Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

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Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

Alexander Gorodnik, Frédéric Paulin

To cite this version:

Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homoge- neous varieties and diagonal flows. 2011. �hal-00635948v2�

Counting orbits of integral points in families of affine homogeneous varieties

and diagonal flows

Alexander Gorodnik Frédéric Paulin June 19, 2013

Abstract

In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces. ¹

1 Introduction

In this paper, we investigate the asymptotic distribution of integral points on families of homogeneous algebraic varieties using dynamical systems techniques. LetLbe a reductive algebraic group in GL_n(C) defined over Q, and let X_v = Lv with v ∈ Qⁿ be a Zariski- closed set. Borel-Harish-Chandra’s finiteness theorem [BHC, Theo. 6.9] says that the number of orbits of L(Z) in X_v(Z) is finite. We investigate the asymptotic behaviour of this number where the orbits are counted with suitable weights. Namely, to each orbit [u]∈L(Z)\X_v(Z), we associate the Siegel weight

w([u]) = vol StabL(u)(Z)\StabL(u)(R) vol L(Z)\L(R)

and consider

β(v) = X

[u]∈L(Z)\X_v(Z)

w([u]).

This quantity appears naturally in the study of spatial distribution of integral points.

Given a cone Ω ⊂Rⁿ and a sequence of colinear vectors v_n ∈ Qⁿ such that X_v

n(R)∩Ω is bounded with boundary of measure zero and StabL(v_n) is a maximal subgroup of L, it was shown in [EO], when L is semi-simple, that

|X_v

n(Z)∩Ω| ∼vol(Ω)·β(v_n) asβ(v_n)→ ∞. (1)

1Keywords: integral point, homogeneous variety, Siegel weight, counting, decomposable form, norm form, diagonalisable flow, mixing, exponential decay of correlation. AMS codes:37A17, 37A45, 14M17, 20G20, 14G05, 11E20

Under some additional conditions on the varietiesX_v

n, the asymptotics of|X_v

n(Z)∩Ω|can be also computed using the Hardy–Littlewood circle method, so that the quantities β(v_n) can be expressed in terms of local densities (see [BR, Oh1]). However, we emphasise that the asymptotic formula (1) and the Hardy–Littlewood method do not apply in general. In particular, it might happen that β(v_n) does not converge to ∞ even though β(v_n) does on average. Contrarily to the last two references, we also consider non semi-simple cases, which are useful for some examples.

The aim of this paper is to develop a direct argument that establishes an asymptotic formula for the quantitiesβ(v)on average. Our methods use flows on homogeneous spaces and dynamical systems techniques. We show that sums over β(v) can be interpreted as volumes of intersections of two transversal submanifolds in a suitable homogeneous space.

Then we prove that one of these submanifolds becomes equidistributed and deduce an asymptotic formula for the sums of β(v)’s.

Now we proceed to state our result precisely. LetLbe a reductive linear algebraic group defined and anisotropic over Q, let π:L→GL(V)be a rational linear representation ofL defined overQand letΛbe aZ-lattice inV(Q) invariant underL(Z). For every v∈V(Q) whose orbit X_v under L is Zariski-closed in V, the number of orbits of L(Z)inX_v∩Λ is finite. We will count these orbits using appropriate weights. For every u ∈ X_v(Q) with stabiliser L_u inL, define theSiegel weight ofu as

wL,π(u) = vol L_u(Z)\L_u(R) vol L(Z)\L(R) ,

using Weil’s convention for the normalisation of the measures on L_u(R)(depending on the choice of a left Haar measure on L(R) and of a L(R)-invariant measure on X_v(R), see Section 2). These weights generalise the ones occuring in Siegel’s weight formula when L is an orthogonal group (see for instance [Sie1, ERS], and [Vos, Chap. 5] for general L).

A particular case of the main results of this paper is the following one.

Theorem 1 Let G be a simply connected reductive linear algebraic group defined overQ, without nontrivial Q-characters. Let Pbe a maximal parabolic subgroup of Gdefined over Q, and let P= AMU be a relative Langlands decomposition of P, such that A(R)₀ is a one-parameter subgroup (a_s)_s∈R, withλ= log det (Ada₁)_|U>0, where Uis the Lie algebra of U(R). Let ρ : G → GL(V) be a rational representation of G defined over Q such that there exists v₀ ∈ V(Q) whose stabiliser in G is MU. Let L be a reductive algebraic subgroup of Gdefined and anisotropic over Q. Assume that LP is Zariski-open in Gand that for every s∈R, the orbitX_s=ρ(La_s)v₀ is Zariski-closed inV.

Let Λ be a Z-lattice in V(Q) invariant under G(Z), and let Λ^prim be the subset of indivisible elements of Λ. Assume ρ to be irreducible over C. Then there exist c, δ > 0 such that, as t tends to +∞,

X

0≤s≤t

X

[x]∈L(Z)\(X_s∩Λ^prim)

wL,ρ_|L(x) =c e^λt+ O(e^(λ−δ)t).

More precisely, let G,P,A,M,U,V,L, ρ, v₀,(a_s)_s∈R be as above (ρ not necessarily irreducible). Endow G(R) with a left-invariant Riemannian metric, for which the Lie algebras of MU(R) and A(R) are orthogonal, and the orthogonal of the Lie algebra of P(R) is contained in the Lie algebra of L(R).

Theorem 2 There exists δ >0 such that, as ttends to +∞, X

0≤s≤t

X

[x]∈L(Z)\(ρ(L(R)as)v0∩ρ(G(Z))v0)

wL,ρ_|L(x)

= vol MU(Z)\MU(R)

vol(a^Z₁\A(R)₀)

λvol(G(Z)\G(R) e^λt+ O(e^(λ−δ)t).

We will prove a more general version of this result in Section 2 without the maximality condition on P, involving the more elaborate root data of P, and without the simple connectedness assumption on G (up to a slight modification of the Siegel weights), see Theorem 5 and Theorem 15.

We illustrate Theorem 1 by an example, which is new: we give an asymptotic estimate on the (weighted) number of inequivalent integral points on hyperplane sections of affine quadratic surfaces. More examples are given in Section 3.

Corollary 3 Let n ≥ 3, let q : Cⁿ → C be a nondegenerate rational quadratic form, which is isotropic over Q, let ℓ : Cⁿ → C be a nonzero rational linear form, and let L ={g ∈SL_n(C) : q◦g =q, ℓ◦g =ℓ}. For every k∈Q, let Σ_k be the set of primitive x∈Zⁿsuch thatq(x) = 0andℓ(x) =k. Assume that the restriction of q to the kernel ofℓ is nondegenerate and anisotropic over Q. Then there exist c=c(q, ℓ)>0andδ=δ(q) >0 such that, as r→+∞,

X

k∈[1,r]

X

[u]∈(L(Z)∩L(R)0)\Σk

vol (L_u(Z)∩L(R)₀)\(L_u∩L(R)₀)

=c rⁿ⁻²+ O r^n−2−δ .

This result fits into the above program (up to a slight modification of the Siegel weights, see Section 2), by taking V = Cⁿ, Λ = Zⁿ, and π : L → GL(V) the inclusion map, noting that L is semisimple, and defined and anisotropic over Q as a consequence of the assumptions (see Section 3.2 for details, where we also explicit c).

Asymptotics of the number of integral points in affine homogeneous varieties has been extensively studied over the last decades using harmonic analysis and dynamical systems techniques. See for instance [ERS, DRS, EM, EMS, GO, Oh1, EO], as well as the surveys [Bab, Oh2]. Our results are quite different, since we are counting whole orbits, weighted by the Siegel weights, of integral points. The asymptotic formula that we obtain here is similar to the classical asymptotic formula for the number of integral quadratic forms averaged over discriminant, proved by Siegel in [Sie2]. More generally, if V is a prehomogeneous vector space, analogous asymptotic formulas can be deduced from the analytic properties of the corresponding zeta functions that were studied by Sato and Shintani in [SS] (see also the monograph [Kim]). However, the dimension of Gv₀ in Theorem 1 is typically much smaller than the dimension of V, and these methods don’t apply. We also note that we do not assume X_v(R) to be an affine symmetric space or that the stabiliser is a maximal subgroup, contrarily to [DRS] and many other references. Another difference with the counting results of [EMS, Oh1, EO] is that these papers are using the dynamics of unipotent flows, as instead we are using here the mixing property with exponential decay of correlations of diagonalisable flows, in the spirit of [KM1] (see also [EM, BO]). We are using the proof of the main result of [PP] as a guideline.

Acknowledgment. We thank Jean-Louis Colliot-Thélène for discussions of our decomposable form result. The second author thanks the University of Bristol for a very fruitful short stay where this paper was conceived. The first author thanks the University Paris-Sud 11 for a month of invited professor where this paper was completed.

2 Counting Siegel weights

Here are a few notational conventions. By linear algebraic groupG^′ defined over a subfield kofC, we mean a subgroup ofGL_N(C)for someN ∈Nwhich is a closed algebraic subset ofM_N(C) defined overk, and we defineG^′(Z) =G^′∩GL_N(Z). For every linear algebraic groupG^′ defined overR, we denote by G^′(R)₀ the identity component of the Lie group of real points ofG^′. We denote by log the natural logarithm.

Let us first recall Weil’s normalisation of measures on homogeneous spaces. Let G^′ be a unimodular real Lie group, endowed with a transitive smooth left action of G^′ on a smooth manifold X^′, with unimodular stabilisers. A triple (ν_G^′, ν_X^′,(ν_G^′_x)_x∈X^′) of a left Haar measure ν_G^′ onG^′, a left-invariant (Borel, positive, regular) measureν_X^′ onX^′ and of a left Haar measureν_G^′_x on the stabiliserG^′_x of every x∈X^′, iscompatible if, for every x∈X^′, for everyf :G^′ →Rcontinuous with compact support, withf_x :X^′→Rthe map (well) defined by gx7→R

h∈G^′_xf(gh)dν_G^′_x(h) for every g∈G, we have Z

G^′

f dν_G^′ = Z

X^′

fxdν_X^′ .

Weil proved (see for instance [Wei, §9]) that, for every left-invariant measure ν_X^′ on X^′, then

• for every left Haar measure ν_G^′ on G^′, there exists a unique compatible triple (ν_G^′, ν_X^′,(ν_G^′_x)_x∈X^′).

• for every x₀ ∈ X^′, for every left Haar measure ν₀ on G^′_x0, there exists a unique compatible triple (ν_G^′, ν_X^′,(ν_G^′_x)x∈X^′) withν_G^′_x

0 =ν₀.

The following remark should be well-known, though we did not found a precise refer- ence.

Lemma 4 If(ν_G^′, ν_X^′,(ν_G^′_x)x∈X^′)is a compatible triple, then for every ℓ∈G^′ andx∈X^′, with i_ℓ :h7→ℓhℓ⁻¹ the conjugation by ℓ, we have

ν_G^′

ℓx = (i_ℓ)∗ν_G^′_x .

Proof. Let x∈X^′,ℓ∈G^′,H^′ =G^′_x and H^′′ =G^′_ℓx =ℓH^′ℓ⁻¹. Using the left invariance of ν_X^′ for the first inequality and the bi-invariance of the Haar measure on G^′ for the last

one, we have, for everyf :G^′→R continuous with compact support, Z

g^′ℓx∈X^′

Z

h^′′∈H^′′

f(g^′h^′′)d(i_ℓ)_∗ν_G^′_x(h^′′)dν_X^′(g^′ℓx)

= Z

g^′ℓx∈X^′

Z

h^′′∈H^′′

f(ℓg^′h^′′)d(i_ℓ)∗ν_G^′_x(h^′′)dν_X^′(g^′ℓx)

= Z

g^′ℓx∈X^′

Z

h^′∈H^′

f(ℓg^′ℓh^′ℓ⁻¹)dν_G^′_x(h^′)dν_X^′(g^′ℓx)

= Z

gx∈X^′

Z

h^′∈H^′

f ◦i_ℓ(gh^′)dν_G^′_x(h^′)dν_X^′(gx)

= Z

G^′

f◦i_ℓdν_G^′ = Z

G^′

f dν_G^′ .

The result then follows by uniqueness.

In order to deal with non simply connected groups, we introduce a modified version of the Siegel weights.

Let L^′ be a reductive linear algebraic group defined and anisotropic over Q, let π : L^′ → GL(V^′) be a rational linear representation of L^′ defined over Q, let v ∈ V^′(Q) be such that its orbit X^′

v under L^′ is Zariski-closed in V^′, let u ∈ X^′

v(Q) and let L^′

u be the stabiliser of uinL^′. We define the modified Siegel weightof uas

w^′L^′,π(u) = vol (L^′

u(Z)∩L^′(R)₀)\(L^′

u∩L^′(R)₀)

vol (L^′(Z)∩L^′(R)0)\L^′(R)0 , (2) using Weil’s convention for the normalisation of the measures on L^′_u(R)(depending on the choice of a left Haar measure onL^′(R) and of aL^′(R)-invariant measure onX^′_v(R)). Note that the denominator of the standard Siegel weight wL^′,π(u) is an integral multiple (de- pending only onL^′) of the denominator of the modified one, since(L^′(Z)∩L^′(R)₀)\L^′(R)₀ is a connected component of L^′(Z)\L^′(R). But the ratio of the numerator of the Siegel weight by the numerator of the modified one may depend on u.

Let us now describe the framework of our main result. LetGbe a connected reductive linear algebraic group defined over Q. Let P be a (proper) parabolic subgroup of G defined over Q (see for instance [BJ, §III.1], [Spr, §5.2])). Recall that a linear algebraic group defined over Qis Q-anisotropicif it contains no nontrivial Q-split torus.

Recall that there exist a (nontrivial) maximal Q-split torus S in G (contained in P and unique modulo conjugation by an element of P(Q)), such that if Φ^C = Φ^C(G,S) is the root system of G relative toS (seen contained in the set of characters of S), if g^C_β is the root space ofβ ∈Φ^C, then there exist a unique set of simple roots ∆ = ∆P inΦ^C and a unique proper subset I = IP of ∆, such that, with Φ^C₊ the set of positive roots of Φ^C defined by∆andΦ^C_I the set of roots ofΦthat are linear combinations of elements ofI, if A is the identity component of \

α∈I

kerα ,

which is a Q-split subtorus of S, if U is the connected algebraic subgroup of G defined over Qwhose Lie algebra is

u^C= M

β∈Φ^C₊−Φ^C_I

g^C_β ,

thenPis the semi-direct product of its unipotent radical U and of the centraliser ofAin G. Note that A is one-dimensional if P is a maximal (proper) parabolic subgroup of G defined over Q(that is, if ∆−I is a singleton).

Let g be the Lie algebra of G(R). Using the multiplicative notation on the group of characters of S, for everyα ∈∆, we definem_α=m_α,P∈N by

Y

β∈Φ^C₊−Φ^C_I

β^{dimR(gCβ∩g)}= Y

α∈∆

α^mα .

Let (α^∨)α∈∆−I in A(R)₀^∆−I be such that logβ(α^∨) is equal to 1 if α = β and to 0 otherwise. Let Λ^∨ be the lattice in A(R)₀ generated by {α^∨ : α ∈ ∆−I}. For every element T = (tα)α∈∆−I of [0,+∞[^∆−I, let

A_T ={a∈A(R)₀ : ∀α∈∆−I, 0≤log(α(a))≤t_α}.

Recall that by the definition of a relative Langlands decomposition of the parabolic subgroup P defined over Q, there exists a connected reductive algebraic subgroup M of Pdefined over Qwithout nontrivialQ-characters such thatAMis the centraliser ofAin G. In particular, AMis a Levi subgroup of Pdefined overQ,AcentralisesMand is the largest Q-split subtorus of the centre ofAM,AMnormalises U, and

P=AMU.

For every Lie groupG^′ endowed with a left Haar measure, for every discrete subgroup Γ^′ ofG^′, we endow Γ^′\G^′ with the unique measure such that the canonical covering map G^′ →Γ^′\G^′ locally preserves the measures.

In what follows, we will need a normalisation of the Haar measures, which behaves appropriately when passing to some subgroups. We will start with a Riemannian metric onG(R), take the induced Riemannian volumes on the real points of the various algebraic subgroups of G defined overQ that will appear, which will give us the choices necessary for using Weil’s normalisation to define the Siegel weights.

The main result of this paper is the following one.

Theorem 5 LetGbe a connected reductive linear algebraic group defined over Q, without nontrivialQ-characters. LetG=G(R)₀ andΓ =G(Z)∩G. LetPbe a parabolic subgroup of G defined over Q, and let P=AMU be a relative Langlands decomposition of P. Let ρ : G → GL(V) be a rational representation of G defined over Q such that there exists v₀ ∈V(Q) whose stabiliser in G is H=MU. Let L be a reductive algebraic subgroup of G defined and anisotropic over Q.

Assume that LP is Zariski-open in G and that for every a∈A, the orbit ρ(La)v₀ is Zariski-closed in V. Endow G(R) with a left-invariant Riemannian metric, for which the Lie algebras of H(R) and A(R) are orthogonal, and the orthogonal of the Lie algebra of P(R) is contained in the Lie algebra of L(R).

Then there exists δ > 0 such that, as T = (t_α)_α∈∆−I ∈ [0,+∞[^∆−I and min_α∈∆−It_α

tends to +∞, X

a∈AT

X

[x]∈(L(R)0∩Γ)\(ρ(L(R)0a)v0∩ρ(Γ)v0)

w^′L,ρ_|L(x) = vol (H∩Γ)\(H∩G)

vol(Λ^∨\A(R)₀) vol(Γ\G)

Y

α∈∆−I

e^mαtα m_α

1 + O(e^{−δminα∈∆−Itα}) .

Proof. Let us start by fixing the notation that will be used throughout the proof of Theorem 5, and by making more explicit the above-mentionned conventions about the various volumes that occur in the asymptotic formula.

Consider the connected real Lie group G=G(R)₀, its (closed) Lie subgroups A=A(R)0, H =H∩G, L=L(R)0, M =M∩G, P =P∩G, U =U(R). We haveH =M U andP =AM U =M U A, since Aand U are connected. Note thatLis also connected, but H andM are not necessarily connected. We denote by

a, g, h, l, m, p, u

the Lie algebras of the real Lie groups A, G, H, L, M, P, U respectively, endowed with the restriction of the scalar product on g defined by the Riemannian metric of G. Since L is Q-anisotropic, so is L∩P. Since the map L∩P→P/H≃A is defined overQ andA is a Q-split torus, this implies that the identity component of L∩P is contained in H. In particular

l∩h=l∩p. (3)

Note that g=l+psince LP is Zariski-open inG. We have assumed thata is orthogonal to hand that the orthogonalp^⊥ofpis contained inl. In particular, with qthe orthogonal of l∩hin h, we have the following orthogonal decompositions

g=p^⊥ ⊕^⊥ (l∩h)⊕^⊥ q⊕^⊥a, h= (l∩h)⊕^⊥ q, l=p^⊥⊕^⊥ (l∩h), p=h⊕^⊥ a. (4) The left-invariant Riemannian metric onGinduces a left Haar measureω_GonG, and a left-invariant Riemannian metric on every Lie subgroupG^′ ofG, hence a left Haar measure

ω_G^′

on G^′ (which is the counting measure if G^′ is discrete). Note thatA, G, H, L, M, U, L∩H are unimodular: indeed A, G, L, M are reductive and U is unipotent; furthermore, L∩H is the stabiliser of v₀ in L, the orbit of v₀ under L is affine and hence L∩His reductive by [BHC, Theo. 3.5]. But P is not unimodular.

The map A×M ×U → P defined by (a, m, u) 7→ amu is a smooth diffeomorphism (see for instance [BJ, page 273]). We will denote by dω_Adω_H the measure on P which is the push-forward of the product measure by the diffeomorphism(a, h)7→ah. SinceAnor- malisesH, the measuredω_Adω_H is left-invariant byP, so thatdω_P(ah)anddω_A(a)dω_H(h) are proportional. Since these measures are induced by Riemannian metrics, and since a and hare orthogonal, we hence have

dω_P(ah) =dω_A(a)dω_H(h).

Since A normalises U, the group A acts on the Lie algebra U of U by the adjoint repre- sentation. The roots of this linear representation of A are exactly the restrictions to A of the elements β in Φ^C₊−Φ^C_I, with root spaces g^C_β ∩g and a set of simple roots is the set of restrictions of the elements of ∆−I to A (see for instance [BJ, Rem. III.1.14]).

Since A is connected, these roots have value in ]0,+∞[. The map A→R^∆−I defined by a7→(log(α(a)))_α∈∆−I is hence a smooth diffeomorphism. We will denote by Q

α∈∆−Idt_α

the measure on A which is the push-forward of the product Lebesgue measure by the inverse of this diffeomorphism. By invariance, there exists a constant c_A>0such that

dω_A=c_A Y

α∈∆−I

dt_α.

By the definition of Λ^∨, we have c_A= Vol(Λ^∨\A).

Let Γ = G(Z) ∩G, which is a discrete subgroup of G acting isometrically for the Riemannian metric of G by left translations. Let Y_G = Γ\G and let π :G→ Y_G = Γ\G be the canonical projection, which is equivariant under the right actions of G. Then Y_G is a connected Riemannian manifold (for the unique Riemannian metric such that π is a local isometry) endowed with the transitive right action of Gby translations on the right.

To simplify the notation, for every Lie subgroup G^′ ofG, define Y_G^′ =π(G^′),

which is a injectively immersed submanifold in YG, endowed with the Riemannian metric induced by Y_G, and identified with (G^′∩Γ)\G^′ by the map induced by the inclusion ofG^′ inG. Note that Y_L and Y_U are connected, but Y_H and Y_M are not necessarily connected.

For every Lie subgroupG^′ ofG, let

µ_G^′

be the Riemannian measure on Y_G^′, which locally is the push-forward of the left Haar measure ω_G^′.

Since Gand the identity component of MU have no nontrivial Q-character, the Rie- mannian manifoldsY_Gand Y_H have finite volume (see [BHC, Theo. 9.4]) and Y_H is closed inY_G(see for instance [Rag, Theo. 1.13]). SinceLis reductive andQ-anisotropic, the sub- manifold Y_L is compact (see [BHC, Theo. 11.6]). Since U is unipotent, the submanifold Y_U is compact (see for instance [BHC, § 6.10]).

For every Lie subgroup G^′ of G such that Y_G^′ has finite measure (that is, such that Γ∩G^′ is a lattice in G^′), we denote by

µ_G′ = µ_G^′ kµ_G^′k

the finite measure µ_G^′ normalised to be a probability measure. In particular,µ_G,µ_H,µ_L, µ_U are well defined.

For everyT = (t_α)α∈∆−I andT^′ = (t^′_α)α∈∆−I in[0,+∞[^∆−I, let A_[T,T′]={a∈A : ∀α∈∆−I, t_α≤log(α(a))≤t^′_α}.

and P_[T,T^′_]=U M A_[T,T^′_]⁻¹ =HA_[T,T^′_]⁻¹. Define Y_{P[T ,T′]} =π(P_[T,T^′_]), which is a subman- ifold with boundary of Y_G, invariant under the right action of H, since A normalises H.

To shorten the notation, we define

A_T =A_[0,T], P_T =P_[0,T]=HA_T⁻¹ and Y_PT =Y_P[0,T] =π(P_T) =Y_HA⁻¹_T , as well asminT = min_α∈∆−It_α≥0, which measures the complexity ofT and will converge to +∞. We will need to estimate the volume of π(P_T) for µ_P.

Lemma 6 For every T = (t_α)_α∈∆−I in[0,+∞[^∆−I, we have µ_P(Y_PT) = vol(Λ^∨\A(R)₀) kµHk Y

α∈∆−I

e^mαtα m_α .

Proof. Denote by du_β the Lebesgue measure on the Euclidean space g_β =g^C_β ∩g. For any order on Φ^C₊ −Φ^C_I, the map from Q

β∈Φ^C₊−Φ^C_I g_β to U defined by (u_β)_β∈Φ^C

+−Φ^C_I 7→

Q

β∈Φ^C₊−Φ^C_I expu_β is a smooth diffeomorphism, and there exists c_U > 0 such that dω_U is the push-forward by this diffeomorphism of the measure c_UQ

β∈Φ^C₊−Φ^C_I du_β.

For everya∈A, if ia :g7→aga⁻¹ is the conjugation bya, then for everyu_β ∈g_β, we have i_a(expu_β) = exp((Ada)(u_β)) = exp(β(a)u_β). Hence

(i⁻¹_a )∗(ωU) = Y

β∈Φ^C₊−Φ^C_I

β(a)^dimgβωU = Y

α∈∆−I

α(a)^mαωU

by the definition of(m_α)α∈∆and since the elements ofI are trivial onA. SinceAcommutes withM, we hence have (i⁻¹_a )_∗(ω_H) =Q

α∈∆−I α(a)^mαω_H. We have, since Ais unimodular,

dω_P(ha⁻¹) =dω_P(a⁻¹aha⁻¹) =dω_A(a⁻¹)dω_H(aha⁻¹) =dω_A(a)d((i⁻¹_a )_∗ω_H)(h). Since Γ∩P = Γ∩H (see for instance the lines following Proposition III.2.21 in [BJ, page 285]) and A∩H={e}, we have π(Ha)6=π(Ha^′) if a6=a^′. Hence

µ_P(Y_PT) = Z

y∈YH

Z

a∈AT

dµ_P(ya⁻¹) = Z

AT

Y

α∈∆−I

α(a)^mαdω_A(a) Z

YH

dµ_H (5)

=kµHkc_A Y

α∈∆−I

Z _tα

0

e^mαs ds .

Since m_α >0, the result follows.

To simplify the notation, we write ρ(g)x =gx for every g∈ G and x ∈V, we define v_a=av₀ for every a∈A, and we denote byL_x =G_x∩Lthe stabiliser of x inL for every x∈V(R).

Since we have a left Haar measureω_Lon L andω_L∩H on L∩H, Weil’s normalisation gives a L-invariant measure on the homogeneous space L/(L∩H), and hence a left Haar measure on the stabilisersℓ(L∩H)ℓ⁻¹ for everyℓinL, as explained above. As announced, the modified Siegel weightsw^′L,ρ_|L(·)are defined using this Weil’s normalisation, as follows.

For everyℓ∈Land a∈A, ifx=ℓav₀, since H=M U is normalised by A and is the stabiliser of v0 inG, we have

L_x=L∩Stab_Gx=L∩(ℓHℓ⁻¹) =ℓ(L∩H)ℓ⁻¹. (6) Note that

L(R)₀∩L(Z) =L(R)₀∩G(Z) =L(R)₀∩G(R)₀∩G(Z) =L∩Γ, (7) and similarly L(R)₀∩L_x(Z) =L_x∩Γ for every x ∈Lv_a∩Γv₀. Hence the denominator of the modified Siegel weight w^′L,ρ_|L(x) is equal to vol (L∩Γ)\L

= vol(Y_L), using the

measureµ_LonY_Linduced by the Haar measureω_LonL. Its numerator vol (L_x∩Γ)\Lx

is defined by using the measure on (L_x ∩Γ)\Lx induced by the left Haar measure on L_x =ℓ(L∩H)ℓ⁻¹ given by Weil’s normalisation.

We now proceed to the proof of Theorem 5.

Step 1. The first step of the proof is the following group theoretic lemma, which relates the counting function of modified Siegel weights to the counting function of volumes of orbits ofL∩H. We denote with square brackets the (left or right) appropriate orbit of an element.

Lemma 7 For every a∈A, there exists a bijection between finite subsets

Θ_a : (L∩Γ)\(Lv_a∩Γv₀) −→ (Y_L∩Y_Ha⁻¹)/(L∩H) such that for every x∈Lv_a∩Γv₀, if[y] = Θ_a([x]), then

vol (L_x∩Γ)\Lx

= vol y(L∩H)

. (8)

In particular, for every T ∈[0,+∞[^∆−I, we have X

a∈AT

X

[x]∈(L∩Γ)\(Lva∩Γv0)

w^′L,ρ|L(x) = X

a∈AT

X

[y]∈(YL∩YHa⁻¹)/(L∩H)

vol y(L∩H) vol(Y_L) . Proof. Fix a∈ A. First note that the groups L∩Γ and L∩H do preserve the subsets Lv_a∩Γv₀ of V(R) and Y_L∩Y_Ha⁻¹ of Y respectively for their left and right action, since A normalises H. The finiteness of the set (L∩Γ)\(Lva∩Γv₀) follows from Borel-Harish- Chandra’s finiteness theorem as in the introduction. Also recall that H = M U is the stabiliser of v₀ inG.

Define

Θa: [ℓva]7→[π(ℓ)].

Let us prove that this map is well defined and bijective. Let ℓ, ℓ^′ ∈L.

We have ℓv_a ∈ Γv₀ if and only if there exists γ ∈ Γ such that ℓav₀ = γv₀, that is, if and only if there exist γ ∈ Γ and h ∈ H such that ℓ = γha⁻¹, that is, if and only if π(ℓ) ∈ Y_L∩Y_Ha⁻¹. This proves that Θ_a has values in (Y_L∩Y_Ha⁻¹)/(L∩H) and is surjective.

Let us prove thatΘ_a does not depend on the choice of representatives and is injective.

We have [ℓ^′v_a] = [ℓv_a]if and only if there exists γ ∈L∩Γsuch that ℓ^′av₀ =γℓav₀, hence if and only if there exist γ ∈ L∩Γ and h ∈ H such that ℓ^′a = γℓah. Note that this equation implies that aha⁻¹ ∈ L if and only if γ ∈ L. Since A normalises H, we hence have [ℓ^′v_a] = [ℓv_a]if and only ifℓ^′ ∈Γℓ(L∩H), that is if and only if [π(ℓ)] = [π(ℓ^′)].

To prove the second assertion, letℓ∈Lbe such thatx=ℓva∈Γv0and lety =π(ℓ), so thatΘ_a([x]) = [y]. The orbit of y under L∩H inY_Gis the image by the locally isometric map π of the Riemannian submanifold ℓ(L∩H) of G. The left translation by ℓ⁻¹ is an isometry (hence is volume preserving) from ℓ(L∩H) to (L∩H). By Lemma 4, the map g 7→ℓgℓ⁻¹ fromL∩H to ℓ(L∩H)ℓ⁻¹, which is equal toL_x by Equation (6), is measure preserving. Therefore the map ϕ: [z]7→ [zℓ] from (L_x∩Γ)\Lx to y(L∩H) is a measure preserving bijection. This proves the volume equality of Equation (8).

The last claim follows from the other ones, since the numerator of the modified Siegel weight w_L^′_,ρ

|L(x) isvol (L_x∩Γ)\Lx

.

Step 2. The second step of the proof is an equidistribution result, in the spirit of [KM1], saying that the piece Y_PT of orbit ofP equidistributes in Y_G asminT →+∞.

For every smooth Riemannian manifoldZ and q∈N, we denote by C_c^q(Z)the normed vector space of C^q maps with compact support on Z, with normk · k_q.

Proposition 8 There exist q ∈ N and κ > 0 such that for every f ∈ C_c^q(Y_G) and T = (tα)α∈∆−I ∈[0,+∞[^∆−I, we have, as minT tends to +∞,

1 µ_P(Y_PT)

Z

Y_PT

f dµ_P = Z

YG

f dµ_G+ O e^−κminT kfk_q .

To prove the proposition, we will use the disintegration formula already seen in the proof of Lemma 6

Z

Y_PT

f dµ_P = Z

AT

Z

YH

f(ya⁻¹)dµ_H(y) Y

α∈∆−I

α(a)^mα

dω_A(a).

This formula indicates that the proposition would follow from (an averaging of) the equidis- tribution of the translates Y_Ha⁻¹, which is established in Proposition 9 below. To state this proposition, we need to introduce additional notation.

The linear algebraic groupGdecomposes as an almost direct product G=Z(G)G₁· · ·G_s

whereZ(G)is the centre ofG, andG₁, . . . ,G_sareQ-simple connected algebraic subgroups of G. The maximalQ-split torusS decomposes as an almost direct product

S=S₁· · ·S_s

where S_i is a maximal Q-split torus inG_i. We also get an almost direct product decom- position

G=Z(G)G₁· · ·G_s, (9) where Z(G) is the centre of G (which is equal to Z(G)₀ since G is connected and G is Zariski-dense in G) and G_i = G_i(R)₀ for 1 ≤ i ≤ s. Since G has no nontrivial Q- character, and since M is the centraliser of A, this gives corresponding almost direct product decompositions of the Lie groups A=A₁. . . A_s (this one being a direct product), U = U₁. . . U_s, M = Z(G)M₁. . . M_s, H = Z(G)H₁. . . H_s. The set of simple roots ∆ decomposes as a disjoint union

∆ = ∆₁⊔ · · · ⊔∆_s

where ∆_i is a set of simple roots of G_i relatively to S_i, and the positive (closed) Weyl chamber A⁺ inA associated to∆decomposes as

A⁺=A⁺₁ · · ·A⁺_s,

where A⁺_i is the positive (closed) Weyl chamber in A∩G_i associated to ∆_i.

For1≤i≤sand a∈A_i, we define

Ei(a) = exp − max

α∈∆i−Ilogα(a)

>0 (10)

if ∆_i−I 6=∅, and E_i(a) = 0, otherwise. For every κ >0, we also define E^κ(a) =

Xs i=1

E_i(a_i)^κ

for every a∈A⁺ witha₁∈A⁺₁,· · · , a_s ∈A⁺_s anda=a₁· · ·a_s.

Proposition 9 There exist q ∈N andκ >0 such that for allf ∈C_c^q(Y_G) anda∈A⁺, Z

YH

f(ya⁻¹)dµ_H(y) = Z

YG

f dµ_G+ O E^κ(a)kfkq

.

Given a Lie subgroup D of G such that Γ∩D is a lattice in D, we denote by ν_D the normalised right invariant measure on (Γ∩D)\D. Recall that Y_D =π(D) is a closed submanifold ofY_G, and thatµ_Dis the invariant measure onY_Dinduced by the Riemannian metric, with normalised measure µ_D.

We identify (Γ∩H)\H withY_H using the (well defined) maph7→Γh(denoting again by h∈H a representative of a coseth∈(Γ∩H)\H). Since the groups Z(G), H₁,· · · , H_s commute, we also have the map

(Γ∩Z(G))\Z(G)×(Γ∩H₁)\H1· · · ×(Γ∩H_s)\Hs→Y_H

well defined by (h₀, h₁, . . . , h_s) 7→ Γh₀h₁· · ·h_s (using conventions similar to the above one for coset representatives). Then the normalised invariant measures µ_H,µ_H1, . . . , µ_Hs satisfy, for all f ∈C_c(Y_G),

Z

YH

f(y)dµ_H(y) = Z

(Γ∩H)\H

f(Γh)dν_H(h)

= Z

(Γ∩Z(G))\Z(G)×···×(Γ∩Hs)\Hs

f(Γh₀h₁· · ·h_s)dν_Z(G)(h₀)· · ·dν_Hs(h_s). (11) We will prove Proposition 9 by using an inductive argument on the number of factors. We start by analysing the distribution of Y_Uia⁻¹ in Lemma 10 and then the distribution of Y_Hia⁻¹ in Lemma 11.

LetD be a product of almost direct factors of G in the decomposition (9). For every f ∈C⁰

c(Y_G), we define a mapP_Df :Y_G→C by (P_Df)(Γg) =

Z

(Γ∩D)\D

f(Γdg)dν_D(d)

which does not depend on the choice of the representative ofΓg, by the right invariance of νD under D. Note that P_Df is continuous and invariant under the right action of D.

Lemma 10 There existq∈Nandκ1>0such that for everyi∈ {1, . . . , s}withAi6={1}, for every f ∈C_c^q(Y_G) and a∈A⁺_i ,

Z

Y_Ui

f(ya⁻¹)dµ_Ui(y) = (P_G

if)(Γe) + O E_i(a)^κ1kf|_YGik_q .

Proof. For 1 ≤ i ≤ s, we consider the unitary representation of the group G_i on the orthogonal complement of the space of G_i-invariant (hence constant on Y_Gi) functions in the Hilbert space L²(Y_Gi, µ_Gi), whose scalar product we denote byh·,·i_YGi (using the nor- malised measureµ_Gi). We note that for everyf ∈C_c(Y_G), the functionf|Y_Gi−(P_G

if)(Γe) belongs to this space.

We say that a unitary representation of a connected real semisimple Lie group G^′ has the strong spectral gap property if the restriction to every noncompact simple factor of G^′ is isolated from the trivial representation for the Fell topology (see for instance [Cow], [BdlHV, Appendix], [KM2, Appendix] for equivalent definitions and examples, and compare for instance with [Nev, KS] for variations on the terminology). We claim that the above unitary representation ofG_i has the strong spectral gap property. Indeed, ifG_i is simply connected and Γi is a congruence subgroup in Gi, then the strong spectral gap property onΓ_i\Giis a direct consequence of the propertyτ proved in [Clo], see Theorem 3.1 therein. By [KM2, Lemma 3.1], this also implies, when G_i is simply connected, the strong spectral property on Γi\Gi for subgroups Γi that are commensurable with congruence subgroups, and, in particular, for arithmetic subgroups of G_i. Now let p_i :Ge_i →G_i be a simply connected cover of G_i, and letGe_i =Ge_i(R). ThenY_Gi ≃p⁻¹_i (Γ∩G_i)\Ge_i, and the strong spectral gap property for L²(Y_Gi, µ_Gi) follows from the above arguments.

Applying [KM1, Theorem 2.4.3], we deduce that there existq ∈N andκ^′₁, C >0 such that for everyi∈ {1, . . . , s} such that A_i 6={e}, for every φ∈C_c^q(Y_Gi) and a∈A⁺_i ,

(f|Y_Gi −(P_G

if)(Γe))◦a⁻¹, φ

Y_Gi ≤C E_i(a)^κ′1 kf|Y_Gikqkφkq, (12) where Ei(a) is defined in Equation (10).

Let P_i⁻ denote the parabolic subgroup in G_i opposite to U_i. The product map U_i× P_i⁻→G_i is a diffeomorphism between neighbourhoods of the identities. SinceY_Ui =π(U_i) is compact, ifǫ >0is small enough, there exists an openǫ-neighbourhoodΩǫof the identity inP_i⁻such that the product mapY_Ui×Ωǫ→Y_Gi is a diffeomorphism onto its imageY_UiΩ_ǫ. We have (see also Lemma 13)

∀y∈Y_Ui, ∀p∈Ω_ǫ, dµ_Gi(yp) =dµ_Ui(y)dω(p),

for a suitably normalised smooth measure ω on Ω_ǫ. There exists σ >0 (depending on q) such that for every ǫ > 0 small enough, there exists a nonnegative function ψ_ǫ ∈C_c^q(Ω_ǫ)

satisfying Z

Ωǫ

ψ_ǫdω= 1 and kψǫkq= O(ǫ^−σ). Define aC^q function φ_ǫ:Y_Gi →[0,+∞[supported onY_UiΩ_ǫ by

∀y ∈Y_Ui, ∀p∈Ω_ǫ, φ_ǫ(yp) =ψ_ǫ(p).

Then Z

Y_Gi

φ_εdµ_Gi = 1 and kφǫkq= O(ǫ^−σ). Since for all a∈A⁺_i and p∈Ω_ǫ,

d(apa⁻¹, e) = O(ǫ),

we obtain

f|Y_Gi ◦a⁻¹, φ_ǫ

Y_Gi = Z

Y_Ui×Ωǫ

f(ypa⁻¹)ψ_ǫ(p)dµ_Ui(y)dω(p)

= Z

Y_Ui

f(ya⁻¹)dµ_Ui(y) + O

ǫkf|_YGik₁ . Since P_G

if isGi-invariant, P_G

if, φǫ

Y_Gi = (P_G

if)(Γe) Z

Y_Gi

φεdµ_Gi

= (P_G

if)(Γe).

Combining these estimates with (12) (we may assume that q≥1), we conclude that Z

Y_Ui

f(ya⁻¹)dµ_Ui(y) = (P_G

if)(Γe) + O (ǫ+E_i(a)^κ′1ǫ^−σ)kf|Y_Gikq

.

Finally, taking ǫ=E_i(a)^{κ′1/(1+σ)} which is small if alies outside a compact subset of A⁺_i , we deduce that

Z

Y_Ui

f(ya⁻¹)dµ_Ui(y) = (P_G

if)(Γe) + O

E_i(a)^{κ′1/(1+σ)}kf|Y_Gikq

,

as required.

Lemma 11 There exist q ∈ N and κ2 > 0 such that for every f ∈C_c^q(YG) and a ∈A⁺_i , for every i∈ {1, . . . , s}, we have

Z

Y_Hi

f(ya⁻¹)dµ_Hi(y) = (P_G

if)(Γe) + O E_i(a)^κ2kf|_YGik_q .

Proof. We first observe that if A_i = {e}, then H_i =G_i, and the claim of the lemma is obvious. Now we assume that A_i 6={e}in which case Lemma 10 applies.

Let N_i = (Γ∩M_i)\Mi. The space Y_Hi = π(U_iM_i) is a bundle over N_i with fibres isomorphic toYUi, and the invariant measure µ_Hi onYHi decomposes with respect to this structure. Explicitly, for everym∈M_i, the integralsR

Y_Uif(ym)dµ_Ui(y)for allf ∈C_c(Y_G) define aU_i-invariant probability measure onY_Uim, which depends only on the cosetn= [m]

of m inN_i = (Γ∩M_i)\M_i, and theH_i-invariant probability measure on Y_Hi is given by R

Ni

R

Y_Uif(ym)dµ_Ui(y)

dν_Mi([m]) for all f ∈ C_c(Y_G). Hence, denoting again by n any representative of a coset ninN_i, since A centralisesM,

Z

Y_Hi

f(ya⁻¹)dµ_Hi(y) = Z

Ni

Z

Y_Ui

f(yna⁻¹)dµ_Ui(y)dν_Mi(n)

= Z

Ni

Z

Y_Ui

f(ya⁻¹n)dµ_Ui(y)dν_Mi(n).

For m ∈ M_i and f ∈ C_c(Y_G), we consider the function f_m : Y_G → C defined by y7→f(ym). We note that there existc₁, C^′ >0 such that for every f ∈C_c^q(Y_G), we have

kfm|Y_Gikq≤C^′e^c1d(e,m)kf|Y_Gikq.