THE TRACE CLASS CONJECTURE IN THE THEORY OF AUTOMORPHIC

FORMS. II.

WERNER MÜLLER

Abstract. Let G be the group of real points of a reductive algebraic ℚ-group satisfying the assumptions made in [H, Chapter I] and let Γ be an arithmetic subgroup of G. Let R_{Γ} be the right regular representation of G on L^{2}(Γ\G) and denote by R_{Γ}^{d} the restriction of R_{Γ} to the discrete subspace. In this paper we prove that for every integrable rapidly decreasing function f on G, the operator R_{Γ}^{d}(f) is of the trace class.

0. Introduction

Let G be the group of real points of a reductive algebraic ℚ-group G satisfying the assumptions made in [H, Chapter I] and let Γ be an arithmetic subgroup of G. We fix a maximal compact subgroup K of G. Let R_{Γ} be the right regular representation of G on L^{2}(Γ\G) and let L_{d}^{2}(Γ\G) be the smallest closed subspace of L^{2}(Γ\G) which contains all irreducible subrepresentations of R_{Γ}. Let Ĝ be the unitary dual of G. For each class π ∈ G, let (π, H_{π}) denote a representative of π. Then L_{d}^{2}(Γ\G) is an invariant subspace of L^{2}(Γ\G) which decomposes discretely as

L_{d}^{2}(Γ\G) = ⨁_{π∈Ĝ} m_{Γ}(π)H_{π}.  (0.1)

It is well known that the multiplicities m_{Γ}(π) are all finite (see, e.g. [BG]). Denote by R_{Γ}^{d} the restriction of R_{Γ} to the invariant subspace L_{d}^{2}(Γ\G). Let C^{1}(G) be Harish–Chandra’s Schwartz space of integrable rapidly decreasing functions on G (see section 1 for its definition). For each f ∈ C^{1}(G), one can define

Date: November 14, 1996.

1991 Mathematics Subject Classification. Primary: 22E40; Secondary: 58G25.

Key words and phrases. Automorphic forms, spectral decomposition, trace class.

R_{Γ}^{d}(f) = ∫_{G} f(g)R_{Γ}^{d}(g) dg

which is a bounded operator on L_{d}^{2}(Γ\G). In [Mu], we proved that for every K–finite function f ∈ C^{1}(G), R_{Γ}^{d}(f) is a trace class operator. This is important for the application of the Arthur trace formula [A]. The main purpose of this paper is to remove the K–finiteness assumption, i.e. to prove the following theorem: Theorem 0.1. For every f ∈ C^{1}(G), the operator R_{Γ}^{d}(f) is of the trace class. This result settles the trace class conjecture in full generality. A different proof has recently been announced by Ji [J]. For simplicity, we have restricted attention to arithmetic subgroups. However, Theorem 0.1 holds, of course, also for other lattices including non-arithmetic lattices in ℝ-rank one semisimple Lie groups (see the remark at the end of section 8). As in [Mu], the proof of Theorem 0.1 is reduced to the estimation of the eigenvalue counting function for Bochner Laplace operators on the locally symmetric space Γ\G/K. In fact, by passing to a normal subgroup of finite index if necessary, we can always assume that Γ acts without fixed points on the symmetric space X = G/K so that Γ\X is a Riemannian manifold. Let σ: K → GL(V_{σ}) be an irreducible unitary representation of K. Then σ defines a homogeneous vector bundle Ẽ_{σ} over X and an associated locally homogeneous vector bundle E_{σ} = Γ\Ẽ_{σ} over Γ\X. Let ∇ be the push down of the canonical connection on the homogeneous vector bundle Ẽ_{σ} and let Δ_{σ} = ∇^{∗}∇ be the Bochner Laplace operator. This is a second order elliptic differential operator, acting in C^{∞}(Γ\X, E_{σ}). Regarded as operator in L^{2}(Γ\X, E_{σ}) with domain C_{c}^{∞}(Γ\X, E_{σ}), Δ_{σ} is essentially self–adjoint and nonnegative, and we will also denote its unique self–adjoint extension by Δ_{σ}. The spectrum of Δ_{σ} is the union of a pure point spectrum Spec_{pp}(Δ_{σ}): 0 ≤ λ_{1} ≤ λ_{2} ≤ ⋯ and an absolutely continuous spectrum Spec_{ac}(Δ_{σ}). Then Theorem 0.1 is a consequence of Theorem 0.2. Let N_{σ}(λ) be the number of linearly independent eigenfunctions of Δ_{σ} with eigenvalues less than λ. Let Ω_{K} ∈ Z (𝔨_{ℂ}) be the Casimir element of K. Put λ_{σ} = σ(Ω_{K}) and m = dim G. Then there exists C > 0 such that for every σ ∈ K̂, one has

N_{σ}(λ) ≤ C(1 + λ_{σ}^{4m} + λ^{4m}), λ ≥ 0.

This is an improved version of the estimate of the counting function obtained in [Mu] in the sense that the dependence of the right hand side on σ ∈ K̂ has been made explicit. Given π ∈ Ĝ, let λ_{π} = π(Ω) be the eigenvalue of the Casimir operator Ω of G in the Gårding space H_{π}^{∞} of H_{π}. As immediate consequence of Theorem 0.2 we get Corollary 0.3. Let N ≥ 3 dim G. Then

∑_{π∈Ĝ} ∑_{σ∈K̂} m_{Γ}(π) dim (H_{π} ⊗ V_{σ})^{K} (1 + λ_{π}^{2} + λ_{σ}^{2})^{−N} < ∞.

This is easily seen to imply Theorem 0.1. In order to prove Theorem 0.2, we proceed essentially as in [Mu], keeping track of the dependence of the various constants on σ ∈ K̂. We now shall explain our method in more detail. First, we recall that the discrete spectrum has a further decomposition

L_{d}^{2} (Γ\X, E_{σ}) = L_{cus}^{2} (Γ\X, E_{σ}) ⊕ L_{res}^{2} (Γ\X, E_{σ})

into the direct sum of the space of cusp forms L_{cus}^{2}(Γ\X, E_{σ}) and the residual subspace L_{res}^{2}(Γ\X, E_{σ}). We start by establishing the estimate of Theorem 0.2 for cuspidal eigenfunctions, i.e. we estimate the number N_{cus,σ}(λ) of linearly independent eigensections of Δ_{σ} which satisfy the cuspidal condition and have eigenvalues less than λ. Donnelly proved in [D1] that

\overline{lim}_{λ→∞} \frac{N_{cus,σ}(λ)}{λ^{n/2}} ≤ (4π)^{−n/2} \frac{Vol(Γ\G/K)}{Γ(n/2 + 1)} dim(V_{σ})

This implies that N_{cus,σ}(λ) ≤ C(1 + λ^{n/2}) for λ ≥ 0 and some constant C > 0. We need, however, an explicit bound of C in terms of σ. To treat this problem, we modify Donnelly’s method [D1]. More precisely, we use the cut–off Laplacian Δ_{σ,T} introduced in [Mu]. Here T ∈ 𝔞_{0} is the truncation parameter and the domain of the quadratic form which defines Δ_{σ,T} consists of those functions whose constant terms along all proper ℚ parabolic subgroups vanish above height T. Then Δ_{σ,T} has pure discrete spectrum. Since each cuspidal eigenfunctions of Δ_{σ} is in the domain of Δ_{σ,T} , the cuspidal spectrum is a part of the spectrum of Δ_{σ,T}. Now the eigenvalue counting function for Δ_{σ,T} can be estimated using the method of Donnelly [D1] combined with the principle of semigroup domination [DL]. In particular, this leads to an upper bound for N_{cus,σ}(λ) of the form claimed by Theorem 0.2. To deal with the residual discrete spectrum, we proceed as in [Mu], using Langlands’ description of L_{res}^{2} (Γ\X, E_{σ}) (cf. [L1]). It follows from Langlands’ theory of Eisenstein series that L_{res}^{2}(Γ\X, E_{σ}) is spanned by iterated residues of cuspidal Eisenstein series [L1, Chapter 7]. In particular, this gives a decomposition

L_{res}^{2} (Γ\X, E_{σ}) = ⨁_{C} L_{res}^{2} (Γ\X, E_{σ}; C)

where C runs over association classes of proper rational parabolic subgroups P of G. One of the important consequences of Langlands’ theory is that the bound on the higher rank residual spectrum can be reduced to bounds on the number of real poles of the intertwining operator C(s; σ) for the cuspidal rank one Eisenstein series. Using the factorization of det C(s; σ) derived in [Mu, Section 6], this problem can be reduced to the estimation of the rank one residual spectrum. At various stages of this reduction process we have to estimate the dimension of spaces of cusp forms associated to the reductive parts of the ℚ parabolic subgroups that are involved. This is where the bounds on the number of cuspidal eigenfunctions of Δ_{σ} enter again. Finally, the rank one residual discrete spectrum is related to some part of the spectrum of Δ_{σ,T} . Applying the estimates obtained for the counting function of Δ_{σ,T} , the proof of Theorem 0.2 is completed. Theorem 0.1 implies a corresponding adèlic version which we discuss in the final section 8.

Acknowledgement: I would like to thank G. Harder for some valuable discussions concerning the adèlic version of Theorem 0.1.

1. Preliminaries

1.1. Let G be a reductive linear algebraic group defined over ℚ which satisfies the assumptions made in [H, Ch.I,§1]. Let G = G(ℝ) and let Γ be an arithmetic subgroup of G, i.e. Γ is a subgroup of G(ℚ) which is commensurable with ρ(G) ∩ GL_{n}(ℤ) for any injective ℚ-morphism ρ: G → GL_{n}. We note that the assumptions that are imposed on G imply that the intersection of Γ with the center Z of G is cocompact and Γ is commensurable with the product of its intersections with Z and the derived group DG^{0} of the identity component G^{0} of G. We fix a maximal compact subgroup K of G and set X = G/K. By θ we shall denote the Cartan involution of G with respect to K. According to [BS, Proposition 1.6] there is a unique choice for θ which is algebraic. Throughout this paper we shall assume that Γ acts without fixed points on X. This can be always achieved by passing to a normal subgroup of finite index.

1.2. The connected component of a Lie group G will be denoted by G^{0}. The Lie algebra of a Lie group G, H,... is denoted by the corresponding lowercase German letter 𝔤, 𝔥,... By Z(𝔤) we shall denote the center of the universal enveloping algebra U(𝔤) of the complexified Lie algebra 𝔤 ⊗ ℂ. In particular, Z(𝔤) contains the Casimir element Ω_{G} (or simply Ω) of G with respect to an admissible real-valued bilinear form F on 𝔤 (cf. [BG]).

1.3. Let P be parabolic subgroup of G which is defined over ℚ. Let P = P(ℝ). We shall call P a ℚ-parabolic subgroup of G. Let N be the unipotent radical of P and L the Levi ℚ-subgroup of P which is stable under θ. Then P = L · N. Let M = ∩_{χ} ker(χ^{2}) where χ runs over the set X_{ℚ}(L) of rational characters of L. Then M is a ℚsubgroup of L. We also note that M is a reductive linear algebraic ℚ-group which satisfies the same assumptions as G. Furthermore, let S = L ∩ R_{d}P be the greatest torus of the split radical R_{d}P of P stable under θ. Put A_{P} = S(ℝ)^{0}, M_{P} = M(ℝ) and N_{P} = N(ℝ). Then we get a decomposition of P as

P = N_{P} A_{P} M_{P}  (1.1)

(or just NAM, if there is no danger of confusion) which is called Langlands decomposition of P and A_{P} is called the special split component of P. The rank of P is defined to be the dimension of A_{P}. The Weyl group of A_{P} is W (A_{P}) = N_{G}(A_{P})/Z_{G}(A_{P}). Furthermore, we set

Γ_{P} = Γ ∩ P and Γ_{M} = NΓ ∩ M.

Then Γ_{M} is an arithmetic subgroup of M_{P}. We also put

K_{M} = K ∩ M and X_{M} = M/K_{M}.

Observe that K ∩ M = K ∩ P. We have G = P K. Therefore, any x ∈ G has a decomposition

x = namk

with k ∈ K, m ∈ M, a ∈ A, n ∈ N. The factor a is uniquely determined by x. Set

H_{P} (x) = log a.

The roots of (P, A) will be denoted by Φ_{P} and Ψ_{P} will denote the set of simple roots of (P, A). For β ∈ Φ_{P}, let

𝔫_{β} = {Y ∈ 𝔤 | [H, Y] = β(H)Y, H ∈ 𝔞}.

Then

𝔫 = ⨁_{β∈Φ_{P}} 𝔫_{β}.

As usual, let

ρ_{P} = \frac{1}{2} ∑_{β∈Φ_{P}} dim(𝔫_{β})β.

For a given subset F ⊂ Ψ_{P} we denote by P_{F} the ℚ-parabolic subgroup of G which is associated to F. Note that P ⊂ P_{F} and the Lie algebra 𝔞_{F} of the split component of P_{F} consists of all H ∈ 𝔞 such that α(H) = 0 for all α ∈ F.

1.4. Next we recall the definition of Harish-Chandra’s Schwartz space C^{1}(G) of integrable rapidly decreasing functions on G (cf. [BM, p.162], [Wa2, p.230]). The definition involves two spherical functions on G, σ and Ξ. The function σ(g) can be defined as the geodesic distance between the cosets K and gK in the symmetric space G/K. Let P be a minimal parabolic subgroup of G. Then Ξ is the zonal spherical function defined by

Ξ(g) = ∫_{K} e^{−ρ_{P} (H_{P} (gk))}dk.

For a given element Y ∈ U(𝔤), let L(Y) (resp. R(Y)) denote the right (resp. left) invariant differential operators on G associated to Y. Then C^{1}(G) is the space of all functions f ∈ C^{∞}(G) such that for all Y_{1}, Y_{2} ∈ U(𝔤) and r ≥ 0

sup_{g∈G}(1 + σ(g))^{r}Ξ(g)^{−2}|L(Y_{1})R(Y_{2})f(g)| < ∞.

We note that C^{1}(G) is contained in L^{1}(G). This follows from [W2, Proposition 8.3.7.5].

1.5. Let P be a ℚ-parabolic subgroup of G with unipotent radical N_{P}. Let f be a complex-valued, locally bounded, measurable function on Γ\G. The constant term f^{P} of f along P is defined as

f^{P} (x) = ∫_{Γ∩N_{P} \N_{P}} f(nx) dn

where the measure dn is normalized by the condition that the volume of Γ ∩ N_{P} \N_{P} equals 1. The subspace of L^{2}(Γ\G) consisting of all f satisfying f^{P} = 0 for all ℚ-parabolic subgroups P ≠ G is denoted by L_{cus}^{2}(Γ\G). This is the space of cusp forms in L^{2}(Γ\G). It is well known that L_{cus}^{2}(Γ\G) is contained in L_{d}^{2}(Γ\G) (cf. [L1, p. 41], [H, Theorem 2]).

2. Reduction to the semisimple case

Let G and Γ be as in section 1.1. In this section we indicate how the proof of Theorem 0.1 can be reduced to the case of a connected semisimple Lie group with finite center. Let Γ′ ⊂ Γ be a subgroup of finite index. Then L_{d}^{2}(Γ\G) is a subspace of L_{d}^{2}(Γ′\G) and it suffices to prove Theorem 0.1 for Γ′. The converse is also true if we include finite-dimensional unitary representations of Γ. Suppose that Γ′ ⊂ Γ is a normal subgroup and set H = Γ′\Γ. Let Ĥ be the set of irreducible unitary representations of H. For any (ρ, V_{ρ}) ∈ Ĥ, let L_{ρ} → Γ\X be the flat unitary vector bundle defined by Γ → H → GL(V_{ρ}). Then for any σ ∈ K̂, one has

L^{2}(Γ\X, E_{σ} ⊗ L_{ρ}) = (L^{2}(Γ′\X, E_{σ}) ⊗ V_{ρ})^{H} .

Let ∇^{ρ} be the product of the canonical connection ∇ on E_{σ} and the flat connection on L_{ρ}. Let

Δ_{σ,ρ} = (∇^{ρ})^{∗}∇^{ρ}

be the associated Bochner Laplace operator. Then Δ_{σ,ρ} has similar properties as Δ_{σ}. Let N_{Γ,σ}(λ, ρ) be the eigenvalue counting function of Δ_{σ,ρ} and let N_{Γ′,σ}(λ) be the eigenvalue counting function of Δ_{σ} acting in L^{2}(Γ′\X, E_{σ}). Proposition 2.1. We have

N_{Γ′,σ}(λ) = ∑_{ρ∈Ĥ} dim(V_{ρ})N_{Γ,σ}(λ, ρ), λ ≥ 0.

Proof. The proof is an immediate consequence of the following analogue of the Peter-Weyl theorem:

L^{2}(Γ′\X, E_{σ}) = ⨁_{ρ∈Ĥ} (V_{ρ}^{∗} ⊗ L^{2}(Γ\X, E_{σ} ⊗ L_{ρ})) .

□

All our arguments work equally well if we include a flat unitary vector bundle as above. Hence, in order to prove Theorem 0.1, we can always replace Γ by any discrete subgroup of G which is commensurable with Γ. First, this means that we can confine ourself to the case of a connected group G. Furthermore, the lattice Γ is commensurable with the product of its intersections with the center Z of G and with the derived group DG^{0} of the identity component G^{0} of G. Since the intersection of Γ with Z is cocompact in Z, it is sufficient to consider the intersection of Γ with DG^{0}. Thus we can assume that G is a connected semisimple linear algebraic group and Γ ⊂ G(ℚ) is an arithmetic subgroup.

3. Heat kernels on the universal covering

In this section, we study the heat kernels of Bochner Laplace operators on G/K. Let (σ, V_{σ}) be an irreducible unitary representation of K on a complex vector space V_{σ}. Let Ẽ_{σ} = (G × V_{σ})/K be the associated homogeneous vector bundle over X = G/K. We equip Ẽ_{σ} with the G– invariant Hermitian fibre metric which is induced by the inner product in V_{σ}. Let C^{∞}(Ẽ_{σ}), C_{c}^{∞}(Ẽ_{σ}) and L^{2}(Ẽ_{σ}) denote the space of smooth sections, the space of compactly supported smooth sections and the Hilbert space of square integrable sections of Ẽ_{σ}, respectively. Then we have

C^{∞}(Ẽ_{σ}) = (C^{∞}(G) ⊗ V_{σ})^{K}, L^{2}(Ẽ_{σ}) = (L^{2}(G) ⊗ V_{σ})^{K} (3.1)

and similarly for C_{c}^{∞}(Ẽ_{σ}). Let ˜∇ be the canonical connection on Ẽ_{σ}. Recall that ˜∇ is the unique G–invariant connection on Ẽ_{σ} such that if s is a C^{∞} cross–section, Y ∈ 𝔭 and π: G → G/K is the canonical projection, then

˜∇_{π_{∗}(Y)}(s) = \frac{d}{dt}s(exp(tY)K)

where π_{∗} is the differential of π at the identity element e ∈ G. Let Δ̃_{σ} = ˜∇^{∗} ˜∇ be the Bochner Laplace operator of ˜∇. Let R be the right regular representation of G on C^{∞}(G). Then with respect to the identification (3.1), we have

˜∇^{∗} ˜∇ = −R(Ω) ⊗ Id + Id ⊗ (Ω_{K})  (3.2)

where Ω and Ω_{K} are the Casimir elements of G and K, respectively [Mi]. Since σ is irreducible, there exists λ_{σ} ∈ ℝ such that

σ(Ω_{K}) = λ_{σ}Id.  (3.3)

In fact λ_{σ} ≥ 0. Thus by (3.2), we get

Δ̃_{σ} = −R(Ω) ⊗ Id + λ_{σ}Id.  (3.4)

The operator Δ̃_{σ}: C_{c}^{∞}(Ẽ_{σ}) → L^{2}(Ẽ_{σ}) is essentially self–adjoint and we will also denote its unique self–adjoint extension by Δ̃_{σ}. Then Δ̃_{σ} ≥ 0. Let exp(−tΔ̃_{σ}) be the heat semigroup. This is a smoothing operator in L^{2}(Ẽ_{σ}) which commutes with the representation of G on L^{2}(Ẽ_{σ}). Therefore, it is of the form

(e^{−tΔ̃_{σ}}φ)(g) = ∫_{G} h_{t}(g^{−1}g_{1})φ(g_{1})dg_{1}, φ ∈ (L^{2}(G) ⊗ V_{σ})^{K}, g ∈ G, (3.5)

where h_{t}: G → End(V_{σ}) is in L^{2} and satisfies the covariance property

h_{t}(g) = σ(k)h_{t}(k^{−1}gk′)σ(k′)^{−1}, for g ∈ G, k, k′ ∈ K. (3.6)

In order to get estimates for h_{t}, we relate it to the heat kernel of G with respect to the left invariant metric on G as in [BM]. Let X_{1}, ⋯, X_{p} be an orthonormal basis for 𝔭 with respect to B|𝔭 × 𝔭 and let Y_{1}, ⋯, Y_{k} be an orthonormal basis for 𝔨 with respect to −B|𝔨 × 𝔨. Then we have

Ω = ∑_{i=1}^{p} X_{i}^{2} − ∑_{i=1}^{k} Y_{i}^{2} and Ω_{K} = − ∑_{i=1}^{k} Y_{i}^{2}.

Let

P = −Ω + 2Ω_{K} = − ∑_{i=1}^{p} X_{i}^{2} − ∑_{i=1}^{k} Y_{i}^{2}.  (3.7)

Then R(P) is the Laplace operator Δ_{G} on G with respect to the left invariant metric. The heat semigroup e^{−tΔ_{G}} is represented by a smooth kernel p_{t}, i.e.

(e^{−tΔ_{G}}f) (g) = ∫_{G} p_{t}(g^{−1}g′)f(g′)dg′, f ∈ L^{2}(G), g ∈ G, (3.8)

where p_{t} ∈ C^{∞}(G) ∩ L^{2}(G). In fact, p_{t} belongs to L^{1}(G) (see, e.g. [N]) so that (3.8) can be written as

e^{−tΔ_{G}} = R(p_{t}).

Let

Q = ∫_{K} R(k) ⊗ σ(k) dk

be the orthogonal projection of L^{2}(G) ⊗ V_{σ} onto its K–invariant subspace (L^{2}(G) ⊗ V_{σ})^{K}. By (3.4) and (3.7) one has

Δ̃_{σ} = −Q(R(Ω) ⊗ Id)Q + λ_{σ}Id_{L^{2}(Ẽ_{σ})}
= Q(R(P) ⊗ Id)Q − 2Q(R(Ω_{K}) ⊗ Id)Q + λ_{σ}Id_{L^{2}(Ẽ_{σ})}
= Q(Δ_{G} ⊗ Id)Q − λ_{σ}Id_{L^{2}(Ẽ_{σ})}.

Hence, we get

e^{−tΔ̃_{σ}} = Q(e^{−tΔ_{G}} ⊗ Id)Q · e^{tλ_{σ}}

which implies that

h_{t}(g) = e^{tλ_{σ}} ∫_{K} ∫_{K} p_{t}(k^{−1}gk′)σ(kk′^{−1}) dk dk′.  (3.9)

Since p_{t} ∈ L^{1}(G), it follows that

h_{t} ∈ (L^{1}(G) ⊗ End(V_{σ}))^{K×K} ,

i.e. h_{t} ∈ L^{1}(G)⊗End(V_{σ}) and it satisfies the covariance property (3.6). Thus, by (3.9), h_{t} can be estimated in terms of p_{t}. In order to estimate p_{t}, we use the following result of Borel [B1] (see also Borel und Harish– Chandra [BH]): every real semisimple Lie group has a discrete torsion free subgroup Γ with Γ\G compact. On Riemannian manifolds admitting a properly discontinuous group of isometries with compact quotient, Donnelly [D2] has constructed a fundamental solution of the heat equation which is uniquely determined by the properties it satisfies. Thus, let H(t, g, g′) be the fundamental solution of the heat equation ∂/∂t + Δ_{G} constructed by Donnelly [D2]. In addition to the usual properties of a fundamental solution of the heat equation, H(t, g, g′) satisfies

| ; |\frac{∂^{i}}{∂t^{i}} ∇_{g}^{j}∇_{g′}^{k}H(t, g, g′)| ; | ≤ Ct^{−n/2−i−j−k} exp (−\frac{d^{2}(g, g′)}{4t}) (3.10)

for 0 < t ≤ T and 0 ≤ i, j, k ≤ 1. Here d(g, g′) is the geodesic distance of g, g′ ∈ G with respect to the left invariant metric on G. It follows from (3.10) that H(t, g, g′) represents e^{−tΔ_{G}} and therefore, it coincides with p_{t}, i.e.

p_{t}(g^{−1}g′) = H(t, g, g′).

This implies that p_{t} satisfies

| ; |\frac{∂^{i}}{∂t^{i}} ∇^{j}p_{t}(g)| ; | ≤ Ct^{−n/2−i−j} exp (−\frac{d^{2}(g, 1)}{4t})  (3.11)

for 0 < t ≤ T, 0 ≤ i, j ≤ 1 and g ∈ G. By (3.9) and (3.11), we obtain Theorem 3.1. For σ ∈ K̂, let h_{t}^{σ}(g) be the kernel of exp(−tΔ̃_{σ}). Let ρ be the geodesic distance on X = G/K. There exists C > 0 such that for every σ ∈ K̂ one has

| ; |\frac{∂^{i}}{∂t^{i}} ∇^{j}h_{t}^{σ}(g)| ; | ≤ C(1 + λ_{σ}^{i})e^{tλ_{σ}}t^{−n/2−i−j} exp (−\frac{ρ^{2}(gK, K)}{4t})(3.12)

for 0 < t ≤ T, 0 ≤ i, j ≤ 1 and g ∈ G.

Proof. First we note that one has

d(g, 1) ≥ ρ(gK, K)  (3.13)

for any g ∈ G. Let Y ∈ 𝔭, ∥ Y ∥= 1. Since σ is unitary, it follows from (3.9), (3.11) and (3.13) that for 0 ≤ i, j ≤ 1:

| ; |\frac{∂^{i}}{∂t^{i}} ∇_{Y}^{j} h_{t}^{σ}(g)| ; | ≤ C ∫_{K×K} | ; |\frac{∂^{i}}{∂t^{i}} ∇_{Y}^{j} p_{t}(k^{−1}gk′) | ; |dk dk′

= C ∫_{K×K} | ; |\frac{∂^{i}}{∂t^{i}} (∇_{Ad(k′)Y}^{j} p_{t}) (k^{−1}gk′) | ; | dk dk′

≤ C_{1}(1 + λ_{σ}^{i})e^{tλ_{σ}}t^{−n/2−i−j} exp (−\frac{ρ^{2}(gK, K)}{4t}) .

□

One also can use the principal of semigroup domination as in [DL] to estimate h_{t}^{σ}. Let H_{σ}(t, x, y) be the heat kernel for Δ̃_{σ}. Then

H_{σ}(t, gK, g′K) = h_{t}^{σ}(g^{−1}g′).  (3.14)

Let |H_{σ}| be the pointwise norm of the homomorphism

H_{σ}(t, x, y) ∈ Hom ((Ẽ_{σ})_{y}, (Ẽ_{σ})_{x}) .

Let ϵ > 0 and set |H_{σ}|_{ϵ} = (|H_{σ}|^{2} + ϵ)^{1/2} . Then |H_{σ}|_{ϵ}(t, x, y) is a smooth function of x, y ∈ X and t ∈ ℝ^{+}. Since (∂/∂t)H_{σ} = −Δ̃_{σ}H_{σ}, the Bochner formula implies

|H_{σ}|_{ϵ} \frac{∂}{∂t}|H_{σ}|_{ϵ} = \frac{1}{2} \frac{∂}{∂t}(|H_{σ}|_{ϵ})^{2} = ⟨\frac{∂}{∂t}H_{σ}, H_{σ}⟩

= ⟨−Δ̃_{σ}H_{σ}, H_{σ}⟩ = −|∇H_{σ}|^{2} − \frac{1}{2}Δ_{0}|H_{σ}|^{2}

= −|∇H_{σ}|^{2} − \frac{1}{2}Δ_{0}|H_{σ}|_{ϵ}^{2}

where Δ_{0} is the Laplacian on functions of X. Employing Bochner’s formula again, we obtain

|H_{σ}|_{ϵ} \frac{∂}{∂t}|H_{σ}|_{ϵ} = |∇|H_{σ}|_{ϵ}|^{2} − |∇H_{σ}|^{2} − |H_{σ}|_{ϵ}Δ_{0}|H_{σ}|_{ϵ}.

Since |∇|H_{σ}|_{ϵ}| ≤ |∇H_{σ}|, it follows that for each ϵ > 0, one has

(\frac{∂}{∂t} + Δ_{0}) |H_{σ}|_{ϵ} ≤ 0.

Now let ϵ → 0. Then, considered as distribution, we get

(\frac{∂}{∂t} + Δ_{0}) |H_{σ}| ≤ 0.

Using (3.12), one can extend the proof of Theorem 4.3 in [DL] to our setting which implies Proposition 3.2. For every T > 0 there exists C > 0 such that for every σ ∈ K̂, one has

|H_{σ}(t, x, y)| ≤ Ct^{−n/2} exp (−\frac{ρ^{2}(x, y)}{4t})  (3.15)

for 0 < t ≤ T, x, y ∈ X. Compared with (3.12), the improvement is that the factor e^{tλ_{σ}} does not occur in (3.14). We claim that the same is true for derivatives of H_{σ}. This would improve our estimates.

4. The cuspidal spectrum

Let (σ, V_{σ}) be an irreducible unitary representation of K on V_{σ} and let

E_{σ} = Γ\Ẽ_{σ} → Γ\X

be the push down of the homogeneous vector bundle Ẽ_{σ} associated to (σ, V_{σ}). Then the space of C^{∞}-sections of E_{σ} is given by

C^{∞}(Γ\X, E_{σ}) = (C^{∞}(Γ\G) ⊗ V_{σ})^{K},

and similarly, the space of L^{2}-sections of E_{σ} equals

L^{2}(Γ\X, E_{σ}) = (L^{2}(Γ\G) ⊗ V_{σ})^{K}.  (4.1)

Since Δ̃_{σ} is G-invariant, it induces an elliptic operator Δ_{σ} acting on C^{∞}(Γ\X, E_{σ}). If ∇ is the push down of the canonical connection ˜∇ on Ẽ_{σ}, then

Δ_{σ} = ∇^{∗}∇.

This operator is essentially self–adjoint on L^{2}(Γ\X, E_{σ}) and our goal is to study the point spectrum of Δ_{σ}. By (4.1), the subspace of L^{2}(Γ\X, E_{σ}) which is spanned by the L^{2}-eigensections of Δ_{σ} equals

L_{d}^{2}(Γ\X, E_{σ}) = (L_{d}^{2}(Γ\G) ⊗ V_{σ})^{K}.

Let

L_{cus}^{2}(Γ\X, E_{σ}) = (L_{cus}^{2}(Γ\G) ⊗ V_{σ})^{K}.

Then L_{cus}^{2}(Γ\X, E_{σ}) is a subspace of L_{d}^{2}(Γ\X, E_{σ}) and the corresponding part of the spectrum of Δ_{σ} is called the cuspidal spectrum. The main purpose of this section is to study the cuspidal spectrum. To this end, we use the so called cut–off Laplacian Δ_{σ,T} introduced in [Mu] where T is the truncation parameter. First we recall the definition of Δ_{σ,T} . Let P_{1}, …, P_{l} be a set of representatives of the Γ–conjugacy classes of rank one ℚ–parabolic subgroups of G. Let A_{j} be the special split component of P_{j} and let 𝔞_{j} be the Lie algebra of A_{j}. Put

𝔞_{0} = ⨁_{j} 𝔞_{j}.

For each ℚ–parabolic subgroup P of G with special split component A_{P} , there exists a well defined linear map

I_{P} : 𝔞_{0} → 𝔞_{P}

[OW], [Mu]. If P = P_{j}, 1 ≤ j ≤ l, then I_{P_{j}} is the projection of 𝔞_{0} onto the summand 𝔞_{j}. Let ρ_{j} be the half-sum of the positive roots of (P_{j}, A_{j}) and let T_{ρ} ∈ 𝔞_{0} be the vector whose i–th component equals H_{ρ_{i}}– the image of ρ_{i} under the canonical identification of 𝔞_{i}^{∗} and 𝔞_{i}. Let t ≫ 0 and put T = tT_{ρ}. Let H^{1}(Γ\X, E_{σ}) be the Sobolev space, i.e. the completion of C_{c}^{∞}(Γ\X, E_{σ}) with respect to the norm

∥ φ ∥_{H^{1}}^{2}= ∫_{Γ\X}(|φ(x)|^{2} + |∇φ(x)|^{2}) dx.

For T as above, let H_{T}^{1}(Γ\X, E_{σ}) be the subspace of H^{1}(Γ\X, E_{σ}) defined by

H_{T}^{1}(Γ\X, E_{σ}) = {φ ∈ H^{1}(Γ\X, E_{σ}) | φ^{P_{i}}(a_{i}m_{i}) = 0
for log(a_{i}) > T_{i}, i = 1, …, m}.

Let H_{T} be the closure of H_{T}^{1}(Γ\X, E_{σ}) in L^{2}(Γ\X, E_{σ}). Consider the quadratic form

q(φ) =∥ ∇φ ∥^{2}, φ ∈ H_{T}^{1}(Γ\X, E_{σ}).

Since H_{T}^{1}(Γ\X, E_{σ}) is a closed subspace of H^{1}(Γ\X, E_{σ}), the quadratic form q is represented by a nonnegative self–adjoint operator Δ_{σ,T} . This is our cut–off Laplacian. If G = SL(2, ℝ) and σ = 1, this operator was first introduced by Lax and Phillips [LP]. In [Mu, Theorem 3.23] we proved that the spectrum of Δ_{σ,T} is discrete consisting of a sequence of eigenvalues 0 ≤ λ_{1} < λ_{2} < ⋯ of finite multiplicity. Furthermore, if φ is a cuspidal eigenfunction of Δ_{σ}, i.e. φ ∈ L_{cus}^{2}(Γ\X, E_{σ}) and Δ_{σ}φ = λφ, then it follows immediately from the description of Δ_{σ,T} [Mu, p. 490] that φ is contained in the domain of Δ_{σ,T} and Δ_{σ,T} φ = λφ. Thus the cuspidal spectrum Spec_{cus}(Δ) of Δ is contained in the spectrum of Δ_{σ,T}. Let E(λ_{i}) be the eigenspace for the eigenvalue λ_{i} of Δ_{σ,T} and let

N_{T,σ}(λ) = ∑_{λ_{i}≤λ} dim E(λ_{i}).  (4.2)

Then, in order to estimate the counting function N_{cus,σ}(λ) of the cuspidal spectrum, it is sufficient to estimate N_{T,σ}(λ). By [Mu, Theorem 3.23], we have

N_{T,σ}(λ) ≤ C(1 + λ^{n/2}), λ ≥ 0,

where n = dim X. The main purpose of this section is to determine explicitly the dependence of C on σ ∈ K̂. To this end, we introduce a covering of Γ\X by special neighborhoods constructed in [BS] and use Neumann bracking as in [Mu]. We briefly recall the definition of these neighborhoods. For details we refer to [BS], [Z]. Let P be a ℚ parabolic subgroup of G with special split component A and Langlands decomposition P = NAM. Let K_{M} = K ∩ M and set X_{M} = M/K_{M}. There is a canonical isomorphism

μ_{P} : N × X_{M} × A ≃ X  (4.3)

and a canonical action of P on N ×X_{M} ×A which is defined as follows: Let p = nam where n ∈ N, a ∈ A and m ∈ M. Then

p · (n_{1}, z, a_{1}) = (nman_{1}a^{−1}m^{−1}, mz, aa_{1}).

With this definition, μ_{P} commutes with the action of P. Set

e(P) = N × X_{M}, Γ_{P} = Γ ∩ P and e′(P) = Γ_{P} \e(P).

There is a canonical fibration

π_{P} : e′(P) → Γ_{M}\X_{M}

with fibre Γ ∩ N\N. Given τ ∈ 𝔞, put

A_{τ} = {a ∈ A | α(log a) > α(τ), α ∈ Ψ_{P} } .

Then for an open subset Ỹ of e(P) and τ ∈ 𝔞, put

W̃(Ỹ, τ) = μ_{P} (Ỹ × A_{τ}).

Now assume that Y ⊂ e′(P) is an open subset and Ỹ its inverse image under the canonical projection e(P) → e′(P). Then W̃(Ỹ, τ) is Γ_{P} – invariant and we put

W (Y, τ) = Γ_{P} \W̃(Ỹ, τ).

Let π: X → Γ\X be the canonical projection. Then, for τ ∈ 𝔞 sufficiently large, we have W (Y, τ) = π(W̃(Ỹ, τ)) and μ_{P} induces an isomorphism

μ′_{P} : Y × A_{τ} ≃ W (Y, τ).

(see [Mu, Lemma 3.16]). An open set in Γ\X of the form W (Y_{P} , τ_{P}) with Y_{P} ⊂ e′(P) a relatively compact open subset and τ_{P} ∈ 𝔞_{P} is called a special neighborhood. Note that for P = G, W (Y_{P} , τ_{P}) = Y_{P} is a relatively compact open subset of Γ\X. Let ω_{M} ⊂ Γ_{M}\X_{M} be a relatively compact open subset and put Y_{P} = π_{P}^{−1}(ω_{M}). Then W̃(Ỹ_{P} , τ_{P}) is N_{P} –invariant and therefore, the cuspidal condition makes sense on W (Y_{P} , τ_{P}) (see [Mu, p. 493]). Hence, for any such W = W (Y_{P} , τ_{P}) we can define the space L_{cusp}^{2}(W, E_{σ}) of cusp forms in L^{2}(W, E_{σ}). To define this space, let P_{F} = N_{F} A_{F} M_{F} , F ⊂ Ψ_{P} , be a ℚ parabolic subgroup that belongs to P, i.e. N_{F} ⊂ N, A_{F} ⊂ A and M_{F} ⊃ M. Given φ ∈ L^{2}(W, E_{σ}), the function

φ^{P_{F}} (x) = ∫_{Γ∩N_{F} \N_{F}} φ(nx) dn

is well defined and belongs to L^{2}(W, E_{σ}). Then

L_{cus}^{2}(W, E_{σ}) = {φ ∈ L^{2}(W, E_{σ}) | φ^{P_{F}} = 0 for all F ⊂ Ψ_{P} } .

Furthermore, let H^{1}(W, E_{σ}) be the Sobolev space of W and let

H_{cus}^{1}(W, E_{σ}) = L_{cus}^{2}(W, E_{σ}) ∩ H^{1}(W, E_{σ}).

In [Mu, Proposition 3.18], we constructed a finite covering of Γ\X by special neighborhoods which is adapted to our purpose. Namely, let Q_{i} = N_{i}A_{i}M_{i} (1 ≤ i ≤ q) be a set of representatives of the Γ–conjugacy classes of ℚ parabolic subgroups of G. Then there exist τ_{i} ∈ 𝔞_{i} and relatively compact open subsets ω_{i} ⊂ Γ_{M_{i}}\X_{M_{i}}, 1 ≤ i ≤ q, such that the following conditions hold:

1) Let Y_{i} = π_{Q_{i}}^{−1}(ω_{i}). Then the canonical projection Γ_{P} \X → Γ\X induces an inclusion W (Y_{i}, τ_{i}) → Γ\X and

{W (Y_{i}, τ_{i})| i = 1, …, q}

is a covering of Γ\X.

2) Set W_{i} = W (Y_{i}, τ_{i}), i = 1, …, q. The map

φ ⟼ (φ|W_{1}, …, φ|W_{q})

defines an embedding

H_{T}^{1}(Γ\X, E_{σ}) ⊂ ⨁_{i=1}^{q} H_{cus}^{1}(W_{i}, E_{σ}).  (4.4)

In general, W_{i} is a manifold with corners. In order to be able to apply Donnelly’s method, we have to smooth out corners. This can be done in a way that preserves the invariance under N_{i}, i.e. the resulting manifold W′_{i} satisfies N_{i}W′_{i} = W′_{i} which gurantees that the cuspidal condition still makes sense for W′_{i}. Now let W be any of the smooth manifolds W′_{i}, i = 1, …, q. Let Δ_{W,σ} be the self–adjoint operator in L^{2}(W, E_{σ}) which is defined by the quadratic form φ ⟼∥ ∇φ ∥^{2} on the Sobolev space H^{1}(W, E_{σ}). Note that this is the self–adjoint extension of the Bochner Laplace operator ∇^{∗}∇ which satisfies Neumann boundary conditions where a section s of E_{σ} over W is said to satisfy Neumann boundary conditions if ∇_{ν}s(x) = 0 for x ∈ ∂W. Here ν is the unit normal vector field along ∂W. Then L_{cus}^{2}(W, E_{σ}) which is defined as above, is an invariant subspace for Δ_{W,σ} and it follows as in [D1] that Δ_{W,σ} has pure discrete spectrum in L_{cus}^{2}(W, E_{σ}). Let N_{W,σ}(λ) be the number of linearly independent cuspidal eigenfunctions of Δ_{W,σ} with eigenvalues less than λ. Using (4.4), it follows by Neumann bracketing that there exists C > 0 such that for all σ ∈ K̂, one has

N_{T,σ}(λ) ≤ C ∑_{i=1}^{q} N_{W′_{i},σ}(λ), λ ≥ 0.  (4.5)

It remains to estimate N_{W,σ}(λ). We have to distinguish two cases. First suppose that W is a relatively compact domain in Γ\X with smooth boundary. This corresponds to a special neighborhood which is defined with respect to Q_{i} = G. Then N_{i} = {0} so that no cuspidal condition has to be imposed. By Corallary 4.5 of [DL] one has

Tr(e^{−tΔ_{W,σ}}) ≤ dim(V_{σ})(1 + C_{1}Vol(W)t^{−n/2})

for 0 < t ≤ T and C_{1} > 0 independent of σ. Let 0 ≤ μ_{1} ≤ μ_{2} ≤ ⋯ be the eigenvalues of Δ_{W,σ}. Then we can estimate the counting function as follows

\begin{aligned}N_{W,σ}(λ) & = ∑_{μ_{i}≤λ} 1 ≤ e ∑_{μ_{i}≤λ} e^{−μ_{i}/λ} ≤ e Tr(e^{−Δ_{W,σ}/λ}) \\ & ≤ e dim(V_{σ})(1 + C_{2}λ^{n/2}) \end{aligned}

Let λ_{σ} = σ(Ω_{K}) be the eigenvalue of the Casimir operator Ω_{K} in V_{σ}. By Weyl’s dimension formula there exists C_{3} > 0 such that

dim(V_{σ}) ≤ C_{3}(1 + λ_{σ}^{l})  (4.6)

for all σ ∈ K̂ where l ≤ \frac{1}{2} dim K. Thus we proved Proposition 4.1. Let W ⊂ Γ\X be a compact submanifold having a smooth boundary. Let (σ, V_{σ}) be an irreducible unitary representation of K with Casimir eigenvalue λ_{σ}. Let Δ_{W,σ} be the Bochner Laplace operator ∇^{∗}∇ on C^{∞}(W, E_{σ}|W) with Neumann boundary conditions and let N_{W,σ}(λ) be the eigenvalue counting function for Δ_{W,σ}. There exists C > 0 such that for every σ ∈ K̂ one has

N_{W,σ}(λ) ≤ C(1 + λ_{σ}^{p} + λ^{n}), λ ≥ 0,  (4.7)

where n = dim X and p = dim K. Next we have to consider one of the noncompact special neighborhoods W′_{i}. To estimate the counting function in this case, we proceed as in [D1] using Theorem 3.1 and keeping track of the constants. We provide some details. Let P be a fixed proper ℚ parabolic subgroup of G with special split component A_{P} and Langlands decomposition P = N_{P} A_{P} M_{P} . Let W = W (Y_{P} , τ_{P}) be a special neighborhood with respect to P. Thus τ_{P} ∈ 𝔞_{P} and Y_{P} = π_{P}^{−1}(ω_{P}) where ω_{P} ⊂ Γ_{M_{P}} \X_{M_{P}} is a relatively compact open subset. Then W ⊂ Γ_{P} \X. First we construct an interior parametrix for the Neumann heat kernel on W′ satisfying the cuspidal conditions. The interior parametrix is obtained by restricting the corresponding heat kernel on Γ_{P} \X to W. Let H_{σ}(t, x, y) be the heat kernel for Δ̃_{σ} on X. Then the fundamental solution F_{σ}(t, x, y) for Δ_{σ} on Γ_{P} \X is given by

F_{σ}(t, x, y) = ∑_{γ∈Γ_{P}} H_{σ}(t, x, γy).  (4.8)

By Proposition 3.2, we have

∑_{γ∈Γ_{P}} |H_{σ}(t, x, γy)| ≤ Ct^{−n/2} ∑_{γ∈Γ_{P}} e^{−ρ^{2}(x,γy)/4t}

By [Wa1, Theorem 9] this series is uniformly convergent on compact subsets. Using Theorem 3.1 and (3.14), the same holds for all derivatives up to order 1. Thus, F_{σ}(t, x, y) is a smooth kernel of (t, x, y) ∈ ℝ^{+} × X × X which represents exp(−tΔ_{σ}) on Γ_{P} \X. Now we have to modify F_{σ}(t, x, y) by projecting onto the cuspidal subspace. Let Ψ be the set of simple roots of (P,A). Given any subset θ ⊂ Ψ, let P_{θ} = N_{θ}A_{θ}M_{θ} be the ℚ parabolic subgroup belonging to P which is defined by θ. A smooth section of E_{σ} over Γ_{P} \X may be identified with a C^{∞}–function φ: Γ_{P} \G → V_{σ} satisfying

φ(gk) = σ(k)^{−1}φ(g), g ∈ G, k ∈ K.

Set

(T (P, θ)φ)(g) = ∫_{Γ∩N_{θ}\N_{θ}} φ(ng) dn.

Then T (P, θ)φ is left invariant under Γ_{P} · N_{θ}. In particular, T (P, θ) is a projection on C^{∞}(Γ_{P} \X, E_{σ}). For any subset θ ⊂ Ψ set

L_{θ} = T (P, Ψ − θ).

and let

\begin{aligned}\overline{F}_{σ}(t, x, y) & = (∏_{α∈ψ} (1 − L_{α}(y))) F_{σ}(t, x, y) \\ & = ∑_{θ⊂Ψ}(−1)^{|θ|}L_{θ}(y)F_{σ}(t, x, y). \end{aligned}  (4.9)

Here the projectors L_{θ}(y) act on the third argument y of F_{σ}(t, x, y). By Lemma 5.1 in [D1] together with the isometry invariance of H_{σ}(t, x, y) it follows that

L_{θ}(x)\overline{F}_{σ}(t, x, y) = 0, θ ⊂ Ψ.

Thus \overline{F}_{σ}(t, x, y) is the fundamental solution for Δ_{σ} on Γ_{P} \X satisfying the cuspidal conditions. The heat kernel \overline{F}_{σ}(t, x, y) can be estimated as in [D1]. Let log: A_{P} →𝔞_{P} be the inverse of the exponential map. Let pr_{A_{P}} : N_{P} × X_{M_{P}} × A_{P} → A_{P} be the projection and let r_{P} : X → 𝔞_{P} be defined by

r_{P} (x) = log (pr_{A_{P}} (μ_{P}^{−1}(x)))  (4.10)

where μ_{P} is the isomorphism (4.3). Furthermore, for x, y ∈ Γ_{P} \X we set

ρ_{N}(x, y) = inf_{n_{1},n_{2}∈N} ρ(xn_{1}, yn_{2}).

Thus ρ_{N}(x, y) is the distance of the N-orbits through x and y, respectively. In other words, if we denote by π : Γ_{P} \X → Γ_{P} N\X the canonical projection, then ρ_{N}(x, y) is the geodesic distance of π(x), π(y). Then, with a slight modification, the analogous statement to Lemma 5.5 in [D1] is Lemma 4.2. For any fixed simple root α, let

F_{σ,α}(t, x, y) = (1 − L_{α}(y)) F_{σ}(t, x, y).

Let T > 0 be fixed. There exists C > 0 such that for every σ ∈ K̂ and 0 < t ≤ T one has

|F_{σ,α}(t, x, y)| ≤ C(1 + λ_{σ})e^{tλ_{σ}}t^{−n/2} min (e^{−α(r(x))}, e^{−α(r(y))})

× max (∏_{β∈Φ} e^{n_{β}β(r(x))}, ∏_{β∈Φ} e^{n_{β}β(r(y))}) .

× max_{\begin{matrix}ν∈Ψ \\ n_{ν}=1 \end{matrix}} max (ν(r(x)), ν(r(y))) exp (−\frac{ρ_{N}^{2}(x, y)}{32t}) ,

uniformly for x, y in any Siegel domain 𝔖_{c}(P). Here λ_{σ} = σ(Ω_{K}) and r = r_{P}. Moreover, n_{β} is the dimension of the root space 𝔫_{β} and n_{ν} is the multiplicity of the simple root ν.

Proof. We identify x, y ∈ Γ_{P} \X with points x, y ∈ X which realize the geodesic distance from x to y. Let ψ = Ψ − {α}. Then by definition

F_{σ,α}(t, x, y) = ∑_{γ∈Γ_{P}} {H_{σ}(t, x, γy) − ∫_{Γ∩N_{ψ}\N_{ψ}} H_{σ}(t, nx, γy) dn} .

where we have identified Γ ∩ N_{ψ}\N_{ψ} with a fundamental domain of Γ ∩ N_{ψ} in N_{ψ}. Using (3.15), we can estimate the term coming from γ = 1. This implies

\begin{aligned}|F_{σ,α}(t, x, y) − ∑_{\begin{matrix}γ∈Γ_{P} \\ γ≠1 \end{matrix}}\{H_{σ}(t, x, γy) − ∫_{Γ∩N_{ψ}\N_{ψ}} & H_{σ}(t, nx, γy) dn\}| \\ & ≤ Ct^{−n/2}e^{−ρ^{2}(x,y)/4t}. \end{aligned}

Now use the mean value theorem combined with (3.12) and (3.14) to estimate the terms with γ ≠ 1. Then we get

|F_{σ,α}(t, x, y)| ≤ C_{2}(1 + λ_{σ})e^{tλ_{σ}}t^{−n/2} ∑_{\begin{matrix}γ∈Γ_{P} \\ γ≠1 \end{matrix}} (1 + \frac{ρ_{N}(x, γy)}{t}) e^{−ρ_{N}^{2} (x,γy)/8t}

× min(diam(x), diam(y)) + C_{1}t^{−n/2}e^{−ρ_{N}^{2} (x,y)/4t}.

Here diam(x) is the diameter of Γ∩N_{ψ}\N_{ψ} at x. As in [D1], this implies the lemma.

□

Since the projections L_{β}(y) are L^{∞}–bounded, it follows from (4.9) that

|\overline{F}_{σ}(t, x, y)| ≤ C(1 + λ_{σ})e^{tλ_{σ}}t^{−n/2} min_{α∈Ψ} min (e^{−α(r(x))/2}, e^{−α(r(y))/2})

× max (∏_{β∈Φ} e^{n_{β}β(r(x))}, ∏_{β∈Φ} e^{n_{β}β(r(y))}) e^{−ρ_{N}^{2} (x,y)/32t}. (4.11)

uniformly for x, y ∈ 𝔖_{c}(P), 0 < t ≤ T, and σ ∈ K̂. Now let W = W (Y_{ρ}, τ_{ρ}) be a special neighborhood as above and let W′ be obtained from W by smoothing out corners such that NW′ = W′. We use \overline{F}_{σ}(t, x, y) as an interior parametrix and apply the single layer potential construction to modify \overline{F}_{σ}(t, x, y) as in [D1, section 6], [F, Chapt. 5], yielding a fundamental solution Q_{σ}(t, x, y) for the heat equation \frac{∂}{∂t} + Δ_{σ} on W′ which satisfies Neumann boundary conditions and cuspidal interior conditions. The kernel Q_{σ}(t, x, y) can be estimated similarly to (4.11):

|Q_{σ}(t, x, y)| ≤ C_{1}(1 + λ_{σ})e^{tλ_{σ}}t^{−n/2} min_{α∈Ψ} min (e^{−α(r(x))/2}, e^{−α(r(y))/2})

× max (∏_{β∈Φ} e^{n_{β}β(r(x))}, ∏_{β∈Φ} e^{n_{β}β(r(y))}) exp (−\frac{C_{2}ρ_{N}^{2}(x, y)}{t})

× (1 + C_{3} exp (−C_{4}(ρ^{2}(x, ∂W′) + ρ^{2}(y, ∂W′))/t))

(4.12)

for 0 < t ≤ T. However, since ρ_{N} and not ρ occurs in the estimate (4.12), it is not obvious that Q_{σ}(t, x, y) satisfies Neumann boundary conditions. In order to verify that it does satisfy Neumann boundary conditions, we proceed as follows. First we take the full heat kernel F_{σ}(t, x, y) and employ the single layer potential construction to get the fundamental solution F_{σ,N}(t, x, y) for the heat equation \frac{∂}{∂t} + Δ_{σ} on W′ which satisfies Neumann boundary conditions. Let \overline{F}_{σ,N}(t, x, y) be defined by (4.9) with F_{σ} replaced by F_{σ,N}. Since the boundary of W′ fibres through N ∩ Γ\N, it follows from the iterative construction of the boundary correction term that

Q_{σ}(t, x, y) = \overline{F}_{σ,N}(t, x, y)

which shows that Q_{σ}(t, x, y) indeed satisfies Neumann boundary conditions. This implies Theorem 4.3. Let W′ be as above. Given an irreducible unitary representation (σ, V_{σ}) of K, let Δ_{W′,σ} be the Bochner Laplace operator with Neumann boundary conditions and cuspidal interior conditions on W′.

Then exp(−tΔ_{W′,σ}) is a trace class operator for t > 0 and there exists C > 0 such that for every (σ, V_{σ}) we have

Tr(e^{−tΔ_{W′,σ}}) ≤ C(1 + λ_{σ})e^{tλ_{σ}} dim(V_{σ})(1 + t^{−n/2})

for 0 < t ≤ T.

Proof. By construction, Q_{σ}(t, x, y) represents the heat semigroup exp(−tΔ_{W′,σ}). Hence, using (4.12) we get

Tr(e^{−tΔ_{W′,σ}}) = ∫_{W′} Tr Q_{σ}(t, x, x)dx ≤ dim(V_{σ}) ∫_{W′} |Q_{σ}(t, x, x)| dx

≤ C_{5}(1 + λ_{σ})e^{tλ_{σ}} dim(V_{σ})t^{−n/2} ∫_{r(W′)} min_{α∈Ψ}(e^{−α(r)/2}) dr

≤ C(1 + λ_{σ}) dim(V_{σ})e^{tλ_{σ}}t^{−n/2}

for 0 < t ≤ T.

□

Corollary 4.4. Let N_{W′,σ}(λ) be the counting function for Δ_{W′,σ}. There exists C > 0 such that

N_{W′,σ}(λ) ≤ C(1 + λ_{σ}^{m/2+1} + λ^{m/2+1}), σ ∈ K̂,

where λ_{σ} = σ(Ω_{K}) and m = dim G.

Proof. By Theorem 4.3, we have

Tr (e^{−t(Δ_{W′,σ}+λ_{σ})}) ≤ C(1 + λ_{σ}) dim(V_{σ})(1 + t^{−n/2}) (4.13)

for 0 < t ≤ 1. Let 0 ≤ μ_{1} ≤ μ_{2} ≤ ⋯ be the eigenvalues of Δ_{W′,σ} where each eigenvalue is repeated according to its multiplicity. Using (4.13) together with (4.6), we get

N_{W′,σ}(λ) = ∑_{μ_{i}≤λ} 1 = ∑_{μ_{i}+λ_{σ}≤λ+λ_{σ}} 1

≤ e ∑_{μ_{i}+λ_{σ}≤λ+λ_{σ}} e^{−(μ_{i}+λ_{σ})/(λ+λ_{σ})}

≤ e Tr (e^{−(Δ_{W′,σ}+λ_{σ})/(λ+λ_{σ})})

≤ C_{1}(1 + λ_{σ}) dim(V_{σ})(1 + (λ + λ_{σ})^{n/2})

≤ C_{2}(1 + λ_{σ}^{m/2+1} + λ^{m/2+1})

for λ ≥ 0 and m = dim G.

□

Combining (4.5), Proposition 4.1 and Corollary 4.4, we obtain Theorem 4.5. Let N_{T,σ}(λ) be the eigenvalue counting function for the cut–off Laplacian Δ_{σ,T} . There exists C > 0 such that for every σ ∈ K̂ one has

N_{T,σ}(λ) ≤ C(1 + λ_{σ}^{m/2+1} + λ^{m/2+1}), λ ≥ 0,

where λ_{σ} = σ(Ω_{K}) and m = dim G. Since the cuspidal spectrum Spec_{cus}(Δ_{σ}) of Δ_{σ} is contained in the spectrum of Δ_{σ,T} this implies Corollary 4.6. Given any irreducible unitary representation (σ, V_{σ}) of K, let N_{cus,σ}(λ) be the number of eigenvalues, counted with multiplicity, less than λ of the Bochner Laplace operator Δ_{σ} = ∇^{∗}∇, acting in the space of cusp forms L_{cus}^{2}(Γ\G, σ). There exists C > 0 such that for every σ ∈ K̂:

N_{cus,σ}(λ) ≤ C(1 + λ_{σ}^{m/2+1} + λ^{m/2+1}), λ ≥ 0,

where m = dim G.

5. The residual spectrum

Let L_{res}^{2}(Γ\G) be the orthogonal complement of the space cusp forms L_{cus}^{2}(Γ\G) in L_{d}^{2}(Γ\G). This is the residual subspace. Thus the discrete spectrum has a further decomposition which is given by

L_{d}^{2}(Γ\G) = L_{cus}^{2}(Γ\G) ⊕ L_{res}^{2}(Γ\G).  (5.1)

Let (σ, V_{σ}) be any irreducible unitary representation of K. Then it follows that L_{d}^{2}(Γ\X, E_{σ}) has a corresponding decomposition. Let

L_{res}^{2}(Γ\G, σ) = (L_{res}^{2}(Γ\G) ⊗ V_{σ})^{K}.  (5.2)

This space can be identified with a subspace of L_{d}^{2}(Γ\X, E_{σ}) which we denote by L_{res}^{2}(Γ\X, E_{σ}). Thus (5.1) induces the decomposition

L_{d}^{2}(Γ\X, E_{σ}) = L_{cus}^{2}(Γ\X, E_{σ}) ⊕ L_{res}^{2}(Γ\X, E_{σ}).  (5.3)

To complete the proof of Theorem 0.2, we have to estimate the number of eigenvalues of Δ_{σ} in the residual subspace L_{res}^{2}(Γ\X, E_{σ}). To this end we use Langlands’ description of the residual spectrum [L1] and proceed as in [Mu]. We may simply repeat the arguments given in [Mu] keeping track of the constants at each step. For the convenience of the reader we include some details. To begin with we recall some facts from Langlands’ theory of Eisenstein series [L1], [H]. Let P be any ℚ–parabolic subgroup of G with special split component A_{P} and Langlands decomposition P = N_{P} A_{P} M_{P} . Let K_{M} = K ∩ M and let σ_{M}: K_{M} → GL(V_{σ}) be the restriction of σ to K_{M}. Given a character χ: Z(𝔪) → ℂ, set

L_{cus}^{2}(Γ_{M}\M, σ, χ) =

{φ ∈ L_{cus}^{2}(Γ_{M}\M, σ_{M}) | Dφ = χ(D)φ for all D ∈ Z(𝔪)}. (5.4)

This is a finite dimensional Hilbert space of automorphic forms. Let 𝔞^{∗} be the dual of the Lie algebra 𝔞 and let

(𝔞^{∗})^{+} = {λ ∈ 𝔞^{∗} | ⟨λ, α⟩ > 0 for all α ∈ Ψ_{P} } .

Let Φ ∈ L_{cus}^{2}(Γ_{M}\M, σ, χ) be given. Then we extend Φ to a function Φ: Γ_{P} · N_{P} \G → V_{σ} by

Φ(namk) = σ(k)^{−1}Φ(m).  (5.5)

Let Λ ∈ 𝔞_{ℂ}^{∗} and suppose that Re(Λ) ∈ ρ_{P} + (𝔞^{∗})^{+}. Then the Eisenstein series attached to Φ is defined as

E(P |A, Φ, Λ, g) = ∑_{γ∈Γ_{P} \Γ} e^{(Λ+ρ_{P})(H_{P} (γg))}Φ(γg).

This series is absolutely and uniformly convergent on compact subsets of (ρ_{P} +(𝔞^{∗})^{+} +\sqrt{−1}𝔞^{∗})×G. As a function of Λ, it can be continued to a meromorphic function of Λ ∈ 𝔞_{ℂ}^{∗} and all singularities of E(P |A, Φ, Λ) lie along hyperplanes. Let P_{i}, i = 1, 2, be two ℚ parabolic subgroups of G with special split components A_{i} and Langlands decomposition P_{i} = N_{i}A_{i}M_{i}. Then (P_{1}, A_{1}) and (P_{2}, A_{2}) are said to be associate if there exists g ∈ G(ℚ) such that Ad(g)𝔞_{1} = 𝔞_{2}. The set of all such isomorphisms is denoted by W (𝔞_{1}, 𝔞_{2}). Set W (𝔞) = W (𝔞, 𝔞). Let χ ∈ Ẑ(𝔪_{1}) and Φ ∈ L_{cus}^{2}(Γ_{M_{1}}\M_{1}, σ, χ). We denote the constant term of E(P_{1}|A_{1}, Φ, Λ) along P_{2} by E^{P_{2}}(P_{1}|A_{1}, Φ, Λ). If (P_{1}, A_{1}) and (P_{2}, A_{2}) are not associate and rankP_{2} ≥ rankP_{1}, then

E^{P_{2}}(P_{1}|A_{1}, Φ, Λ) = 0.

If (P_{1}, A_{1}) and (P_{2}, A_{2}) are associated, then

E^{P_{2}}(P_{1}|A_{1}, Φ, Λ, x) = ∑_{w∈W (𝔞_{1},𝔞_{2})}e^{(wΛ+ρ_{2})(H_{2}(x))}(c_{P_{2}|P_{1}}(w, Λ)Φ)(x)(5.6)

where ρ_{2} = ρ_{P_{2}}, H_{2} = H_{P_{2}} and

c_{P_{2}|P_{1}}(w, Λ): L_{cus}^{2}(Γ_{M_{1}}\M_{1}, σ, χ) → L_{cus}^{2}(Γ_{M_{2}}\M_{2}, σ, ^{w}χ)

is a linear operator which is a meromorphic function of Λ ∈ 𝔞_{ℂ}^{∗}. This operator is called an intertwining operator.

It is convenient to combine the intertwining operators into matrices of such operators. If O ∈ Ẑ(𝔪)/W (A), put

L_{cus}^{2}(Γ_{M}\M, σ, O) = ⨁_{χ∈O} L_{cus}^{2}(Γ_{M}\M, σ, χ).

Furthermore, the relation of association is an equivalence relation, hence breaks up the set of all ℚ–parabolic subgroups of G into equivalence classes. Fix one such class C. Choose a complete set of representatives P_{1}, …, P_{r} of the G(ℚ) conjugacy classes C_{1}, …, C_{r} in C. For each 1 ≤ i ≤ r, let P_{il}, l = 1, …, r_{i}, be a complete set of representatives of the Γ–conjugacy classes in C_{i}. Then

{P_{il} | 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}}

is a set of representatives for C/Γ. Let y_{il} ∈ K be such that

P_{il} = y_{il}P_{i}y_{il}^{−1}.

Let A_{i} be a split component of P_{i}, 1 ≤ i ≤ r, and set A_{il} = y_{il}A_{i}y_{il}^{−1}. Then A_{il} is a split component of P_{il}. Let P_{il} = N_{il}A_{il}M_{il} be the Langlands decomposition of P_{il}. Let

O = {O_{il} | 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}}  (5.7)

be a set of associate orbits with O_{il} ∈ Ẑ(𝔪_{il})/W (A_{il}). For each 1 ≤ i ≤ r, let O_{i} = {O_{il} | 1 ≤ l ≤ r_{i}} and set

E_{cus}(σ, O_{i}) = ⨁_{l} L_{cus}^{2} (Γ_{M_{il}}\M_{il}, σ, O_{il}) .  (5.8)

Given w ∈ W (𝔞_{i}, 𝔞_{j}) and Λ ∈ 𝔞_{i,ℂ}^{∗}, define C_{ji}(w, Λ_{i}) to be the matrix

(c_{P_{jk}|P_{il}}(y_{jk}wy_{il}^{−1},^{y_{il}}Λ))

where 1 ≤ l ≤ r_{i}, 1 ≤ k ≤ r_{j}. This matrix defines a linear operator

C_{ji}(w, Λ): E_{cus}(σ, O_{i}) → E_{cus}(σ, O_{j}).  (5.9)

For Φ_{i} = {Φ_{il} | 1 ≤ l ≤ r_{i}} ∈ E_{cus}(σ, O_{i}) and Λ ∈ 𝔞_{i,ℂ}^{∗}, define

E(P_{i}, Φ_{i}, Λ) = ∑_{l=1}^{r_{i}} E(P_{il}|A_{il}, Φ_{il}, ^{y_{il}}Λ).  (5.10)

Then all Eisenstein series E(P_{i}, Φ_{i}, Λ), 1 ≤ i ≤ r, Φ_{i} ∈ E_{cus}(σ, O_{i}) and intertwining operators C_{ji}(w, Λ), 1 ≤ i, j ≤ r, w ∈ W (𝔞_{i}, 𝔞_{j}) are meromorphic functions of Λ ∈ 𝔞_{ℂ}^{∗} all of whose singularities lie along hyperplanes. Moreover, the following functional equations hold

E(P_{i}, Φ_{i}, Λ) = E(P_{j}, C_{ji}(w, Λ)Φ_{i}, wΛ)
C_{ji}(w_{1}w_{2}, Λ) = C_{jk}(w_{1}, w_{2}Λ)C_{ki}(w_{2}, Λ)

for 1 ≤ i, j, k ≤ r, w ∈ W (𝔞_{i}, 𝔞_{j}), w_{1} ∈ W (𝔞_{j}, 𝔞_{k}) and w_{2} ∈ W (𝔞_{k}, 𝔞_{i}). For the proof of these statements see [L1] or [H, Theorems 8,9]. Using the inner product formula for truncated Eisenstein series [L2], we obtain Proposition 5.1. The singular hyperplanes of any E(P_{i}, Φ, Λ), 1 ≤ i ≤ r, Φ ∈ E(σ, O_{i}), are contianed in the union of the singular hyperplanes of C_{ji}(w, Λ): E_{cus}(σ, O_{i}) → E_{cus}(σ, O_{j}), w ∈ W (𝔞_{i}, 𝔞_{j}), 1 ≤ i, j ≤ r. Next we recall the iterated residue construction. Let P = NAM be a ℚ–parabolic subgroup of G. Given α ∈ Φ_{P} and c ∈ ℝ, set

H(α, c) = {Λ ∈ 𝔞_{ℂ}^{∗} | ⟨Λ, α⟩ = c} .

An affine subspace H ⊂ 𝔞_{ℂ}^{∗} is called admissible if H is the intersection of such hyperplans. Suppose that H_{1} ⊃ H_{2} are two admissible affine subspaces of 𝔞_{ℂ}^{∗} such that H_{2} is of codimension one in H_{1}. Let F (Λ) be a meromorphic function on H_{1} whose singularities lie along hyperplans which are admissible subspaces of 𝔞_{ℂ}^{∗}. Choose a real unit vector Λ_{0} in H_{1} which is normal to H_{2}. Then we can define a meromorphic function Res_{H_{2}}F on H_{2} by

Res_{H_{2}}F (Λ) = \frac{δ}{2πi} ∫_{0}^{1} F (Λ + δe^{2πiθ}Λ_{0})dθ

where δ > 0 is sufficiently small so that F (Λ+zΛ_{0}) has no singularities for 0 < |z| < 2δ. The singularities of Res_{H_{2}}F lie on the intersections of H_{2} with the singular hyperplanes of F that are different from H_{2}. Now consider a complete flag

𝔞_{ℂ}^{∗} = H_{p} ⊃ H_{p−1} ⊃ ⋯ ⊃ H_{1} ⊃ H_{0} = {Λ}, p = dim 𝔞^{∗},

of affine admissible subspaces of 𝔞_{ℂ}^{∗} and let Λ_{i} ∈ H_{i} be a real unit vector which is normal to H_{i−1}, i = 1, …, p. We call F = {H_{i}, Λ_{i}} an admissible flag. Let F be a meromorphic function on 𝔞_{ℂ}^{∗} whose singularities lie along admissible hyperplanes of 𝔞_{ℂ}^{∗}. Then we define F_{i} inductively by

F_{p} = F, F_{i} = Res_{H_{i}}F_{i+1}, i = 0, …, p − 1.

Set

Res_{F}F = F_{0}.

This is the iterated residue of F.

Let χ ∈ Ẑ(𝔪) and Φ ∈ L_{cus}^{2}(Γ_{M}\M, σ, χ). It follows from [H, Lemma 118] that all singular hyperplanes of E(P |A, Φ, Λ) are admissible. Let H(α_{i}, c_{i}), i = 0, …, p − 1, p = dim 𝔞, be a set of singular hyperplanes of E(P |A, Φ, Λ) such that ∩_{i=0}^{p−1}H(α_{i}, c_{i}) = {Λ_{0}}. Set H_{i} = ∩_{j≥i}H(α_{i}, c_{i}), i = 0, …, p − 1, and H_{p} = 𝔞_{ℂ}^{∗}. Choose real unit vectors Λ_{i} ∈ H_{i} normal to H_{i−1}. Then F = {H_{i}, Λ_{i}} is an admissible flag. Let φ ∈ C_{c}^{∞}(𝔞) and let φ̂(Λ) be its Fourier transform. This is a holomorphic function on 𝔞_{ℂ}^{∗}. Put

Ψ = Res_{F}[E(P |A, Φ, Λ)φ̂(Λ)].  (5.11)

It is clear that Ψ depends only on the derivatives of φ̂ at Λ_{0}. Let ^{+}𝔞^{∗} be the positive cone in 𝔞^{∗} spanned by the simple roots of (P, A). If Λ_{0} ∈ ^{+}𝔞^{∗} then Ψ is square integrable and satisfies [H, Lemma 118]

R(Ω)Ψ = (|Λ_{0}|^{2} − |ρ_{P} |^{2} + χ(Ω_{M})) Ψ.

By (3.4) it follows that Ψ is also an eigenfunction of Δ_{σ}:

Δ_{σ}Ψ = (|ρ_{P} |^{2} − |Λ_{0}|^{2} − χ(Ω_{M}) + λ_{σ}) Ψ  (5.12)

and by construction, Ψ is not a cusp form. This implies that Ψ is contained in L_{res}^{2}(Γ\X, E_{σ}). The main statement of Langlands’ theory asserts that this construction gives all of the residual spectrum. More precisely, we have [L1, Theorem 7.1] Theorem 5.2. The residual subspace L_{res}^{2}(Γ\X, E_{σ}) is spanned by iterated residues of the form (5.11) at singular points Λ_{0} ∈ {Λ ∈ +𝔞_{P}^{∗} | |Λ| ≤ |ρ_{P} |} where P = NAM runs over a set of representatives of the Γ–conjugacy classes of proper ℚ–parabolic subgroups of G, Φ ∈ L_{cus}^{2}(Γ_{M}\M, σ, χ), χ ∈ Ẑ(𝔪) and φ ∈ C_{0}^{∞}(𝔞). Thus each association class C of proper ℚ–parabolic subgroups defines a subspace L_{res}^{2}(Γ\X, E_{σ}; C) of L_{res}^{2}(Γ\X, E_{σ}) consisting of all iterated residues associated to parabolic groups P ∈ C. In this way, we get a decomposition

L_{res}^{2}(Γ\X, E_{σ}) = ⨁_{C} L_{res}^{2}(Γ\X, E_{σ}; C)

where C runs over associtation classes of proper ℚ–parabolic subgroups of G. Let N_{C,σ}(λ) be the eigenvalue counting function of Δ_{σ} acting in the space L_{res}^{2}(Γ\X, E_{σ}; C). It follows from Theorem 5.2 that in order to estimate N_{C,σ}(λ), it is sufficient to count the number of iterated residues associated to C with eigenvalues less than λ. Let O = {O_{il}} be a set of associated orbits as in (5.7). Then −χ(Ω_{M}) has the same value μ(O) for all χ ∈ O_{il}, 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}. Since the singular points Λ_{0} that contribute to the residual spectrum satisfy |Λ_{0}| ≤ |ρ_{P} |, it follows from 5.12 that the eigenvalue of an iterated residue 5.11 with Φ ∈ L_{cus}^{2}(Γ_{M_{il}}\M_{il}, O_{il}) is bounded from below by μ(O) + λ_{σ}. Thus it remains to count all intersections of singular hyperplanes of C_{ji}(w, Λ): E_{cus}(σ, O_{i}) → E_{cus}(σ, O_{j}), w ∈ W (𝔞_{i}, 𝔞_{j}), such that the intersection consists of a single point which lies in {Λ ∈ ^{+}𝔞_{i}^{∗} | |Λ| ≤ |ρ_{P_{i}}|}, together with the orders of the singularities of C_{ji}(w, Λ) along the corresponding singular hyperplanes. More precisely, let 𝔥 = {H_{1}, ..., H_{p}}, p = dim 𝔞_{i}, be a set of singular hyperplanes of C_{ji}(w, Λ) such that

⋂_{k} H_{k} = {Λ_{0}} and Λ_{0} ∈ {Λ ∈ ^{+}𝔞_{i}^{∗} | |Λ| ≤ |ρ_{P_{i}}|}. (5.13)

Let m(H_{k}) denote the order of the pole of C_{ji}(w, Λ) at H_{k} and put

m(𝔥) = ∑_{k=1}^{p} m(H_{k}).

Let S_{ji}(w) be the set of all systems 𝔥 of singular hyperplanes of C_{ji}(w, Λ) such that (5.13) holds. Put

m_{ji}(w, σ, O) = ∑_{𝔥∈S_{ji}(w)}m(𝔥)

and

m(σ, O) = ∑_{i,j} ∑_{w∈W (𝔞_{i},𝔞_{j})}m_{ji}(w, σ, O).

Then by Proposition 5.1 we get

N_{C,σ}(λ) ≤ ∑_{i,l} ∑_{\begin{matrix}O \\ μ(O)+λ_{σ}≤λ \end{matrix}} m(σ, O) dim(L_{cus}^{2}(Γ_{M_{il}}\M_{il}, σ, O_{il})).(5.14)

First, we estimate the dimension of the spaces of cusp forms that enter (5.14). Given an irreducible unitary representation (τ, V_{τ}) of K_{M}, let [σ: τ] = dim(V_{σ} ⊗ V_{τ})^{K_{M}} . Then

σ_{M} = ⨁_{τ∈K̂_{M}}[σ: τ]τ.

Let Ω_{K_{M}} ∈ Z(𝔨_{M}) be the Casimir element of K_{M}. For each τ ∈ K̂_{M}, one has

τ(Ω_{K_{M}}) = λ_{τ}Id,

where λ_{τ} ≥ 0. Let P_{τ} be the orthogonal projection of V_{σ} onto V_{τ}. Then we have

σ(Ω_{K_{M}}) = ∑_{τ∈K̂_{M}}[σ: τ]λ_{τ}P_{τ}.

Next note that Ω_{K} = Ω_{K_{M}} + Ω′ where Ω′ = − ∑_{i=1}^{q} Y_{i}^{2} and Y_{1}, …, Y_{q} is an orthonormal basis of 𝔨_{M}^{⊥}. Thus σ(Ω′) is a symmmetric nonnegative operator in V_{σ}. Since σ(Ω_{K}) = λ_{σ}Id, this implies

λ_{σ} ≥ λ_{τ} for all τ ∈ K̂ with [σ: τ] > 0.  (5.15)

Note that μ(O) + λ_{τ} is an eigenvalue of the Bochner Laplace operator in L^{2}(Γ_{M}\X_{M}, E_{τ}). Hence by Corollary 4.6 and (5.15), we get

\begin{aligned}∑_{i,l} ∑_{\begin{matrix}O \\ μ(O)+λ_{σ}≤λ \end{matrix}} & dim(L_{cus}^{2}(Γ_{M_{il}}\M_{il}, σ, O_{il})) \\ & ≤ C dim(V_{σ})(1 + λ_{σ}^{m_{1}} + λ^{m_{1}}) \\ & ≤ C(1 + λ_{σ}^{q} + λ^{q}) \end{aligned}  (5.16)

where m_{1} = \frac{1}{2} dim M_{il} + 1, q = \frac{1}{2} dim M_{il} + \frac{1}{2} dim K + 1 and C > 0 is independent of σ. It remains to bound m(σ, O), i.e. the number of intersections of real singular hyperplanes of the intertwining operators C_{ji}(w, Λ) that are reduced to a point in the bounded domain {Λ ∈ ^{+}𝔞^{∗} | |Λ| ≤ |ρ_{P} |}. The first step is the reduction to rank one parabolic groups. We proceed as in [Mu, section 8], using that the intertwining operators for higher rank parabolic subgroups can be expressed as products of intertwining operators of rank one parabolic subgroups of the Levi components of the higher rank groups (see [H, chapter V]). Let C be an association class of ℚ–parabolic subgroups of G. Let P = NAM be any element in C. Denote by L the set of Weyl chambers in 𝔞. There is a one-to-one correspondence between C/G(ℚ) and L/W (A) (cf. [H, chapter V, section 4]). To each C ∈ L, one can associate a unique P_{C} ∈ L with P_{C} = N_{C}AM. Let L = {C_{1}, …, C_{q}} and set P_{i} = P_{C_{i}}, i = 1, …, q. This set contains a set of representatives for C/G(ℚ). Let C_{i} be the G(ℚ)–conjugacy class containing P_{i}, 1 ≤ i ≤ q. Note that all P_{i} have the same split component A. We choose a set of representatives P_{il}, 1 ≤ l ≤ q_{i}, for the Γ–conjugacy classes in C_{i}.

Let y_{il} ∈ G(ℚ) such that P_{il} = y_{il}P_{i}y_{il}^{−1} and A_{il} = y_{il}Ay_{il}^{−1}. Then A_{il} is a split component of P_{il}. Let P_{il} = N_{il}A_{il}M_{il} be the corresponding Langlands decomposition. Let O = {O_{il} | 1 ≤ i ≤ q, 1 ≤ l ≤ q_{i}} be a set of associate orbits where O_{il} ∈ Ẑ(𝔪_{il})/W (A_{il}). Then for each w ∈ W (𝔞), we can define intertwining operators

C_{ji}(w, Λ): E_{cus}(σ, O_{i}) → E_{cus}(σ, O_{j})

as above. As explained in Chapter V, section 4 of [H], the functional equation implies that there exists k, 1 ≤ k ≤ q, such that

C_{ji}(w, Λ) = C_{jk}(1, wΛ)C_{ki}(w, Λ)

and C_{ki}(w, Λ) is entire. Hence, we only have to consider C_{ji}(1, Λ). Furthermore, by Lemma 117 in [H] there exists a sequence i = i_{1}, …, i_{p} = j, 1 ≤ i_{l} ≤ q, p ≤ q, such that the chambers C_{i_{l}} and C_{i_{l+1}} are adjacent for all l = 1, …, p − 1 and

C_{ji}(1, Λ) = C_{i_{p}i_{p−1}}(1, Λ) ⋯ C_{i_{3}i_{2}}(1, Λ)C_{i_{2}i_{1}}(1, Λ).  (5.17)

Therefore, it suffices to bound the number of singular hyperplanes of C_{ji}(1, Λ) for adjacent chambers C_{i} and C_{j}. In the proof of Lemma 116 in [H] it is shown that for adjacent chambers, the entries of C_{ji}(w, Λ) can be expressed in terms of intertwining operators for rank one parabolic groups. Assume that i = 1, and j = 2. Since C_{1} and C_{2} are adjacent, there exists a unique ℚ–parabolic subgroup P′ = N′A′M′ of G such that (P′, A′) dominates (P_{1}, A) and (P_{2}, A), i.e. P′ ⊃ P_{i} and A_{i} ⊃ A′, i = 1, 2. Moreover, \overline{(𝔞)^{+}} = \overline{C_{1}} ∩ \overline{C_{2}}. Therefore, we have that rank(P′) = rank(P_{1}) − 1. By Lemma 2 in [H], P_{i}, i = 1, 2, determines a ℚ–parabolic subgroup ^{∗}P_{i} = ^{∗}N_{i}^{∗}A_{i}^{∗}M_{i} of M′ such that

∗M_{i} = M_{i}, A_{i} = ^{∗}A_{i}A′, N_{i} = ^{∗}N_{i}N′, i = 1, 2.  (5.18)

In particular, dim ^{∗}A_{i} = 1, i.e. ^{∗}P_{i}, i = 1, 2, are rank one ℚ–parabolic subgroups of M′. Let ^{∗}C_{i} be the M′(ℚ) conjugacy class of ℚ–parabolic subgroups of M′ containing^{∗}P_{i}. Associated to^{∗}C_{1} and^{∗}C_{2}, there are intertwining operators ^{∗}C_{21}(^{∗}w,^{∗} Λ) where ^{∗}w ∈ W (^{∗}𝔞_{1}) and ^{∗}Λ ∈ (𝔞_{1})_{ℂ}^{∗}. These operators act in spaces E_{cus}^{∗}(σ, O_{i}) which are similar to (5.8). Now each entry of C_{21}(1, Λ) is equal to the product of a holomorphic function and some entry of ^{∗}C_{21}(w,^{∗} Λ), w ∈ W (^{∗}𝔞_{1}), where ^{∗}Λ is the restriction of Λ to ^{∗}𝔞_{1} ⊂ 𝔞_{1}. For details see [H, pp. 124–125], [Mu, pp. 524–525]. This implies Lemma 5.3. There exists c > 0 which is independent of σ and O, such that for any adjacent chambers C_{i} and C_{j}, the number of real singular hyerplanes of C_{ji}(1, Λ) which intersect {Λ ∈ ^{+}𝔞_{i}^{∗} | |Λ| ≤ |ρ_{P} |} is bounded by the number of real poles of ^{∗}C_{ji}(w,^{∗} Λ), w ∈ W (^{∗}𝔞_{i}), in the interval [−c, |ρ|]. Thus, using (5.17), we have reduced our problem to the rank one case. Namely, let m_{1}(σ, O) be the maximum of the number of poles, counted with their order, of the intertwining operators^{∗}C_{ji}(w,^{∗} Λ), w ∈ W (^{∗}𝔞_{i}), in the interval [−c, |ρ|] where c ≫ 0. Let p = dim 𝔞_{i}. Then

m(σ, O) ≤ m_{1}(σ, O)^{p}.  (5.19)

6. The rank one residual spectrum

In this section we study the poles of rank one cuspidal Eisenstein series in the half–plane Re(s) > 0. The corresponding residues span a subspace of L_{res}^{2}(Γ\X, E_{σ}) which is the direct sum of all L_{res}^{2}(Γ\X, E_{σ}; C) where C runs over association classes of rank one ℚ–parabolic subgroups. Let C be an association class of rank one (i.e. maximal) ℚ–parabolic subgroups of G. Let P = NAM be an element of C and let P^{−} = N^{−}AM be the opposite ℚ–parabolic subgroup. Then any element of C is G(ℚ)–conjugate either to P or to P^{−}. Moreover, P and P^{−} are G(ℚ)–conjugate if and only if −1 ∈ W (A). Let P_{1}, ⋯, P_{r} be a set of representatives of the G(ℚ)–conjugacy classes in C. Then r = 1 or 2. Let C_{i} be the G(ℚ) conjugacy class containing P_{i}. Let P_{il}, l = 1, …, r_{i}, be a set of representatives of the Γ–conjugacy classes in C_{i}. Then {P_{il} | 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}} is a set of representatives of the Γ–conjugacy classes in C. Let

{O_{il} | 1 ≤ i ≤ r, 1 ≤ l ≤ r_{i}}

be a set of associated orbits with O_{il} ∈ Ẑ(𝔪_{il})/W (A_{il}). For O_{i} = {O_{il} | 1 ≤ l ≤ r_{i}}, let E_{cus}(σ, O_{i}) be defined by (5.8) and let C_{ji}(w, Λ) be the corresponding intertwining operators (5.9). Set

E_{cus}(σ, O) = ⨁_{i=1}^{r} E_{cus}(σ, O_{i}).  (6.1)

Let α_{i} be the simple root of (P_{i}, A_{i}), i = 1, …, r, and put λ_{i} = α_{i}/|α_{i}|. Then ρ_{i} = |ρ|λ_{i} (note that |ρ_{1}| = |ρ_{2}| which we denote by |ρ|). If r = 1, then W (A_{1}) = {±1}. Define

C(s) = C_{11}(−1, sλ_{1}), s ∈ ℂ.  (6.2)

If r = 2, then W (A_{i}) = {1}, i = 1, 2, and W (𝔞_{1}, 𝔞_{2}) = {w}. Define

C(s) = (\begin{matrix}0 & C_{12}(w^{−1}, sλ_{2}) \\ C_{21}(w, sλ_{1}) & 0 \end{matrix}) .  (6.3)

In either case

C(s): E_{cus}(σ, O) → E_{cus}(σ, O)  (6.4)

is a linear operator that is a meromorphic function of s ∈ ℂ and satisfies

C(s)C(−s) = Id, C(s)^{∗} = C(\overline{s}), s ∈ ℂ.  (6.5)

The poles of C(s) in the half–plane Re(s) ≥ 0 are all simple and contained in the interval (0, | ρ|] (see [H, chapter IV]). For Φ ∈ E_{cus}(σ, O) and Φ = {Φ_{i} | 1 ≤ i ≤ r} set

E(Φ, s) = ∑_{i=1}^{r} E(P_{i}, Φ_{i}, sλ_{i})

where the Eisenstein series on the right hand side are defined by (5.10). Then each E(Φ, s) is a meromorphic function of s ∈ ℂ and satisfies the following functional equation:

E(C(s)Φ, −s) = E(Φ, s), s ∈ ℂ.

The poles of E(Φ, s) are contained in the poles of C(s), and all poles of E(Φ, s) in Re(s) > 0 are simple and contained in (0, |ρ|] (cf. [H], Ch. IV, Theorem 7.) For any pole s_{0} ∈ (0, |ρ|] of E(Φ, s), the residue

Ψ = Res_{s=s_{0}}E(Φ, s)

is square integrable and satisfies

−R(Ω)Ψ = (−s_{0}^{2} + |ρ|^{2} + μ)Ψ

where μ = −χ(Ω_{M_{il}}), χ ∈ O_{il}, which is independent of i, l. By (3.4), Ψ is an eigenfunction of Δ_{σ} with eigenvalue −s_{0}^{2} + |ρ|^{2} + μ + λ_{σ}. Then Ψ belongs to L_{res}^{2}(Γ\X, E_{σ}; C) and this space is spanned by all such residues where Φ ∈ E_{cus}(σ, O), s_{0} ∈ (0, |ρ|] is a pole of C(s) and O runs over all orbits. Thus in order to estimate N_{C,σ}(λ) for rank(C) = 1, it is sufficient to bound the number of poles of C(s) in (0, |ρ|]. Using (4.11) in [Mu] and Corollary 3.15 in [Mu], we get Proposition 6.1. Let C(s; σ, O): E_{cus}(σ, O) → E_{cus}(σ, O) be the intertwining operator with respect to σ ∈ K̂ and a set of associated orbits O. Let μ(O) be the common eigenvalue of the Casimir elements Ω_{M_{il}} with respect to O_{il}. Then for all σ and O, the number n(σ, O) of poles, counted with their order, of C(s; σ, O) in (0, |ρ|] can be estimated by

n(σ, O) ≤ dim(E_{cus}(σ, O))N_{T,σ}(|ρ|^{2} + μ(O) + λ_{σ})

where N_{T,σ}(λ) is the eigenvalue counting function for the cut–off Laplacian Δ_{σ,T} . Remark. The cut–off Laplacian used in [Mu] is equal to Δ_{σ,T} − λ_{σ}Id. This gives rise to the additional λ_{σ} in the counting function on the right hand side. Now we apply Theorem 4.5 to estimate the counting function N_{T,σ}(λ) and (5.16) to bound dim E_{cus}(σ, O). This leads to Corollary 6.2. Let m = dim G. There exists C > 0 such that

m(σ, O) ≤ C(1 + λ_{σ}^{2m} + (μ(O) + λ_{σ})^{2m})

for all σ ∈ K̂ and orbits O. Note that by (5.15), μ(O) + λ_{σ} ≥ 0. Corollary 6.2 is the improved version of Corollary 3.25 in [Mu]. Corollary 6.3. Let C(s; σ, O) be as above. There exists C > 0 such that the number of poles, counted with multiplicity, of det C(s; σ, O) in the half–plane Re(s) > 0 is bounded by

C(1 + λ_{σ}^{3m} + (μ(O) + λ_{σ})^{3m})

for m = dim G.

Proof. Let s_{0}, Re(s_{0}) > 0, be a pole of det C(s; σ, O). Then s_{0} is also a pole of C(s; σ, O). Since all poles of C(s; σ, O) in Re(s) > 0 are simple, the order of det C(s; σ, O) at s_{0} does not exceed d = dim E_{cus}(σ, O). Using (5.16), we get the desired result.

□

7. The poles of the rank one intertwining operators

Let the notation be the same as above. In this section we investigate the poles of the rank one intertwining operator C(s) = C(s; σ, O) in the half–plane Re(s) ≤ 0. Let μ = μ(O) be the common eigenvalue

−χ(Ω_{M_{ij}}), χ ∈ O_{ij}, 1 ≤ i ≤ r, 1 ≤ j ≤ r_{i}.

We first study the poles of det C(s). The analogous result to [Mu, Theorem 7.1] is

Theorem 7.1. There exists C > 0 such that for all σ ∈ K̂ and all associated orbits O one has

| ∫_{−Λ}^{Λ} \frac{d}{ds} log det C(iλ; σ, O) dλ| ≤ C(Λ^{2m} + μ^{2m} + λ_{σ}^{2m})

for Λ ≥ 0 and m = dim G.

Proof. We fix a truncation parameter T = tT_{ρ} with t ≫ 0 as in section 3. Recall that T_{ρ} ∈ 𝔞_{0} is the vector whose i–th component equals H_{ρ_{i}}. Set

C_{t}(s) = e^{−2st|ρ|}C(s).  (7.1)

By the functional equation (6.5) for C(s), it follows that C_{t}(iλ) is unitary for λ ∈ ℝ and hence, can be diagonalized. For Λ ≥ 0, let n(Λ) be the number of points w ∈ [0, Λ] such that C_{t}(iw) has at least one eigenvalue equal to −1. Using (3.7) in [Mu], we obtain

| ∫_{−Λ}^{Λ} \frac{d}{ds} log detC(iλ; σ, O) dλ|

≤ | ∫_{−Λ}^{Λ} \frac{d}{ds} log det C_{t}(iλ; σ, O) dλ|

+ 4t|ρ|Λ dim E_{cus}(σ, O) ≤ 4 dim E_{cus}(σ, O) (2πn(Λ) + t|ρ|Λ),

(7.2)

for Λ ≥ 0. Let w ∈ [0, Λ] and assume that C_{t}(iw) has at least one eigenvalue equal to −1. Thus there exists Φ ∈ E_{cus}(σ, O), Φ ≠ 0, such that C(iw)Φ = −e^{2iwt|ρ|}Φ. Let Λ^{T} be the truncation operator [Mu, (3.10)]. Put φ = Λ^{T} E(Φ, w). By (3.4) and [Mu, Lemma 3.14] we get

Δ_{σ,T} φ = (w^{2} + |ρ|^{2} + μ + λ_{σ}) φ.

Now observe that different values of w give linearly independent eigenfunctions. Hence

n(Λ) ≤ N_{T,σ}(Λ^{2} + |ρ|^{2} + μ + λ_{σ}).  (7.3)

Using Corollary 4.6 and (5.16), the claimed estimate follows from (7.2). □

In [Mu, section 6] we established some key results concerning the analytic properties of the rank one intertwining operators. In particular we have Proposition 7.2. Let v_{1}, …, v_{l} ∈ [0, |ρ|] be the poles of det C(s) in the half–plane Re(s) ≥ 0, and let η_{k}, k ∈ ℕ, be the poles of det C(s) in the half–plane Re(s) ≤ 0, both repeated according to their multiplicity. We fix a truncation parameter T = tT_{ρ}, t ≫ 0. Then for λ ∈ ℝ, one has

\frac{d}{ds} log det C(iλ) = log q + ∑_{j=1}^{l} \frac{2v_{j}}{λ^{2} + v_{j}^{2}}

+ ∑_{k=1}^{∞} \frac{2 Re(η_{k})}{(Re(η_{k}))^{2} + (λ − Im(η_{k}))^{2}} .

(7.4)

where log q ≤ 2(t + 1) dim E_{cus}(σ, O) + 1. We now can proceed as in [Mu, p. 520] to estimate the real poles of det C(s) in Re(s) < 0. Note that for any v > 0, one has

∫_{−1}^{1} \frac{v}{v^{2} + λ^{2}} dλ < ∫_{−∞}^{∞} \frac{1}{1 + λ^{2}} dλ = π.

Hence, by (7.4) and Theorem 7.1, we get

|2 log q + ∫_{−1}^{1} ∑_{k=1}^{∞} \frac{2 Re(η_{k})}{(Re(η_{k}))^{2} + (λ − Im(η_{k}))^{2}} dλ|
≤ 2πl + C(1 + μ^{2n} + λ_{σ}^{m}).

Here l is the number of poles, counted with multiplicity, of detC(s) in Re(s) ≥ 0. Thus l can be estimated by Corollary 6.3 which leads to

|2 log q + ∫_{−1}^{1} ∑_{k=1}^{∞} \frac{2 Re(η_{k})}{(Re(η_{k}))^{2} + (λ − Im(η_{k}))^{2}} dλ|
≤ C_{1}(1 + μ^{3m} + λ_{σ}^{3m}).

(7.5)

We distinguish two cases:

1) q ≥ 1. Then we have 0 ≤ log q ≤ 2(t + 1) dim(E_{cus}(σ, O)) + 1. Since Re(η_{k}) < 0 for all k, (7.5) together with (5.16) leads to

∫_{−1}^{1} ∑_{k=1}^{∞} \frac{| Re(η_{k})|}{(Re(η_{k}))^{2} + (Im(η_{k}) − λ)^{2}} dλ ≤ C_{2}(1 + μ^{3m} + λ_{σ}^{3m}). (7.6)

2) q < 1. Then log q < 0. Since

∑_{k=1}^{∞} \frac{Re(η_{k})}{(Re(η_{k}))^{2} + (Im(η_{k}) − λ)^{2}} < 0,

(7.6) holds in this case too. Let C > 0 and denote by N(c, σ, O) the number of poles, counted with multiplicity, of det C(s; σ, O) in [−c, 0]. Then

\begin{aligned}N(c, σ, O) ∫_{−1/c}^{1/c} \frac{1}{1 + λ^{2}} dλ & ≤ ∑_{\begin{matrix}η_{k}∈ℝ \\ −c≤η_{k}≤0 \end{matrix}} ∫_{−1}^{1} \frac{|η_{k}|}{|η_{k}|^{2} + λ^{2}} dλ \\ & ≤ ∫_{−1}^{1} ∑_{k=1}^{∞} \frac{| Re(η_{k})|}{(Re(η_{k}))^{2} + (λ − Im(η_{k}))^{2}} dλ. \end{aligned}

and by (7.6), this implies

N(c, σ, O) ≤ C_{2}(1 + μ^{3m} + λ_{σ}^{3m})

for a constant C_{2} > 0 that is independent of σ, O. Combined with Corollary 6.3, we get Theorem 7.3. Let c > 0. There exists C > 0 such that for all σ ∈ K̂ and orbits O, the number of poles, counted with multiplicity, of det C(s; σ, O) in [−c, |ρ|] is bounded by C(1 + μ^{3m} + λ_{σ}^{3m}) where μ = μ(O) is the eigenvalue of the Casimir operator and m = dim G. Choose an orthonormal basis Φ_{1}, …, Φ_{d} in E(σ, O) and set

C_{ij}(s; σ, O) = ⟨C(s; σ, O)Φ_{i}, Φ_{j}⟩, i, j = 1, …, d.

Let s_{0} be a pole of C(s; σ, O) and let ν_{ij}(s_{0}; σ, O) be the order of C_{ij}(s; σ, O) in s_{0}. Set

ν(s_{0}; σ, O) = max_{i,j} ν_{ij}(s_{0}; σ, O).

If s_{0} is not a pole of C(s; σ, O), set ν(s_{0}; σ, O) = 0.

Corollary 7.4. Let c > 0. There exists C > 0 such that for all σ ∈ K̂ and O, one has

∑_{−c≤s_{0}≤|ρ|} ν(s_{0}; σ, O) ≤ C(1 + μ^{3m} + λ_{σ}^{3m}), μ = μ(O).

Proof. We proceed exactly as in the proof of Corollary 7.6 in [Mu], using Corollary 4.6 in place of Theorem 9.1 in [D1].

□

Using the description of the residual subspace L_{res}^{2}(Γ\X, E_{σ}) together with (5.14), (5.16) and (5.19), we obtain an estimate for N_{res,σ}(λ) of the form claimed by Theorem 0.2. Together with Corollary 4.6, this completes the proof of Theorem 0.2.

8. The trace class conjecture

To begin with, we prove Corollary 0.3. First note that for any π ∈ Ĝ and any σ ∈ K̂, one has

− λ_{π} + λ_{σ} ≥ 0, if dim(H_{π} ⊗ V_{σ})^{K} ≠ 0.  (8.1)

Indeed, by (3.4), −λ_{π} + λ_{σ} is an eigenvalue of Δ_{σ} ≥ 0. Let σ ∈ K̂ be fixed. By (8.1) and Theorem 0.2 we get

∑_{\begin{matrix}π∈Ĝ \\ λ_{π}^{2}+λ_{σ}^{2}≤λ^{2} \end{matrix}} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K}

≤ ∑_{\begin{matrix}π∈Ĝ \\ −λ_{π}+λ_{σ}≤\sqrt{2}λ \end{matrix}} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K} = N_{σ}(\sqrt{2}λ)

≤ C(1 + λ_{σ}^{4m} + λ^{4m}).

(8.2)

Since λ_{σ} ≤ λ, (8.2) leads to

∑_{σ∈K̂} ∑_{\begin{matrix}π∈Ĝ \\ λ_{π}^{2}+λ_{σ}^{2}≤λ^{2} \end{matrix}} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K} ≤ C(1 + λ^{4m}) ∑_{\begin{matrix}σ∈K̂ \\ λ_{σ}≤λ \end{matrix}} 1. (8.3)

Let 𝔥 ⊂ 𝔨 be a Cartan subalgebra of 𝔨 and let Φ_{c} be the set of roots of (𝔨_{ℂ}, 𝔥_{ℂ}). Let Φ_{c}^{+} ⊂ Φ_{c} be a set positive roots. By L ⊂ 𝔥_{ℂ}^{∗} we denote the additive subgroup of integral linear functions, i.e. λ ∈ L iff λ(H_{α}) ∈ ℤ for all α ∈ Φ_{c}. Let L^{+} ⊂ L be the dominant integral linear functions with respect to Φ_{c}^{+}. Recall that the irreducible finite– dimensional representations of 𝔨_{ℂ} are parametrized by L^{+}. If (σ_{μ}, E_{μ}) is an irreducible representation of 𝔨_{ℂ} with highest weight μ ∈ L^{+}, then the eigenvalue of the Casimir operator Ω_{K} on E_{μ} is given by

λ_{σ_{μ}} =∥ μ + ρ_{c} ∥^{2} − ∥ ρ_{c} ∥^{2}

where ρ_{c} = \frac{1}{2} ∑_{α∈Φ_{c}^{+}} α. Furthermore, L is a lattice in the real vector space F ⊂ 𝔥_{ℂ}^{∗} spanned by the roots. Let l = dim F. Then we get

∑_{\begin{matrix}σ∈K̂ \\ λ_{σ}≤λ \end{matrix}} 1 ≤ C ∑_{\begin{matrix}μ∈L \\ ∥μ∥^{2}≤λ \end{matrix}} 1 ≤ C ( ∑_{μ∈L−\{0\}} ∥ μ ∥^{−2l}) λ^{l} = C_{1}λ^{l}.

Combined with (8.3), we obtain

∑_{σ∈K̂} ∑_{\begin{matrix}π∈Ĝ \\ λ_{π}^{2}+λ_{σ}^{2}≤λ^{2} \end{matrix}} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K} ≤ C_{2}(1 + λ^{4m+l}). (8.4)

Let N ≥ m. Using (8.4), we get

∑_{π∈Ĝ} ∑_{σ∈K̂} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K}(1 + λ_{π}^{2} + λ_{σ}^{2})^{−N}

= ∑_{k=0}^{∞} ∑_{\begin{matrix}π,σ \\ k^{2}≤λ_{π}^{2}+λ_{σ}^{2}<(k+1)^{2} \end{matrix}} m_{Γ}(π) dim(H_{π} ⊗ V_{σ})^{K}(1 + λ_{π}^{2} + λ_{σ}^{2})^{−N}

≤ C ∑_{k=0}^{∞}(1 + (k + 1)^{(4m+l)})(1 + k^{2})^{−N} < ∞.

We are now ready to prove Theorem 0.1. Given π ∈ Ĝ and σ ∈ K̂, let H_{π}(σ) be the σ–isotypical subspace of H_{π}, i.e., H_{π}(σ) is the sum of all irreducible subspaces of the representation (π|K, H_{π}) of K that are equivalent to σ. Note that

dim H_{π}(σ) = dim(H_{π} ⊗ V_{σ})^{K}

For all (π, σ) such that H_{π}(σ) ≠ {0}, we choose an orthonormal basis e_{π,σ,i}, i ∈ I(π, σ), for H_{π}(σ). Each e_{π,σ,i} belongs to the Gårding space H_{π}^{∞} of H_{π} and the Casimir operators Ω and Ω_{K} of G and K, respectively, act on e_{π,σ,i} by

R(Ω)e_{π,σ,i} = λ_{π}e_{π,σ,i}, R(Ω_{K})e_{π,σ,i} = λ_{σ}e_{π,σ,i}.

Let C^{1}(G) be Harish-Chandra’s Schwartz space of integrable rapidly decreasing functions on G (cf. section 1), and let f ∈ C^{1}(G). Then the Hilbert–Schmidt norm of R_{Γ}^{d}(f) is given by

∥ R_{Γ}^{d}(f) ∥_{HS}^{2} = ∑_{σ∈K̂} ∑_{π∈Ĝ} m_{Γ}(π) ∑_{i∈I(π,σ)}(R_{Γ}(f)e_{π,σ,i}, R_{Γ}(f)e_{π,σ,i})

= ∑_{σ∈K̂} ∑_{π∈Ĝ} m_{Γ}(π)(1 + λ_{π}^{2} + λ_{σ}^{2})^{−N}

× ∑_{i∈I(π,σ)} ∫_{G} ((Ω^{2} + Ω_{K}^{2} + Id)^{N}f) (g)(R_{Γ}(g)e_{π,σ,i}, R_{Γ}(f)e_{π,σ,i}) dg.

By Corollary (0.3) we obtain

∥ R_{Γ}^{d}(f) ∥_{HS}^{2}≤ C ∥ (Ω^{2} + Ω_{K}^{2} + Id)^{N}f ∥_{L^{1}(G)}∥ f ∥_{L^{1}(G)} .

for some constant C > 0. Thus R_{Γ}^{d}(f) is a Hilbert–Schmidt operator. The proof shows that the same conclusion holds for any f ∈ C_{c}^{p}(G) with p ≥ N. Fix an integer p ≥ 3 dim G and let f ∈ C^{1}(G). According to Lemma 4.5 in [W1] there exist h ∈ C^{1}(G), μ_{1} ∈ C_{c}^{p}(G) and μ_{2} ∈ C_{c}^{∞}(G) such that

f = h ∗ μ_{1} + f ∗ μ_{2}.

Thus

R_{Γ}^{d}(f) = R_{Γ}^{d}(h) ∘ R_{Γ}^{d}(μ_{1}) + R_{Γ}^{d}(f) ∘ R_{Γ}^{d}(μ_{2}).

All operators on the right hand side are Hilbert–Schmidt operators and hence, R_{Γ}^{d}(f) is a trace class operator. This completes the proof of Theorem (0.1). Remark. For simplicity, we have restricted attention to arithmetic subgroups Γ. But in fact, this is no restriction at all. First, let G be a connected semisimple Lie group with finite center and suppose that the ℝ-rank of G is one. Let Γ be a discrete subgroup of G such that the volume of Γ\G with respect to any invariant measure is finite. Then for any f ∈ C^{1}(G), the restriction of R_{Γ}(f) to the subspace of cusp forms L_{cus}^{2}(Γ\G) is of the trace class [W1, Theorem 4.3]. Moreover, it is not difficult to estimate the residual spectrum in the rank one case by the methods we used to deal with the rank one spectrum in general. As above, this implies that R_{Γ}^{d}(f) is of the trace class. On the other hand, if G is a connected semisimple ℝ-group with rank_{ℝ}G(ℝ) ≥ 2 and Γ is an irreducible lattice in G(ℝ), then by Margulis [Ma, Chapter IX], Γ is an arithmetic subgroup in the following sense: There exists a cannected non-commutative almost ℚ-simple ℚ-group H and an ℝ-epimorphism τ : H → G such that the Lie group (ker τ)(ℝ) is compact and the subgroups τ(H(ℤ)) and Γ are commensurable. It is not difficult to extend Theorem 0.1 to such lattices.

9. The adèlic version

In this section we denote by G a connected reductive linear algebraic group defined over ℚ. We fix an embedding of G into GL_{n}. For a given place v of ℚ, let G(ℚ_{v}) be the group of ℚ_{v}–rational points of G. Let 𝔸 be the ring of adèles of ℚ and let G(𝔸) be the corresponding adèle–valued group. If f stands for the set of finite places of ℚ and 𝔸_{f} is the corresponding ring of finite adèles, then

G(𝔸) = G(ℝ) × G(𝔸_{f}).

For any prime p, let

G(ℤ_{p}) = GL_{n}(ℤ_{p}) ∩ G(ℚ_{p}).

This is an open compact subgroup of G(ℚ_{p}). Then any open compact subgroup K′_{f} of G(𝔸_{f}) contains a subgroup of finite index K_{f} of the form

K_{f} = ∏_{p} K_{p}

where K_{p} is an open compact subgroup of G(ℚ_{p}) and for almost all primes p, one has K_{p} = G(ℤ_{p}). For any such K_{f}, the double coset space

G(ℚ)\G(𝔸)/G(ℝ)K_{f}

is known to be finite [B2, Section 5]. If x_{1} = 1, x_{2}, ..., x_{l} is a set of representatives in G(𝔸_{f}) of the cosets, the groups

Γ_{i} = (G(ℝ) × x_{i}K_{f}x_{i}^{−1}) ∩ G(ℚ), 1 ≤ i ≤ l,

are arithmetic subgroups of G(ℝ) and

G(ℚ)\G(𝔸)/K_{f} ≅ ⨆_{i=1}^{l}(Γ_{i}\G(ℝ)).  (9.1)

Let Z be the ℚ–split component of the center of G and let Z(ℝ)^{0} be the connected component of 1 in Z(ℝ). Let dg be the Tamagawa measure on G(𝔸) (cf. [A1, p.329]). Then Z(ℝ)^{0}G(ℚ)\G(𝔸) has finite volume with respect to dg. Furthermore, it follows from (9.1) that

L^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸))^{K_{f}} ≅ ⨁_{i=1}^{l} L^{2}(Z(ℝ)^{0}Γ_{i}\G(ℝ))  (9.2)

as G(ℝ)–modules. Let G^{1} = Z\G and G_{1} = Z(ℝ)^{0}\G(ℝ). Then G^{1}(ℝ) is the quotient of G_{1} by a finite group and Γ_{i}, 1 ≤ i ≤ l, is a discrete subgroup of G_{1} of finite covolume whose intersection with the center of G_{1} is cocompact. Thus (G_{1}, Γ_{i}), 1 ≤ i ≤ l, satisfy the assumptions made in section 1.1 and therefore, our results apply to this setting. Let R be the right regular representation of G(𝔸) on the Hilbert space L^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸)) and let L_{d}^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸)) be the smallest closed invariant subspace that contains all irreducible subrepresentations of R. Let { \atop G(𝔸)} be the unitary dual of G(𝔸). Then

L_{d}^{2} (Z(ℝ)^{0}G(ℚ)\G(𝔸)) = ⨁_{π∈{ \atop G(𝔸)}}m(π)H_{π}.

Any π ∈ { \atop G(𝔸)} that occurs in R is a restricted tensor product π = ⊗_{v}π_{v} of irreducible representations π_{v} of the local group G(ℚ_{v}). Thus

π = π_{∞} ⊗ π_{f}

where π_{∞} and π_{f} are irreducible representations of G(ℝ) and G(𝔸_{f}), respectively. Let H_{π_{f}} be the representation space of π_{f} and let H_{π_{f}}^{K_{f}} denote the subspace of K_{f}-invariant vectors in H_{π_{f}}. This is a finitedimensional subspace and

H_{π}^{K_{f}} = H_{π_{∞}} ⊗ H_{π_{f}}^{K_{f}} .

Hence it follows that as G(ℝ)-modules, we have

L_{d}^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸))^{K_{f}} ≅ ⨁_{i=1}^{l} L_{d}^{2}(Z(ℝ)^{0}Γ_{i}\G(ℝ))  (9.3)

Let

h = ∏_{p} h_{p}  (9.4)

be a function on G(𝔸_{f}) which satisfies the following properties:

1): For any prime p, the function h_{p} is locally constant with com-

pact support, 2): For almost all primes p, the function h_{p} is the characteristic

function of G(ℤ_{p}).

The space of all linear combinations of such functions will be denoted by H(G(𝔸_{f})). This is the Hecke algebra of locally constant functions of compact support on G(𝔸_{f}). Let R_{f} denote the restriction of the regular representation to G(𝔸_{f}). Then for any h ∈ H(G(𝔸_{f})) we can define the Hecke operator

R_{f}(h) = ∫_{G(𝔸_{f})} h(g)R_{f}(g) dg.

Since R_{f} is unitary, R_{f}(h) is a bounded operator on the Hilbert space L^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸)). Moreover, there exists an open compact subgroup K_{f} of G(𝔸_{f}), which depends on h, such that h is right K_{f}- invariant. Hence R_{f}(h) maps the space L^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸)) into the subspace of K_{f}-invariant functions. Let h_{∞} ∈ C^{1}(G(ℝ)) and h_{f} ∈ H(G(𝔸_{f})). Set

h = h_{∞} × h_{f}  (9.5)

and let

R(h) = ∫_{G(𝔸)} h(g)R(g) dg.

Then R(h) = R_{∞}(h_{∞}) · R_{f}(h_{f}) and R_{∞}(h_{∞}) commutes with the orthogonal projection P_{f} of L^{2}(Z(ℝ)^{0}G(ℚ)\G(𝔸)) onto the subspace of K_{f}-invariant functions. Thus

R(h) = P_{f}R_{∞}(h_{∞})P_{f} R_{f}(h_{f}).

Let R^{d} denote the restriction of R to the discrete subspace (9.3). By (9.3) and Theorem 0.1 it follows that P_{f}R_{∞}^{d}(h_{∞})P_{f} is a trace class operator. Since R_{f}(h_{f}) is bounded, R^{d}(h) is also a trace class operator. Let C^{1}(G(𝔸)) be the space of all linear combinations of functions of the form (9.5). Then we have proved Theorem 9.1. For any h ∈ C^{1}(G(𝔸)), the operator R^{d}(h) is a trace class operator. In [Mu] we proved Theorem 9.1 under the additional assumption that h is K-finite for a maximal compact subgroup of G(𝔸). This assumption has now been eliminated. One can also include characters of Z(ℝ)^{0}. Let ξ be a quasi-character of Z(ℝ)^{0}. Denote by L^{2}(G(ℚ)\G(𝔸), ξ) the space of functions ϕ on G(ℚ)\G(𝔸) such that

ϕ(zg) = ξ(z)ϕ(g), z ∈ Z(ℝ)^{0}, g ∈ G(𝔸)

and which are square integrable modulo Z(ℝ)^{0}. Here the latter condition is well defined, since G(𝔸) is the direct product of Z(ℝ)^{0} with a normal subgroup G(𝔸)^{1} which contains G(ℚ). Then G(𝔸) acts by right translations on L^{2}(G(ℚ)\G(𝔸), ξ). We denote this representation by R_{ξ}. Let R_{ξ}^{d} be the restriction of R_{ξ} to the corresponding discrete subspace. Then as above, it follows that for any h ∈ C^{1}(G(𝔸)), the operator R_{ξ}^{d}(h) is a trace class operator.

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Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

E-mail address: mueller@rhein.iam.uni-bonn.de