ETA INVARIANTS AND REGULARIZED DETERMINANTS FOR ODD DIMENSIONAL HYPERBOLIC MANIFOLDS WITH CUSPS

JINSUNG PARK

Abstract. We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg trace formula, we show an equality between the eta invariant and the Selberg zeta function of odd type. The complete analysis of the unipotent orbital integral allows us to show the vanishing of the unipotent contribution in this relation. We also show that the eta invariant and the Selberg zeta function of odd type satisfy certain functional equation. These results generalize the earlier work of John Millson (see [12]) to hyperbolic manifolds with cusps. We also get the corresponding relation and the functional equation for the regularized determinant and the Selberg zeta function of even type where the unipotent factor plays a non-trivial role.

0. Introduction

In the seminal paper [12], Millson proved a formula for the relation between the eta invariant of the odd signature operator and the Selberg zeta function of odd type for compact hyperbolic manifolds of (4n−1) dimension . The proof of Millson’s formula is an application of the Selberg trace formula, which expresses a relation between the spectral data and the geometric data. In [7], the corresponding work was done for the analytic torsion and the Ruelle zeta function for odd dimensional hyperbolic manifolds. These results are generalized to the case of compact locally symmetric spaces of higher ranks in [13], [14] .

It would be interesting problems to extend aforementioned results to the noncompact manifolds, in particular, to the noncompact locally symmetric spaces with finite volumes. However, we encounter several serious difficulties, when we discuss the extension of those results to the noncompact locally symmetric spaces. First of all, the heat operator of the Laplacian is not of trace class. Hence we can not use the trace of the heat operator as in the compact case. In [18] and [19], Müller introduced the relative trace to overcome this kind of difficulty and defined the relative eta invariant and the relative regularized determinant.

2000 Mathematics Subject Classification. Primary:58J28, 58J52, Secondary:11M36.

In this paper, we follow Müller’s approach to study the (relative) spectral invariants for noncompact hyperbolic manifolds with finite volumes. If the continuous spectrum of the Laplacian has a nonzero lower bound, the relative trace behaves as the usual trace in the compact case. This is the situation in the case of the Laplacian acting on functions. The regularized determinants of the Laplacian acting on functions and their relations with Selberg zeta functions are extensively studied for 2, 3-dimensional manifolds with cusps in [4], [5], [10], [11], [17]. However, if the continuous spectrum of the Laplacian reaches zero, we need more advanced analysis of the relative trace and the scattering theory appears on the scene. In this paper we study the spectral invariants of Dirac operators and Dirac Laplacians for odd dimensional hyperbolic manifolds with cusps. The continuous spectrum of these operators reaches the zero in this case, and we need the analysis of the scattering operator in order to treat the spectral invariants.

Now, we explain our result more precisely. Let X = Γ\Spin(2n + 1, 1)/Spin(2n + 1) be a (2n + 1)-dimensional hyperbolic manifold with cusps. We consider the Dirac operator D on the spin bundle E which is a locally homogeneous vector bundle defined by the spin representation τ_{s} of the maximal compact subgroup Spin(2n+1) of Spin(2n+1, 1). Following [12], we decompose the space of sections for E into two subspaces which are given by the half spin representations σ_{±} of Spin(2n) ⊂ Spin(2n + 1). Note that the representations σ_{+}, σ_{−} are unramified. Let K_{t} be a family of functions over G = Spin(2n + 1, 1) given by taking the local trace of the integral kernel of e^{−tD̃^{2}} or D̃e^{−tD̃^{2}} where D̃ is the lifting of D over the universal covering space of X. The Selberg trace formula applied to the scalar kernel function K_{t} gives

∑_{σ=σ_{±}} ∑_{λ_{k}∈σ_{p}^{±}} K̂_{t}(σ, iλ_{k}) − \frac{1}{4π} ∫_{−∞}^{∞} Tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)π_{Γ}(σ_{+}, iλ)(K_{t})) dλ

= I_{Γ}(K_{t}) + H_{Γ}(K_{t}) + U_{Γ}(K_{t})

where σ_{p} := σ_{p}^{+} ∪σ_{p}^{−} gives the point spectrum of D , C_{Γ}(σ_{+}, iλ) is the intertwining operator and I_{Γ}(·), H_{Γ}(·), U_{Γ}(·) are the identity, hyperbolic and unipotent orbital integrals. Following [15], [18], [19], we define an operator D_{0} , which is determined by D, with the property that e^{−tD^{2}} − e^{−tD_{0}^{2}} and De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}} are the trace class operators and show that the spectral side of the Selberg trace formula is equal to the relative traces Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}), Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}). Then we have

Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}})
= ∑_{σ=σ_{±}} ∑_{λ_{k}∈σ_{p}^{±}} K̂_{t}(σ, iλ_{k}) − \frac{1}{4π} ∫_{−∞}^{∞} Tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)π_{Γ}(σ_{+}, iλ)(K_{t})) dλ
= I_{Γ}(K_{t}) + H_{Γ}(K_{t}) + U_{Γ}(K_{t})

if K_{t} is give by e^{−tD̃^{2}}. A similar formula holds for Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) with the corresponding kernel function K_{t}. If K_{t} is given by D̃e^{−tD̃^{2}}, we can easily see that I_{Γ}(K_{t}) = 0 by the Fourier transformation of K_{t}. For the unipotent orbital integral, we use result from [9] to compute it explicitly. By this explicit computation, we can show that U_{Γ}(K_{t}) = 0 for K_{t} determined by D̃e^{−tD̃^{2}}. This fact simplifies many steps in our analysis and gives the vanishing of the contribution from the unipotent terms in the relation of the eta invariant and the Selberg zeta function of odd type for hyperbolic manifolds with cusps .

Following Müller’s approach in [17], [18], [19], we define the eta function and the zeta function using the relative traces Tr(e^{−tD^{2}} −e^{−tD_{0}^{2}}), Tr(De^{−tD^{2}} −D_{0}e^{−tD_{0}^{2}}). The continuous spectrum of D is the whole real line in our case, hence the large time contribution of the relative trace can also give the poles of the eta function and the zeta function. Therefore, we need to consider separately the small time contribution and the large time contribution for the meromorphic extensions of the eta function and the zeta function. We use the analytic expansion of the intertwining operator along the imaginary axis to get the large time contribution and we analyze all terms in the geometric side of the Selberg trace formula for the small time contribution. We prove the following pole structures of the eta and zeta functions,

Theorem 0.1. The eta function η_{D}(z) and the zeta function ζ_{D^{2}}(z) have the following pole structures

Γ(z + 1/2)η_{D}(z) = ∑_{k=0}^{∞} \frac{−2γ_{k}}{z − 2k − 2} + K(z),

Γ(z)ζ_{D^{2}}(z) = ∑_{k=−n}^{∞} \frac{β_{k}}{z + k − \frac{1}{2}} + \frac{β′_{0}}{(z − \frac{1}{2})^{2}} − \frac{h}{z} + ∑_{k=0}^{∞} \frac{−γ′_{k}}{z − k − \frac{1}{2}} + H(z)

where β_{k}, β′_{0} are locally computable constants, γ_{k}, γ′_{k} are constants which are determined by the intertwining operator C_{Γ}(σ_{+}, iλ), h is the number of the zero eigenvalues of D and K(z), H(z) are holomorphic functions . In particular, η_{D}(z) and ζ_{D^{2}}(z) are regular at z = 0.

It follows from Theorem 0.1 that we can define the eta invariant by

η(D) := η_{D}(0).

Following [12], we use the Selberg trace formula to show how the eta invariant is related to the Selberg zeta function of odd type. As we mentioned above, we show that all unipotent terms are vanishing in the Selberg trace formula applied to the odd heat kernel function. As the conclusion, the equality between the relative trace and the orbital integrals for the odd kernel function is exactly the same as in the case of compact hyperbolic manifolds. Therefore we can get the same formula for the eta invariant and the Selberg zeta function of odd type as in compact hyperbolic manifolds. On the other hand, in the functional equation for the eta invariant and the Selberg zeta function of odd type, the term determined by the intertwining operator appears.

Theorem 0.2. For the Dirac operator D on a (2n + 1)-dimensional hyperbolic manifold with cusps, the following equalities hold

η(D) = \frac{1}{πi} log Z_{H}^{o}(n),
Z_{H}^{o}(n + s)Z_{H}^{o}(n − s) = exp(2πiη(D)) (\frac{det C_{+}(s)C_{−}(0)}{det C_{−}(s)C_{+}(0)})^{−2^{n−1}}  for s ∈ ℂ

where Z_{H}^{o}(s) is the Selberg zeta function of odd type and C_{±}(s) is a linear operator given

by C_{Γ}(σ_{+}, s) = (\begin{matrix}0 & C_{−}(s) \\ C_{+}(s) & 0 \end{matrix}).

We can define the zeta function ζ_{D^{2}}(z, s) where D^{2} + s^{2} is the shifted Dirac Laplacian with a positive real number s . Now the continuous spectrum of D^{2}+s^{2} does not reach 0 and the large time contribution does not create any poles of ζ_{D^{2}}(z, s). Therefore we can see that ζ_{D^{2}}(z, s) is regular at z = 0 by Theorem 0.1. It follows that the regularized determinant

Det(D^{2}, s) := exp(−ζ′_{D^{2}}(0, s))

is well-defined. We show that Det(D^{2}, s) can be extended to a meromorphic function of s on the whole complex plane. As such a meromorphic function, we have Det(D^{2}, s) ≠ Det(D^{2}, −s). This is because the meromorphic extension of Det(D^{2}, s) involves the intertwining operator C_{Γ}(σ_{+}, s) which is not an even function of s. We use the Selberg trace formula to prove the relation between Det(D^{2}, s) and the geometric data which consists of the Selberg zeta function of even type, the identity factor and the unipotent factor. We also get the functional equation for Det(D^{2}, s) and the Selberg zeta function of even type where the unipotent factor plays a non-trivial role.

Theorem 0.3. For the Dirac Laplacian D^{2} on a (2n + 1)-dimensional hyperbolic manifold with cusps, the following equalities hold for any s ∈ ℂ,

Det(D^{2}, s) = Z_{H}^{e}(s + n)Γ(s + \frac{1}{2})^{−2^{n}κ} exp(2π ∫_{0}^{s} 2p(σ_{+}, iλ) + 2P_{U}(iλ) dλ),
Det(D^{2}, s)^{2} = (det C_{Γ}(σ_{+}, s))^{−2^{n−1}} Z_{H}^{e}(n + s)Z_{H}^{e}(n − s)(Γ(s + \frac{1}{2})Γ(−s + \frac{1}{2}))^{−2^{n}κ}

where Z_{H}^{e}(s) is the Selberg zeta function of even type, p(σ_{+}, s) is Plancherel measure for σ_{+}, P_{U}(s) is an even polynomial of degree 2n − 4 and κ denotes the number of the cusps of X .

If we compare Theorem 0.2 with Theorem 0.3, we can see that the eta invariant does not contain defect terms determined by cusps. But the defect terms, which play a non-trivial role in the functional equation, appear in the case of the regularized determinant.

The paper has the following structure. In Section 1, we define the Dirac operator D for the locally homogeneous vector bundle which is defined by the spin representation of maximal compact subgroup Spin(2n + 1). We introduce an operator D_{0} which is used to define the relative trace. In Section 2, we introduce the Selberg trace formula for nontrivial locally homogeneous vector bundles over noncompact locally symmetric spaces of ℝ-rank 1. In Section 3, we use results from [9] to analyze unipotent terms. In Section 4, we derive the relation between the relative trace and the spectral side of the Selberg trace formula following [15], [18]. In Section 5, we define the eta function and the zeta function and prove Theorem 0.1. In Sections 6 and 7 we prove Theorem 0.2 and Theorem 0.3 using results of Sections 2, 3 and 4.

Acknowledgments. The author wants to express his gratitude to Werner Müller and Werner Hoffmann for their helpful comments on this paper. The author also wants to express his gratitude to Krzysztof Wojciechowski and Masato Wakayama for their concern on this paper and encouragement when he started this work. The author also thanks for the advice of Morten Skarsholm Risager. A part of this work was done during the author’s stay at ICTP and MPI. He wishes to express his thanks to ICTP and MPI for the financial support and their hospitality.

1. Dirac operators on odd dimensional hyperbolic manifolds with cusps

Let G be a noncompact connected semisimple Lie group with finite center Z_{G} and let K be a maximal compact subgroup of G. We assume that the ℝ-rank of the noncompact symmetric space G/K is 1. Let Γ be a discrete subgroup of G such that Γ is torsion free and the volume of Γ\G is finite. Let G = KAN be the Iwasawa decomposition, then there is a Γ-cuspidal minimal parabolic subgroup P of G with Langlands decomposition P = MAN, where M is the centralizer of A in K. Let M^{∗} be the normalizer of A in K and W (A) = M^{∗}/M be the Weyl group of the restricted roots for the pair (G, A). The order of W (A) is 2 since the dimension of A is 1. The set of the restricted roots consists of at most two roots {α_{1}, α_{2}} with α_{2} = 2α_{1} if there are two different elements. The root space corresponding to α_{1}, α_{2} is denoted by 𝔫_{1}, 𝔫_{2} respectively.

From now on, we assume that G = Spin(2n + 1, 1) and K = Spin(2n + 1). Let X̃ denote the noncompact symmetric space determined by G/K = Spin(2n + 1, 1)/Spin(2n + 1). We denote the Lie algebra of G by 𝔤 . We have a decomposition

𝔤 = 𝔨 ⊕ 𝔭 = 𝔨 ⊕ 𝔞 ⊕ 𝔫

where 𝔨, 𝔞, 𝔫 are the Lie algebras of K, A, N and 𝔭 can be identified with the tangent space of G/K at the identity coset. We observe that 𝔫 = 𝔫_{1} and 𝔫_{2} = 0 in this case. The Killing form C( , ) provides us with an invariant metric on G/K given by ⟨X, Y ⟩ = \frac{1}{4n}C(X, θY), where θ is the Cartan involution. This metric gives us the constant curvature −1. We construct a basis {X_{i} : 1 ≤ i ≤ 2n + 1} for 𝔭. First, we choose H ∈ 𝔞 with α_{1}(H) = 1. Let N_{1}, ⋯, N_{2n} be the root vectors for the root α_{1} such that

[N_{i}, θ(N_{j})] = 0 if i ≠ j, [N_{j}, θ(N_{j})] = −2H.

Then, we define X_{i} in the orthogonal complement of 𝔞 in 𝔭 using the formula

X_{i} = \frac{1}{2}(N_{i} − θ(N_{i}))

for i = 1, ⋯, 2n. Let X_{2n+1} = H. The basis {X_{i} : 1 ≤ i ≤ 2n + 1} is an orthonormal basis for (𝔭, ⟨ , ⟩). We define Y_{i} ∈ 𝔨 for i = 1, ⋯, 2n as follows

Y_{i} = \frac{1}{2}(N_{i} + θ(N_{i})).

We have the equalities N_{i} = X_{i} + Y_{i}, [X_{i}, Y_{i}] = −H and X_{i}, Y_{i}, H have the unit length with respect to the inner product ⟨ , ⟩.

We use the spin representation (τ_{s}, V_{τ_{s}}) of Spin(2n + 1) to define a homogeneous vector bundle Ẽ over Spin(2n + 1, 1)/Spin(2n + 1) by Ẽ = G × V_{τ_{s}}/ ∼ where

(g, v) ∼ (g′, v′) if (g′, v′) = (gk, τ_{s}(k^{−1})v)

for g, g′ ∈ G, k ∈ K. The action of the group G = Spin(2n + 1, 1) on the section s of Ẽ is given by the standard formula

g · s(x) = gs(g^{−1}x) for g ∈ G , x ∈ G/K.

This procedure allows us to associate every unitary representation τ of K with a unitary representation of G. We denote this induced representation of G by Ind_{K}^{G}τ. If we restrict the spin representation τ_{s} of K = Spin(2n + 1) to M = Spin(2n), then this representation decomposes into two half spin representations σ_{+}, σ_{−} of Spin(2n) . Two representations (σ_{+}, H_{σ_{+}}), (σ_{−}, H_{σ_{−}}) are unramified and wσ_{+} = σ_{−}, wσ_{−} = σ_{+} for the nontrivial element w ∈ W (A).

Let H ∈ 𝔞 be the frame vector with α_{1}(H) = 1 as above. Then the spinor bundle Ẽ decomposes into the eigenspaces for the eigenvalues i, −i of c(H). Since the Clifford multiplication c(·) is K-equivariant, in particular M-equivariant, we have

τ_{s}(m)c(X)τ_{s}(m^{−1}) = c(mXm^{−1})

for m ∈ M and X ∈ 𝔭. Since the group M is the centralizer of A in K, c(mHm^{−1}) = c(H). Hence the restriction of τ_{s} to M preserves each ±i-eigenspace of c(H) and this means that the eigenspaces of c(H) are the representation spaces of σ_{+}, σ_{−} of M = Spin(2n).

We define the Dirac operator D̃ : C^{∞}(X̃, Ẽ) → C^{∞}(X̃, Ẽ) by

D̃ = ∑_{i=1}^{2n+1} c(X_{i})∇_{X_{i}}

where {X_{i} : 1 ≤ i ≤ 2n + 1} is the orthonormal frame constructed above and c(·) is the Clifford multiplication.

We consider a locally symmetric space given by X = Γ\Spin(2n + 1, 1)/Spin(2n + 1) where Γ is a discrete subgroup of Spin(2n + 1, 1) with unipotent elements and no elliptic elements. We assume that Γ ∩ P = Γ ∩ N so that Γ ∩ P \N = Γ ∩ N\N. Then X is a (2n + 1)-dimensional hyperbolic manifold with cusps. We denote by E the quotient space of Γ\Ẽ. This is a locally homogeneous vector bundle over X. Moreover the Dirac operator D̃ can be pushed down to X. We denote this operator by D.

For the Iwasawa decomposition G = KAN, we can identify A with the real line ℝ by the map a ⟶ |α_{1}|^{−1}α_{1}(log a). For a Γ-cuspidal minimal parabolic subgroup P, we define P^{+} := MA^{+}N where A^{+} is the inverse image of the above identification of the nonnegative real part ℝ^{+}. We denote by P_{1}, ⋯, P_{κ} the representatives of the set of Γ-cuspidal parabolic subgroups, which are not Γ-conjugate to each other. We have the corresponding decomposition for P_{i}. Hence there exists the identification map I_{i} for each cuspidal part

I_{i} : ℝ^{+} × T^{2n} → Γ ∩ P_{i}\P_{i}^{+}/K ∩ M_{i} = Γ ∩ N_{i}\P_{i}^{+}/M_{i}

and under this map we have the induced metric dg^{2}(r, x) = dr^{2} + e^{−2r}dn^{2}(x) for (r, x) ∈ ℝ^{+} × T^{2n}.

By the identification map I_{i}, we may regard that the restriction of the Dirac operator D to Γ ∩ P_{i}\P_{i}^{+}/K ∩ M_{i} is defined over ℝ^{+} × T^{2n}. Let B be the Dirac operator acting on C^{∞}(T^{2n}, E|_{T^{2n}}) given by ∑_{i=1}^{2n} c(X_{i})c(H)∇_{X_{i}}. Note that B has the property c(H)B = −Bc(H). We have the following expression for D over cusps

(1)  D = c(H)(∇_{H} + B − nI)

(see (7) in [2]). Let 0 < μ_{1} ≤ μ_{2} ≤ μ_{3} ≤ ⋯ → ∞ be positive eigenvalues of B, each eigenvalue repeated according to its multiplicity with corresponding eigensections ψ_{j}. We denote by E_{μ_{j}} the eigenspace corresponding to μ_{j}. We decompose

L^{2}(ℝ^{+} × T^{2n}, E|_{ℝ^{+}×T^{2n}}, dg^{2}(r, x))

into

L^{2}(ℝ^{+}, E_{0}, e^{−2nr}dr) ⊕ ⊕_{j=1}^{∞}L^{2}(ℝ^{+}, E_{μ_{j}} ⊕ c(H)E_{μ_{j}}, e^{−2nr}dr).

Note that E_{0} ≅ V_{τ_{s}} is a symplectic vector space with a symplectic structure ⟨c(H) , ⟩. We fix a Lagrangian subspace L of E_{0} ≅ V_{τ_{s}} such that E_{0} = L ⊕ c(H)L. Then the map

α_{j}ψ_{j} + α_{−j}c(H)ψ_{j} → e^{−nr} (\begin{matrix}α_{j} \\ α_{−j} \end{matrix})

gives a unitary equivalence

L^{2}(ℝ^{+}, E_{0}, e^{−2nr}dr) ⊕ ⊕_{j=1}^{∞}L^{2}(ℝ^{+}, E_{μ_{j}} ⊕ c(H)E_{μ_{j}}, e^{−2nr}dr)

≅ ⊕_{j=1}^{d(σ_{+})}L^{2}(ℝ^{+}, ℂ^{2}, dr) ⊕ ⊕_{j=1}^{∞}L^{2}(ℝ^{+}, ℂ^{2}, dr)

under which the Dirac operator D over the cusp decomposes as follows

(2)  D → ⊕_{i=1}^{d(σ_{+})}D_{0} ⊕ ⊕_{j=1}^{∞}D_{μ_{j}}

where

D_{μ} = (\begin{matrix}0 & −\frac{d}{dr} + μ \\ \frac{d}{dr} + μ & 0 \end{matrix}) .

Hence the Dirac Laplacian D^{2} is transformed into ⊕_{μ∈spec(B)}D_{μ}^{2} where D_{μ}^{2} = −\frac{d^{2}}{dr^{2}} + μ^{2} on L^{2}(ℝ^{+}, ℂ, dr).

Now we consider the operator

c(H)(\frac{d}{dr} − nI) : C_{0}^{∞}(ℝ^{+}, E_{0}) → C_{0}^{∞}(ℝ^{+}, E_{0})

whose L^{2}-extension ( with respect to e^{−2nr}dr ) is transformed to d(σ_{+})-copies of D_{0} in (2). The operator

c(H)(\frac{d}{dr} − nI) : C_{0}^{∞}(ℝ^{+}, E_{0}, L) := \{ ϕ ∈ C_{0}^{∞}(ℝ^{+}, E_{0}) | ϕ(0) ∈ L \} → L^{2}(ℝ^{+}, E_{0}, e^{−2nr}dr)

is essentially self adjoint. By the natural embedding and the identification map I_{i}, we have the embedding map J_{i} defined by

(3)  J_{i} : ℝ^{+} ↪ ℝ^{+} × T^{2n} ≅ Γ ∩ N_{i}\P_{i}^{+}/M_{i} ↪ X.

By the embedding map J_{i} we can consider κ-copies of L^{2}(ℝ^{+}, E_{0}, L, e^{−2nr}dr) as a subspace of L^{2}(X, E) and κ-copies of c(H)(\frac{d}{dr} − nI) can be extended to the self adjoint operator D_{0} on L^{2}(X, E) by zero map over the orthogonal complement of this subspace in L^{2}(X, E). We can see easily that D_{0} has no point spectrum. Let D be the unique selfadjoint extension of D on L^{2}(X, E).

Proposition 1.1. For t > 0, operators e^{−tD^{2}} −e^{−tD_{0}^{2}} and De^{−tD^{2}} −D_{0}e^{−tD_{0}^{2}} are trace class operators.

Proof. Let F_{0} be the union of Γ ∩ N_{i}\P_{i}^{+}/M_{i} for i = 1, ⋯, κ. Pick f ∈ C^{∞}(X) such that 0 < f ≤ 1, f(x) = 1 for x ∈ X − F_{0} and f((r, ·)) = e^{− \frac{r}{4}} for (r, ·) ∈ F_{0} with r ≥ 1. Denote by U_{f} an operator in L^{2}(X, E) defined by the multiplication by f. Then we may write

e^{−tD^{2}} − e^{−tD_{0}^{2}} =(e^{− \frac{t}{2} D^{2}} − e^{− \frac{t}{2} D_{0}^{2}}) ∘ U_{f}^{−1} ∘ U_{f} ∘ e^{− \frac{t}{2} D^{2}}
+ e^{− \frac{t}{2} D_{0}^{2}} ∘ U_{f} ∘ U_{f}^{−1} ∘ (e^{− \frac{t}{2} D^{2}} − e^{− \frac{t}{2} D_{0}^{2}}).

The heat kernel estimates for e^{−tD^{2}} and e^{−tD_{0}^{2}} show that (e^{− \frac{t}{2} D^{2}} −e^{− \frac{t}{2} D_{0}^{2}})∘U_{f}^{−1}, U_{f} ∘e^{− \frac{t}{2} D^{2}}, e^{− \frac{t}{2} D_{0}^{2}} ∘ U_{f} and U_{f}^{−1} ∘ (e^{− \frac{t}{2} D^{2}} − e^{− \frac{t}{2} D_{0}^{2}}) are Hilbert-Schmidt operators. The composition of Hilbert-Schmidt operators is of trace class, hence e^{−tD^{2}} − e^{−tD_{0}^{2}} is a trace class operator. The remaining case is proved in the same way. □

2. The Selberg trace formula

Let R_{Γ} be the right regular representation of G on L^{2}(Γ\G) and f be a right K-finite function in the Harish-Chandra L^{p}-Schwartz space C^{p}(G) where 0 < p < 1. The trace of the restriction R_{Γ}(f) to the discrete part L_{d}^{2}(Γ\G) of L^{2}(Γ\G) can be written as

T r(R_{Γ}(f)|_{L_{d}^{2}(Γ\G)}) = I_{Γ}(f) + H_{Γ}(f) + U_{Γ}(f) + S_{Γ}(f) + T_{Γ}(f).

Here

I_{Γ}(f) = Vol(Γ\G)f(1),
H_{Γ}(f) = ∑_{{γ:hyperbolic}} Vol(Γ_{γ}\G_{γ}) ∫_{G_{γ}\G} f(x^{−1}γx)dx,
U_{Γ}(f) = \frac{1}{|α_{1}|}\{C_{1}(Γ)T_{1}(f) + C_{2}(Γ)T_{2}(f) + C′_{1}(Γ)T′_{1}(f)\},
S_{Γ}(f) = \frac{1}{4πi} ∑_{σ∈M̂} ∫_{Re(ν)=0} Tr(C_{Γ}(σ, ν)^{−1} \frac{d}{dν} C_{Γ}(σ, ν)π_{Γ}(σ, ν)(f))dν,
T_{Γ}(f) = −\frac{1}{4} ∑_{σ∈M̂} Tr(C_{Γ}(σ, 0)π_{Γ}(σ, 0)(f))

where Γ_{γ}, G_{γ} are centralizers of γ in Γ, G respectively, constants C_{1}(Γ), C_{2}(Γ), C′_{1}(Γ) and tempered distributions T_{1}, T_{2}, T′_{1} will be discussed in the next section and C_{Γ}(σ, ν) denote the intertwining operator. We refer to [1], [21], [22], [23] for detailed expositions of the Selberg trace formula. The various invariant measures that we use are normalized as in

The kernel K̃_{t}(x : y) of the integral operator e^{−tD̃^{2}} (or D̃e^{−tD̃^{2}}) over G/K is a section of Ẽ ⊠ Ẽ^{∗}, the external tensor product of Ẽ and Ẽ^{∗} over G × G. The bundle E is a trivial bundle, hence the kernel K̃_{t}(x : y) is an element of (C^{∞}(G × G) ⊗ End(V_{τ_{s}}))^{K}, that is an endomorphism valued function on G × G, invariant under K. It follows that there exists a function K̃_{t}(x) : G → End(V_{τ_{s}}) such that

K̃_{t}(x : y) = K̃_{t}(x^{−1}y) and K̃_{t}(k_{1}^{−1}xk_{2}) = τ_{s}(k_{1}^{−1})K̃_{t}(x)τ_{s}(k_{2})

for x, y ∈ G, k_{1}, k_{2} ∈ K. We define the local trace of K̃_{t}(x) to be the scalar function on G by K_{t}(x) := tr(K̃_{t}(x)). We denote by K̂_{t}(σ, ν) the Fourier transform of K_{t} for the unitary principal representation π_{σ,ν} of G, that is,

K̂_{t}(σ, ν) := Tr(π_{σ,ν}(K_{t})).

By the Plancherel theorem, we have

Ind_{K}^{G}τ_{s} = ∫_{Re(ν)=0} π_{σ_{+},ν} p(σ_{+}, ν) dν ⊕ ∫_{Re(ν)=0} π_{σ_{−},ν} p(σ_{−}, ν) dν

where p(σ_{±}, ν) is the Plancherel measure. Hence K̂_{t}(σ, ν) is zero unless σ = σ_{±}. From now on, we will denote by K_{t}^{e}, K_{t}^{o} the scalar functions corresponding to e^{−tD̃^{2}}, D̃e^{−tD̃^{2}}. By (4.5) in [13], we have

Proposition 2.1. For λ ∈ ℝ, we have

K̂_{t}^{e}(σ_{±}, iλ) = e^{−tλ^{2}},  K̂_{t}^{o}(σ_{+}, iλ) = λe^{−tλ^{2}},  K̂_{t}^{o}(σ_{−}, iλ) = −λe^{−tλ^{2}}.

Since the representation σ_{±} is unramified, the intertwining operator C_{Γ}(σ_{+}, ν) = C_{Γ}(σ_{−}, ν) acting on H_{Γ}(σ_{+}, ν) = H_{Γ}(σ_{−}, ν) switches the subspaces induced by the representations σ_{+}, σ_{−} . Hence we have the following form of C_{Γ}(σ_{+}, ν),

C_{Γ}(σ_{+}, ν) = (\begin{matrix}0 & C_{−}(ν) \\ C_{+}(ν) & 0 \end{matrix})

with respect to the decomposition of H_{Γ}(σ_{+}, ν). We refer to p. 166–167 in [1] or p. 9–10 in [22] for more details on C_{Γ}(σ, ν), H_{Γ}(σ, ν) . Therefore we have

C_{Γ}(σ_{+}, ν)^{−1} \frac{d}{dν} C_{Γ}(σ_{+}, ν) = (\begin{matrix}C_{−}(−ν)\frac{d}{dν} C_{+}(ν) & 0 \\ 0 & C_{+}(−ν)\frac{d}{dν} C_{−}(ν) \end{matrix})

and

T_{Γ}(·) = −\frac{1}{4}Tr(C_{Γ}(σ_{+}, 0)π_{Γ}(σ_{+}, 0)(·)) = 0.

Now the Selberg trace formulas applied to K_{t}^{e}, K_{t}^{o} are given by

∑_{σ=σ_{±}} ∑_{λ_{k}∈σ_{p}^{±}} K̂_{t}^{e}(σ, iλ_{k}) − \frac{1}{4π} ∫_{−∞}^{∞} Tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)π_{Γ}(σ_{+}, iλ)(K_{t}^{e})) dλ (4)

= ∑_{λ_{k}∈σ_{p}} e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} e^{−tλ^{2}}tr (C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ
= I_{Γ}(K_{t}^{e}) + H_{Γ}(K_{t}^{e}) + U_{Γ}(K_{t}^{e})

∑_{σ=σ_{±}} ∑_{λ_{k}∈σ_{p}^{±}} K̂_{t}^{o}(σ, iλ_{k}) − \frac{1}{4π} ∫_{−∞}^{∞} Tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)π_{Γ}(σ_{+}, iλ)(K_{t}^{o})) dλ (5)

= ∑_{λ_{k}∈σ_{p}} λ_{k}e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr(C_{−}(−iλ)\frac{d}{d(iλ)}C_{+}(iλ)) dλ

+ \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr(C_{+}(−iλ)\frac{d}{d(iλ)}C_{−}(iλ)) dλ

= ∑_{λ_{k}∈σ_{p}} λ_{k}e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ
= I_{Γ}(K_{t}^{o}) + H_{Γ}(K_{t}^{o}) + U_{Γ}(K_{t}^{o})

where d(σ_{+}) is the degree of σ_{+}. Although we know d(σ_{+}) = 2^{n−1}, we will keep to use the notation d(σ_{+}) instead of 2^{n−1}. This indicates the origin of the constant factor. And

tr_{s}(C_{Γ}(σ_{+}, −ν)\frac{d}{dν} C_{Γ}(σ_{+}, ν)) := tr(C_{−}(−ν)\frac{d}{dν} C_{+}(ν)) − tr(C_{+}(−ν)\frac{d}{dν} C_{−}(ν)).

For I_{Γ}(K_{t}), we have

I_{Γ}(K_{t}) = Vol(Γ\G)( ∫_{−∞}^{∞} K̂_{t}(σ_{+}, iλ)p(σ_{+}, iλ) dλ + ∫_{−∞}^{∞} K̂_{t}(σ_{−}, iλ)p(σ_{−}, iλ) dλ)

where p(σ_{±}, iλ) is the Plancherel measure which is an even polynomial with respect to λ. The equalities K̂_{t}^{e}(σ_{±}, iλ) = e^{−tλ^{2}} and p(σ_{+}, iλ) = p(σ_{−}, iλ) give

(6)  I_{Γ}(K_{t}^{e}) = 2Vol(Γ\G) ∫_{−∞}^{∞} e^{−tλ^{2}}p(σ_{+}, iλ) dλ

and K̂_{t}^{o} is odd with respect to λ so that

(7)  I_{Γ}(K_{t}^{o}) = 0.

It is well known that

(8) H_{Γ}(K_{t}^{e}) = \frac{1}{\sqrt{4πt}} ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} + \overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} ,

(9) H_{Γ}(K_{t}^{o}) = \frac{2πi}{(4πt)^{\frac{3}{2}}} ∑_{γ:hyperbolic} l(C_{γ})^{2}j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}}

where l(C_{γ}) is the length of the closed geodesic C_{γ}, j(γ) is the positive integer such that γ = γ_{0}^{j(γ)} for a primitive element γ_{0}, D(γ) = e^{nl(C_{γ})}|det(Ad(m_{γ}a_{γ})^{−1} − I|_{𝔫})| for the element m_{γ}a_{γ} ∈ MA^{+} which is conjugate to γ and χ_{σ} is the character of σ.

3. Unipotent terms

In this Section we compute unipotent terms U_{Γ}(K_{t}^{e}), U_{Γ}(K_{t}^{o}). We use formula obtained by Hoffmann (see [9]) to compute these terms explicitly. By this explicit computation and Proposition 2.1, it follows that U_{Γ}(K_{t}^{o}) = 0. This simplifies many steps related to the application of the Selberg trace formula for K_{t}^{o}.

The unipotent term for a right K-finite function f in C^{p}(G), 0 < p < 1 is given by

U_{Γ}(f) = \frac{1}{|α_{1}|}\{C_{1}(Γ)T_{1}(f) + C_{2}(Γ)T_{2}(f) + C′_{1}(Γ)T′_{1}(f)\}.

Here α_{1} is the unique positive restricted root for (𝔤, 𝔞), the constants C_{1}(Γ), C_{2}(Γ), C′_{1}(Γ) which depend on Γ are computed in [3] and

T_{1}(f) = \frac{1}{A(𝔫_{1})} ∫_{N} ∫_{K} f(k^{−1}nk)dkdn
T_{2}(f) = \frac{|α_{1}|}{2}\{∫_{G/G_{n_{0}}} f(xn_{0}x^{−1})dx + ∫_{G/G_{n{−1 \atop 0}}} f(xn_{0}^{−1}x^{−1})dx\}
T′_{1}(f) = \frac{m_{1} + 2m_{2}}{A(𝔫_{1})} ∫_{N_{1}} ∫_{N_{2}} ∫_{K} f(k^{−1}n_{1}n_{2}k) log | log(n_{1})|dkdn_{1}dn_{2}

where n_{0} is a representative of the nontrivial unipotent orbit in 𝔫_{2}, m_{i} = dim(𝔫_{𝔦}) and A(𝔫_{1}) is the volume of the unit sphere in 𝔫_{1} and G_{n} is the centralizer of an element n in G.

In our case G = Spin(2n + 1, 1), the second term T_{2}(f) = 0 and m_{2} = 0 in the third term T′_{1}(f) since N = N_{1}. The following formula for C′_{1}(Γ) results from the computations in [3] ,

(10)  \frac{C′_{1}(Γ)}{|α_{1}|} \frac{m_{1}}{A(𝔫_{1})} = \frac{κ}{2}.

The first term T_{1} for K_{t} is given by

T_{1}(K_{t}) = \frac{1}{2πA(𝔫_{1})}( ∫_{−∞}^{∞} K̂_{t}(σ_{+}, iλ) dλ + ∫_{−∞}^{∞} K̂_{t}(σ_{−}, iλ) dλ).

We have K̂_{t}^{e}(σ_{±}, iλ) = e^{−tλ^{2}} and K̂_{t}^{o}(σ_{±}, iλ) = ±λe^{−tλ^{2}}, which implies

(11)  T_{1}(K_{t}^{e}) = \frac{1}{πA(𝔫_{1})} ∫_{−∞}^{∞} e^{−tλ^{2}} dλ and T_{1}(K_{t}^{o}) = 0.

The third term T′_{1}(f) is more complicated and we need to introduce some notations. Let T_{M} be a Cartan subgroup in M , so that T = T_{M} · A is a Cartan subgroup of G. Let Σ_{M} denote the set of positive roots for (𝔪_{ℂ}, 𝔱_{𝔪}_{ℂ}). Let ρ_{Σ_{M}} be the half sum of elements in Σ_{M}. We denote by Σ_{A} the set of positive roots of (𝔤_{ℂ}, 𝔱_{ℂ}) which do not vanish on 𝔞_{ℂ}. The union of Σ_{M} with Σ_{A} gives the set of positive roots for (𝔤_{ℂ}, 𝔱_{ℂ}) denoted by Σ_{G}. Let H_{α} ∈ 𝔱_{ℂ} be the coroot corresponding to α ∈ ±Σ_{G} ,that is, α(H_{α}) = 2, α′(H_{α}) ∈ ℤ for all α, α′ ∈ ±Σ_{G} and Π := ∏_{α∈Σ_{M}} H_{α} be the element of the symmetric algebra S(𝔱_{𝔪}_{ℂ}). We denote the simple reflection corresponding to α by s_{α} for α ∈ Σ_{G}. Now we define

I(f) := \frac{1}{2π} ∑_{σ∈M̂} ∫_{−∞}^{∞} Ω(σ, −iλ)f̂(σ, iλ) dλ

where

Ω(σ, iλ) = 2d(σ)ψ(1) − \frac{1}{2} ∑_{α∈Σ_{A}} λ(H_{α})\frac{Π(s_{α}λ_{σ})}{Π(ρ_{Σ_{M}})}×(ψ(1 + λ_{σ}(H_{α})) + ψ(1 − λ_{σ}(H_{α})))

where d(σ) is the degree of σ, ψ is the logarithmic derivative of the Gamma function and λ_{σ} − ρ_{Σ_{M}} is the highest weight of (σ, iλ) ∈ M̂ × i𝔞.

The principal series representation (π_{σ,ν}, H_{σ,ν}) depends on the Langlands decomposition and we denote the dependence on P by (π_{σ,ν}(P), H_{σ,ν}(P)). The intertwining operator

J_{P̄|P} (σ, ν) : H_{σ,ν}(P) → H_{σ,ν}(P̄)

is defined by

(J_{P̄|P} (σ, ν)ϕ) = ∫N̄ ϕ(xn̄) dn̄

and satisfies

J_{P̄|P} (σ, ν)π_{σ,ν}(P) = π_{σ,ν}(P̄)J_{P̄|P} (σ, ν).

The restriction to K defines an isomorphism from H_{σ,ν}(P) to H_{σ}(P). Then J_{P̄|P} (σ, ν) can be considered as a family of operators from H_{σ}(P) to H_{σ}(P̄). Here H_{σ}(P) is a space of all measurable functions v : K → H_{σ} such that

v(km) = σ(m)^{−1}v(k)

for all m ∈ M, k ∈ K. Let

J_{P} (σ, ν : f) := −Tr(π_{σ,ν}(f)J_{P̄|P} (σ, ν)^{−1}∂_{ν}J_{P̄|P} (σ, ν))

where the derivative ∂_{ν} is taken with respect ν for the family of operators J_{P̄|P} (σ, ν) acting on H_{σ}(P) . Then there exits the Harish Chandra C-function C_{τ} (σ, ν) such that

T_{τ} J_{P̄|P} (σ, ν)^{−1}∂_{ν}J_{P̄|P} (σ, ν) = C_{τ} (σ, ν)^{−1}∂_{ν}C_{τ} (σ, ν)T_{τ}

where T_{τ} is the projection to the τ-isotypic component of H_{σ,ν}(P). We refer to [6] for more detail. We have the following proposition for C_{τ_{s}}(σ_{±}, ν),

Proposition 3.1. For the half spin representation σ_{±} of Spin(2n),

C_{τ_{s}}(σ_{+}, iλ) = C_{τ_{s}}(σ_{−}, iλ) = \frac{(2n − 1)!}{(n − 1)!} \frac{Γ(iλ + \frac{1}{2})}{Γ(iλ + n + \frac{1}{2})}.

Proof. This is a special case of the theorem 8.2 in [6]. □

The weighted orbital integral T′_{1} is given by

T′_{1}(f) = \frac{m_{1}}{A(𝔫_{1})}(I(f) + \frac{1}{2π} p.v. ∑_{σ∈M̂} d(σ) ∫_{−∞}^{∞} J_{P} (σ, iλ : f) dλ + ∑_{σ∈M̂} d(σ)\frac{n(σ)}{2}f̂(σ, 0))

where p.v. means the Cauchy principal value and 2n(σ) is the order of the zero of p(σ, ν) at ν = 0. Using n(σ_{±}) = 0 and Proposition 3.1 we obtain

T′_{1}(K_{t}^{e}) = \frac{m_{1}}{2πA(𝔫_{1})} ∫_{−∞}^{∞} e^{−tλ^{2}}(Ω(σ_{+}, −iλ) + Ω(σ_{−}, −iλ) − 2d(σ_{+})\frac{d}{d(iλ)} log C_{τ_{s}}(σ_{+}, iλ)) dλ,

T′_{1}(K_{t}^{o}) = \frac{m_{1}}{2πA(𝔫_{1})} ∫_{−∞}^{∞} λe^{−tλ^{2}}(Ω(σ_{+}, −iλ) − Ω(σ_{−}, −iλ)) dλ.

Now we consider Ω(σ_{±}, iλ) in the following proposition,

Proposition 3.2. For the half spin representation σ_{±} of Spin(2n), the following equalities hold

Ω(σ_{+}, iλ) = Ω(σ_{−}, iλ)

= −\frac{d(σ_{+})}{2}(ψ(iλ − n + \frac{1}{2}) + ψ(−iλ − n + \frac{1}{2}) + ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2})) + P^{n}(λ)

where P^{n}(λ) is an even polynomial of 2n − 4 degree for n ≥ 2 and P^{1}(λ) is a constant.

Proof. The case for n = 1 can be computed by the same way as in the other cases for n ≥ 2. Hence we assume that n ≥ 2 in the following proof. The highest weight of the half spin representation σ_{±} of Spin(2n) ⊂ Spin(2n + 1) is given by

\frac{1}{2}(e_{2} + e_{3} + ⋯ + e_{n} ± e_{n+1})

with respect to the standard basis {e_{i}}. This implies

iλe_{1} + σ_{±} + ρ_{Σ_{M}} = iλe_{1} + (n − \frac{1}{2})e_{2} + (n − \frac{3}{2})e_{3} + ⋯ + \frac{3}{2}e_{n} ± \frac{1}{2}e_{n+1}.

The positive roots α ∈ Σ_{A} are given by e_{1} − e_{j}, e_{1} + e_{j} for 2 ≤ j ≤ n + 1. Then we can see that s_{(e_{1}−e_{j})}(iλe_{1} + σ_{±} + ρ_{Σ_{M}}) is

iλe_{j} + (n − \frac{1}{2})e_{2} + ⋯ + (n − j + \frac{3}{2})e_{1} + ⋯ + \frac{3}{2}e_{n} ± \frac{1}{2}e_{n+1} if 2 ≤ j ≤ n,

iλe_{n+1} + (n − \frac{1}{2})e_{2} + ⋯ + \frac{3}{2}e_{n} ± \frac{1}{2}e_{1} if j = n + 1

and s_{(e_{1}+e_{j})}(iλe_{1} + σ_{±} + ρ_{Σ_{M}}) is

− iλe_{j} + (n − \frac{1}{2})e_{2} + ⋯ − (n − j + \frac{3}{2})e_{1} + ⋯ + \frac{3}{2}e_{n} ± \frac{1}{2}e_{n+1} if 2 ≤ j ≤ n,

− iλe_{n+1} + (n − \frac{1}{2})e_{2} + ⋯ + \frac{3}{2}e_{n} ∓ \frac{1}{2}e_{1} if j = n + 1.

These give us the following formula,

Π(s_{(e_{1}±e_{j})}(iλe_{1} + σ_{±} + ρ_{Σ_{M}}))

(12)  =C_{j}(λ^{2} + (n − \frac{1}{2})^{2}) · (λ^{2} + (n − \frac{3}{2})^{2}) ⋯ (λ^{2} + (n − j + \frac{5}{2})^{2})

· (−λ^{2} − (n − j + \frac{1}{2})^{2}) ⋯ (−λ^{2} − (\frac{3}{2})^{2}) · (−λ^{2} − (\frac{1}{2})^{2})

where

C_{j} = ∏_{1≤k<l≤n,k≠j,l≠j}((n − k + \frac{1}{2})^{2} − (n − l + \frac{1}{2})^{2})

= ∏_{1≤k<l≤n,k≠j,l≠j}(l − k)(2n − k − l + 1)

for 2 ≤ j ≤ n + 1. In particular, we obtain

Π(s_{e_{1}±e_{j}}(iλe_{1} + σ_{+} + ρ_{Σ_{M}})) = Π(s_{e_{1}±e_{j}}(iλe_{1} + σ_{−} + ρ_{Σ_{M}})) Π(s_{e_{1}−e_{j}}(iλe_{1} + σ_{±} + ρ_{Σ_{M}})) = Π(s_{e_{1}+e_{j}}(iλe_{1} + σ_{±} + ρ_{Σ_{M}}))

for 2 ≤ j ≤ n + 1. From now on, we denote by P_{j}(λ) this common 2(n − 1) degree even polynomial of λ in (12) for 2 ≤ j ≤ n + 1.

On the other hand, (iλe_{1} + σ_{±} + ρ_{Σ_{M}})(H_{α}) is given by

iλ − (n − j + \frac{3}{2}) if α = e_{1} − e_{j}, 2 ≤ j ≤ n

iλ ∓ \frac{1}{2} if α = e_{1} − e_{n+1}

iλ + (n − j + \frac{3}{2}) if α = e_{1} + e_{j}, 2 ≤ j ≤ n

iλ ± \frac{1}{2} if α = e_{1} + e_{n+1}.

18 JINSUNG PARK

Then the pair (ψ(1 + (iλe_{1} + σ_{±} + ρ_{Σ_{M}})(H_{α})), ψ(1 − (iλe_{1} + σ_{±} + ρ_{Σ_{M}})(H_{α}))) is given by

ψ(iλ − n + j − \frac{1}{2}), ψ(−iλ + n − j + \frac{5}{2}) for e_{1} − e_{j}, 2 ≤ j ≤ n

ψ(iλ + \frac{1}{2}), ψ(−iλ + \frac{3}{2}) for e_{1} − e_{n+1}, σ = σ_{+}

ψ(iλ + \frac{3}{2}), ψ(−iλ + \frac{1}{2}) for e_{1} − e_{n+1}, σ = σ_{−}

ψ(iλ + n − j + \frac{5}{2}), ψ(−iλ − n + j − \frac{1}{2}) for e_{1} + e_{j}, 2 ≤ j ≤ n

ψ(iλ + \frac{3}{2}), ψ(−iλ + \frac{1}{2}) for e_{1} + e_{n+1}, σ = σ_{+}

ψ(iλ + \frac{1}{2}), ψ(−iλ + \frac{3}{2}) for e_{1} + e_{n+1}, σ = σ_{−}.

Now we compare the following two sets {Π(s_{α}(iλe_{1} + σ_{+} + ρ_{Σ_{M}})), 1 + (iλe_{1} + σ_{+} + ρ_{Σ_{M}})(H_{α}), 1 − (iλe_{1} + σ_{+} + ρ_{Σ_{M}})(H_{α})}, {Π(s_{α}(iλe_{1} + σ_{−} + ρ_{Σ_{M}})), 1 + (iλe_{1} + σ_{−} + ρ_{Σ_{M}})(H_{α}), 1 − (iλe_{1} + σ_{−} + ρ_{Σ_{M}})(H_{α})}

and see that they are equal to each other so that Ω(σ_{+}, iλ) = Ω(σ_{−}, iλ). We compute the exact form of Ω(σ_{+}, iλ) = Ω(σ_{−}, iλ). Using the relations ψ(z + 1) = \frac{1}{z} + ψ(z), we have

ψ(iλ − n + j − \frac{1}{2}) + ψ(−iλ − n + j − \frac{1}{2}) + ψ(iλ + n − j + \frac{5}{2}) + ψ(−iλ + n − j + \frac{5}{2})

=\frac{1}{λ^{2} + (\frac{1}{2})^{2}} + ⋯ + \frac{2(n − j + \frac{1}{2})}{λ^{2} + (n − j + \frac{1}{2})^{2}} + \frac{−2(n − j + \frac{5}{2})}{λ^{2} + (n − j + \frac{5}{2})^{2}} + ⋯ + \frac{−2(n − \frac{1}{2})}{λ^{2} + (n − \frac{1}{2})^{2}}

+ ψ(iλ − n + \frac{1}{2}) + ψ(−iλ − n + \frac{1}{2}) + ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2}).

Now using the formula (see the last line of p.95 in [9]),

∑_{α∈Σ_{A}} Π(s_{α}λ_{σ}) = 2Π(λ_{σ}),

we decompose

\frac{1}{2} ∑_{α∈Σ_{A}} \frac{Π(s_{α}λ_{σ})}{Π(ρ_{Σ_{M}})}×(ψ(1 + λ_{σ}(H_{α})) + ψ(1 − λ_{σ}(H_{α})))

into

\frac{d(σ_{±})}{2}(ψ(iλ − n + \frac{1}{2}) + ψ(−iλ − n + \frac{1}{2}) + ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2}))

and

\frac{1}{4Π(ρ_{Σ_{M}})} ∑_{j=2}^{n+1} P_{j}(λ)R_{j}(λ)

where

R_{j}(λ) = \frac{1}{λ^{2} + (\frac{1}{2})^{2}} + ⋯ + \frac{2(n − j + \frac{1}{2})}{λ^{2} + (n − j + \frac{1}{2})^{2}} + \frac{−2(n − j + \frac{5}{2})}{λ^{2} + (n − j + \frac{5}{2})^{2}} + ⋯ + \frac{−2(n − \frac{1}{2})}{λ^{2} + (n − \frac{1}{2})^{2}} .

We can see, from the definitions of P_{j}(λ) and R_{j}(λ), that P_{j}(λ)R_{j}(λ) is an even polynomial of degree 2n − 4. This ends the proof. □

Now we have the following corollary

Corollary 3.3.

U_{Γ}(K_{t}^{o}) = 0, U_{Γ}(K_{t}^{e}) = \frac{2}{2π} ∫_{−∞}^{∞} e^{−tλ^{2}}(P_{U}(λ) + Q(λ)) dλ

where P_{U}(λ) is an even polynomial of degree 2n − 4 and Q(λ) is given by

−κ\frac{d(σ_{+})}{2}(ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2})).

Proof. The first claim follows easily from (10), (11), Proposition 3.1 and 3.2. For the second claim, Q(λ) is a priori given by

− \frac{κd(σ_{+})}{2}(\frac{1}{2}(ψ(iλ − n + \frac{1}{2}) + ψ(−iλ − n + \frac{1}{2}))

− \frac{1}{2}(ψ(iλ + n + \frac{1}{2}) + ψ(−iλ + n + \frac{1}{2})) + (ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2}))).

We use the relation ψ(z + 1) = \frac{1}{z} + ψ(z), then we can reduce the above formula to the claimed one for Q(λ). □

4. Relative Traces and Spectral Sides

In this Section we study relations of Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}), Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) with the spectral sides of the Selberg trace formulas applied to the test functions K_{t}^{e}, K_{t}^{o}. The formula of this type was proved by Müller in the case of the Laplacian acting on functions over the manifold with cusps (see [15]) and for the Dirac operator over the manifold with cylindrical end (see [18]). Following [15], [18], we prove the corresponding formula for the Dirac operator acting on spinor bundle over the hyperbolic manifold with cusps.

We can identify L^{2}(X, E) with the space

(L^{2}(Γ\G) ⊗ V_{τ_{s}})^{K} = { f ∈ L^{2}(Γ\G) ⊗ V_{τ_{s}} | f(xk) = τ_{s}(k)^{−1}f(x) for k ∈ K, x ∈ G }.

Then the decomposition of L^{2}(Γ\G) = L_{d}^{2}(Γ\G) ⊕ L_{c}^{2}(Γ\G) allows us to decompose

(13)  (L^{2}(Γ\G) ⊗ V_{τ_{s}})^{K} = (L_{d}^{2}(Γ\G) ⊗ V_{τ_{s}})^{K} ⊕ (L_{c}^{2}(Γ\G) ⊗ V_{τ_{s}})^{K}.

Recall that the continuous part (L_{c}^{2}(Γ\G) ⊗ V_{τ_{s}})^{K} is spanned by the wave packets with the Eisenstein series. Therefore we need to know how D acts on the Eisenstein series. For Φ ∈ (H_{Γ}(σ_{+}, ν) ⊗ V_{τ_{s}})^{K}, the Eisenstein series attached to Φ is defined by

E(Φ : ν : x) := ∑_{γ∈Γ∩P \Γ} Φ(γx).

Assume that Φ_{j,±} is in the ±i-eienspace of c(H) of (H_{Γ}(σ_{+}, ν)⊗V_{τ_{s}})^{K} for j = 1, ⋯, d(σ_{+}). Then the Eisenstein series E(Φ_{j,±} : iλ : x) for ν = iλ satisfies

(14)  DE(Φ_{j,±} : iλ : x) = ±λE(Φ_{j,±} : iλ : x).

For ϕ ∈ C_{0}^{∞}(X, E), the decomposition (13) provides us with the formula

(15)  ϕ(x) = ∑_{k}(ϕ, ϕ_{k})ϕ_{k}(x)

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} E(Φ_{j,+} : iλ : x) ∫_{X} E(Φ_{j,+} : −iλ : y)ϕ(y) dy dλ

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} E(Φ_{j,−} : iλ : x) ∫_{X} E(Φ_{j,−} : −iλ : y)ϕ(y) dy dλ

where {ϕ_{k}} is the orthonormal basis of (L_{d}^{2}(Γ\G) ⊗ V_{τ_{s}})^{K}. Since D preserves the decomposition of (13), we may assume that each ϕ_{k} is the eigensections of D. Therefore (14) and (15) implies

Dϕ(x) = ∑_{λ_{k}} λ_{k}(ϕ, ϕ_{k})ϕ_{k}(x)

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λE(Φ_{j,+} : iλ : x) ∫_{X} E(Φ_{j,+} : −iλ : y)ϕ(y) dy dλ

− \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λE(Φ_{j,−} : iλ : x) ∫_{X} E(Φ_{j,−} : −iλ : y)ϕ(y) dy dλ.

The integral kernels of e^{−tD^{2}}, De^{−tD^{2}} are given by the formulas

K_{t}^{e}(x : y) := ∑_{γ∈Γ} K̃_{t}^{e}(γx̃ : ỹ), K_{t}^{o}(x : y) := ∑_{γ∈Γ} K̃_{t}^{o}(γx̃ : ỹ)

where x = Γx̃, y = Γỹ, x̃, ỹ ∈ G and K̃_{t}^{e}, K̃_{t}^{o} are integral kernels of e^{−tD̃^{2}}, D̃e^{−tD̃^{2}}. The following equalities hold, when we consider those kernels as distributions

(16)  K_{t}^{e}(x : y) = ∑_{λ_{k}∈σ_{p}} e^{−tλ_{k}^{2}}ϕ_{k}(x) ⊗ ϕ_{k}^{∗}(y)

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} e^{−tλ^{2}}E(Φ_{j,+} : iλ : x) ⊗ E(Φ_{j,+} : −iλ : y)^{∗} dλ

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} e^{−tλ^{2}}E(Φ_{j,−} : iλ : x) ⊗ E(Φ_{j,−} : −iλ : y)^{∗} dλ,

(17) K_{t}^{o}(x : y) = ∑_{λ_{k}∈σ_{p}} λ_{k}e^{−tλ_{k}^{2}}ϕ_{k}(x) ⊗ ϕ_{k}^{∗}(y)

+ \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λe^{−tλ^{2}}E(Φ_{j,+} : iλ : x) ⊗ E(Φ_{j,+} : −iλ : y)^{∗} dλ

− \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λe^{−tλ^{2}}E(Φ_{j,−} : iλ : x) ⊗ E(Φ_{j,−} : −iλ : y)^{∗} dλ.

We can regard [R, ∞) × T^{2n} ⊂ ℝ^{+} × T^{2n} as a subset of X by the map I_{i} for i = 1, ⋯, κ. We denote this subset by F_{i,R} and let F_{R} := ∪_{i=1}^{κ}F_{i,R} and X_{R} := X − F_{R}. The constant term of the Eisenstein series E(Φ : iλ : x) over F_{i,0} has the following form,

(18)  e^{(−iλ+n)r}Φ_{i} + e^{(iλ+n)r}((C_{Γ}(σ_{+}, iλ) ⊗ I)Φ)_{i}.

Here Φ_{i} is the component of Φ over F_{i,0} and note that the operator C_{Γ}(σ_{+}, iλ) ⊗ I acts on (H_{Γ}(σ_{+}, iλ) ⊗ V_{τ_{s}})^{K}. From now on, we assume that ||Φ_{i}|| = 1 for i = 1, ⋯, κ. We now discuss the Maass-Selberg relation in our context.

Proposition 4.1. (Maass-Selberg) We have

(19)  ∫_{X_{R}} |E(Φ_{j,±} : iλ : x)|^{2}dx = 2κR − tr(C_{±}(iλ)\frac{d}{d(iλ)}C_{∓}(−iλ)) + O(e^{−cR})
= 2κR − tr(C_{∓}(−iλ)\frac{d}{d(iλ)}C_{±}(iλ)) + O(e^{−cR})

where C_{±}(iλ)_{ik} is a component of C_{±}(iλ) and c is a positive constant.

Proof. We will consider only the case of Φ_{+} so that we use the notation Φ instead of Φ_{+} in the following proof. The case of Φ_{−} can be done by the same way. It follows from the Green’s formula that we have an equality

(λ − λ′)⟨E(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{X_{R}}
(20)  = ⟨DE(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{X_{R}} − ⟨E(Φ : iλ : x), DE(Φ : iλ′ : x)⟩_{X_{R}}
= ⟨c(H)E(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{∂X_{R}}.

By (18), we have

⟨c(H)E(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{∂F_{i,R}}
= i⟨ e^{(−iλ+n)R}Φ_{i} , e^{(−iλ′+n)R}Φ_{i} ⟩_{∂F_{i,R}}

− i ⟨ e^{(iλ+n)R}((C_{Γ}(σ_{+}, iλ) ⊗ I)Φ)_{i} , e^{(iλ′+n)R}((C_{Γ}(σ_{+}, iλ′) ⊗ I)Φ)_{i} ⟩_{∂F_{i,R}} + O(e^{−cR})

= ie^{−i(λ−λ′)R} − ie^{i(λ−λ′)R} ∑_{k=1}^{κ} C_{+}(iλ)_{ik}C_{−}(−iλ′)_{ki} + O(e^{−cR}).

Now we use the functional equation

C_{Γ}(σ_{+}, iλ)C_{Γ}(σ_{+}, −iλ) = (\begin{matrix}0 & C_{−}(iλ) \\ C_{+}(iλ) & 0 \end{matrix}) (\begin{matrix}0 & C_{−}(−iλ) \\ C_{+}(−iλ) & 0 \end{matrix}) = I,

which implies the following equality,

⟨c(H)E(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{∂F_{i,R}}

(21)  = ie^{−i(λ−λ′)R} − ie^{i(λ−λ′)R}

− ie^{i(λ−λ′)R} ∑_{k=1}^{κ} C_{+}(iλ)_{ik}C_{−}(−iλ′)_{ki}

+ ie^{i(λ−λ′)R} ∑_{k=1}^{κ} C_{+}(iλ)_{ik}C_{−}(−iλ)_{ki} + O(e^{−cR}).

Combining (20) and (21), we have

(λ − λ′)⟨E(Φ : iλ : x), E(Φ : iλ′ : x)⟩_{X_{R}} = iκe^{−i(λ−λ′)R} − iκe^{i(λ−λ′)R} (22)

+ ie^{i(λ−λ′)R} ∑_{i,k=1}^{κ} C_{+}(iλ)_{ik}(C_{−}(−iλ)_{ki} − C_{−}(−iλ′)_{ki}) + O(e^{−cR}).

If we pass to the limit λ → λ′ after dividing each side of (22) by λ − λ′, we get the equality (19).

□

The next task is to consider the corresponding formula for D_{0}. We fix a Lagrangian subspace L in V_{τ_{s}} as before. For the orthonormal basis {ψ_{j}} ∈ L, we define

ϕ_{j,+} := \frac{1}{\sqrt{2}}(ψ_{j} − ic(H)ψ_{j}), ϕ_{j,−} := \frac{1}{\sqrt{2}}(ψ_{j} + ic(H)ψ_{j})

and

e(ϕ_{j,+} : iλ : x) := e^{(n−iλ)r}ϕ_{j,+} + e^{(n+iλ)r}ϕ_{j,−},
e(ϕ_{j,−} : iλ : x) := e^{(n−iλ)r}ϕ_{j,−} + e^{(n+iλ)r}ϕ_{j,+}.

Note that e(ϕ_{j,±} : iλ : x) lies in C^{∞}(ℝ^{+}, V_{τ_{s}}, L) and we have

c(H)(\frac{d}{dr} − nI)e(ϕ_{j,+} : iλ : x) = λe(ϕ_{j,+} : iλ : x),

c(H)(\frac{d}{dr} − nI)e(ϕ_{j,−} : iλ : x) = −λe(ϕ_{j,−} : iλ : x).

As before, by the map J_{i} we regard e(ϕ_{j,±} : iλ : x) lying in C^{∞}(X, E) and define a section E(ϕ_{j,±} : iλ : x) in C^{∞}(X, E) by the sum of these for j = 1, ⋯, d(σ_{+}). For ϕ ∈ C_{0}^{∞}(X, E), we have

(23)  D_{0}ϕ = \frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λE(ϕ_{j,+} : iλ : x) ∫_{X} E(ϕ_{j,+} : −iλ : y)ϕ(y)dy dλ

−\frac{1}{4π} ∑_{j=1}^{d(σ_{+})} ∫_{−∞}^{∞} λE(ϕ_{j,−} : iλ : x) ∫_{X} E(ϕ_{j,−} : −iλ : y)ϕ(y)dy dλ.

The following equality is proved in the same way as in the proof of Proposition 4.1,

(24)  ∫_{X_{R}} |E(ϕ_{j,±} : iλ : x)|^{2}dx = 2κR + O(e^{−cR})

for some positive constant c. It follows from (19) and (24) that we have

∫_{X} |E(Φ_{j,±} : iλ : x)|^{2} − |E(ϕ_{j,±} : iλ : x)|^{2}dx (25)

= lim_{R→∞} ∫_{X_{R}} |E(Φ_{j,±} : iλ : x)|^{2} − |E(ϕ_{j,±} : iλ : x)|^{2}dx
= − tr(C_{∓}(−iλ)\frac{d}{d(iλ)}C_{±}(iλ)).

Finally, the following proposition is the result of (4), (5), (7), (16), (17), (25) and Corollary 3.3,

Proposition 4.2. We have

Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) = ∫_{X} |K_{t}^{e}(x : x)|^{2} − |k_{t}^{e}(x, x)|^{2}dx (26)

= ∑_{λ_{k}} e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} e^{−tλ^{2}}tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ
= I_{Γ}(K_{t}^{e}) + H_{Γ}(K_{t}^{e}) + U_{Γ}(K_{t}^{e}),

Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) = ∫_{X} |K_{t}^{o}(x : x)|^{2} − |k_{t}^{o}(x, x)|^{2}dx (27)

= ∑_{λ_{k}} λ_{k}e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ
= ∑_{λ_{k}} λ_{k}e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr(C_{−}(−iλ)\frac{d}{d(iλ)}C_{+}(iλ)) dλ

+ \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr(C_{+}(−iλ)\frac{d}{d(iλ)}C_{−}(iλ)) dλ

= H_{Γ}(K_{t}^{o})

where k_{t}^{e}(x, y), k_{t}^{o}(x, y) are the integral kernels of e^{−tD_{0}^{2}}, D_{0}e^{−tD_{0}^{2}}.

Remark 4.3. Although the definition of D_{0} depends on the choices of the Lagrangian subspace L and the embedding in (3), the relative traces in the left hand sides of (26), (27) do not depend on these choices. First, we observe that the right hand side in (24) does not depend on the choice of the Lagrangian subspace L. Hence the relative traces do not depend on the choice of L. Second, for the embedding in (3), we may use the different embedding ℝ^{+} → ℝ^{+} × T^{2n} given by r → (r + r_{0}, ·) for a positive r_{0}. If we recall the proof of (24), we can see that the boundary contribution of {r_{0}} × T^{2n} is trivial in the step corresponding to (20) since the boundary condition is given by the Lagrangian subspace L. Hence the relative traces are independent on the choice of the embedding in (3) too.

5. Meromorphic continuations of the eta and the zeta functions

In this Section, we prove Theorem 0.1, which provides us with the pole structures of the eta function and the zeta function over ℂ.

We follow [18] and [19] and define

η_{D,1}(z) := \frac{1}{Γ(z + 1/2)} ∫_{0}^{1} t^{\frac{z−1}{2}} Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt,

ζ_{D^{2},1}(z) := \frac{1}{Γ(z)} ∫_{0}^{1} t^{z−1} [ Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) − h ] dt

for Re(z) ≫ 0 and

η_{D,2}(z) := \frac{1}{Γ(z + 1/2)} ∫_{1}^{∞} t^{\frac{z−1}{2}} Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt,

ζ_{D^{2},2}(z) := \frac{1}{Γ(z)} ∫_{1}^{∞} t^{z−1} [ Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) − h ] dt

for Re(z) ≪ 0 where h is the number of zero eigenvalues of D^{2}. To introduce the eta invariant and the regularized determinant we need to study the meromorphic extension of η_{D,i}(z) and ζ_{D^{2},i}(z) near z = 0 for i = 1, 2. The difficulty not present for the closed manifold is the presence of the continuous spectrum. Moreover, the continuous spectrum of D is equal to the whole real line in our case. Hence the meromorphic extensions of η_{D,2}(z) and ζ_{D^{2},2}(z) have the nontrivial poles. We use analytic expansion of C_{Γ}(σ_{+}, iλ) along the imaginary axis and the analysis of the geometric terms in the left hand side of the Selberg trace formula for the meromorphic extensions of η_{D,i}(z) and ζ_{D^{2},i}(z) over ℂ.

We start with η_{D,1}(z). It follows from (9) and (27) that we have

Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) = H_{Γ}(K_{t}^{o}) (28)

=\frac{2πi}{(4πt)^{\frac{3}{2}}} ∑_{γ:hyperbolic} l(C_{γ})^{2}j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} .

The number c_{0} := min_{{γ:hyperbolic}}l(C_{γ}) is a positive real number, hence

(29)  \frac{2πi}{(4πt)^{\frac{3}{2}}} ∑_{γ:hyperbolic} l(C_{γ})^{2}j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} ∼ ae^{−c_{0}^{2}/t}

as t → 0 for a constant a. Now (28) and (29) give

Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) ∼ ae^{−c_{0}^{2}/t} as t → 0.

This means that η_{D,1}(z) can be extended to the whole complex plane without poles.

For ζ_{D^{2},1}(z), we use (26) to get

Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) = I_{Γ}(K_{t}^{e}) + H_{Γ}(K_{t}^{e}) + U_{Γ}(K_{t}^{e}).

Recall that

(30)  H_{Γ}(K_{t}^{e}) = \frac{1}{\sqrt{4πt}} ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} + \overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} .

As H_{Γ}(K_{t}^{o}), we have

(31)  H_{Γ}(K_{t}^{e}) ∼ a′e^{−c_{0}^{2}/t} as t → 0

for a constant a′. We can see that H_{Γ}(K_{t}^{e}) does not give any poles in the meromorphic continuation of ζ_{D^{2},1}(z). Next we consider I_{Γ}(K_{t}^{e}) and U_{Γ}(K_{t}^{e}). An elementary computation shows

(32)  I_{Γ}(K_{t}^{e}) = ∑_{k=0}^{n} a_{k}t^{−k− \frac{1}{2}}

for some constants a_{k}. By Corollary 3.3, we have

U_{Γ}(K_{t}^{e}) = \frac{2}{2π} ∫_{−∞}^{∞} e^{−tλ^{2}}(P_{U}(λ) + Q(λ)) dλ

where P_{U}(λ) is the even polynomial of degree 2n − 4 and Q(λ) is given by

−κ\frac{d(σ_{+})}{2}(ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2})).

An elementary computation leads to

(33)  ∫_{−∞}^{∞} e^{−tλ^{2}}P_{U}(λ) dλ = ∑_{k=0}^{n−2} b_{k}t^{−k− \frac{1}{2}}

for some constants b_{k}. Using the relations

ψ(x + 1) = \frac{1}{x} + ψ(x), ψ(x) + ψ(x + \frac{1}{2}) = 2(ψ(2x) − log 2),

we can reduce Q(λ) to the sum of following terms:

ψ(iλ + 1) + ψ(−iλ + 1), ψ(2iλ + 1) + ψ(−2iλ + 1).

We use the Stirling-Binet formula to deal with the terms ψ(iλ + 1) + ψ(−iλ + 1)

Re(ψ(iλ + 1)) = log λ + ∑_{k=1}^{N} α_{k}λ^{−2k} + R_{N}(λ)

where α_{k} are constants and |R_{N}(λ)| ≤ M_{N}/λ^{2N+1} for some constant M_{N} and λ ≥ \frac{1}{2}. This implies the following expansion

∫_{1}^{∞} e^{−tλ^{2}}(ψ(iλ + 1) + ψ(−iλ + 1)) dλ (34)

∼ ∑_{k=0}^{∞} c_{k}t^{k− \frac{1}{2}} + d_{0}t^{− \frac{1}{2}} log t  as t → 0

for some constants c_{k} and d_{0}. We deal with terms ψ(2iλ + 1) + ψ(−2iλ + 1) in a similar way. By (31), (32), (33), (34), we have

I_{Γ}(K_{t}^{e}) + H_{Γ}(K_{t}^{e}) + U_{Γ}(K_{t}^{e}) ∼ ∑_{k=−n}^{∞} β_{k}t^{k− \frac{1}{2}} + β′_{0}t^{− \frac{1}{2}} log t as t → 0

for constants β_{k}, β′_{0}. Therefore ζ_{D^{2},1}(z) is well defined for Re(z) > n + \frac{1}{2} and we can extend ζ_{D^{2},1}(z) to a meromorphic function on ℂ, with poles determined by

(35)  Γ(z)ζ_{D^{2},1}(z) = ∑_{k=−n}^{∞} \frac{β_{k}}{z + k − \frac{1}{2}} + \frac{β′_{0}}{(z − \frac{1}{2})^{2}} − \frac{h}{z} + H_{1}(z)

where β_{k}, β′_{0} are constants and H_{1}(z) is a holomorphic function.

To deal with the meromorphic extensions of η_{D,2}(z) and ζ_{D^{2},2}(z), we consider the right hand sides of the following equations

Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) = ∑_{λ_{k}} λ_{k}e^{−tλ_{k}}

− \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} λe^{−tλ^{2}}tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ,

Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) − h = ∑_{λ_{k}≠0} e^{−tλ_{k}^{2}}

− \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} e^{−tλ^{2}}tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ.

The discrete eigenvalues give

∑_{λ_{k}} λ_{k}e^{−tλ_{k}^{2}} ∼ e^{−ct}, ∑_{λ_{k}≠0} e^{−tλ_{k}^{2}} ∼ e^{−ct} as t → ∞

for a positive constant c. The operator C_{Γ}(σ_{+}, iλ) is analytic along the imaginary axis and

tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) = −tr_{s}(C_{Γ}(σ_{+}, iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, −iλ)),
tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) = tr(C_{Γ}(σ_{+}, iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, −iλ)),

hence we have the following analytic expansion at λ = 0

tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) = ∑_{k=0}^{∞} f_{2k+1}λ^{2k+1},

tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) = ∑_{k=0}^{∞} g_{2k}λ^{2k}

for some constants f_{2k+1}, g_{2k}. Therefore we have

(36)  ∫_{−1}^{1} λe^{−tλ^{2}}tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ ∼ ∑_{k=0}^{∞} γ_{k}t^{−(k+ \frac{3}{2} )},

(37)  ∫_{−1}^{1} e^{−tλ^{2}}tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ ∼ ∑_{k=0}^{∞} γ′_{k}t^{−(k+ \frac{1}{2} )}

as t → ∞. The equality (36) shows that η_{D,2}(z) is well defined for Re(z) < 1 and extends to the whole complex plane with the following pole structure

(38)  Γ(z + 1/2)η_{D,2}(z) = ∑_{k=0}^{∞} \frac{−2γ_{k}}{z − 2k − 2} + K_{2}(z)

for the constants γ_{k} and a holomorphic function K_{2}(z). The equation (37) implies that the ζ_{D^{2},2}(z) is well defined for Re(z) < \frac{1}{2} and extends to the whole complex plane with the poles determined by the equality

(39)  Γ(z)ζ_{D^{2},2}(z) = ∑_{k=0}^{∞} \frac{−γ′_{k}}{z − k − \frac{1}{2}} + H_{2}(z)

for the constants γ′_{k} and a holomorphic function H_{2}(z).

We define the eta and zeta functions by

η_{D}(z) := η_{D,1}(z) + η_{D,2}(z), ζ_{D^{2}}(z) := ζ_{D^{2},1}(z) + ζ_{D^{2},2}(z).

The equalities (35), (38), (39) give the following theorem

Theorem 5.1. The poles of the eta function η_{D}(z) and the zeta function ζ_{D^{2}}(z) are determined by the following equations

Γ(z + 1/2)η_{D}(z) = ∑_{k=0}^{∞} \frac{−2γ_{k}}{z − 2k − 2} + K(z),

Γ(z)ζ_{D^{2}}(z) = ∑_{k=−n}^{∞} \frac{β_{k}}{z + k − \frac{1}{2}} + \frac{β′_{0}}{(z − \frac{1}{2})^{2}} − \frac{h}{z} + ∑_{k=0}^{∞} \frac{−γ′_{k}}{z − k − \frac{1}{2}} + H(z)

where K(z) and H(z) are holomorphic. In particular, η_{D}(z) and ζ_{D^{2}}(z) are regular at z = 0.

6. Eta invariants, Zeta functions of odd type and functional equations

In this Section, we study the eta invariant and its relation with the Selberg zeta function of odd type. We use the Selberg trace formula to prove a generalization of Millson’s theorem in [12] for hyperbolic manifolds with cusps. We showed the vanishing of the unipotent term in our situation (see Corollary 3.3), hence we obtain the same formula as in the case of Millson in [12]. We also prove the functional equation for the eta invariant and the Selberg zeta function of odd type.

By the result in Section 5, the eta function η_{D}(z) is regular at z = 0. In particular, this implies that the we can put z = 0 in the following equality,

η_{D}(z) = \frac{1}{Γ(\frac{z+1}{2})} ∫_{0}^{∞} t^{\frac{z−1}{2}} Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt.

We define the eta invariant of D by

η(D) := η_{D}(0) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} t^{− \frac{1}{2}} Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt.

Let us recall that

ETA INVARIANT, REGULARIZED DETERMINANT 31

Tr(De^{−tD^{2}}−D_{0}e^{−tD_{0}^{2}}) = \frac{2πi}{(4πt)^{\frac{3}{2}}} ∑_{γ:hyperbolic} l(C_{γ})^{2}j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})}−\overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} .

Using the elementary equality ∫_{0}^{∞} e^{−s^{2}t} \frac{e^{−r^{2}/4t}}{(4πt)^{\frac{3}{2}}} dt = \frac{e^{−sr}}{4πr}, we have

∫_{0}^{∞} e^{−s^{2}t}Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt

=\frac{i}{2} ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{−sl(C_{γ})}
=\frac{i}{2} ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}|det(Ad(m_{γ}a_{γ})^{−1} − I|_{𝔫})|^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{−(s+n)l(C_{γ})}.

We define the Selberg zeta function of odd type by

Z_{H}^{o}(s) := exp(− ∑_{γ:hyperbolic} j(γ)^{−1}|det(Ad(m_{γ}a_{γ})^{−1} − I|_{𝔫})|^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{−sl(C_{γ})})

for Re(s) ≫ 0. We will show that Z_{H}^{o}(s) has the meromorphic extension over ℂ and Z_{H}^{0}(s) is regular at s = n in Proposition 6.2. Now we have

∫_{0}^{∞} e^{−s^{2}t}Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt = \frac{i}{2} \frac{d}{ds} log Z_{H}^{o}(s + n).

Following the argument in p. 27 of [12], we use the equality,

t^{\frac{z−1}{2}} = \frac{2}{Γ(\frac{1−z}{2} )} ∫_{0}^{∞} s^{−z}e^{−s^{2}t} ds

for Re(z) < 1. Then we have

η_{D}(z) = \frac{2}{Γ(\frac{z+1}{2})Γ(\frac{1−z}{2} )} ∫_{0}^{∞} ∫_{0}^{∞} s^{−z}e^{−s^{2}t} ds Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt
= \frac{2}{Γ(\frac{z+1}{2})Γ(\frac{1−z}{2} )} ∫_{0}^{∞} s^{−z} ∫_{0}^{∞} e^{−s^{2}t}Tr(De^{−tD^{2}} − D_{0}e^{−tD_{0}^{2}}) dt ds
= \frac{i}{Γ(\frac{z+1}{2})Γ(\frac{1−z}{2} )} ∫_{0}^{∞} s^{−z} \frac{d}{ds} log Z_{H}^{o}(s + n) ds.

If we evaluate the above equality at z = 0, we get the following theorem, Theorem 6.1. For the Dirac operator D over a (2n + 1)-dimensional hyperbolic manifold with cusps, the eta invariant η(D) and the Selberg zeta function of odd type Z_{H}^{o}(s) satisfy the following equality

(40)  η(D) = \frac{1}{πi} log Z_{H}^{o}(n).

Let us consider the meromorphic extension of Z_{H}^{o}(s) over ℂ. We select a smooth odd function g(u) such that |g(u)| = 1 if |u| > c_{0}, g(u) = 0 near 0 and ∫_{0}^{∞} g′(u)du = 1 where c_{0} = min_{{γ:hyperbolic}}l(C_{γ}). We define

\{ \begin{matrix}H_{s}(σ_{+}, λ) := ∫_{−∞}^{∞} g(u)e^{−s|u|}e^{iλu}du, \\ H_{s}(σ_{−}, λ) := −H_{s}(σ_{+}, λ) \end{matrix}

for a complex parameter s. By the Paley-Wiener theorem, ( see the theorem 2.2 in [12] ), there exists f_{s} over G with f̂_{s}(σ_{±}, iλ) = H_{s}(σ_{±}, λ), f̂_{s}(σ, iλ) = 0 if σ ≠ σ_{±}. We apply the Selberg trace formula to the one parameter family of functions f_{s} for Re(s) ≫ 0 on G, then we get

∑_{λ_{j}∈σ_{p}^{+}} H_{s}(σ_{+}, λ_{j}) + ∑_{λ_{j}∈σ_{p}^{−}} H_{s}(σ_{−}, λ_{j}) (41)

− \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ

= ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{−sl(C_{γ})}
= \frac{d}{ds} log Z_{H}^{o}(s + n).

We use the above equality to get the meromorphic extension of

Z(s) := \frac{d}{ds} log Z_{H}^{o}(s + n)

over ℂ and to investigate its poles.

Discrete eigenvalue term: The integration by part gives

H_{s}(σ_{+}, λ) = ±\frac{1}{s − iλ} ∫_{0}^{∞} g′(u)e^{−(s−iλ)u}du ∓ \frac{1}{s + iλ} ∫_{0}^{∞} g′(u)e^{−(s+iλ)u}du.

This equality provides the meromorphic extension of H_{s}(σ_{±}, λ_{j}) over ℂ and we see that H_{s}(σ_{±}, λ_{j}) has the simple pole at iλ_{j}, −iλ_{j} with residue ±m_{j}, ∓m_{j} for λ_{j} ∈ σ_{p}^{±} where m_{j} is the multiplicity of λ_{j}.

Scattering term: We consider the scattering term

(42)  −\frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ.

First we observe

Φ(z) := tr_{s}(C_{Γ}(σ_{+}, −z)\frac{d}{dz} C_{Γ}(σ_{+}, z)) = \frac{d}{dz} ( log det C_{+}(z) − log det C_{−}(z) ).

Using the functional equation C_{Γ}(σ_{+}, z)C_{Γ}(σ_{+}, −z) = Id and the general formula for \frac{d}{dz} log det C_{Γ}(σ_{+}, z) given in [16], we have

det C_{+}(z) = det C_{+}(0) p_{+}^{z} ∏_{Re(q_{+})<0} \frac{z + q_{−}}{z − q_{+}} ,

det C_{−}(z) = det C_{−}(0) p_{−}^{z} ∏_{Re(q_{−})<0} \frac{z + q_{+}}{z − q_{−}}

for some constants p_{+}, p_{−}. Hence Φ(z) has the following form over ℂ,

Φ(z) = − ∑_{Re(q_{+})<0} \frac{1}{z − q_{+}} + \frac{1}{z + q_{+}} + ∑_{Re(q_{−})<0} \frac{1}{z − q_{−}} + \frac{1}{z + q_{−}} + log p_{+} − log p_{−}. (43)

We observe that the infinite sum in the right hand side of (43) converges by the lemma 6.6 in [16]. Now we consider the contour integral,

L_{R} := \frac{1}{4πi} ∫_{L_{R}} H_{s}(σ_{+}, z) tr_{s}(C_{Γ}(σ_{+}, −iz)\frac{d}{d(iz)}C_{Γ}(σ_{+}, iz))dz

where L_{R} = [ −R, R ] ∪ { Re^{iθ} | 0 ≤ θ ≤ π }. As in the proposition 3.10 of [8], we apply the Cauchy integral formula and obtain

lim_{R→∞} L_{R} = i( ∑_{Re(q_{+})<0} \frac{1}{s − q_{+}} ∫_{0}^{∞} g′(u)e^{−(s−q_{+})u}du − ∑_{Re(q_{−})<0} \frac{1}{s − q_{−}} ∫_{0}^{∞} g′(u)e^{−(s−q_{−})u}du)

for Re(s) ≫ 0 and the right hand side gives us the meromorphic extension over ℂ. On the other hand, as in the proposition 3.10 in [8], we also show that

lim_{R→∞} L_{R} = \frac{1}{4πi} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)tr_{s}(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ.

Therefore the scattering term has simple poles at q_{±} for Re(q_{±}) < 0 with residues ±d(σ_{+})b_{±} where b_{±} denotes the order of the pole of det C_{Γ}(σ_{+}, z), q_{±}.

Combining the contributions of H_{s}(σ_{+}, λ_{j})’s and the scattering term, we see that Z(s) has a meromorphic extension over ℂ and its simple poles are located at ±iλ_{j}, ∓iλ_{j} for λ_{j} ∈ σ_{p}^{±} and q_{±} for Re(q_{±}) < 0 with residues m_{j}, −m_{j} and ±d(σ_{+})b_{±}. In particular, we can see that Z(s) is regular at s = 0. This implies the following proposition,

Proposition 6.2. The Selberg zeta function of odd type Z_{H}^{o}(s + n) has the meromorphic extension over ℂ and is regular at s = 0.

Remark 6.3. The zeros of Z_{H}^{o}(s + n) are located at ±iλ_{j} for λ_{j} ∈ σ_{p}^{±}, q_{+} for Re(q_{+}) < 0 and their orders are m_{j}, d(σ_{+})b_{+}. The poles of Z_{H}^{o}(s + n) are located at ∓iλ_{j} for λ_{j} ∈ σ_{p}^{±}, q_{−} for Re(q_{−}) < 0 and their orders are m_{j}, d(σ_{+})b_{−}.

Let us study the functional equation of η(D) and Z_{H}^{o}(s). We set

R(s) := Z(s) − Z(−s) + d(σ_{+})Φ(s).

Then R(s) is an odd entire function of s. Now let h(s) be an odd function which decreases sufficiently rapidly as Im(s) → ∞ in the strip { s ∈ ℂ | |Re(s)| < n + ϵ, ϵ > 0 }. Consider the contour integral,

L_{T} := \frac{1}{2πi} ∫_{L_{T}} h(s)Z(s)ds

where L_{T} is the rectangle with the corners a+iT, a−iT, −a+iT, −a−iT with n < a < n+ϵ . Then we have

lim_{T →∞} L_{T} = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞} h(s)(Z(s) − Z(−s))ds + 2\frac{1}{2πi} ∫_{−a+i∞}^{−a−i∞} h(s)Z(s)ds.

We apply the Cauchy integral theorem to get the equality

(44)  lim_{T →∞} L_{T} = 2 ∑_{λ_{j}} m_{j}h(iλ_{j}) + ∑_{−a<Re(q_{k})<0} d(σ_{+})b_{k}h(q_{k}).

Here we used the notations q_{k}, b_{k} instead of q_{±}, ±b_{±}. By the simple pole of Z(s) located at q_{k} with residue d(σ_{+})b_{k} between Re(s) = −a and Re(s) = 0, we have

(45) \frac{1}{2πi} ∫_{−a+i∞}^{−a−i∞} h(s)Z(s)ds = \frac{1}{2πi} ∫_{i∞}^{−i∞} h(λ)Z(λ) dλ + ∑_{−a<Re(q_{k})<0} d(σ_{+})b_{k}h(q_{k}).

The simple poles at −q_{k} of Z(−s) between Re(s) = a and Re(s) = 0 produce the following equality

\frac{1}{2πi} ∫_{a−i∞}^{a+i∞} h(s)(Z(s) − Z(−s))ds (46)

=\frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)(Z(λ) − Z(−λ)) dλ + ∑_{0<Re(−q_{k})<a} d(σ_{+})b_{k}h(−q_{k}).

By (44), (45) and (46), we have

(47)  2 ∑_{λ_{j}} m_{j}h(iλ_{j}) = 2\frac{1}{2πi} ∫_{i∞}^{−i∞} h(λ)(Z(λ) + \frac{d(σ_{+})}{2}Φ(λ)) dλ

+ \frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)(Z(λ) − Z(−λ) + d(σ_{+})Φ(λ)) dλ.

If we exchange λ → iλ, we see that the first term of the right hand side in (47) is equal to

(48) 2\frac{1}{2π} ∫_{−∞}^{∞} h(iλ) ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})e^{−iλl(C_{γ})} dλ

+ 2\frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} h(iλ)Φ(iλ) dλ

= 2( ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} − \overline{χ_{σ_{−}}(m_{γ})})\frac{1}{2π} ∫_{−∞}^{∞} h(iλ)e^{−iλl(C_{γ})}dλ

+ \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} h(iλ)Φ(iλ) dλ ).

By (47), (48) and the Selberg trace formula applied to h(iλ), we have

\frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)R(λ) dλ = 0

for any odd holomorphic h(s) satisfying suitable growth conditions. Since R(s) is an entire function, this implies that R(s) = 0 over ℂ. Hence we have the equality

\frac{d}{ds} log(Z_{H}^{o}(n + s)Z_{H}^{o}(n − s)) = −d(σ_{+})Φ(s)

for any s ∈ ℂ. Recalling the equalities,

Φ(s) = \frac{d}{ds}( log det C_{+}(s) − log det C_{−}(s) ),

η(D) = \frac{1}{πi} log Z_{H}^{o}(n),

we get the following theorem,

Theorem 6.4. We have

Z_{H}^{o}(n + s)Z_{H}^{o}(n − s) = exp(2πiη(D)) (\frac{det C_{+}(s)C_{−}(0)}{det C_{−}(s)C_{+}(0)})^{−d(σ_{+})} for s ∈ ℂ.

Remark 6.5. In the above equality, Z_{H}^{o}(n + s)Z_{H}^{o}(n − s) has the zeros at q_{+}, −q_{+} of orders d(σ_{+})b_{+} for Re(q_{+}) < 0 and the poles at q_{−}, −q_{−} of orders d(σ_{+})b_{−} for Re(q_{−}) < 0. These zeros and poles coincide with the zeros and poles of (\frac{det C_{+}(s)C_{−}(0)}{det C_{−}(s)C_{+}(0)})^{−d(σ_{+})}.

7. Cusp contributions for regularized determinants and functional equations

In this Section, we compute the unipotent factor in the relation between the regularized determinant and the Selberg zeta function of even type. We also prove the functional equation for the regularized determinant and the Selberg zeta function of even type where the unipotent factor plays a nontrivial role. This equation is the even type counterpart of the functional equation for the eta invariant and the Selberg zeta function of odd type proved in Section 6.

We define the zeta function with the factor e^{−ts^{2}} for a positive real number s by

ζ_{D^{2}}(z, s) := \frac{1}{Γ(z)} ∫_{0}^{∞} t^{z−1} Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) e^{−ts^{2}}dt.

Note that ζ_{D^{2}}(z, 0) = ζ_{D^{2}}(z) if the kernel of D^{2} is trivial. We proved that ζ_{D^{2}}(z) is regular at z = 0 in Section 5. Due to the factor s^{2}, the continuous spectrum of D^{2} + s^{2} has minimum s^{2} and the large time part of the zeta function ζ_{D^{2}}(z, s) does not produce any poles. Therefore ζ_{D^{2}}(z, s) is regular at z = 0. We define the regularized determinant of D^{2} + s^{2} by

Det(D^{2}, s) := exp(−ζ′_{D^{2}}(0, s))

for a positive real number s. Now we observe that ζ_{D^{2}}(0, s) = 0 by the results of Section 5. Then we have

\frac{d}{dz} |_{z=0} ζ_{D^{2}}(z, s) = ∫_{0}^{∞} t^{−1} Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) e^{−ts^{2}}dt.

Recall that

Tr(e^{−tD^{2}} − e^{−tD_{0}^{2}}) (49)

= ∑_{λ_{k}} e^{−tλ_{k}^{2}} − \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} e^{−tλ^{2}}tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ
=I_{Γ}(K_{t}^{e}) + H_{Γ}(K_{t}^{e}) + U_{Γ}(K_{t}^{e}).

We denote the Mellin transform of each term of the right side of (49) by

I(z, s) = ∫_{0}^{∞} t^{z−1}I_{Γ}(K_{t}^{e})e^{−ts^{2}}dt,
H(z, s) = ∫_{0}^{∞} t^{z−1}H_{Γ}(K_{t}^{e})e^{−ts^{2}}dt,
U(z, s) = ∫_{0}^{∞} t^{z−1}U_{Γ}(K_{t}^{e})e^{−ts^{2}}dt.

Hence we have

(50)  − log Det(D^{2}, s) = (I(z, s) + H(z, s) + U(z, s)) |_{z=0} .

Note that each term I(z, s), H(z, s), U(z, s) may not be finite at z = 0. But the sum of these has the finite value at z = 0 since the left hand side of (50) is finite.

We recall that the Selberg zeta function of even type,

Z_{H}^{e}(s) := exp(− ∑_{σ_{±}} ∑_{γ:hyperbolic} j(γ)^{−1}|det(Ad(m_{γ}a_{γ})^{−1} − I|_{𝔫})|^{−1}\overline{χ_{σ}(m_{γ})}e^{−sl(C_{γ})})

is defined for Re(s) ≫ 0 and we will show that this has the meromorphic extension over ℂ. We note the following equality,

t^{z−1} = \frac{1}{Γ(1 − z)} ∫_{0}^{∞}(x(x + 2s))^{−z}e^{−tx(x+2s)}2(x + s) dt

for Re(z) < 0. From (8), we have

H(z, s) = ∫_{0}^{∞} t^{z−1} \frac{1}{\sqrt{4πt}} ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})}+\overline{χ_{σ_{−}}(m_{γ})})e^{− \frac{l(Cγ)^{2}}{4t}} e^{−ts^{2}}dt.

Using the elementary equality ∫_{0}^{∞} \frac{1}{\sqrt{4πt}} exp(−(\frac{r^{2}}{4t} + ts^{2}))dt = \frac{1}{2s}e^{−rs}, we see that H(z, s) is given by

(51)  H(z, s) = \frac{1}{Γ(1 − z)} ∫_{0}^{∞}(x(x + 2s))^{−z} \frac{d}{dx} log Z_{H}^{e}(x + s + n) dx.

It follows from Corollary 3.3 that U(z, s) is the Mellin transform with factor e^{−ts^{2}} of the following terms

\frac{2}{2π} ∫_{−∞}^{∞} e^{−tλ^{2}}(P_{U}(λ) − κ\frac{d(σ_{±})}{2}(ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2}))) dλ

where P_{U}(λ) is the even polynomial of degree 2n − 4. We repeat the method used for H(z, s). We deal with the term ψ(±iλ + \frac{1}{2}) using the Cauchy integral formula

\frac{1}{2π} ∫_{0}^{∞} ∫_{−∞}^{∞} e^{−t(λ^{2}+(s+x)^{2})}ψ(±iλ + \frac{1}{2}) dλ dt

= \frac{1}{2π} ∫_{−∞}^{∞} \frac{1}{λ^{2} + (s + x)^{2}} ψ(±iλ + \frac{1}{2}) dλ = \frac{1}{2(s + x)}ψ(s + x + \frac{1}{2})
= \frac{1}{2(s + x)} \frac{d}{dx} log Γ(s + x + \frac{1}{2}).

This gives us the following equality for a part of U(z, s),

(52)  U_{1}(z, s) = \frac{1}{Γ(1 − z)} ∫_{0}^{∞}(x(x + 2s))^{−z} \frac{d}{dx} log Γ(s + x + \frac{1}{2})^{−2κd(σ_{+})} dx.

We define

U_{2}(z, s) := \frac{2}{2π} ∫_{0}^{∞} t^{z−1}e^{−ts^{2}} ∫_{−∞}^{∞} e^{−tλ^{2}}P_{U}(λ) dλ dt.

Then U_{2}(z, s) is regular at z = 0 and

(53)  U_{2}(0, s) = − log exp(2 ∫_{0}^{s} P_{U}(iλ) dλ)

by the lemma 3 in [7].

For I(z, s), we can treat this as U_{2}(z, s) since p(σ_{+}, λ) = p(σ_{−}, λ) is an even polynomial of λ. By (50), (51), (52) and (53), we get the following theorem

Theorem 7.1. The following equality holds for any s ∈ ℂ,

(54)  Det(D^{2}, s) = Z_{H}^{e}(s + n)Γ(s + \frac{1}{2})^{−2κd(σ_{+})} exp(2 ∫_{0}^{s} P_{I}(iλ) + P_{U}(iλ) dλ)

where P_{I}(λ) = 2πV ol(Γ\G)p(σ_{+}, λ).

Proof. The equality (54) holds for positive real numbers s, and the right hand side gives the meromorphic extension over ℂ by Proposition 7.2. □

We consider an even smooth function g(|u|) where g(u) is the given function in Section 6. We set

H_{s}(σ_{±}, λ) = ∫_{−∞}^{∞} g(|u|)e^{−s|u|}e^{iλu}du

for a complex parameter s. The integration by parts gives

(55)  H_{s}(σ_{±}, λ) = \frac{1}{s − iλ} ∫_{0}^{∞} g′(u)e^{−(s−iλ)u}du + \frac{1}{s + iλ} ∫_{0}^{∞} g′(u)e^{−(s+iλ)u}du.

We apply the Selberg trace formula to the one parameter family of functions f_{s} for Re(s) ≫ 0 on G with f̂_{s}(σ_{±}, iλ) = H_{s}(σ_{±}, λ), f̂_{s}(σ, iλ) = 0 if σ ≠ σ_{±}. Then we get

∑_{λ_{j}∈σ_{p}^{+}} H_{s}(σ_{+}, λ_{j}) + ∑_{λ_{j}∈σ_{p}^{−}} H_{s}(σ_{−}, λ_{j}) (56)

− \frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ) tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ

= ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} + \overline{χ_{σ_{−}}(m_{γ})})e^{−sl(C_{γ})}

− \frac{κd(σ_{+})}{2π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)(ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2})) dλ

+ \frac{2}{2π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)(P_{I}(λ) + P_{U}(λ)) dλ.

We have the equality

\frac{d}{ds} log Z_{H}^{e}(s + n) = ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} + \overline{χ_{σ_{−}}(m_{γ})})e^{−sl(C_{γ})},

and now we investigate the other terms in (56) to get the meromorphic extension of \frac{d}{ds} log Z_{H}^{e}(s + n) over ℂ and to determine its poles.

Discrete eigenvalue term: Using (55) as before, we can see that H_{s}(σ_{±}, λ_{j}) has the meromorphic extension over ℂ and has the simple poles at iλ_{j} and −iλ_{j} for λ_{j} ∈ σ_{p}^{±} with the residue m_{j} where m_{j} is the multiplicity of λ_{j}.

Scattering term: Now we consider the scattering term

−\frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ) tr(C_{Γ}(σ_{+}, −iλ)\frac{d}{d(iλ)}C_{Γ}(σ_{+}, iλ)) dλ.

By the general formula given in [16], the function

Ψ(z) := tr(C_{Γ}(σ_{+}, −z)\frac{d}{dz} C_{Γ}(σ_{+}, z))

has the following form over ℂ,

Ψ(z) = ∑_{Re(q_{k})<0} −\frac{1}{z − q_{k}} + \frac{1}{z + q_{k}} + log p

for some constant p. We repeat the method in Section 6 to prove that the scattering term has the meromorphic extension over ℂ and has poles at q_{k} for Re(q_{k}) < 0 with residues d(σ_{+})b_{k} . As before b_{k} denotes the order of the pole of det C_{Γ}(σ_{+}, z), q_{k}.

Identity and Unipotent term: It follows from the proposition 3.9 in [8], that we have an identity

\frac{1}{2π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)(P_{I}(λ) + P_{U}(λ)) dλ = 0.

We now turn our attention to other unipotent terms. As in the proposition 3.7 of [8], we use the Cauchy integral formula to get

\frac{κd(σ_{+})}{2π} ∫_{−∞}^{∞} H_{s}(σ_{+}, λ)(ψ(iλ + \frac{1}{2}) + ψ(−iλ + \frac{1}{2})) dλ = 2κd(σ_{+})ψ(s + \frac{1}{2})

for Re(s) ≫ 0 and the right hand side gives us the meromorphic extension over ℂ of the left hand side.

Considering the equality (56) and the above analysis of other terms, we can conclude

Proposition 7.2. The Selberg zeta function of even type Z_{H}^{e}(s) has the meromorphic extension over ℂ.

Remark 7.3. The zeros of Z_{H}^{e}(s + n)Γ(s + \frac{1}{2})^{−2κd(σ_{+})} are located at iλ_{j}, −iλ_{j} for λ_{j} ∈ σ_{p}^{±} and q_{k} for Re(q_{k}) < 0 and their orders are m_{j}, d(σ_{+})b_{k}.

Now we prove the functional equation for Det(D^{2}, s) and Z_{H}^{e}(s). First let us define

Z(s) := \frac{d}{ds} log Z_{H}^{e}(s + n) − 2κd(σ_{+})ψ(s + \frac{1}{2})

and we can see that

R(s) := Z(s) + Z(−s) + d(σ_{+})Ψ(s)

is an even entire function of s. Now let h(s) be an even function which decreases sufficiently rapidly as Im(s) → ∞ in the strip { s ∈ ℂ | |Re(s)| < n + ϵ, ϵ > 0 }. We follow Section 6 and consider the contour integral

L_{T} := \frac{1}{2πi} ∫_{L_{T}} h(s)Z(s)ds

where L_{T} is the rectangle with the corners a+iT, a−iT, −a+iT, −a−iT with n < a < n+ϵ . Then we have

lim_{T →∞} L_{T} = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞} h(s)(Z(s) + Z(−s))ds + \frac{2}{2πi} ∫_{−a+i∞}^{−a−i∞} h(s)Z(s)ds.

The Cauchy integral theorem gives

(57)  lim_{T →∞} L_{T} = 2 ∑_{λ_{j}} m_{j}h(iλ_{j}) + ∑_{−a<Re(q_{k})<0} d(σ_{+})b_{k}h(q_{k}).

The simple poles at q_{k} of Z(s) in the strip between Re(s) = −a and Re(s) = 0 have residues d(σ_{+})b_{k} and give the following equality,

(58) \frac{1}{2πi} ∫_{−a+i∞}^{−a−i∞} h(s)Z(s)ds = \frac{1}{2πi} ∫_{i∞}^{−i∞} h(λ)Z(λ) dλ + ∑_{−a<Re(q_{k})<0} d(σ_{+})b_{k}h(q_{k}).

Similarly the simple poles at −q_{k} of Z(−s) in a strip between Re(s) = a and Re(s) = 0 have residues −d(σ_{+})b_{k} and provide us with

\frac{1}{2πi} ∫_{a−i∞}^{a+i∞} h(s)(Z(s) + Z(−s))ds (59)

=\frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)(Z(λ) + Z(−λ)) dλ + ∑_{0<Re(−q_{k})<a} −d(σ_{+})b_{k}h(−q_{k}).

We use (57), (58) and (59) to obtain

2 ∑_{λ_{j}} m_{j}h(iλ_{j}) = 2\frac{1}{2πi} ∫_{i∞}^{−i∞} h(λ)(Z(λ) + \frac{d(σ_{+})}{2}Ψ(λ)) dλ (60)

+ \frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)(Z(λ) + Z(−λ) + d(σ_{+})Ψ(λ)) dλ.

We replace λ by iλ in the first term of the right hand side of the equality (60). Then this term is equal to

(61) 2 ∑_{γ:hyperbolic} l(C_{γ})j(γ)^{−1}D(γ)^{−1}(\overline{χ_{σ_{+}}(m_{γ})} + \overline{χ_{σ_{−}}(m_{γ})}) \frac{1}{2π} ∫_{−∞}^{∞} h(iλ)e^{−iλl(C_{γ})} dλ

− 2\frac{1}{2π} ∫_{−∞}^{∞} h(iλ)(2κd(σ_{+})ψ(iλ + \frac{1}{2})) dλ + 2\frac{d(σ_{+})}{4π} ∫_{−∞}^{∞} h(iλ)Ψ(iλ) dλ.

By (60), (61) and the Selberg trace formula applied to h(iλ), we have

\frac{1}{2πi} ∫_{−i∞}^{i∞} h(λ)(R(λ) + 4P_{I}(−iλ) + 4P_{U}(−iλ)) dλ = 0

for any even holomorphic h(s) satisfying suitable growth conditions. Since

R(s) + 4P_{I}(−is) + 4P_{U}(−is) = R(s) + 4P_{I}(is) + 4P_{U}(is)

is an entire function, we have

Z(s) + Z(−s) + 4P_{I}(is) + 4P_{U}(is) = −d(σ_{+})Ψ(s)

over ℂ. Now we get the following equality

\frac{d}{ds} log(Z_{H}^{e}(n + s)Γ(s + \frac{1}{2})^{−2κd(σ_{+})}) + 4P_{I}(is) + 4P_{U}(is)

= \frac{d}{ds} log(Z_{H}^{e}(n − s)Γ(−s + \frac{1}{2})^{−2κd(σ_{+})}) − d(σ_{+})\frac{d}{ds} log(det C_{Γ}(σ_{+}, s)).

This leads to the following formula,

Z_{H}^{e}(n + s)Γ(s + \frac{1}{2})^{−2κd(σ_{+})} exp(4 ∫_{0}^{s} P_{I}(iλ) + P_{U}(iλ) dλ) (62)

= (det C_{Γ}(σ_{+}, s))^{−d(σ_{+})} Z_{H}^{e}(n − s)Γ(−s + \frac{1}{2})^{−2κd(σ_{+})}.

This equality holds a priori up to constant C. As in the proof of the lemma 4.3 in [8], we multiply s^{−2m_{0}} to both sides of (62) where m_{0} is the multiplicity of the zero eigenvalue of D. Then this remove the spectral zero of Z_{H}^{e}(n ± s) at s = 0. If we compare the remaining parts s = 0, we can see that such a constant C should be equal to 1. From (54), we have

Det(D^{2}, s)^{2} = Z_{H}^{e}(s + n)^{2}Γ(s + \frac{1}{2})^{−4κd(σ_{+})} exp(4 ∫_{0}^{s} P_{I}(iλ) + P_{U}(iλ) dλ).

Finally we combine (62) and this equality, then we have

Theorem 7.4. For any s ∈ ℂ, we have

Det(D^{2}, s)^{2} = (det C_{Γ}(σ_{+}, s))^{−d(σ_{+})} Z_{H}^{e}(n + s)Z_{H}^{e}(n − s)(Γ(s + \frac{1}{2})Γ(−s + \frac{1}{2}))^{−2κd(σ_{+})}. (63)

Remark 7.5. The right hand side of (63) has the zeros at iλ_{j}, −iλ_{j} of order 2m_{j} for λ_{j} ∈ σ_{p}^{±} and at q_{k} of order 2d(σ_{+})b_{k} for Re(q_{k}) < 0 . These are the zeros of Det(D^{2}, s)^{2} as we expected.

References

[1] Dan Barbasch and Henri Moscovici, L^{2}-index and the Selberg trace formula, J. Funct. Anal., 53, (1983) 151-201.

[2] Christian Bär, The Dirac operator on Hyperbolic manifolds of Finite volume, J. Diff. Geom., 54, (2000) 439-488.

[3] David Lee DeGeorge, On a Theorem of Osborne and Warner. Multiplicities in the Cuspidal Spectrum , J. Funct. Anal., 48, (1982) 81-94.

[4] Isaac Efrat, Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys., 119, (1988) 443-451.

[5] Isaac Efrat, Erratum: “Determinants of Laplacians on surfaces of finite volume”, Comm. Math. Phys., 138 , (1991) 607.

[6] Masaaki Eguchi, Shin Koizumi and Masaichi Mamiuda, The expression of the Harish-Chandra Cfunctions of semisimple Lie groups Spin(n, 1), SU(n, 1), J. Math. Soc. Japan , 51, (1999) 955-985.

[7] David Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math., 84, (1986) 523-540.

[8] Ramesh Gangolli and Garth Warner, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J., 78, (1980) 1-44.

[9] Werner Hoffmann, The Fourier transforms of weighted orbital integrals on semisimple groups of real rank one, J. Reiner Angew. Math., 489, (1997) 53-97.

[10] Shin-ya Koyama, Determinant expression of Selberg zeta functions.I, Trans. Amer. math. Soc., 324, (1991) 149-168.

[11] Shin-ya Koyama, Determinant expression of Selberg zeta functions.II, Trans. Amer. math. Soc., 329, (1992) 755-772.

[12] John Millson, Closed geodesic and the η-invariant, Ann. of Math., 108, (1978) 1-39.

[13] Henri Moscovici and Robert Stanton, Eta invariants of Dirac operators on locally symmetric manifolds, Invent. Math., 95, (1989) 629-666.

[14] Henri Moscovici and Robert Stanton, R-torsion and zeta functions for locally symmetric manifolds, Invent. Math., 105, (1991) 185-216.

[15] Werner Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr., 111, (1983) 197-288.

[16] Werner Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math., 130, (1989) 473-529.

[17] Werner Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math., 109, (1992) 265-305.

[18] Werner Müller, Eta invariants and Manifolds with boundary, J. Diff. Geom., 40, (1994) 311-377.

[19] Werner Müller, Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys., 192, (1998) 309-347.

[20] Peter Sarnak, Determinants of Laplacians, Comm. Math. Phys., 110, (1987) 113-120.

[21] M. Scott Osborne and Garth Warner, Multiplicities of the Integrable Discrete Series: The case of a Nonuniform Lattice in an R-rank One Semisimple Group, J. Funct. Anal., 30, (1978) 287-320.

[22] Peter Sarnak and Masato Wakayama, Equidistribution of Holonomy about Closed Geodesics, Duke Math. J., 100, (1999) 1-57.

[23] Garth Warner, Selberg’s trace formula for nonuniform lattices: The R-rank one case, Adv. Math. Suppl. Stud., 6, (1979) 1-142.

Max Planck Institute of Mathematics, Vivatgasse 7, D-53111 Bonn, Germany

E-mail address: jspark@mpim-bonn.mpg.de