 as follows: (x, y) := x1 y1 + x2 y2 + · · · + xn yn − xn+1 yn+1 . The inner product space Rn+1 together with the Lorentzian inner product ( , ) is called Lorentzian (n + 1)-space and is also denoted by Rn,1 . The norm in Rn+1 associated with ( , ) is defined to be the complex number 1

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∥x∥ = (x, x) 2 . Here ∥x∥ is either positive real number, zero, or positive imaginary. This norm is also called the Lorentzian norm. A function φ : Rn+1 → Rn+1 is a Lorentz transformation if and only if (φ(x), φ(y)) = (x, y)

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for all x, y ∈ Rn+1 .

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Definition 2.1. A vector x ∈ Rn+1 is called

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(1) space-like if ∥x∥ > 0, (2) light-like if ∥x∥ = 0, or (3) time-like if ∥x∥ is imaginary.

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If a vector x ∈ Rn+1 is light-like or time-like, x is said to be positive (resp. negative) if and only if xn+1 > 0 (resp. xn+1 < 0).

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Definition 2.2. A vector subspace V of Rn+1 is called

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(1) space-like if and only if every nonzero vector in V is space-like, (2) time-like if and only if V has a time-like vector, or (3) light-like otherwise.

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Two vectors x and y in Rn+1 are called Lorentz orthogonal if and only if (x, y) = 0. In such cases, if both x and y are nonzero vectors, then one of them must be time-like while the other space-like. In addition, two Lorentz orthogonal vectors x and y are called Lorentz orthonormal if and only if ∥x∥2 = ±1 and ∥y∥ = ∓1. We define the hyperbolic n-space as the hyperboloid model. Let F n ⊂ Rn+1 be the sphere of unit imaginary radius, i.e. F n := { x ∈ Rn+1 | ∥x∥2 = −1 }.

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Then F is disconnected. The subset of all x ∈ F such that xn+1 > 0 (resp. xn+1 < 0) is called the positive (resp. negative) sheet of F n . The hyperboloid model of hyperbolic n-space is defined as the positive sheet of F n . We denote it by F+n . Then, for two vectors x, y ∈ F+n , the hyperbolic distance between x and y is written as follows: n

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cosh dF+n (x, y) = −(x, y),

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where (, ) is the Lorentzian inner product. This hyperbolic distance function defines a hyperbolic metric on F+n . Then the hyperbolic geodesic and the hyperbolic m-plane in F+n are defined as follows. Definition 2.3. A hyperbolic geodesic of F+n is the intersection of F+n with a 2-dimensional time-like vector subspace of Rn+1 . Definition 2.4. A hyperbolic m-plane of F+n is the intersection of F+n with a (m + 1)dimensional time-like vector subspace of Rn+1 . Let V be the (m + 1)-dimensional time-like vector subspace of Rn+1 . Then the intersection ∩ V is a hyperbolic m-plane. Then for any x ∈ F+n there exist orthonormal vectors x1 and x2 such that

F+n

x = cosh(t)x1 + sinh(t)x2 ,

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where x1 ∈ F+n ∩ V , ∥x2 ∥ = 1 and t is the hyperbolic distance from x to x1 .

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2.2. The conformal ball model

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Next, we define the conformal ball model of the hyperbolic n-space. Let B n be the open unit ball in Rn :

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B n := {x = (x1 , . . . , xn ) ∈ Rn | |x|2 = x12 + · · · + xn2 < 1}. n

Let P1 be the stereographic projection of B onto

F+n

defined by

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1 + |x| (x + en+1 ), 1 − |x|2 where | | is the Euclidean norm of x. The metric d B n on B n is defined by P1 (x) = x +

d B n (x, y) = dF+n (P1 (x), P1 (y)). Then P1 is an isometry from B n to F+n with this metric d B n . The hyperbolic line element and the hyperbolic volume element of B n associated to d B n are given as 2|dx| 2n d x 1 · · · d x n and . 1 − |x|2 (1 − |x|2 )n Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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Then the hyperbolic Laplace–Beltrami operator associated with the hyperbolic line element is given by ∆=

n n ∑ ∑ ∂2 n−2 1 ∂ 2 + (1 − |x|2 )2 (1 − |x| ) . xi 2 4 2 ∂ xi ∂ xi i=1 i=1

2.3. The upper-half space model We introduce another model of hyperbolic n-space, namely the upper-half space model. Let U n be the upper-half space of Rn i.e. U n = { x ∈ Rn | xn > 0 }. The isometry from B n to U n is given by the stereographic projection P2 of Rn . The metric on dU n on U n is defined by dU n (x, y) = d B n (P2−1 (x), P2−1 (y)). The hyperbolic line element and the hyperbolic volume element of U n associated to dU n are given as |dx| d x1 · · · d xn and . xn xn n Then the hyperbolic Laplace–Beltrami operator associated with the hyperbolic line element is given by ) ( 2 ∂2 ∂ ∂ 2 + · · · + 2 − (n − 2)xn . ∆ = xn 2 ∂ xn ∂ xn ∂ x1 2.4. Orthogonal group O(n, 1) A real (n + 1) × (n + 1) matrix A is said to be Lorentzian if and only if the corresponding linear transformation A : Rn+1 → Rn+1 is Lorentzian. A Lorentzian matrix A is said to be positive (resp. negative) if and only if A transforms positive time-like vectors into positive (resp. negative) time-like vectors. The set of all Lorentzian matrices forms a group with the ordinary matrix multiplication. We let ⏐ { ( ) ( )} ⏐ 1 1 G = O(n, 1) := g ∈ G L(n + 1, R) ⏐⏐ t g n g= n −1 −1 be the orthogonal group of signature (n, 1). Here 1n denotes the n × n unit matrix. Then any element of G is a Lorentzian matrix and G is naturally isomorphic to the group of all Lorentz transformations of Rn+1 . Immediately, G acts F n transitively and preserves the Lorentz inner product so that we can naturally identify G with the isometry group of F n . Let P O(n, 1) be the set of all positive matrices in G. Then P O(n, 1) acts on F+n transitively. Let K be the stabilizer of en+1 in G. Then K is a maximal compact subgroup of G. By definition, the determinant of g ∈ G is equal to +1 or −1. We denote the connected component of G (resp. K ) containing the unit element by G 0 (resp. K 0 ). Then G 0 acts on F+n transitively and naturally identifies the orientation preserving isometries on F+n . K 0 is the stabilizer of en+1 in G 0 and a maximal compact subgroup of G 0 . Then the quotient space G 0 /K 0 is naturally identified with F+n . Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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2.5. Eisenstein series associated to cusps

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Let Γ ⊂ G 0 be a cofinite discrete subgroup of G 0 and ν ∈ Rn−1 ∪ {∞} be a cusp. We define the stabilizer-group of ν by Γν := {M ∈ Γ | Mν = ν}.

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Choose A ∈ G 0 such that Aν = ∞. For any x ∈ U , we write its coordinates x = (x1 , . . . , xn ).

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Then, for any x ∈ U n and s ∈ C with sufficiently large Re(s), the Eisenstein series associated to ν is defined as ∑ E ν (x, s) := xn (AMx)s .

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M∈Γν \Γ

The Eisenstein series E ν (x, s) converges absolutely and locally uniformly for any x ∈ U n and s ∈ C with Re(s) > n − 1 and it defines a Γ -invariant function where it converges. We set B1 , . . . , Bh ∈ G 0 so that η1 = B1−1 ∞ , . . . , ηh = Bh−1 ∞ ∈ Rn−1 ∪ {∞} are the representatives for Γ -classes of cusps of Γ . For ν = 1, . . . , h and x ∈ U n , the Eisenstein series associated to ην is defined ∑ E ν (x, s) = xn (Bν Mx)s .

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M∈Γην \Γ

Then E ν (x, s) has the Fourier expansions at the cusp of the form E ν (Bν−1 x, s) = xns + φνν (s)xnn−1−s + · · · for ν = 1, . . . , h and the case ν ̸= µ E ν (Bµ−1 x, s)

=

φνµ (s)xnn−1−s

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where the functions φνµ (s) are described as certain Dirichlet series. Using the above notation, we define ⎛ ⎞ E 1 (x, s) ⎜ ⎟ E(x, s) := ⎝ ... ⎠ ,

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Φ(s) := (φνµ (s)), where ν is the row index and µ the column index. The matrix Φ(s) is called the scattering matrix for E(x, s). Then the following facts are known about E(x, s) and Φ(s). Both Φ(s) and E(x, s) have meromorphic continuations to all of C in the following sense. There is a holomorphic function g : C → C with g ̸= 0 such that for every ν = 1, . . . , h the product g(s)E ν (x, s) can be continued to a function on U n × C which is real analytic in x and holomorphic in s ∈ C. Then the continued functions satisfy the following functional equation E(x, n − 1 − s) = Φ(n − 1 − s)E(x, s) Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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and Φ(s)Φ(n − 1 − s) = 1h , where 1h is the h × h unit matrix. Furthermore, the continued components E ν (x, s) of E(x, s) satisfy the following differential equation (−∆ − s(n − 1 − s))E ν (x, s) = 0,

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if s is not a pole of E ν (x, s). The Eisenstein series E ν (x, s) and the entries of scattering matrix Φ(s) have no poles in } except possibly finitely many points in the semi-open interval {s ∈ C | Re(s) > n−1 2 ( n−1 , n − 1] on the real line. These poles are simple. Furthermore, E ν (x, s) has a simple pole 2 at s = n − 1. Both the entries of E(·, s) and of Φ(s) have no poles for s ∈ C with Re(s) = n−1 2 and Φ(s) is a unitary matrix on this line.

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2.6. Domain of Laplace–Beltrami operator

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Let Γ ⊂ G 0 be a cofinite subgroup of G 0 . We denote by L 2 (Γ \U n ) the set of all Γ -invariant measurable functions f : U n → C which satisfy ∫ | f |2 dv < ∞, FΓ

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where FΓ denotes a fundamental domain of Γ . For f, g ∈ L 2 (Γ \U n ), the function f g¯ is Γ -invariant. Hence the definition ∫ ⟨ f, g⟩ := f g¯ dv (7) FΓ

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makes sense and ⟨·, ·⟩ is an inner product on L 2 (Γ \U n ). The space L 2 (Γ \U n ) is a Hilbert space through the inner product ⟨·, ·⟩. For any f ∈ L 2 (Γ \U n ), we have the following lemma. Proposition 2.5. Every f ∈ L 2 (Γ \U n ) has the following spectral expansion associated to −∆ ∑ f (x) = ⟨ f, em ⟩em (x) m∈D

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( )⟩ ( ) ∫ ∞⟨ n−1 n−1 1 ∑ f, E ν ·, + it · E ν x, + it dt, + 4π ν : cusps −∞ 2 2

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where D ⊂ N is an index set for a complete orthonormal set of eigenfunctions (en )n∈D for −∆ ∫ in L 2 (Γ \U n ) and ⟨ f, E ν (·, n−1 + it)⟩ is defined by FΓ f (y)E ν (y, n−1 + it)dv(y). The series 2 2 of the right hand side of (8) converges in the norm of the L 2 (Γ \U n ). Besides, if f ∈ C l0 (Γ \U n ) ∩ L 2 (Γ \U n ) for a positive integer l0 > 0 such that l0 > n2 and −∆l f ∈ L 2 (Γ \U n ) for any 0 ≤ l ≤ ⌊ n+1 ⌋ + 1, the spectral expansion (8) of f converges 4 uniformly and locally uniformly on Γ \U n . Especially, if f and −∆l f are smooth and bounded on Γ \U n for any 0 ≤ l ≤ ⌊ n+1 ⌋ + 1, the spectral expansion (8) of f converges uniformly and 4 locally uniformly on Γ \U n .

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Proof. See [9] p. 103 in Chapter 7, [1] p. 268 in Chapter 6 or [17]. □

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3. Hyperbolic Eisenstein series

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Let V be the space-like vector subspace of dim V = k and V ⊥ be the orthogonal complement space of V . The dimension of V ⊥ is n − k + 1. Then F+n ∩ V ⊥ is the hyperbolic (n − k)-plane. Let σ = σV ∈ O(n + 1) be the involution such that { −1 on V σ = 1 on V ⊥ . Then F+n ∩ V ⊥ is the fixed point set of σ in F+n . Let G σ be the centralizer of σ in G i.e.

Let Γ ⊂ G be a cofinite discrete subgroup of G i.e. the quotient has finite volume and Γσ be the intersection of Γ with G σ . We assume the following assumption σΓσ = Γ.

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In addition, we assume the quotient Γ \(F+n ∩ V ⊥ ) is compact. Without loss of generality, we may assume the vector subspace V and V ⊥ in Rn+1 as follows V

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k+1≤i ≤n+1 } 1 ≤ i ≤ k }.

Then the intersection F+n ∩ V ⊥ is identified with

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Dσ = { x ∈ U n | x = (0, . . . , 0, xk+1 , . . . , xn ), xn > 0 }.

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We introduce the partial polar coordinate on U . It is defined as follows. If 2 ≤ k ≤ n − 1, ⎧ x1 = eρ cos ϕ0 sin ϕ1 , ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ xi = eρ cos ϕ0 · · · cos ϕi−1 sin ϕi , 2 ≤ i ≤ k − 1, ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎨ .. xk = eρ cos ϕ0 · · · cos ϕk−2 cos ϕk−1 , ⎪ ⎪ ⎪ ⎪ ⎪ xk+1 = xk+1 , ⎪ ⎪ ⎪ ⎪. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xn−1 = xn−1 , and ⎪ ⎪ ⎩ xn = eρ sin ϕ0 ,

where ⎛

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G σ = { g ∈ G | σ gσ = g }.

V ={ x ∈ Rn+1 | xi = 0,

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ρ = log x12 + · · · + xk2 + xn2 , ⎜ π ⎜ ⎜ 0 < ϕ0 < , ⎜ 2 ⎜ π ⎜ − < ϕ < π , 1 ≤ i ≤ k − 2, and i ⎝ 2 2 0 ≤ ϕk−1 < 2π.

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Y. Irie / Indagationes Mathematicae xxx (xxxx) xxx

If k = 1, ⎧ x1 = eρ cos ϕ0 , ⎪ ⎪ ⎪ ⎪ ⎪ x2 = x2 , ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ xn−1 = xn−1 , and ⎪ ⎪ ⎪ ⎩ xn = eρ sin ϕ0 , where (

ρ = log

√

(11)

x12 + xn2 ,

0 < ϕ0 < π. Under these coordinates, the hyperbolic line element dσ and the hyperbolic volume element dv are given by ⎧ ⎫ ⎛ ⎞ k−1 i−1 n−1 ⎨ 2⎬ ∑ ∏ ∑ d x 1 i ⎝ (cos ϕ j )2 ⎠ dϕi2 + dρ 2 + dϕ02 + , dσ 2 = (sin ϕ0 )2 ⎩ e2ρ ⎭ i=1

dv =

1 e(n−k−1)ρ (sin ϕ

0)

n

·

k−2 ∏

j=0

i=k+1

(cos ϕi )k−1−i · dρdϕ0 · · · dϕk−1 d xk+1 · · · d xn−1 .

i=0

We define restricted volume element dv1 and dv2 as follows ) (k−2 ∏ dv1 = (cos ϕi )k−1−i · dϕ1 · · · dϕk−1 , i=1

dv2 = 5

6

7

8

9

10

11 12

13 14 15

1 e(n−k−1)ρ

· dρd xk+1 · · · d xn−1 .

Then we can write (cos ϕ0 )k−1 dv = dϕ0 dv1 dv2 . (sin ϕ0 )n Furthermore, the Laplace–Beltrami operator is given by ) {( 2 k−1 n−1 2 ∑ ∑ ∂ 1 ∂2 ∂2 2 2ρ ∂ ∆ = (sin ϕ0 ) + 2+ · + e ∏i−1 2 ∂ϕ 2 ∂ρ 2 ∂ϕ0 ∂ xi2 i j=0 (cos ϕ j ) i=1 i=k+1 ( ∂ ∂ − (n − k − 1) + ((k − 1) tan ϕ0 + (n − 2) cot ϕ0 ) ∂ρ ∂ϕ0 )} k−1 ∑ 1 ∂ + (k − i − 1) tan ϕi ∏i−1 . 2 ∂ϕ i j=0 (cos ϕ j ) i=1 Under above coordinates, we define the generalized hyperbolic Eisenstein series associated to σ as follows. Definition 3.1. Let x ∈ U n and s ∈ C with sufficiently large Re(s). Then the hyperbolic Eisenstein series associated to the involution σ is defined as follows ∑ E hyp,σ (x, s) := (sin ϕ0 (ηx))s . (12) η∈Γσ \Γ Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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Y. Irie / Indagationes Mathematicae xxx (xxxx) xxx

Let dhyp (x, Dσ ) be the hyperbolic distance from x to Dσ . Then we have

1

sin ϕ0 (x) · cosh(dhyp (x, Dσ )) = 1

2

for any x ∈ U . Using this formula, we can write the Eisenstein series associated to σ as ∑ cosh(dhyp (ηx, Dσ ))−s . (13) E hyp,σ (x, s) = n

3

4

η∈Γσ \Γ

Proposition 3.2. The Eisenstein series associated to σ converges absolutely and locally uniformly for any x ∈ U n and s ∈ C with Re(s) > n − 1 and satisfies the following differential shift equation (−∆ + s(s − n + 1))E hyp,σ (x, s) = s(s − n + k + 1)E hyp,σ (x, s + 2).

(14)

5 6 7

8

In order to prove Proposition 3.2, we need the help of a definition and several lemmas.

9

Definition 3.3. Let T > 0 be a positive real number. Then we define the counting function associated to σ as follows.

11

Nσ (T ; x, Dσ ) := ♯{η ∈ Γσ \Γ | dhyp (ηx, Dσ ) < T },

(15)

where ♯ is the cardinality of the set. By using the counting function defined above, we can write the hyperbolic Eisenstein series associated to σ as the Stieltjes integrals, namely ∫ ∞ E hyp,σ (x, s) = cosh(u)−s d Nσ (u; x, Dσ ). (16)

10

12

13 14 15

16

0

Lemma 3.4. Let r > 0 be the injective radius at x, i.e. for any γ1 , γ2 ∈ Γ , Bhyp (γ1 x, r ) ∩ Bhyp (γ2 x, r ) = ∅. For positive real numbers u, T0 and r such that u > T0 > r , we have the following inequality of the counting function Nσ (u; x, Dσ ) ≤ Nσ (T0 ; x, Dσ ) volhyp (Γσ \Dσ ) 1 · vol(S k−1 ) · + n−1 volhyp (Bhyp (x, r )) × {(cosh(u + r ))n−1 − (cosh(T0 − r ))n−1 }, where S k−1 is the unit (k − 1)-dimensional sphere.

17

Proof. We define the subset V (T ) ⊂ Γσ \U n by

18

V (T ) := {x ∈ Γσ \U | dhyp (x, Γσ \Dσ ) < T }. n

The hyperbolic distance from any x ∈ U n to Γσ \Dσ is given by sin ϕ0 (x). Hence the hyperbolic volume of V (T ) is given by following integral ∫ ∫ π ∫ π ∫ 2π ∫ π 2 2 2 (cos ϕ0 )k−1 ··· dϕ0 dv1 dv2 . n Γσ \Dσ ϕ1 =− π2 ϕk−2 =− π2 ϕk−1 =0 ϕ0 =ϕ0 (x) (sin ϕ0 ) Since the restricted volume element dv2 is the hyperbolic volume element on hyperbolic n − k-plane Dσ , we have ∫ dv2 = volhyp (Γσ \Dσ ). Γσ \Dσ Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

19

20 21

22

23 24

25

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In addition, ∫ π

∫

2

ϕ1 =− π2

∫ =

··· π 2

− π2

π 2

∫

ϕk−2 =− π2

∫ ···

π 2

− π2

∫

2π

ϕk−1 =0 2π

dv1

(cos ϕ1 )k−2 (cos ϕ2 )k−3 · · · cos ϕk−2 dϕ1 · · · dϕk−2 dϕk−1

0

= vol(S k−1 ), 1 2

3

4

5

6 7 8 9

10

where vol(S k−1 ) is the Euclidean volume of the (k − 1)-dimensional unit sphere. It is given explicitly as follows. ⎧ ( ) ( ) k−2 1 1 1 π 2 ⎪ k−2 ⎪ ⎪2 · · ··· k : even, · ⎨ k−2 k−4 2 2 ( ) ( ) k−3 ⎪ π 2 1 1 1 ⎪ ⎪ · ··· · k : odd. ⎩ 2k−2 · k−2 k−4 3 2 Therefore, the hyperbolic volume of V (T ) is given by ∫ π 2 (cos ϕ0 )k−1 dϕ0 , volhyp (V (T )) = vol(S k−1 ) · volhyp (Γσ \Dσ ) · n ϕ0 (T ) (sin ϕ0 ) where sin ϕ0 (T ) cosh(T ) = 1. Let r > 0 be the injective radius at x and Bhyp (x, r ) := {y ∈ U n | dhyp (x, y) < r } be the hyperbolic ball with center x and hyperbolic radius r in U n . For positive real numbers u and T0 such that u > T0 > r , we define {ηk }k ⊂ Γσ \Γ as the maximal set such that ηx ∈ V(u) \ V(T0 ). Then it follows that ⋃ Bhyp (ηk x, r ) ⊂ V (u + r ) \ V (T0 − r ). k

From this inclusion of the set, the following inequality holds ( ) ( ) ⋃ ∑ volhyp Bhyp (ηk x, r ) = volhyp Bhyp (ηk x, r ) k

k

=

∑

volhyp (Bhyp (ηk x, r ))

k

=

∑

volhyp (Bhyp (x, r ))

k

≤ volhyp (V (u + r ) \ V (T0 − r )) = volhyp (V (u)) − volhyp (V (T0 )).

(17)

For u > T0 > 0, we have ∫ π ∫ π 2 (cos ϕ0 )k−1 2 (cos ϕ0 )k−1 dϕ − dϕ0 0 n n ϕ(u) (sin ϕ0 ) ϕ(T0 ) (sin ϕ0 ) ∫ ϕ(T0 ) (cos ϕ0 )k−1 dϕ0 = (sin ϕ0 )n ϕ(u) ∫ ϕ(T0 ) cos ϕ0 ≤ dϕ0 (sin ϕ0 )n ϕ(u) ( ) 1 1 1 = − n − 1 (sin ϕ(u))n−1 (sin ϕ(T0 ))n−1 Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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13

) 1 ( (cosh(u))n−1 − (cosh(T0 ))n−1 . n−1 Hence we have the following inequality =

volhyp (V (u)) − volhyp (V (T0 )) ≤ vol(S k−1 ) · volhyp (Γσ \Dσ ) ·

{ } 1 · (cosh u)n−1 − (cosh T0 )n−1 . n−1

(18)

From (17) and (18), we have ♯{η ∈ Γσ \Γ | T0 ≤ dhyp (ηx, Dσ ) < u } = k = ≤ vol(S k−1 ) ·

∑

( ) volhyp Bhyp (x, r ) ( ) vol Bhyp (x, r )

k

volhyp (Γσ \Dσ ) vol(Bhyp (x, r ))

( ) 1 · (cosh(u + r ))n−1 − (cosh(T0 − r ))n−1 . n−1 By the definition of the set, we also have ×

(19)

♯{η ∈ Γσ \Γ | ηx ∈ V (u) \ V (T0 )} = ♯{η ∈ Γσ \Γ | ηx ∈ V (u)} − ♯{η ∈ Γσ \Γ | ηx ∈ V (T0 )}.

(20)

Therefore, from (19) and (20), we have the following inequality ♯{η ∈ Γσ \Γ | dhyp (ηx, Dσ ) < u } volhyp (Γσ \Dσ ) · vol(Bhyp (x, r )) ( ) · (cosh(u + r ))n−1 − (cosh(T0 − r ))n−1 .

≤ ♯{η ∈ Γσ \Γ | dhyp (ηx, Dσ ) < T0 } + vol(S k−1 ) · 1 n−1 From above, the assertion of the lemma holds. ×

□

1

The following inequality plays an important role in the analysis of the hyperbolic Eisenstein series. Lemma 3.5. Let F be a real-valued, smooth, decreasing function defined for u > 0 and let g1 , g2 be real-valued, non decreasing functions defined for u ≥ a > 0 and satisfying g1 (u) ≤ g2 (u) for u ≥ a. Then, the following inequality of Stieltjes integrals holds when both integrals exist: ∫ ∞ ∫ ∞ F(a)g1 (a) + F(u)dg1 (u) ≤ F(a)g2 (a) + F(u)dg2 (u). (21) a

Proof. See [3] Section 2.6 or [4] Section 2.7.

2 3

4 5 6

7

a

□

Lemma 3.6. Let s ∈ C be a complex number with Re(s) > n − 1. Then, for any ε > 0, there exists a sufficient large T0 > 0 such that ⏐∫ ∞ ⏐ ⏐ ⏐ −s ⏐ (cosh u) d Nσ (u; x, Dσ )⏐⏐ < ε. (22) ⏐ T0

Proof. We define real valued functions F(u), g1 (u) and g2 (u) as follows: F(u) := (cosh u)−Re(s) , Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

8

9 10

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g1 (u) := Nσ (u; x, Dσ ), g2 (u) := Nσ (T0 ; x, Dσ ) + vol(S k−1 ) ·

volhyp (Γσ \Dσ ) vol(Bhyp (x, r ))

( ) 1 · (cosh(u + r ))n−1 − (cosh(T0 − r ))n−1 . n−1 Then both g1 and g2 are non-decreasing and g1 (u) ≤ g2 (u) for u ≥ T0 > 0. Elementary calculations imply that ) volhyp (Γσ \Dσ ) ( · (cosh(u + r ))n−2 sinh(u + r ) du. dg2 (u) = vol(S k−1 ) · vol(Bhyp (x, r )) ×

1 2

3

By using Lemma 3.5, we have ⏐∫ ∞ ⏐ ⏐ ⏐ −s ⏐ (cosh u) d Nσ (u; x, Dσ )⏐⏐ ⏐ T0 ∫ ∞ (cosh u)−Re(s) d Nσ (u; x, Dσ ) ≤ T0 ∫ ∞ (cosh u)−Re(s) dg2 + (cosh T0 )−Re(s) {g2 (T0 ) − g1 (T0 )} . ≤

(23)

T0

The second term of (23) has the following estimate (cosh T0 )−Re(s) {g2 (T0 ) − g1 (T0 )} volhyp (Γσ \Dσ ) vol(Bhyp (x, r )) × {(cosh(T0 + r ))n−1 − (cosh(T0 − r ))n−1 } volhyp (Γσ \Dσ ) = vol(S k−1 ) · vol(Bhyp (x, r )) × (cosh T0 )−Re(s) · {(cosh(T0 + r ))n−1 − (cosh(T0 − r ))n−1 } volhyp (Γσ \Dσ ) ( e T0 )−Re(s) ( (T0 +r ) )n−1 · · e ≤ vol(S k−1 ) · vol(Bhyp (x, r )) 2 volhyp (Γσ \Dσ ) Re(s) (n−1−Re(s))T0 +(n−1)r = vol(S k−1 ) · ·2 ·e . vol(Bhyp (x, r )) ≤ (cosh T0 )−Re(s) · vol(S k−1 ) ·

(24)

We calculate the first integral as ∫ ∞ (cosh u)−Re(s) dg2 T0 ∫ ∞ volhyp (Γσ \Dσ ) = (cosh u)−Re(s) · vol(S k−1 ) · vol(Bhyp (x, r )) T0 ( ) × (cosh(u + r ))n−2 sinh(u + r ) du volhyp (Γσ \Dσ ) = vol(S k−1 ) · vol(Bhyp (x, r )) ∫ ∞ × (cosh u)−Re(s) · (cosh(u + r ))n−2 sinh((u + r ))du T0 ( u+r ) ∫ ∞ ( u )−Re(s) ( u+r )n−2 volhyp (Γσ \Dσ ) e e k−1 ≤ vol(S ) · · · e · du vol(Bhyp (x, r )) T0 2 2 Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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≤ vol(S k−1 ) ·

volhyp (Γσ \Dσ ) Re(s)−1 ·2 vol(Bhyp (x, r ))

∫

15

∞

e(n−1−Re(s))u+(n−1)r du.

(25)

T0

The integral in (25) converges for Re(s) > n − 1 and we have (25) = −2Re(s)−1 · vol(S k−1 ) ·

volhyp (Γσ \Dσ ) e(n−1−Re(s))T0 +(n−1)r · . vol(Bhyp (x, r )) n − 1 − Re(s)

1

(26)

2

Summing up (24) and (26), we have ⏐∫ ∞ ⏐ ⏐ ⏐ −s ⏐ (cosh u) d Nσ (u; x, Dσ )⏐⏐ ⏐ T0

volhyp (Γσ \Dσ ) e(n−1−Re(s))T0 +(n−1)r · vol(Bhyp (x, r )) n − 1 − Re(s) vol (Γ \D hyp σ σ) + 2Re(s) · vol(S k−1 ) · · e(n−1−Re(s))T0 +(n−1)r vol(Bhyp (x, r )) volhyp (Γσ \Dσ ) = 2Re(s)−1 · vol(S k−1 ) · vol(Bhyp (x, r )) ( ) 1 × e(n−1−Re(s))T0 +(n−1)r · 2 − . n − 1 − Re(s) ≤ −2Re(s)−1 · vol(S k−1 ) ·

(27)

Therefore, for any ε > 0, by choosing { ( volhyp (Γσ \Dσ ) 1 T0 > − log ε + log vol(S k−1 ) · Re(s) − n + 1 vol(Bhyp (x, r )) ( ))} 1 × 2Re(s)−1 e(n−1)r 2 + , Re(s) − n + 1 we have that the last term of (24) is less than ε. □

3

Proof of Proposition 3.2. From Lemma 3.6, the hyperbolic Eisenstein series E hyp,σ (x, s) converges absolutely for s ∈ C with Re(s) > n − 1. This convergence is locally uniformly and absolutely from the definition of E hyp,σ (x, s). Furthermore, direct calculation shows that E hyp,σ (x, s) satisfies the differential equation (14). Hence we complete the proof of Proposition 3.2. □ 4. Spectral expansion

4 5 6 7 8

9

4.1. Some lemmas

10

Lemma 4.1. For any s ∈ C with Re(s) > n − 1, the hyperbolic Eisenstein series E hyp,σ (x, s) is bounded as a function of x ∈ Γ \U n . If Γ is not cocompact and ν is a cusp such that ν = A(xn ∞) for some A ∈ G, then we have the estimate −1

|E hyp,σ (x, s)| = O(xn (A x)

−Re(s)

)

(28)

11 12 13

14

as x → ν.

15

Proof. If Γ \U n is compact, the boundedness of the hyperbolic Eisenstein series E hyp,σ (x, s) is clear. We consider the case that Γ \U n is non compact. Without loss of generality, we may assume that the cusp of Γ \U n is xn ∞. Let S0 = {x | xn = r0 } be the holosphere such

17

Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

16

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INDAG: 660

16 1 2

3

4

5

6

7

8 9 10

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that every point of Dσ has xn -part less than r0 . We consider x ∈ U n with xn > r0 . Then dhyp (x, S0 ) = log(xn /r0 ). We recall the counting function (15) Nσ (T ; x, Dσ ) = {η ∈ Γσ \Γ | dhyp (ηx, Dσ ) < T } and define the new counting function Nσ′ (T ; S0 , Dσ ) by Nσ′ (T ; S0 , Dσ ) := {η ∈ Γγ \Γ | dhyp (ηS0 , Dσ ) < T }. Every element of the set {η ∈ Γγ \Γ | dhyp (ηx, Dσ ) < T } corresponds to a geodesic path from ηx to Dσ of length less than T . It necessarily intersects the holosphere ηS0 . For η ∈ {η ∈ Γγ \Γ | dhyp (ηx, Dσ ) < T }, trivially dhyp (ηS0 , Dσ ) < T − log(xn /r0 ). Thus we have {η ∈ Γγ \Γ | dhyp (ηx, Dσ ) < T } ⊂ {η ∈ Γγ \Γ | dhyp (ηS0 , Dσ ) < T − log(xn /r0 )}. We set d := log(xn /r0 ). Recalling the Stieltjes integral representation of the hyperbolic Eisenstein series (16), we have the following estimate ∫ ∞ ⏐ ⏐ ⏐ E hyp,σ (x, s)⏐ ≤ cosh(u)−Re(s) d Nσ (u; x, Dσ ) 0 ∫ ∞ cosh(u)−Re(s) d Nσ′ (u − d; S0 , Dσ ). (29) ≤ d

Since e /2 ≤ cosh(u) holds for any u ∈ R, we have ∫ ∞ cosh(u)−Re(s) d Nσ′ (u − d; S0 , Dσ ) d ( ) ∫ xn −Re(s) ∞ −uRe(s) ′ e d Nσ (u; S0 , Dσ ). ≤ 2r0 0 u

12 13 14 15

(30)

The integral on the right hand side of (30) converges when Re(s) > n − 1. This is shown by using the same method with the proof of convergence of the hyperbolic Eisenstein series in the proof of Proposition 3.2. Therefore, from (29) and (30), we obtain the assertion of the lemma. □ Lemma 4.2. Let ⟨·, ·⟩ be the inner product in L 2 (Γ \U n ) and ψ be the real-valued, smooth, bounded function on FΓ = Γ \U n . Assume ε > 0 to be the sufficiently small. Then we have the following estimate

16

⟨E hyp,σ (x, s), ψ(x)⟩ (∫ ) 1 Γ ((s − n + 1)/2) Γ (k/2) = vol(S k−1 ) · ψ(x)dv + O(ε) · 2 Γ ((s − n + k + 1)/2) Γσ \Dσ as s → ∞. Proof. Since Γ is the cofinite discrete subgroup of G 0 , the quotient Γ \U n has finite volume. Therefore, ψ ∈ L 2 (Γ \U n ). From Lemma 4.1, we have E hyp,σ (x, s) ∈ L 2 (Γ \U n ). Then we have ⟨E hyp,σ (x, s), ψ(x)⟩ Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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∫ E hyp,σ (x, s)ψ(x)dv ⎛ ⎞ ∑ ⎝ (sin ϕ0 (ηx))s ⎠ ψ(x)dv

= FΓ

∫ = FΓ

η∈Γσ \Γ

∫

(sin ϕ0 (x))s ψ(x)dv

= FΓσ

∫

∫

π 2

∫

= Γσ \Dσ

∫

ϕ0 =0

S k−1

∫

π 2

∫

= Γσ \Dσ

ϕ0 =0

S k−1

(sin ϕ0 (x))s ψ(x)dv (sin ϕ0 (x))s ψ(x) ·

(cos ϕ0 )k−1 · dϕ0 dv1 dv2 , (sin ϕ0 )n

(31)

where S k−1 = (ϕ1 , . . . , ϕk−1 ). For 0 < ε < π/2 such that sin(π/2 − ε) < aε < 1, we have ∫

∫

Γσ \Dσ

+

π 2

∫

(31) =

π −ε 2

S k−1

(sin ϕ0 (x))s ψ(x) ·

O(aεs−n )

(32) π 2

as s → ∞. Furthermore, for any ∫ Γσ \U n−k

(cos ϕ0 )k−1 · dϕ0 dv1 dv2 (sin ϕ0 )n

ψ(x)dv2 =

∫

− ε < ϕ0 <

π 2

and any Θ ∈ S k−1 , we have

ψ(x)dv2 + O(ε)

1

(33)

Γσ \Dσ

as s → ∞. From (31)–(33), for any sufficiently small ε > 0, we have ⟨E hyp,σ (x, s), ψ(x)⟩ (∫ ∫ π ∫ 2 = π −ε 2

S k−1

) ψ(x)dv2 + O(ε)

Γσ \Dσ

× (cos ϕ0 )k−1 (sin ϕ0 )s−n dv1 dϕ0 + O(aεs−n ) ) ψ(x)dv2 + O(ε)

(∫ = Γσ \Dσ

π 2

∫ × = vol(S

k−1

π −ε 2

∫ S k−1

(∫

(cos ϕ0 )k−1 (sin ϕ0 )s−n dv1 dϕ0 + O(aεs−n )

) ψ(x)dv2 + O(ε)

)· Γσ \Dσ π 2

∫ × = vol(S k−1 ) ·

π −ε 2

(cos ϕ0 )k−1 (sin ϕ0 )s−n dϕ0 + O(aεs−n )

(∫

) ψ(x)dv2 + O(ε)

Γσ \Dσ

π 2

∫ ×

(cos ϕ0 )k−1 (sin ϕ0 )s−n dϕ0

(34)

0

Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

2

INDAG: 660

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as s → ∞. For complex numbers s1 , s2 with Re(s1 ) > 0 and Re(s2 ) > 0, we have the following formula ∫ 1 Γ (s1 )Γ (s2 ) . x s1 −1 (1 − x)s2 −1 d x = Γ (s1 + s2 ) 0 Applying this formula to the integral in (34), we have ⟨E hyp,σ (x, s), ψ(x)⟩ (∫ ) 1 Γ ((s − n + 1)/2) Γ (k/2) ψ(x)dv2 + O(ε) · = vol(S k−1 ) · 2 Γ ((s − n + k + 1)/2) Γσ \Dσ

4

as s → ∞. We obtain the assertion of the lemma.

5

4.2. Spectral expansion

6 7

8

□

Theorem 4.3. For any s ∈ C with Re(s) > n − 1, the hyperbolic Eisenstein series E hyp,σ (x, s) admits the following spectral expansion: ∑ E hyp,σ (x, s) = am,σ (s)em (x) m∈D

( ) ∫ ∞ n−1 1 ∑ a n−1 (s)E par,ν x, + iµ dµ. + 4π ν:cusps −∞ 2 +iµ,σ 2

9

10 11

12

13

14

(35)

Then the series on right hand side converges absolutely and locally uniformly. The coefficients am,σ (s) and a n−1 +iµ,σ (s) are given by 2 ( ) k 1 am,σ (s) = vol(S k−1 ) · Γ 2 2 (( ) ) (( ) ) n−1 Γ s − 2 + µm /2 Γ s − n−1 − µm /2 2 × Γ (s/2) Γ ((s − n + k + 1) /2) ∫ em dv2 (36) × Γσ \Dσ

15

16

and a n−1 +iµ,σ (s) = 2

17

18

19

20 21 22 23 24

where µ2m =

( n−1 )2 2

( ) 1 k vol(S k−1 ) · Γ 2 2 (( ) ) (( ) ) Γ s − n−1 + iµ /2 Γ s − n−1 − iµ /2 2 2 × Γ (s/2) Γ ((s − n + k + 1) /2) ( ) ∫ n−1 E par,ν x, × + iµ dv2 , 2 Γσ \Dσ

(37)

− λm and λm is the eigenvalue of the eigenfunction em .

Proof. The hyperbolic Eisenstein series E hyp,σ (x, s) is a bounded and smooth function on Γ \U n by Definition 3.1 and Lemma 4.1. Since the hyperbolic Eisenstein series E hyp,σ (x, s) satisfies the differential shift equation (14), −∆l f are bounded and smooth on Γ \U n for any 0 ≤ l ≤ ⌊ n+1 ⌋ + 1. Hence, from Proposition 2.5, the hyperbolic Eisenstein series has the 4 spectral expansion (35) and it converges absolutely and locally uniformly. Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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In order to give the coefficients am,γ (s), we calculate the inner product ⟨E hyp,σ , em ⟩, which converges by asymptotic bound proved in Lemma 4.2. Furthermore, we know the asymptotic bounds for eigenfunctions of the Laplacen–Beltrami operator ∆ by Lemma 4.2. We have the following relation by using the differential equation (14) λm am,σ (s) = λm ⟨E hyp,σ , em ⟩ = ⟨E hyp,σ , λm em ⟩ = ⟨−∆E hyp,σ , em ⟩ = −s (s − n + 1) am,σ (s) + s(s − n + k + 1)am,σ (s + 2), which implies the relation

1

s(s − n + 1) + λm am,σ (s). s(s − n + k + 1) )2 ( − λm we set the function g(s) as For µm with µ2m = n−1 2 am,σ (s + 2) =

g(s) =

Γ ((s −

2

3

+ µm )/2)Γ ((s − n−1 − µm )/2) 2 . Γ (s/2)Γ ((s − n + k + 1)/2) n−1 2

4

Then g(s) satisfies the relation g(s + 2) =

5

s(s − n + 1) + λm g(s). s(s − n + k + 1)

6

From this, we conclude that the quotient am,γ (s)/g(s) is invariant under s ↦→ s + 2. Furthermore, it is bounded in a vertical strip. Therefore, the quotient am,σ (s)/g(s) is constant, so we have am,σ (s) = bm,σ ·

Γ ((s −

+ µm )/2)Γ ((s − − µm )/2) Γ (s/2)Γ ((s − n + k + 1)/2) n−1 2

n−1 2

(38)

for some constant bm,σ . From Lemma 4.2, for sufficiently small ε > 0, we have following estimate am,σ (s) = ⟨E hyp,σ (x, s), em (x)⟩ ) (∫ Γ ((s − n + 1)/2) Γ (k/2) 1 em (x)dv2 + O(ε) · = vol(S k−1 ) · 2 Γ ((s − n + k + 1)/2) Γσ \Dσ

1 1 log(2π ) − log(s) + s log(s) − s + o(1) 2 2 as s → ∞. From (38)–(40), by comparing the order as s → ∞, we obtain ( ) ∫ 1 k k−1 bm,σ = vol(S ) · Γ · em (x) dv2 2 2 Γσ \Dσ

8 9

10

11 12

13

(39)

as s → ∞. In order to determine the constant bm,σ , we use the Stirling’s asymptotic formula for the gamma function, which states that for a complex number s ∈ C such that |args| < π −ε, the gamma function Γ (s) satisfies the following asymptotic formula log Γ (s) =

7

(40)

as claimed. We obtain the coefficients a n−1 +iµ,σ (s) by the same method as the discrete case. We 2 complete the proof of Theorem 4.3. □ Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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15 16 17

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4.3. Meromorphic continuation From the spectral expansion of the hyperbolic Eisenstein series E hyp,σ (x, s), we can prove the meromorphic continuation of E hyp,σ (x, s) and the location of the possible poles with the residues. Namely, we obtain the following theorem. Theorem 4.4. The hyperbolic Eisenstein series E hyp,σ (x, s) has a meromorphic continuation to all s ∈ C. The possible poles of the continued function are located at the following points: )2 ( − λm for the eigenvalue λm , with (a) s = n−1 ± µm − 2n ′ , where n ′ ∈ N and µ2m = n−1 2 2 residues [ ] Ress= n−1 ±µm −2n ′ E hyp,σ (x, s) 2

′

1 (−1)n Γ (k/2)Γ (±µm − n ′ ) = vol(S k−1 ) · 2 n ′ ! · Γ (( n−1 ± µm − 2n ′ )/2)Γ ((− n−1 ± µm − 2n ′ + k)/2) 2 2 ∫ × em (x)dv2 · em (x). Γσ \Dσ

(b) s = ρν − 2n ′ , where n ′ ∈ N and ω = ρν is a pole of the Eisenstein series E par,ν (x, ω) with Re(ρν ) < n−1 , with residues 2 [ ] Ress=ρν −2n ′ E hyp,σ (x, s) m

=

∑ (−1) j Γ (k/2)Γ (ρν − 2n ′ + j − (n − 1)/2) 1 vol(S k−1 ) · 2 j! · Γ ((ρν − 2n ′ )/2)Γ ((ρν − 2n ′ + k + 1)/2) j=0 ∫ ∑[ × CTω=ρν −2n ′ +2 j E par,ν (x, ω) · Resω=ρν −2n ′ +2 j E par,ν (x, ω)dv2 Γσ \Dσ

ν:cusps

+ Resω=ρν −2n ′ +2 j E par,ν (x, ω) · 7 8 9

∫ Γσ \Dσ

] CTω=ρν −2n ′ +2 j E par,ν (x, ω)dv2 ,

where CTω E par,ν (x, ω) denotes the constant term of the Laurent expansion of the Eisen− 2 − 2m + 2n ′ < stein series E par,ν at ω and m ∈ N is the real number such that n−1 2 n−1 ′ Re(ρν ) ≤ 2 − 2m + 2n . (c) s = n − 1 − ρν − 2n ′ , where n ′ ∈ N and ω = ρν is a pole of the Eisenstein series E par,ν (x, ω) with Re(ρν ) ∈ ( n−1 , n − 1], with residues 2 [ ] 1 Ress=n−1−ρν −2n ′ E hyp,σ (x, s) = vol(S k−1 ) 2 m ∑ (−1) j Γ (k/2)Γ ((n − 1)/2 − ρν − 2n ′ + j) × j! · Γ ((n − 1 − ρν − 2n ′ )/2)Γ ((−ρν − 2n ′ + k)/2) j=m−⌊ n−1 4 ⌋ ∫ ∑[ × CTω=ρν +2n ′ −2 j E par,ν (x, ω) · Resω=ρν +2n ′ −2 j E par,ν (x, ω)dv2 Γσ \Dσ

ν:cusps

+ Resω=ρν +2n ′ −2 j E par,ν (x, ω) ·

∫ Γσ \Dσ

] CTω=ρν +2n ′ −2 j E par,ν (x, ω)dv2 ,

Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

INDAG: 660

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where CTω E par,ν (x, ω) denotes the constant term of the Laurent expansion of the Eisen+ 2m − 2n ′ < stein series E par,ν at ω and m ∈ N is the real number such that n−1 2 n−1 ′ Re(ρν ) ≤ 2 + 2m − 2n + 2. Proof. First, we give the meromorphic continuation of the hyperbolic Eisenstein series E hyp,σ (x, s). The explicit formula (36) shows that the series in (35) arising from the discrete spectrum have the meromorphic continuation to all s ∈ C. We give the meromorphic continuation of the continuous spectrum in (35). We consider the meromorphic continuation of the following integral ∫ Γ ((s − n + 1 + z)/2)Γ ((s − z)/2) I n−1 (s) = 2 Γ (s/2)Γ ((s − n + k + 1)/2) Re(z)= n−1 (∫ 2 ) × E par,ν (x, z)dv2 · E par,ν (x, z)dt, (41)

1 2 3

4 5 6 7 8

9

10

Γσ \Dσ

where t denotes the imaginary part of z. The function I n−1 (s) is holomorphic for s ∈ C with 2 Re(s) > n−1 . Let ε > 0 be sufficiently small such that E par,ν (x, s) has no poles in the strip 2 n−1 < Re(s) < n−1 + ε. For s ∈ C with n−1 < Re(s) < n−1 + ε, we have by the residue 2 2 2 2 theorem Γ (s − (n − 1)/2) I n−1 (s) = I n−1 +ε (s) − 4πi 2 2 Γ (s/2)Γ ((s − n + k + 1)/2) (∫ ) × E par,ν (x, s)dv2 · E par,ν (x, s). (42)

11 12 13 14

15

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Γσ \Dσ

The right-hand side of (42) is a meromorphic function for n−1 − ε < Re(s) < n−1 + ε and 2 2 it gives the meromorphic continuation of the integral I n−1 (s) to the strip n−1 − ε < Re(s) < 2 2 n−1 n−1 n−1 + ε. Now, for 2 − ε < Re(s) < 2 using the residue theorem once again, we have 2 Γ (s − (n − 1)/2) I n−1 +ε (s) = I n−1 (s) + 4πi 2 2 Γ (s/2)Γ ((s − n + k + 1)/2) (∫ ) × E par,ν (x, n − 1 − s)dv2 · E par,ν (x, n − 1 − s).

18 19

20

(43)

Γσ \Dσ

Then, from (42) and (43), the following expression Γ (s − (n − 1)/2) I n−1 (s) + 4πi 2 Γ (s/2)Γ ((s − n + k + 1)/2) (∫ ) E par,ν (x, n − 1 − s)dv2 · E par,ν (x, n − 1 − s) × Γσ \Dσ

Γ (s − (n − 1)/2) − 4πi Γ (s/2)Γ ((s − n + k + 1)/2) (∫ ) × E par,ν (x, s)dv2 · E par,ν (x, s)

17

(44)

Γσ \Dσ

coincides with the right-hand side of (42) on n−1 − ε < Re(s) < n−1 . Furthermore, it gives 2 2 n−1 n−1 a meromorphic function for 2 − 2 < Re(s) < 2 . Therefore (44) gives the meromorphic − 2 < Re(s) ≤ n−1 . By continuing this continuation of the integral I n−1 (s) to the strip n−1 2 2 2 n−1 process, that is, by applying the residue theorem to I n−1 (s) for 2 −2m < Re(s) ≤ n−1 −2m+ε 2 2

Please cite this article as: Y. Irie, Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.07.005.

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INDAG: 660

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Y. Irie / Indagationes Mathematicae xxx (xxxx) xxx

−2m −ε < Re(s) ≤ n−1 −2m inductively, the meromorphic continuation and I n−1 +ε (s) for n−1 2 2 2 of the integral I n−1 (s) to the strip n−1 − 2 − 2m < Re(s) ≤ n−1 − 2m is given by 2 2 2

m ∑ 4πi · (−1)l Γ (s + l − (n − 1)/2) I n−1 (s) + · 2 l! Γ (s/2)Γ ((s − n + k + 1)/2) (∫ l=0 ) E par,ν (x, n − 1 − 2l − s)dv2 · E par,ν (x, n − 1 − 2l − s) × Γσ \Dσ m ∑ Γ (s + l − (n − 1)/2) 4πi · (−1)l · − l! Γ (s/2)Γ ((s − n + k + 1)/2) l=0 (∫ ) × E par,ν (x, s + 2l)dv2 · E par,ν (x, s + 2l).

(45)

Γσ \Dσ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

In order to determine the poles, we consider the spectral expansion (35). First, we consider the discrete part of the spectral expansion. The poles arising from discrete part of (35) are all derived from Γ functions. Hence the poles are located at s = n−1 ± µm − 2l, where l ∈ N 2 ( n−1 )2 2 − λm for the eigenvalue λm . We can also calculate their residues from the and µm = 2 residues of Γ function. Therefore we obtain (a). Next, we consider the continuous spectrum of (35). We work from their meromorphic continuation (45). Then only poles can arise from the summands E par,ν (x, s + 2l) and E par,ν (x, n − 1 − 2l − s), where l = 0, . . . , m. The poles arising from E par,ν (x, s + 2l) are located at sν,l = ρν − 2l, where l = 0, . . . , m and ρν is a pole of E par,ν (x, s) with n−1 − 2 − 2m + 2l < Re(ρν ) ≤ n−1 − 2m + 2l. The poles arising from 2 2 ′ E par,ν (x, n −1−2l −s) are located at sν,l = n −1−ρν −2l, where l = 0, . . . , m and ρν is a pole + 2m − 2l ≤ Re(ρν ) < n−1 + 2m − 2l + 2; in this case there are only of E par,ν (x, s) with n−1 2 2 n−1 poles for m − ⌊ 4 ⌋ ≤ l ≤ m, because the parabolic Eisenstein series E par,ν (x, s) has no poles in {s ∈ C | Re(s) > n−1 } except possibly finitely many simple poles in the segment ( n−1 , n −1] 2 2 on the real line. The residues can be easily derived from (45). Thereby we complete the proof of Theorem 4.4. □

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Remark 4.5. The poles given in Theorem 4.4 might coincide in parts. If this is the case, the corresponding residues need to be summed up.

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[7], [8], [13], [14], [15] Acknowledgments

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I would like to express my deepest gratitude to Professor Yasuro Gon. He guided me patiently as my supervisor in doctoral study and research over the years. I would also like to thank my colleagues for their comments in discussions and seminars.

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