WEAK WEYL’S LAW FOR CONGRUENCE SUBGROUPS

JEAN-PIERRE LABESSE AND WERNER MÜLLER

1. Introduction

Let G be a connected and simply connected semisimple algebraic group defined over ℚ and let Z_{G} be its center. Let Γ be an arithmetic subgroup of G(ℝ). For simplicity we assume in the introduction that Γ is torsion free. We denote by K_{∞} a maximal compact subgroup of G(ℝ). We endow G(ℝ) and K_{∞} with Haar measures so that the symmetric space X = G(ℝ)/K_{∞} has the canonical measure, i.e., the measure defined using the Killing form. Let

HΓ_{cusp} = L2_{cusp}(Γ\G(ℝ))

denote the closure in L^{2}(Γ\G(ℝ)) of the subspace spanned by all Γ-cuspidal automorphic forms. We denote by ρ_{∞} the right regular representation of G(ℝ) in HΓ_{cusp}. Let (σ, V_{σ}) be an irreducible unitary representation of K_{∞}. Set

HΓ_{cusp}(σ) = (HΓ_{cusp} ⊗ V_{σ})^{K_{∞}}.

Let Ω be the Casimir operator for G(ℝ). Then ρ_{∞}(Ω) ⊗ Id induces a selfadjoint operator in the Hilbert space HΓ_{cusp}(σ) which has pure point spectrum. We denote by Λ_{cusp}(σ) the set of eigenvalues of ρ_{∞}(Ω)⊗Id in HΓ_{cusp}(σ). Let N Γ_{cusp}(T, σ) be the counting function of the cuspidal spectrum Λ_{cusp}(σ), i.e., N Γ_{cusp}(T, σ) is the number of eigenvalues λ of ρ_{∞}(Ω) ⊗ Id in HΓ_{cusp}(σ), counted with their multiplicities, with |λ| ≤ T. We shall also consider the discrete subspace

HΓ_{disc} = L2_{disc}(Γ\G(ℝ)),

and we define HΓ_{disc}(σ) and the corresponding eigenvalue counting function N Γ_{disc}(T, σ) similarly. It is an important question in the theory of automorphic forms to understand the asymptotic behavior of the counting function as T → ∞. Let d be the dimension of the symmetric space X = G(ℝ)/K_{∞} and let Γ denote the Gamma function. Let

c_{σ}(Γ) = \frac{dim(σ) vol(Γ\X)}{(4π)^{d/2}Γ(d/2 + 1)}

Date: April 14, 2004.

be Weyl’s constant. A conjecture of Sarnak [Sa] states that the counting function of the cuspidal spectrum satisfies Weyl’s law, i.e.,

(1.1)  lim_{T →∞} \frac{N Γ_{cusp}(T, σ)}{T^{d/2}} = lim_{T →∞} \frac{N Γ_{disc}(T, σ)}{T^{d/2}} = c_{σ}(Γ).

This conjecture has been established in some special cases. First of all, it was Selberg [Se] who proved it for congruence subgroups of SL(2, ℤ) and σ = 1. Other cases for which the conjecture has been established are Hilbert modular groups [Ef], congruence subgroups of SO(n, 1) [Rez], SL(3, ℤ) [Mil], and in particular, the conjecture was proved in [Mu2] for principal congruence subgroups of SL(n, ℤ) and arbitrary σ.

A possible approach to prove (1.1) is through the Selberg trace formula applied to a function constructed from the heat kernel of the corresponding Bochner-Laplace operator. Let E_{σ} → X be the homogeneous vector bundle associated to σ and let Δ_{σ} be the elliptic differential operator induced by −R(Ω) ⊗ Id in C^{∞}(X, E_{σ}), where R denotes the right regular representation of G(ℝ) in C^{∞}(G(ℝ)). Using the heat kernel of Δ_{σ}, one produces a 1-parameter family of functions h_{t} on G(ℝ), depending on t > 0, such that for any irreducible unitary representation (π, H(π)) of G(ℝ) one has

(1.2)  Tr π(h_{t}) = e^{tπ(Ω)} dim Hom_{K_{∞}}(H(π), V_{σ}^{∗}).

Given λ ∈ Λ_{cusp}(σ), let m(λ) denote its multiplicity. Then it follows from (1.2) that

Tr (ρ_{∞}(h_{t})|HΓ_{cusp}) = ∑_{λ∈Λ_{cusp}(σ)} m(λ)e^{tλ}.

The idea is now to study the asymptotic behavior of the left hand side as t → 0. Using the properties of the heat kernel, it follows that

(1.3)  h_{t}(1) ∼ \frac{dim(σ)}{(4πt)^{d/2} vol(K_{∞})}

as t → 0. A good control of the trace formula should allow to prove that as t → 0, the constant term in the asymptotic expansion of

t ⟼ t^{d/2} Tr (ρ(h_{t})|HΓ_{cusp})

is the contribution of the unit element to the trace formula:

t^{d/2} Tr (ρ(h_{t})|HΓ_{cusp}) = t^{d/2} vol(Γ\G(ℝ))h_{t}(1) + o(1)

(we recall that we have assumed that Γ is torsion free). The desired estimate is then an immediate consequence of (1.3) combined with the Tauberian theorem. The second named author has been able to apply successfully this method for Γ a principal congruence subgroup for G = SL(n). However, many technical difficulties are still to overcome in order to prove (1.1) for a general G.

The purpose of this paper is to prove a weaker result which holds for every G. Recall that an upper bound, which holds for arbitrary G and Γ, is already known thanks to Donnelly [Do]:

lim sup_{T →∞} \frac{N Γ_{cusp}(T, σ)}{T^{d/2}} ≤ c_{σ}(Γ).

By working with a simple form of the trace formula, we shall get, for a general G, a lower bound that depends on the choice of a set S of primes containing at least two finite primes. For every such set S we shall define a certain constant c_{S}(Γ) ≤ 1, which is non zero for Γ deep enough. Let Z_{Γ} = Γ ∩ Z_{G}(ℝ). Note that Z_{Γ} ⊂ K_{∞}. Then our main result is the following theorem. Theorem 1.1. Let G be an almost simple connected and simply connected semisimple algebraic group defined over ℚ such that G(ℝ) is non compact. Then for every congruence subgroup Γ and every σ such that σ|_{Z_{Γ}} = Id we have

c_{σ}(Γ)c_{S}(Γ) ≤ lim inf_{T →∞} \frac{N Γ_{cusp}(T, σ)}{T^{d/2}}.

This will turn out to be very easy to prove but the price to pay for simplicity is that this lower bound is far from being sharp; in particular the lower bound is non trivial, i.e. c_{S}(Γ) > 0, only for discrete subgroups Γ that are deep enough (see below for a more precise statement). We note that a weaker form of this result is due to Piatetski-Shapiro [PS] and for G = SL(2, ℝ) also to Venkov [Ve]. In [PS] it is proved that for every Γ there exists a subgroup of finite index Γ′ such that the space of Maass cusp forms with respect to Γ′ is infinite dimensional.

Theorem 1.1 is a consequence of Theorem 4.2. To get the lower bound we shall prove a sharp estimate for a certain strict subspace HΓ,S_{cusp} of HΓ_{cusp} that will be defined via the adèlic picture. From now on, for simplicity of exposition, let us assume that G is absolutely almost simple and that G(ℝ) is non compact. We consider a discrete subgroup Γ such that

Γ = K_{fin} ∩ G(ℚ),

where

K_{fin} = ∏ K_{p}

is a decomposable open compact subgroup of the group of finite adèles G(𝔸_{fin}). Let S be a finite set of primes of cardinality at least 2 and let

L2_{cusp}(G(ℚ)\G(𝔸), S) ⊂ L2_{cusp}(G(ℚ)\G(𝔸))

be the G(𝔸)-module generated by cusp forms that are orthogonal to the trivial representation of G(𝔸) and on which G_{S} = G(ℚ_{S}) acts by the Steinberg representation. Then we define

HΓ,S_{cusp} = L2_{cusp}(G(ℚ)\G(𝔸), S)^{K_{fin}}

and

HΓ_{cusp}(σ, S) = (L2_{cusp}(G(ℚ)\G(𝔸), S) ⊗ V_{σ})^{K} = (HΓ,S_{cusp} ⊗ V_{σ})^{K_{∞}} .

Let Λ_{cusp}(σ, S) be the spectrum of ρ_{∞}(Ω) ⊗ Id in HΓ_{cusp}(σ, S). It consists of a discrete set of eigenvalues. Given λ ∈ Λ_{cusp}(σ, S) denote by m(λ) the multiplicity of the eigenvalue λ. Then it follows from (1.2) that

Tr(ρ_{∞}(h_{t})|HΓ,S_{cusp}) = ∑_{λ∈Λ_{cusp}(σ,S)} m(λ)e^{tλ}.

Using the trace formula, we shall prove in 4.2 that

Tr(ρ_{∞}(h_{t})|HΓ,S_{cusp}) ∼ c_{S}(Γ)\frac{dim σ vol(Γ\X)}{(4πt)^{d/2}}

as t → 0. For T ≥ 0 let Λ_{cusp}(σ, S; T) be the subset of all λ ∈ Λ_{cusp}(σ, S) with |λ| ≤ T. Set

N Γ_{cusp}(T, σ, S) = ∑_{λ∈Λ_{cusp}(σ,S;T)} m(λ).

If c_{S}(Γ) ≠ 0 the Tauberian theorem yields the estimate:

lim_{T →∞} \frac{N Γ_{cusp}(T, σ, S)}{T^{d/2}} = c_{σ}(Γ)c_{S}(Γ).

This is our Theorem 4.2. Now clearly N Γ_{cusp}(T, σ, S) ≤ N Γ_{cusp}(T, σ) and hence

c_{σ}(Γ)c_{S}(Γ) ≤ lim inf_{T →∞} \frac{N Γ_{cusp}(T, σ)}{T^{d/2}}.

We still have to observe that c_{S}(Γ) ≠ 0 for subgroups Γ such that at primes p ∈ S the open compact subgroup K_{p} is small enough, i.e., if it is a subgroup of some minimal parahoric subgroup I_{p} of G_{p} = G(ℚ_{p}).

2. A simple trace formula

From now on we shall work in the adèlic picture. We choose a Haar measure, for example the Tamagawa measure, on G(𝔸) which is given as the product of measures on each G_{p} = G(ℚ_{p}). We fix a decomposable open compact subgroup K_{fin} of G(𝔸_{fin}). Set

K = K_{∞}K_{fin}.

Then K is a compact subgroup of G(𝔸). Let f ∈ C_{c}^{∞}(G(𝔸)) be K-finite. The Arthur-Selberg trace formula in its invariant form is the equality

∑_{𝔬∈O}I_{𝔬}(f) = I(f) = ∑_{χ∈𝔛}I_{χ}(f)

of a geometric and a spectral expansion for an invariant linear form f ⟼ I(f) (see [Ar]).

Let Σ be some large enough set of primes, containing in particular the real place and all ramified primes. We shall consider functions f that are decomposable

f = ⊗f_{p}

and such that for p ∉ Σ, f_{p} is the characteristic function of a hyperspecial maximal compact subgroup divided by its volume. We put, as usual,

f^{Σ} = ⊗_{p∉Σ} f_{p}

and

f_{Σ} = ⊗_{p∈Σ} f_{p}

so that f = f_{Σ} ⊗ f^{Σ}. The main observation is the following result of Arthur: when applied to a decomposable function such that f_{p} is cuspidal, in the sense of Arthur, at two primes p ∈ S the trace formula is “simple”. Proposition 2.1. If f is cuspidal at two places, the linear form I(f) has a discrete spectral expansion:

I(f) = ∑ a(π) tr π(f)

where the sum is over certain subrepresentations of parabolically induced representations from representations in the discrete spectrum of Levi subgroups. In particular, if π occurs in the discrete spectrum for G(𝔸) then a(π) = m(π), where m(π) denotes the multiplicity with which π occurs in the discrete spectrum. The geometric expansion is given by a linear combination of ordinary orbital integrals (over G_{Σ})

I(f) = ∑_{γ∈𝔊_{Σ}} a(γ, Σ)O_{γ}(f_{Σ}),

where 𝔊_{Σ} is a set of representatives of conjugacy classes with elliptic semisimple part, O_{γ}(f_{Σ}) is the orbital integral of f_{Σ} and a(γ, Σ) is some scalar. If γ is elliptic we have

a(γ, Σ)O_{γ}(f_{Σ}) = a(γ)O_{γ}(f),

where the scalar a(γ) is independent of Σ . Moreover if the support of f_{Σ} remains in a fixed compact set then, in the sum over 𝔊_{Σ} , only a finite number of elliptic conjugacy classes in G(ℚ) may give a non trivial contribution.

Proof. This is the content of [Ar, Corollary 7.2]. We omit the sum over t ≥ 0 that occurs in [Ar] since the sum is now known to be absolutely convergent by [Mu1]. □

We shall denote by 1(f) the operator defined by f in the trivial representation; this is a scalar operator in a one dimensional space:

1(f) = ∫_{G(𝔸)} f(x) dx.

Let p be a finite place. The existence of (normalized) pseudo-coefficients of the Steinberg representation is a particular case of a general result due to Kazdan, but, up to a sign, they can also be viewed as Euler-Poincaré functions at non archimedean places and an explicit construction has been given by Kottwitz [Ko]. A pseudo-coefficient of the Steinberg representation is a K_{p}-finite function f_{p} ∈ C_{c}^{∞}(G_{p}) such that

Tr π_{p}(f_{p}) = 0

for all irreducible unitary representations π_{p} unless either π_{p} is the Steinberg representation of G_{p} in which case

Tr π_{p}(f_{p}) = 1

or π_{p} = 1_{p} the trivial representation in which case

1_{p}(f_{p}) = (−1)^{q}

for some integer q. Moreover, [Ko, Theorem 2] shows that orbital integrals O_{γ}(f_{p}) of non elliptic elements γ vanish.

Let S be a finite set of finite primes. Let

L2_{cusp}(G(ℚ)\G(𝔸), S) ⊂ L2_{cusp}(G(ℚ)\G(𝔸))

be the G(𝔸)-module generated by cusp forms that are orthogonal to the trivial representation of G(𝔸) and on which

G_{S} := ∏_{p∈S} G(ℚ_{p})

acts by the Steinberg representation. Let ρ denote the regular representation of G(𝔸) in

L2_{cusp}(G(ℚ)\G(𝔸)).

Proposition 2.2. Let G be a simply connected almost simple algebraic group defined over ℚ. Assume that S is a set of finite primes containing at least two primes. Consider

f = f_{∞} ⊗ f_{fin,S} ⊗ f_{S},

where f_{∞} ∈ C_{c}^{∞}(G_{∞}), f_{fin,S} is a locally constant compactly supported function on G(𝔸S_{fin}), the group of finite adèles outside S, and f_{S} is a pseudo-coefficient of the Steinberg representation of G_{S}. Then one has

∑_{γ∈𝔊_{e}} a(γ)O_{γ}(f) = Tr(ρ(f)|L_{cusp}^{2}(G(ℚ)\G(𝔸), S)) + 1(f),

where 𝔊_{e} is a set of representatives of elliptic conjugacy classes in G(ℚ).

Proof. By a standard argument, automorphic representations in the discrete spectrum with a Steinberg component at some place p must be cuspidal. In fact Langlands description of the discrete spectrum implies that representations that are in the non cuspidal discrete spectrum arise from residues of Eisenstein series for non unitary parameters and hence are non tempered at all places, while the Steinberg representation is tempered. Moreover, automorphic representations with a Steinberg component at some place p cannot be subrepresentations of parabolically induced representations from representations in the discrete spectrum of Levi subgroups for a proper parabolic subgroup. Suppose that G_{p} is non compact. Since our group G is absolutely almost simple and simply connected, the strong approximation theorem holds and hence G(ℚ)G_{p} is dense in G(𝔸). Therefore an automorphic representations that is trivial at p is necessarily trivial. On the other hand, if the group G_{p} is compact then all automorphic representations are cuspidal. Hence an automorphic representations that is trivial at one place is either trivial or cuspidal. We now use Proposition 2.1 on the simple form of the trace formula with Σ large enough, containing S. The above discussion shows that the spectral expansion of I(f) is given by the sum of the trace in the cuspidal spectrum plus the trace in the trivial representation:

I(f) = Tr(ρ(f)|L_{cusp}^{2}(G(ℚ)\G(𝔸), S)) + 1(f).

The geometric expansion of I(f) is a linear combination of orbital integrals (over G_{Σ}) and, as recalled above, [Ko, Theorem 2] shows that orbital integrals of non elliptic elements vanish at p ∈ S. Hence

I(f) = ∑_{γ∈𝔊_{e}} a(γ)O_{γ}(f),

where 𝔊_{e} is a set of representatives of elliptic conjugacy classes in G(ℚ). □

Let

HΓ,S_{cusp} = L_{cusp}^{2}(G(ℚ)\G(𝔸), S)^{K_{fin}}

be the Hilbert subspace of L2_{cusp}(G(ℚ)\G(𝔸)) generated by vectors of automorphic representations that are Steinberg at p ∈ S and that are K_{fin}-invariant. Let e_{fin} be the characteristic function of K_{fin}, our chosen open compact subgroup at finite adèles divided by its volume:

e_{fin} = \frac{1}{vol(K_{fin})}χ_{K_{fin}}.

Let

K_{S} = ∏_{p∈S} K_{p}, K_{fin,S} = ∏_{p∉S∪{∞}} K_{p}.

Let e_{S} (resp. e_{fin,S}) be the characteristic function of K_{S} (resp. K_{fin,S}) divided by its volume. Then

e_{fin} = e_{S} ⊗ e_{fin,S}.

Corollary 2.3. Assume that f is of the form

f = f_{∞} ⊗ f_{S} ⊗ e_{fin,S}

where f_{S} is a pseudo-coefficient of the Steinberg representation of G_{S}. Let c(K_{S}) be the dimension of the space of K_{S} fixed vectors in the Steinberg representation of G_{S} and let

b(f_{∞}, S) = Tr 1(f_{∞} ⊗ f_{S}),

where 1 is the trivial representation of G_{∞} × G_{S}. Then one has

Tr (ρ_{∞}(f_{∞})|HΓ,S_{cusp}) = c(K_{S}) ( ∑_{γ∈𝔊_{e}} a(γ)O_{γ}(f) − b(f_{∞}, S)) .

Proof. We first observe that for any admissible representation τ of G(𝔸_{fin}), the operator τ(e_{fin}) is the projection on the subspace of K_{fin}-invariant vectors. Hence,

Tr(ρ_{∞}(f_{∞})|HΓ,S_{cusp}) = Tr(ρ(f_{∞} ⊗ e_{fin})|L2_{cusp}(G(ℚ)\G(𝔸), S)).

However, if π_{S} is the Steinberg representation of G_{S} then

Tr π_{S}(e_{S}) = c(K_{S}) Tr π_{S}(f_{S})

and hence we get

Tr(ρ_{∞}(f_{∞})|HΓ,S_{cusp}) = c(K_{S}) Tr(ρ(f)|L2_{cusp}(G(ℚ)\G(𝔸), S)).

The assertion is then an immediate consequence of Proposition 2.2. □

3. The heat kernel

In this section we establish a number of facts about the heat kernel of a Bochner-Laplace operator on the Riemannian symmetric space X = G(ℝ)/K_{∞}. Let E_{σ} → X be the homogeneous vector bundle associated to σ. Let Ω ∈ Z(𝔤_{ℂ}) be the Casimir element of G(ℝ) and let R be the right regular representation of G(ℝ) on C^{∞}(G(ℝ)). Denote by Δ_{σ} the second order elliptic differential operator induced by −R(Ω) ⊗ Id in C^{∞}(X, E_{σ}) ≅ (C^{∞}(G(ℝ)) ⊗ V_{σ})^{K_{∞}}. If ∇^{σ} denotes the canonical invariant connection on E_{σ}, then

Δ_{σ} = (∇^{σ})^{∗}∇^{σ} − λ_{σ}Id,

where λ_{σ} = σ(Ω_{K_{∞}}) is the Casimir eigenvalue of σ. Hence Δ_{σ} : C_{c}^{∞}(X, E_{σ}) → L^{2}(X, E_{σ}) is essentially selfadjoint and bounded from below. We continue to denote its unique selfadjoint extension by Δ_{σ}. Let exp(−tΔ_{σ}), t ≥ 0, be the associated heat semigroup. The heat operator is a smoothing operator on L^{2}(X, E_{σ}) which commutes with the representation of G(ℝ) on L^{2}(X, E_{σ}). Therefore, it is of the form

(e^{−tΔ_{σ}}φ)(g) = ∫_{G(ℝ)} H_{t}(g^{−1}g_{1})(φ(g_{1}))dg_{1}, g ∈ G(ℝ),

where φ ∈ (L^{2}(G(ℝ)) ⊗ V_{σ})^{K_{∞}} and H_{t}: G(ℝ) → End(V_{σ}) is in L^{2} ∩ C^{∞} and satisfies the covariance property

H_{t}(g) = σ(k)H_{t}(k^{−1}gk′)σ(k′)^{−1}, for g ∈ G(ℝ), k, k′ ∈ K_{∞}.

Let C^{1}(G(ℝ)) be Harish-Chandra’s space of integrable rapidly decreasing functions on G(ℝ). Then

(3.1)  H_{t} ∈ (C^{1}(G(ℝ)) ⊗ End(V_{σ}))^{K_{∞}×K_{∞}}

[BM, Proposition 2.4].

Next we need to estimate the covariant derivatives of the heat kernel. Let ∇ denote the Levi-Civita connection on G(ℝ) with respect to the invariant metric on G(ℝ). Let d(x, y) be the geodesic distance of x, y ∈ X. Define the function r(g) on G(ℝ) by

(3.2)  r(g) = d(gK_{∞}, K_{∞}), g ∈ G(ℝ).

The following proposition is proved in [Mu2, Proposition 2.1]. Proposition 3.1. Let a = dim G(ℝ), l ∈ ℕ_{0} and T > 0. There exist C, c > 0 such that

∥ ∇^{l}H_{t}(g) ∥≤ Ct^{−(a+l)/2} exp (−\frac{cr^{2}(g)}{t})

for all 0 < t ≤ T and g ∈ G(ℝ).

Define the one-parameter family of smooth functions h_{t} on G(ℝ) by

(3.3)  h_{t}(g) = tr H_{t}(g), g ∈ G(ℝ), t ≥ 0,

where tr denotes the trace function on End(V_{σ}). Then it follows from 3.1 that h_{t} belongs to C^{1}(G(ℝ)). In addition, h_{t} is both left and right K_{∞}-finite.

Let π be a unitary irreducible representation of G(ℝ) on the Hilbert space H(π). Then π(h_{t}) is trace class and it follows as in [BM] that

(3.4)  Tr π(h_{t}) = e^{tπ(Ω)} dim Hom_{K_{∞}}(H(π), V_{σ}^{∗}).

4. The main result

First we note that by our assumptions on G, strong approximation holds for G(𝔸) and hence we have

L2_{cusp}(G(ℚ)\G(𝔸)/K_{fin}) ≅ L2_{cusp}(Γ\G(ℝ))

as G(ℝ)-modules. Let h_{t}, t > 0, be the one-parameter family of functions on G(ℝ) defined by (3.3). By [Mu1, Corollary 0.2] the restriction of ρ_{∞}(h_{t}) to L2_{cusp}(Γ\G(ℝ)) is a trace class operator. However, since the support of h_{t} is not compact it is not known that we may insert h_{t} in the trace formula for an arbitrary G (although this is likely since h_{t} is a very rapidly decreasing function at infinity). This difficulty is bypassed by modifying h_{t} in the following way. Let φ ∈ C_{c}^{∞}(ℝ) be such that φ(u) = 1, if |u| ≤ 1/2, and φ(u) = 0, if |u| ≥ 1. Given t > 0, let φ_{t} ∈ C_{c}^{∞}(G(ℝ)) be defined by

φ_{t}(g) = φ(r^{2}(g)/t^{1/2}),

where r(g) is defined by (3.2). Set

h̃_{t}(g) = φ_{t}(g)h_{t}(g), g ∈ G(ℝ), t > 0.

Then h̃_{t} ∈ C_{c}^{∞}(G(ℝ)). Recall that ρ_{∞} denotes the right regular representation of G(ℝ) in

HΓ_{cusp}.

Proposition 4.1. There exist C, c > 0 such that

| Tr(ρ_{∞}(h_{t})) − Tr(ρ_{∞}(h̃_{t}))| ≤ Ce^{−c/\sqrt{t}}

for all 0 < t ≤ 1.

Proof. For π ∈ Π(G(ℝ)) let m_{Γ}(π) denote the multiplicity with which π occurs in the discrete subspace L2_{cusp}(Γ\G(ℝ)) and denote by λ_{π} the Casimir eigenvalue π(Ω) of π. For every π ∈ Π(G(ℝ)) with m_{Γ}(π) ≠ 0 we choose an orthonormal basis e_{π,i}, i ∈ I(π), for the finite-dimensional vector space (H(π) ⊗ V_{σ})^{K_{∞}}. Let Δ_{G} be the Laplace operator on G(ℝ) with respect to the left invariant metric. Note that

Δ_{G} = R(−Ω + 2Ω_{K_{∞}}),

where Ω_{K_{∞}} is the Casimir element of K_{∞}. Let λ_{σ} be the Casimir eigenvalue of σ. We note that

−λ_{π} + 2λ_{σ} ≥ 0

for all π ∈ Π(G(ℝ)) with (H(π) ⊗ V_{σ})^{K_{∞}} ≠ {0} (cf. Lemma 2.6 of [DH]). Let χ_{σ} be the character of σ and let d = dim X. Then for every f ∈ C^{1}(G(ℝ)) with f = f ∗ \overline{χ}_{σ} we get

Tr ρ_{∞}(f) = ∑_{π∈Π(G(ℝ))}m_{Γ}(π) ∑_{i∈I(π)}(ρ_{∞}(f)e_{π,i}, e_{π,i})

= ∑_{π∈Π(G(ℝ))}m_{Γ}(π)(1 − λ_{π} + 2λ_{σ})^{−4d}

× ∑_{i∈I(π)} ∫_{G(ℝ)} ((Id + Δ_{G})^{4d}f)(g)(ρ_{∞}(g)e_{π,i}, e_{π,i}) dg.

By [Mu1, Theorem 0.1] we have

∑_{π∈Π(G(ℝ))}m_{Γ}(π) dim(H(π) ⊗ V_{σ})^{K_{∞}}(1 − λ_{π} + 2λ_{σ})^{−4d} < ∞.

Using this result we get

(4.1)  | Tr ρ_{∞}(f)| ≤ C ∥ (Id + Δ_{G})^{4d}f ∥_{L^{1}(G(ℝ))}

for some constant C > 0, independent of f. Let ψ_{t} = 1 − φ_{t}, t > 0. Then it follows from (4.1) that

| Tr(ρ_{∞}(h_{t})) − Tr(ρ_{∞}(h̃_{t}))| = | Tr(ρ_{∞}(ψ_{t}h_{t})|

≤ C ∥ (Id + Δ_{G})^{4d}(ψ_{t}h_{t}) ∥_{L^{1}(G(ℝ))} . (4.2)

It remains to estimate the right hand side. Let X_{1}, ..., X_{a} be an orthonormal basis of 𝔤(ℝ). Then Δ_{G} = − ∑_{i} X_{i}^{2}. Denote by ∇ the canonical left invariant connection on G(ℝ). Then it follows that there exists C_{1} > 0 such that for all f ∈ C^{∞}(G(ℝ)) we have

(4.3)  |(Id + Δ_{G})^{4d}f(g)| ≤ C_{1} ∑_{j=0}^{8d} ∥ ∇^{j}f(g) ∥, g ∈ G(ℝ).

By Proposition 3.1 there exist constants C, c > 0 such that

∥ ∇^{j}h_{t}(g) ∥≤ Ct^{−(a+j)/2}e^{−cr^{2}(g)/t}, g ∈ G(ℝ),

for j ≤ 8d and 0 < t ≤ 1. Let χ_{t} be the characteristic function of the complement of the interval (−t^{1/4}/\sqrt{2}, t^{1/4}/\sqrt{2}) in ℝ. Recall that ψ_{t}(g) = (1 − φ)(r^{2}(g)/t^{1/2}) and (1 − φ)(u) is constant for |u| ≥ 1. This implies that there exists a constant C_{2} > 0 such that

(4.4)  ∥ ∇^{j}ψ_{t}(g) ∥≤ C_{2}t^{−4d}χ_{t}(r(g)), g ∈ G(ℝ),

for j ≤ 8d and 0 < t ≤ 1. Combining (4.3) and (4.4) we obtain

\begin{aligned}∑_{j=0}^{8d} ∥ ∇^{j}(ψ_{t}h_{t})(g) ∥ & ≤ C_{2}t^{−a/2−8d}χ_{t}(r(g))e^{−c_{2}r^{2}(g)/t} \\ & ≤ C_{3}e^{−c_{3}/\sqrt{t}}e^{−c_{3}r^{2}(g)} \end{aligned}

for all g ∈ G(ℝ) and 0 < t ≤ 1. Finally note that for every c > 0, e^{−cr^{2}(g)} is an integrable function on G(ℝ). Together with (4.2) and (4.3) the proof follows. □

We are now ready to apply the trace formula. Given π ∈ Π(G(ℝ)), let m_{Γ}(π, S) denote the multiplicity with which π occurs in the regular representation of G(ℝ) on HΓ,S_{cusp}. Then it follows from (3.4) that

(4.5)  Tr (ρ_{∞}(h_{t})|HΓ,S_{cusp}) = ∑_{π∈Π(G(ℝ))}e^{tλ_{π}}m_{Γ}(π, S) dim Hom_{K_{∞}}(H(π), V_{σ}^{∗}).

We note that the λ_{π}’s are the eigenvalues of the operator which is induced by ρ_{∞}(Ω) ⊗ Id in HΓ_{cusp}(σ, S) and m_{Γ}(π, S) dim Hom_{K_{∞}}(H(π), V_{σ}^{∗}) is the multiplicity of the corresponding eigenvalue. Let

N Γ_{cusp}(T, σ, S) = ∑_{|λ_{π}|≤T} m_{Γ}(π, S) dim Hom_{K_{∞}}(H(π), V_{σ}^{∗}).

We may now state and prove our main result. Theorem 4.2. Let G be an almost simple, connected and simply connected algebraic group defined over ℚ such that G(ℝ) is non compact. Let d_{S} denote the formal dimension of the Steinberg representation of G_{S} and let

c_{S}(Γ) = c(K_{S})d_{S} vol(K_{S}).

Then, if Z_{Γ} = Γ ∩ Z_{G}(ℝ) ⊂ K_{∞} is such that σ|_{Z_{Γ}} = Id, we have

lim_{T →∞} \frac{N_{cusp}^{Γ}(T, σ, S)}{T^{d/2}} = c_{S}(Γ)\frac{vol(Γ\X)}{(4π)^{d/2}Γ(d/2 + 1)}.

Proof. We apply the trace formula to

f_{t} = h̃_{t} ⊗ e_{fin,S} ⊗ f_{S}.

The assumptions of Corollary 2.3 are satisfied. We first observe that b(h̃_{t}, S) remains bounded. Since G(ℝ) is non compact we have

dim(G(ℝ)/K_{∞}) = d > 0

and hence t^{d/2}b(h̃_{t}, S) → 0 as t → 0. Then Corollary 2.3 yields

t^{d/2} tr(ρ_{∞}(h̃_{t})|HΓ,S_{cusp}) = c(K_{S})t^{d/2} ∑_{γ∈𝔊_{e}} a(γ)O_{γ}(f_{t}) + o(1)

as t → 0. Since the support of f_{Σ} remains in a fixed compact set we already observed in Proposition 2.1 that in the sum over 𝔊_{e} only a finite fixed number of terms may be non zero. Hence it suffices to estimate each term. We first remark that

O_{γ}(f_{t}) = O_{γ}(h̃_{t})O_{γ}(e_{fin,S})O_{γ}(f_{S}).

We shall now use results that are established for the case G = SL(n) in [Mu2], but notation make sense and proofs extend readily for the general case. According to [Mu2, Proposition 7.3], we have

t^{d/2}J_{M}(γ, h̃_{t}) → 0, t → 0,

for M = G and γ ∉ Z_{G}(ℝ), while we know by [Mu2, Lemma 2.3] that t^{d/2}h_{t}(z) has a non zero positive limit for z ∈ Z_{G}(ℝ) :

(4.6)  t^{d/2}h_{t}(z) → \frac{tr σ(z)}{(4π)^{d/2} vol(K_{∞})}.

Since

J_{G}(γ, h) = O_{γ}(h)

it follows that non central elements give a negligible contribution and, since d_{S} = f_{S}(z) , we get

lim_{t→0} t^{d/2} Tr(ρ_{∞}(h̃_{t})|HΓ,S_{cusp}) = \frac{vol(G(ℚ)\G(𝔸)) c_{S}(Γ)}{vol(K_{fin})} lim_{t→0} t^{d/2} ∑_{z∈Z_{Γ}} h_{t}(z)

with

c_{S}(Γ) = c(K_{S}) d_{S} vol(K_{S}) .

We observe that

\frac{vol(G(ℚ)\G(𝔸))}{vol(K_{∞}) vol(K_{fin})} = \frac{vol(Γ\X)}{Card(Z_{Γ})}.

Using (4.6) and the assumption that σ|_{Z_{Γ}} = Id, it follows that the above limit can be rewritten as

lim_{t→0} t^{d/2} tr(ρ_{∞}(h̃_{t})|HΓ,S_{cusp}) = c_{S}(Γ)\frac{dim σ vol(Γ\X)}{(4π)^{d/2}}.

Now, it follows from Proposition 4.1 that

lim_{t→0} t^{d/2} Tr(ρ_{∞}(h̃_{t})|HΓ,S_{cusp}) = lim_{t→0} t^{d/2} Tr(ρ_{∞}(h_{t})|HΓ,S_{cusp}),

and hence we get

lim_{t→0} t^{d/2} Tr(ρ_{∞}(h_{t})|HΓ,S_{cusp}) = c_{S}(Γ)\frac{dim σ vol(Γ\X)}{(4π)^{d/2}}.

We conclude using (4.5) and the Tauberian theorem. □

As explained in the introduction, Theorem 1.1 follows immediately from Theorem 4.2.

Remarks. 1) The constant c_{S}(Γ) is non zero if and only if c(K_{S}) ≠ 0 and this is true whenever K_{S} is small enough so that the space of the Steinberg representation of G_{S} contains non zero K_{S}-invariant vectors. This is the case if K_{p} ⊂ I_{p} a minimal parahoric subgroup for p ∈ S.

2) If the assumption σ|_{Z_{Γ}} = Id in Theorem 4.2 is not satisfied, i.e, if there exists z ∈ Z_{Γ} such that σ(z) ≠ Id, then HΓ_{cusp}(σ) = {0}.

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Institut de Mathématiques de Luminy, Université Aix-Marseille II, Av. de Luminy, Case Postale 907, 13288 Marseille, CEDEX 9, France

E-mail address: labesse@math.jussieu.fr

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

E-mail address: mueller@math.uni-bonn.de