ERGODIC THEORETIC PROOF OF EQUIDISTRIBUTION OF HECKE POINTS

ALEX ESKIN AND HEE OH

1. Introduction

Let G be a connected non-compact ℚ-simple real algebraic group defined over ℚ, that is, the identity component of the group of the real points of a connected ℚ-simple algebraic group which is ℝ-isotropic. Let Γ ⊂ G(ℚ) be an arithmetic subgroup of G. As is well known, Γ has finite co-volume in G [BH]. Denote by μ_{G} the G-invariant Borel probability measure on Γ\G. Two subgroups Γ_{1} and Γ_{2} of G are said to be commensurable with each other if Γ_{1} ∩ Γ_{2} has a finite index both in Γ_{1} and Γ_{2}. The commensurator group Comm(Γ) of Γ is defined as follows:

Comm(Γ) = {g ∈ G : Γ and gΓg^{−1} are commensurable with each other}.

Since Γ is an arithmetic subgroup, Comm(Γ) contains G(ℚ) and in particular, Γ has an infinite index in Comm(Γ).

For an element a ∈ Comm(Γ), the Γ-orbit Γ\ΓaΓ in Γ\G has finitely many points called Hecke points associated with a. We set

deg(a) = #Γ\ΓaΓ.

It is easy to see that deg(a) = [Γ : Γ ∩ a^{−1}Γa].

In this paper, we are interested in the equidistribution problem of the Hecke points Γ\ΓaΓ as deg(a) → ∞. Namely, for any continuous function f in Γ\G with compact support, any x ∈ Γ\G, and for any sequence a_{i} ∈ Comm(Γ) with deg(a_{i}) → ∞,

(1.1)  Does T_{a_{i}}(f)(x) := \frac{1}{deg(a_{i})} ∑_{γ∈Γ\Γa_{i}Γ} f(γx) converge to ∫_{Γ\G} f dμ_{G}?

We remark that T_{a_{i}}(f) is well defined as a function on Γ\G. When G is simple, a_{i} ∈ G(ℚ) and Γ is a congruence subgroup, this was answered in the affirmative way in [COU], based on the adelic interpretation of L^{2}(Γ\G) and an information on the local harmonic analysis of the p-adic groups G(ℚ_{p}). For smooth functions, the methods in [COU] also give rate of the convergence. Several partial results in this direction were known (see [C], [CU], [Sa], [GM], etc.)

It was pointed out by Burger and Sarnak [BS, ”Theorem 5.2”] in 1991 that the equidistribution of Hecke points follows from Ratner’s measure classification theorem [Ra] provided a_{i}’s converge to an element not belonging to Comm(Γ). However Burger and Sarnak did not give a detailed proof of this claim: this was done by Dani and Margulis [DM2, Corollary 6.2] in 1993, where they deduced the assertion from their ergodic results built up on Ratner’s fore-mentioned theorem.

The first author is partially supported by Packard foundation.

The second author partially supported by NSF grants DMS 0070544 and DMS 0333397.

Unlike the method used in [COU], this ergodic theoretic method does not provide a rate information of the equidistribution. However it works for much more general cases, for instance, Γ can be a non-congruence subgroup and a_{i}’s need not be necessarily elements of G(ℚ).

Our main purpose of this paper is to present an ergodic theoretic proof of the following result:

Theorem 1.2. Let G be a connected non-compact ℚ-simple real algebraic group defined over ℚ and Γ ⊂ G(ℚ) an arithmetic subgroup of G. Let {a_{i} ∈ Comm(Γ)} be a sequence such that lim_{i→∞} deg(a_{i}) = ∞. Then for any bounded continuous function f on Γ\G and for any x ∈ Γ\G,

lim_{i→∞} T_{a_{i}}(f)(x) = ∫_{Γ\G} f(g) dμ_{G}(g).

2. Limits of H-invariant measures

Let G be a connected real semisimple algebraic group defined over ℚ and Γ ⊂ G(ℚ) an arithmetic subgroup of G. Let H be a connected real non-compact semisimple ℚ-simple subgroup of G. Then Γ ∩ H is an irreducible (H being ℚ-simple) Zariski dense lattice in H.

Let {g_{m} ∈ G} be a sequence such that g_{m}^{−1}Γg_{m} ∩ H is commensurable with Γ ∩ H. It follows that each Γ\Γg_{m}H is closed [Rag] and there exists the unique H-invariant probability measure, say ν_{m}, in Γ\G supported on Γ\Γg_{m}H.

Let Y denote Γ\G if Γ\G is compact; and otherwise the one point compactification Γ\G ∪ {∞}. The space P(Y) of the probability measures on Y equipped with the weak^{∗}-topology is weak^{∗} compact.

Our basic tools for the proof of Theorem 1.2 are the following two propositions: Denote by H_{N} the unique maximal connected normal subgroup of H with no compact factors.

Proposition 2.1. Suppose that g_{m}H_{N}g_{m}^{−1} is not contained in any proper parabolic ℚ-subgroup of G for each m. Then every weak limit of {ν_{m} : m ∈ ℕ} in P(Y) is supported in Γ\G.

This proposition is shown in [EO, Proposition 3.4] based on theorems of Dani and Margulis ([DM1, Theorem 2] and [DM2, Theorem 6.1]).

Proposition 2.2. Suppose that ν_{m} weakly converges to a measure ν in P(Γ\G) as m → ∞. Then there exists a closed connected subgroup L of G containing H such that

(1) ν is an L-invariant measure supported on Γ\Γc_{0}L for some c_{0} ∈ G;

(2) Γ ∩ c_{0}Lc_{0}^{−1} is a Zariski dense lattice in c_{0}Lc_{0}^{−1} and hence in particular c_{0}Lc_{0}^{−1} is defined over ℚ;

(3) there exist m_{0} ∈ ℕ and a sequence {x_{m} ∈ Γg_{m}H} converging to c_{0} as m → ∞ such that c_{0}Lc_{0}^{−1} contains the subgroup generated by {x_{m}Hx_{m}^{−1} : m ≥ m_{0}}.

This proposition is deduced from the following theorem of Mozes and Shah:

Theorem 2.3 (MS, Theorem 1.1). Let {u_{i}(t)}_{t∈ℝ}, i ∈ ℕ be a sequence of unipotent oneparameter subgroups of G and let {ν_{m} : m ∈ ℕ} be a sequence in P(Γ\G) such that each ν_{i} is an ergodic {u_{i}(t)}-invariant measure. Suppose that ν_{m} → ν in P(Γ\G) and let x ∈ supp(ν). Then the following holds:

(1) supp(ν) = xΛ(ν) where Λ(ν) = {g ∈ G : νg = ν}.

(2) Let g′_{i} → e be a sequence in G such that for every i ∈ ℕ, xg′_{i} ∈ supp(ν_{i}) and the trajectory {xg′_{i}u_{i}(t)} is uniformly distributed with respect to ν_{i}. Then there exists an i_{0} ∈ ℕ such that for all i ≥ i_{0},

supp(ν_{i}) ⊂ supp(ν)g′_{i}.

(3) ν is invariant and ergodic for the action of the subgroup generated by the set {g′_{i}u_{i}(t)g′_{i}^{−1} : i ≥ i_{0}}.

Proof of Proposition 2.2 Since g_{m}^{−1}Γg_{m} ∩ H is an irreducible lattice in H, every noncompact simple factor of H acts ergodically on each Γ\Γg_{m}H with respect to ν_{m}. There exists a unipotent one-parameter subgroup U := {u(t)} in H_{N} not contained in any proper closed normal subgroup of H_{N} (cf. Lemma 2.3 [MS]). Then by Moore’s ergodicity theorem (cf. Theorem 2.1 in [BM]), U acts ergodically with respect to each ν_{m}. Moreover by the Birkhoff ergodic theorem, the following subset R has the zero co-measure in H:

{h ∈ H : Γ\Γg_{m}hu(t) is uniformly distributed in Γ\Γg_{m}H w. r. t. ν_{m} for each m ∈ ℕ}.

Hence for any h ∈ R and for any continuous bounded function f on X with compact support, we have

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(g_{m}hu(t)) dt = ∫_{X} f dν_{m}.

If we set L = Λ(ν), we have that ν is supported on Γ\Γc_{0}L for some c_{0} ∈ supp(ν) by Theorem 2.3(1). There exist γ_{m} ∈ Γ and h_{m} ∈ R such that γ_{m}g_{m}h_{m} → c_{0} as m → ∞. If we set

x_{m} := γ_{m}g_{m}h_{m} and g′_{m} := c_{0}^{−1}x_{m},

then

lim_{m→∞} g′_{m} = e

and Γ\Γc_{0}g′_{m}u(t) = Γ\Γg_{m}h_{m}u(t) is uniformly distributed with respect to ν_{m}. By Theorem 2.3(2), there exists an m_{0} ∈ ℕ such that for all m ≥ m_{0}

supp(ν_{m}) ⊂ supp(ν)g′_{m}, or equivalently Γ\Γg_{m}H ⊂ Γ\Γc_{0}Lg′_{m}.

Hence Γx_{m}Hx_{m}^{−1} ⊂ Γ(c_{0}Lc_{0}^{−1}). By the connectedness of H, we may assume that c_{0}Lc_{0}^{−1} is connected and

{x_{m}Hx_{m}^{−1} : m ≥ m_{0}} ⊂ c_{0}Lc_{0}^{−1}.

By Theorem 2.3(3), the subgroup generated by the set {x_{m}Hx_{m}^{−1} : m ≥ m_{0}} acts ergodically on Γc_{0}Lc_{0}^{−1}. Hence c_{0}Lc_{0}^{−1} is the smallest closed subgroup containing the subgroup generated by the set {x_{m}Hx_{m}^{−1} : m ≥ m_{0}} such that the orbit Γc_{0}Lc_{0}^{−1} is closed. This proves (3). The second claim (2) follows from [MS, Proposition 2.1].

3. Proof of Theorem 1.2

For a G-space X and a subgroup M of G, P(X)^{M} denotes the space of M-invariant Borel probability measures on X. We recall the ergodic theoretic approach suggested in [BS]. Let Δ(G) be the diagonal embedding of G into G × G, that is, Δ(G) = {(g, g) : g ∈ G}. For each ν ∈ P(Γ\G)^{Γ}, the measure ν̃ defined by

ν̃(f) := ∫_{Γ\G} ∫_{Γ\G} f(g, hg) dμ_{G}(g)dν(h)

for any bounded continuous function f on Γ\G×Γ\G is a Δ(G)-invariant probability measure on Γ\G × Γ\G. Moreover the map ν ↦ ν̃ is a homeomorphism from P(Γ\G)^{Γ} to P(Γ\G × Γ\G)^{Δ(G)}. We make the following simple observation:

• For a ∈ Comm(Γ), if we set ν_{a} = \frac{1}{deg(a)} ∑_{y∈Γ\ΓaΓ} δ_{y} where δ_{y} denotes the standard delta measure with support y, then ν̃_{a} is the (unique) Δ(G)-invariant probability measure supported on [(e, a)]Δ(G) ⊂ (Γ × Γ)\(G × G).

• The element a ∈ G is contained in Comm(Γ) if and only if the orbit [(e, a)]Δ(G) is

closed and supports a finite Δ(G)-invariant measure. Set X = (Γ × Γ)\(G × G) and consider its one point compactification X ∪ {∞}. By the

fact that P(X ∪{∞}) is compact with respect to weak ^{∗}-topology and the above observation, it suffices to show that, assuming the sequence {ν̃_{a_{i}}} weakly converging to ν̃ in P(X ∪ {∞}), the limit ν̃ is G×G-invariant and supported on X. Note that for each i, (e, a_{i})Δ(G)(e, a_{i}^{−1})∩ (Γ × Γ) is commensurable with Δ(G) ∩ (Γ × Γ). It follows that (e, a_{i})Δ(G)(e, a_{i}^{−1}) contains a Zariski dense subset contained in Δ(G)(ℚ). This implies that (e, a_{i})Δ(G)(e, a_{i}^{−1}) is a ℚ-subgroup of G × G (cf. [Zi, Prop. 3.18]) and ℚ-simple as well. Moreover, it is easy to see that the unique maximal connected normal subgroup of (e, a_{i})Δ(G)(e, a_{i}^{−1}) with no compact factors cannot be contained in any proper parabolic ℚ-subgroup of G × G for each i. Therefore by Proposition 2.1, ν̃ is supported on X. Also by applying Proposition 2.2, ν̃ is either G×G-invariant or Δ(G)-invariant supported on xΔ(G) for some x ∈ X. Suppose that the latter case happens. Then Proposition 2.2 also says that there exist i and y_{j} ∈ Γ×Γ such that y_{j}(e, a_{j})Δ(G)(e, a_{j}^{−1})y_{j}^{−1} = (e, a_{i})Δ(G)(e, a_{i}^{−1}) for all j ≥ i. That is, (e, a_{i}^{−1})y_{j}(e, a_{j}) belongs to the normalizer of Δ(G) in G × G for all j ≥ i. Since Δ(G) has a finite index in its normalizer in G×G, we have (e, a_{i}^{−1})y_{j}(e, a_{j}) ∈ Δ(G) and hence [(e, a_{i})]Δ(G) = [(e, a_{j})]Δ(G) for infinitely many j. This implies that Γa_{i} = Γa_{j} and hence deg(a_{j}) is constant for infinitely many j. This is a contradiction, since deg(a_{i}) tends to ∞ as i → ∞. Therefore ν̃ is the G × G-invariant probability measure supported on X, as desired.

References

[BH] A. Borel and Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. Math., Vol 75, 1962, pp. 485–535.

[BM] M. B. Bekka and M. Mayer Ergodic theory and Topological Dynamics of group actions on homogeneous spaces, LMS Lecture Note series 269, Cambridge

[BS] M. Burger and P. Sarnak Ramanujan duals II, Invent. Math., Vol 106, 1991, pp. 1–11.

[C] P. Chiu Covering with Hecke points, J. Number theory. Vol 53, 1995, pp. 25-44

[COU] L. Clozel, H. Oh and E. Ullmo Hecke operators and equidistribution of Hecke points, Invent. Math., Vol 144, 2001, pp. 327–351.

[CU] L. Clozel and E. Ullmo Equidistribution des pointes de Hecke, Preprint (1999)

[DM1] S. G. Dani and G. Margulis Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces, Proc. Indian. Acad. Sci. (Math. Sci), Vol 101, No 1, 1991, pp. 1–17.

[DM2] S. G. Dani and G. Margulis Limit distribution of orbits of unipotent flows and values of quadratic forms, Advances in Soviet Math., Vol 16, 1993, pp. 91–137.

[EO] A. Eskin and H. Oh Representations of integers by an invariant polynomial and unipotent flows, Preprint (2002)

[GM] D. Goldstein and A. Mayer On the equidistribution of Hecke points, Form Math., Vol 15, 2003, pp. 165–189.

[MS] S. Mozes and N. Shah On the space of ergodic invariant measures of unipotent flows, Ergod. Th. & Dynam. Sys., Vol 15, 1995, pp. 149–159.

[Ra] M. Ratner On Raghunatan’s conjecture, Ann. Math., Vol 134, 1991, pp. 545–607 Springer-Verlag

[Rag] M. S. Raghunathan Discrete subgroups of Lie groups, Ergebnisse Der Mathematik, Springer-Verlag [Sa] P. Sarnak Diophantine problems and linear groups, Proc. Int. Cong. Math. No. 1. 1991, pp. 459– 471 [Zi] R. Zimmer Ergodic theory and semisimple groups, Birkhäuser, Boston-Basel-Stuttgart

Mathematics Department, University of Chicago, Chicago, IL 60637

   E-mail address: eskin@math.uchicago.edu

Mathematics Department, Princeton University, Princeton, NJ 08544, Current address: Math 253-37, Caltech, Pasadena, CA 91125

   E-mail address: heeoh@its.caltech.edu