ON QUANTUM UNIQUE ERGODICITY FOR Γ\ℍ × ℍ

ELON LINDENSTRAUSS

1. Introduction

The purpose of this note is to point out a connection between the Quantum Unique Ergodicity (QUE) conjecture of Z. Rudnick and P. Sarnak [RS] (or, more precisely, natural higher rank generalizations of this conjecture) and conjectures of H. Furstenberg (unpublished), A. Katok and R. Spatzier [KS] and G. Margulis [M] regarding the scarcity of measures invariant under natural ℝ^{d} actions (d ≥ 2) in the ergodic theory of Lie groups, as well as improve and generalize on a related result of Rudnick and Sarnak (Theorem 1.6) .

Let M be a compact manifold, Δ the Laplacian on M.

Conjecture 1.1 (QUE [RS]). If M is of negative curvature, then for any sequence of eigenfunctions ϕ_{i} of the Laplacian, normalized to have L^{2}-norm 1, such that the eigenvalues λ_{i} tend to −∞, the probability measures |ϕ_{i}(x)|^{2}d vol(x) converge in the weak^{∗} topology to the Riemannian volume vol(M)^{−1}d vol.

(Recall that μ_{i} converge weak^{∗} to μ if for every continuous function with compact support,

∫ fd μ_{i} ⟶ ∫ fd μ

as i → ∞.) A similar conjecture can be stated also in the finite volume case [Sa].

A. Schnirelman [Sc], Y. Colin de Verdière [CdV], and S. Zelditch [Ze] showed for compact manifolds with ergodic geodesic flow (in particular for manifolds of negative curvature) that if one omits a subsequence of density zero of eigenvalues, the corresponding eigenfunctions obey the above conjecture. For the special case of M = Γ\ℍ with Γ a (congruence) arithmetic lattice in P SL_{2}(ℝ) ⊂ Isom(ℍ), and if the ϕ_{i} are assumed to be eigenfunctions of all Hecke operators associated with Γ, it follows from an identity of Watson [Wa] that the Grand Riemann Hypothesis implies that |ϕ_{i}(x)|^{2}d vol(x) converge weak^{∗} to vol(M)^{−1}d vol. In fact, the GRH gives a best possible rate of convergence of these measures. (ℍ denotes the hyperbolic upper half plane; for a definition of the Hecke operators see below.)

Date: February 15, 2001.

Consider now the analogous situation in the higher rank case. Let G be a semi-simple Lie group without compact factors and with finite center, Γ an irreducible lattice, and K a maximal compact subgroup of G. Take the symmetric space G/K as well as the finite volume quotient manifold Γ\G/K. The Laplacian on G/K is G-invariant, but unless G is of rank 1, it does not generate the algebra of all G-invariant differential operators. In this paper we present some evidence to the following conjecture, for the special case G = P SL_{2}(ℝ) × P SL_{2}(ℝ).

Conjecture 1.2. Let G, Γ, and K be as above. Let ϕ_{i} ∈ C^{∞}(Γ\G/K) be a sequence of eigenfunctions of the algebra of G-invariant differential operators, normalized to have L^{2}-norm 1, such that the eigenvalues of the Laplacian tend to −∞. Then the probability measures |ϕ_{i}(x)|^{2}d vol(x) converge weak^{∗} to the normalized Riemannian volume.

This conjecture can be weakened by imposing additional conditions on the sequence ϕ_{i}. First we note that whenever Γ is arithmetic, and in particular in the higher rank case, there are additional natural Ginvariant operators, the Hecke operators, commuting with the invariant differential operators, which can be defined as follows: Take a in the commensurator of Γ, set L ⊂ G to be a set of representatives of the cosets of

Γ\ΓaΓ,

and now define the Hecke operator T_{a} by

T_{a}: f(x) ↦ ∑_{g∈L} f(gx).

A related more number theoretic definition of the Hecke operator can be found in §5.

• One can assume e.g. that the absolute value of the eigenvalues of the ϕ_{i} with respect to any G-invariant differential operator tends to ∞.

• As in the hyperbolic case one may assume that the ϕ_{i} are eigenfunctions of some or all of the Hecke operators.

We now restrict ourselves to the case of the rank 2 locally symmetric manifold M = Γ\ℍ × ℍ with Γ an irreducible lattice in

P SL_{2}(ℝ) × P SL_{2}(ℝ) ⊂ Isom(ℍ × ℍ),

ON QUE FOR Γ\ℍ × ℍ 3

although everything written here can be generalized to products of arbitrarily many upper half planes and probably without much difficulty to products of hyperbolic 2-spaces and 3-spaces. Here the algebra of invariant differentiable operators is generated by the partial Laplacian operators Δ_{1} and Δ_{2}. By partial Laplacian operators we mean the differential operators on ℍ × ℍ defined by

Δ_{1}(g ⊗ f) = Δ_{ℍ}(g) ⊗ f g, f ∈ C^{∞}(ℍ)
Δ_{1}(g ⊗ f) = g ⊗ Δ_{ℍ}(f),

Δ_{ℍ} being the hyperbolic Laplacian on ℍ. The Laplacian on Γ\ℍ × ℍ is Δ_{1} + Δ_{2}. Furthermore, it is clear that Δ_{1} and Δ_{2} commute.

Let ϕ_{i} be a sequence of eigenfunctions of Δ_{1} and Δ_{2} with eigenvalues λ_{1,i} and λ_{2,i} respectively, such that the eigenvalues λ_{1,i} + λ_{2,i} of the complete Laplacian tend to −∞. Assume that |ϕ_{i}(x)|^{2}d vol(x) tends to some probability measure μ′ on M. The micro local lift is the basic construction employed by Schnirelman, Colin de Verdière, and Zelditch in the proof of their result mentioned earlier, and in this particular context will enable us to lift μ′ to a probability measure μ on the unit cotangent bundle S^{∗}M, or more precisely, allows us to construct from a subsequence of the ϕ_{i} a measure μ that projects to μ′. A basic property of the micro local lift is that the measure μ constructed in this way will be invariant under a 1-parameter flow — the geodesic flow.

However, in this case we can do better. As long as both λ_{1,i} and λ_{2,i} tend to −∞, we can lift μ′ to a measure μ on the 6-dimensional space Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ) (whereas S^{∗}M is 7-dimensional). We define this lift in §3, where the main ingredient is a simplified version of Wolpert’s elementary construction of the Zelditch micro local lift for quotients of ℍ. As we show in §4, the measure μ so obtained is invariant under the action from the right by the 2-parameter Cartan subgroup A < P SL_{2}(ℝ) × P SL_{2}(ℝ) where

A = \{((\begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix}) , (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix})) : t, s > 0\} . (1.1)

In particular, we prove the following:

Theorem 1.3. Let ϕ_{i} be a sequence of eigenfunctions of Δ_{1} and Δ_{2} such that the eigenvalues λ_{1,i} and λ_{2,i} both tend to −∞. Then any weak^{∗} limit of |ϕ_{i}(x)|^{2}d vol(x) is the projection of an A invariant measure on Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ).

It is reasonable to believe that this theorem should be true in the more general context considered in Conjecture 1.2.

While there are many measures on S^{∗}M invariant under the geodesic flow, including many ‘wild’ ergodic measures which are supported on sets of fractal Hausdorff dimension, it is expected that the A invariant measures on Γ\P SL_{2}(ℝ)×P SL_{2}(ℝ) are extremely rare – we quote the following conjecture of Margulis (as mentioned earlier in this introduction, there are similar conjectures by other authors):

Conjecture 1.4 ([M, Conjecture 2]). Assume that G is a Lie group, Γ is a lattice in G, H is connected and generated by elements g such that all the eigenvalues of Ad_{G}(g) are real. Let μ be an H-invariant H-ergodic measure on G/Γ. Suppose that for no x ∈ supp μ there exists a closed connected subgroup F < G and a continuous epimorphism of Lie groups ψ : F →→ L such that H ⊂ F, F x is closed in Γ\G, ψ(F_{x}) is closed in L (where F_{x} denotes the stabilizer of x in F), and ψ(H) is a one parameter subgroup of L containing no nontrivial Ad_{L}-unipotent elements.

Then μ is algebraic, i.e. there is some closed connected F < G such that F x is closed in Γ\G and μ is Haar measure on xF ≅ (F ∩ x^{−1}Γx)\F.

For the special case of the action of the diagonal group A on Γ\P SL_{2}(ℝ)× P SL_{2}(ℝ), Conjecture 1.4 implies for uniform Γ

Conjecture 1.5. Any A ergodic invariant probability measure on Γ\P SL_{2}(ℝ)× P SL_{2}(ℝ) is either Haar measure or Haar measure on a compact A orbit.

We present a fairly standard argument deducing the above conjecture for uniform lattices from Conjecture 1.4 in the appendix. In addition to these conjectures, the results of [KS] give substantial restrictions on such measures unconditionally.

In section §5 we consider the question whether algebraic measures supported on compact A-orbit can appear in the decomposition of the A-invariant measure μ we obtain by lifting in Theorem 1.3 into its ergodic components, or what amounts to the same thing, whether the measure of a compact A-orbit (i.e. an invariant torus) can be strictly positive. Note that there are a coutable number of such invariant tori.

Several authors dealt with the analogous question in the rank 1 case; in this case the analogue of a compact A-orbit are the periodic orbits of the geodesic flow on S^{∗}(Γ\ℍ) ≅ Γ\SL_{2}(ℝ) . Rudnick and Sarnak showed in the P SL_{2}(ℝ) case for certain uniform arithmetic lattices that if ϕ_{i} are eigenfunctions of the hyperbolic Laplacian on Γ\ℍ and also of all Hecke operators then the lift of any limiting measure is not supported on a finite set of compact orbits [RS]. Wolpert proved that if the singular support of a limiting measure of lifts of Hecke eigenforms on P SL_{2}(ℤ)\P SL_{2}(ℝ) is compact, then the measure of any periodic orbit is 0 (indeed, under this assumption he proves a somewhat stronger statement) [Wo].

Here we show how to extend Rudnick and Sarnak’s argument to show both for the geodesic flow in the rank 1 case and for the A-action on Γ\P SL_{2}(ℝ)×P SL_{2}(ℝ) that the measure of a compact orbit according to a measure which is the lift of a limiting measure is 0. We do this for a fairly general class of lattices, namely uniform irreducible congruence lattices in SL_{2}(ℝ)^{n}. In particular we obtain the following:

Theorem 1.6. Let ϕ_{j} be a sequence of eigenfunctions on (say) • Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ) of both Δ_{1} and Δ_{2},

• Γ\P SL_{2}(ℝ) of the Laplacian with all eigenvalues tending in absolute value to infinity as j → ∞, and assume further that ϕ_{j} are also eigenfunctions of all the Hecke operators, with Γ a uniform congruence lattice (see section §5 for more details). Let A be as in (1.1) in the former case, and the subgroup of diagonal matricies of P SL_{2}(ℝ) in the latter. Then the measure of any compact A-orbit (i.e. an A-invariant torus or a periodic geodesic) according to the limiting measure μ obtained in Theorem 1.3 is 0.

Corollary 1.7. Conjecture 1.4 implies Conjecture 1.2 for Γ\ℍ × ℍ and Γ a uniform lattice as above.

Another unconditional corollary of Theorem 1.6 can be obtained by combining it with Theorem 1.1 from Lindenstrauss and Weiss [LW] (note that while stated for SL_{n}(ℝ), the proof of this result can be easily seen to work also SL_{2}(ℝ) × SL_{2}(ℝ) , which is in fact simpler than the general case). For the special case of Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ), this theorem implies that the only A invariant closed subset of Γ\P SL_{2}(ℝ)×P SL_{2}(ℝ) that contains a non isolated compact A orbit is Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ). Thus we get the following unconditional result:

Corollary 1.8. Under the conditions of Theorem 1.6, if the support of μ contains a compact A-orbit then

supp μ = Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ).

2. Notations

We let G_{1} ≅ G_{2} ≅ P SL_{2}(ℝ), G = G_{1} × G_{2}. G_{1} and G_{2} can be considered as subsets of G in the obvious way. Take Γ to be an irreducible lattice in G. Let K_{1}, K_{2}, K = K_{1} × K_{2} be the usual maximal compact subgroups of G_{1}, G_{2} and G respecitively. We will use the Iwasawa decomposition to represent every g ∈ SL_{2}(ℝ) × SL_{2}(ℝ) as

g = (g_{1}, g_{2})

g_{j} = (\begin{matrix}y_{j}^{1/2} & 0 \\ 0 & y_{j}^{−1/2} \end{matrix}) (\begin{matrix}1 & x_{j} \\ 0 & 1 \end{matrix}) (\begin{matrix}cos θ_{j} & sin θ_{j} \\ −sin θ_{j} & cos θ_{j} \end{matrix}) j = 1, 2.

Of course, this also parametrizes G.

This representation, in particular, gives rise to a coordinate system on ℍ × ℍ = G/K by (y_{1}, x_{1}, y_{2}, x_{2}) which is identical with the usual coordinates on ℍ × ℍ that is (x_{1} + iy_{1}, x_{2} + iy_{2}). Furthermore, this identification is compatible with the action of Γ on G and ℍ × ℍ, so we can identify functions on Γ\ℍ × ℍ with K-invariant functions on Γ\G.

Let k(θ) = (\begin{matrix}cos θ & sin θ \\ −sin θ & cos θ \end{matrix}) and define

A_{n,m} = {f ∈ C^{∞}(Γ\G) : f(x_{1}k(θ_{1}), x_{2}k(θ_{2})) = e^{2i(nθ_{1}+mθ_{2})}f(x_{1}, x_{2})}

(in particular A_{0,0} is the space of K invariant functions). It is easy to see that the K-finite function, i.e. those functions in the linear span of the A_{n,m} are dense in C(Γ\G).

Of particular importance are the raising and lowering operators

E_{j}^{±} = ±2iy_{j}e^{±2iθ_{j}}(\frac{∂}{∂ x_{j}} ∓ i\frac{∂}{∂ y_{i}}) ∓ ie^{±2iθ_{j}} \frac{∂}{∂θ_{j}}

E_{1}^{+}, for example, takes functions in A_{n,m} to A_{n+1,m}.

Other operators we will need are H_{j} which generate the Cartan flow.

H_{j} = −2y_{j} sin 2θ_{j} \frac{∂}{∂ x_{j}} + 2y_{j}cos 2θ_{j} \frac{∂}{∂ y_{j}} + sin 2θ_{j} \frac{∂}{∂θ_{j}}

i.e.

\frac{d}{dt}f(g_{1}(\begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix}), g_{2})|_{t=0} = H_{1}f.

We note that 2H_{j} = E_{j}^{+} + E_{j}^{−}. We will also use X_{j}, the generators of a two parameter unipotent subgroup of G.

X_{j} = y_{j}cos 2θ_{j} \frac{∂}{∂ x_{i}} + y_{j}sin 2θ_{i} \frac{∂}{∂ y_{j}} + sin^{2}θ_{j} \frac{∂}{∂ θ_{j}} .

Finally, we will use the operators W_{j}

W_{j} = \frac{∂}{∂θ_{j}} .

A last remark is that E_{j}^{−} E_{j}^{+}, or perhaps more importantly the Casimir element of sl_{2}(ℝ),

ω_{j} = E_{j}^{−} E_{j}^{+} − W^{2} − 2iW

coincide with the Laplacian

Δ_{j} = 4y_{j}^{2}(\frac{∂^{2}}{∂ x_{j}^{2}} + \frac{∂^{2}}{∂ y_{j}^{2}} )

on the K-invariant functions.

Let now ϕ_{i} be some sequence of normalized eigenvectors of Δ_{1} and Δ_{2} with eigenvalues −(1+r_{1,i}^{2}) and −(1+r_{2,i}^{2}), such that both r_{1,i} → ∞ and r_{2,i} → ∞ (possibly at very different rates). To eliminate excessive indexing we omit the index i in ϕ, r_{1}, etc.

For any such ϕ we let

ϕ_{0,0}(g) = ϕ(gK)

and define ϕ_{2n,2l} so that they satisfy the following

(ir_{1} + 1 + 2n)ϕ_{2n+2,2l} = E_{1}^{+}ϕ_{2n,2l}
(ir_{1} + 1 − 2n)ϕ_{2n−2,2l} = E_{1}^{−}ϕ_{2n,2l}
(ir_{2} + 1 + 2l)ϕ_{2n,2l+2} = E_{2}^{+}ϕ_{2n,2l}
(ir_{2} + 1 − 2l)ϕ_{2n,2l−2} = E_{2}^{−}ϕ_{2n,2l}

3. Defintion of the Micro Local Lift

We will assign to every Δ_{1}, Δ_{2}-eigenfunction ϕ a functional defined on all K-finite functions as follows:

(∗) I_{ϕ}(f) = lim_{N→∞}⟨f ∑_{n,l=−N}^{N} ϕ_{2n,2l}, ϕ_{0,0}⟩

where ⟨a, b⟩ = ∫_{Γ\G} a(g)\overline{b}(g)dg. Note that for K-finite f the limit in (∗)

is purely formal since

⟨f ∑_{n,l=−N}^{N} ϕ_{2n,2l}, ϕ_{0,0}⟩

is constant for any N ≥ N_{0} if

f ∈ ∑_{n,l=N_{0}}^{N_{0}} A_{n,l}

8 ELON LINDENSTRAUSS

by orthogonality of the weight spaces A_{n,l}. Also note that

I_{ϕ}(f) = ∫_{Γ\G/K} f(x)|ϕ(x)|^{2}dx for f ∈ A_{0,0}.

Theorem 3.1. Set Φ_{N} := \frac{1}{2N+1} ∑_{l,m=−N}^{N} ϕ_{2l,2m}. Then for K-finite f

I_{ϕ}(f) = ⟨fΦ_{N}, Φ_{N}⟩ + O_{f}(N^{−1}) + O_{f}(N max(r_{1}^{−1}, r_{2}^{−1}))

Proof. Suppose

f ∈ ∑_{l,m=−N_{0}}^{N_{0}} A_{l,m}.

By definition,

⟨fΦ_{N}, Φ_{N}⟩ = \frac{1}{(2N + 1)^{2}} ∑_{k,t,l,m=−N}^{N}⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩.

Since

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ ≠ 0

only if max(|l − k|, |m − t|) ≤ N_{0}, we have that ⟨fΦ_{N}, Φ_{N}⟩ = O_{f}(1). Thus without loss of generality, N max(r_{1}^{−1}, r_{2}^{−1}) ≤ 1.

We now observe that for −N ≤ l, m, k, t ≤ N,

(∗∗) ⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ = ⟨fϕ_{2l−2,2m}, ϕ_{2k+2,2t}⟩ + O_{f}(Nr_{1}^{−1}) Indeed

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ =

= \frac{1}{(ir_{1} + 2l − 1)(−ir_{1} − 2k − 1)}⟨fE_{1}^{+}ϕ_{2l−2,2m}, E_{1}^{−}ϕ_{2k+2,2t}⟩

= ⟨f(\frac{E_{1}^{+}}{ir_{1}} )ϕ_{2l−2,2m}, (\frac{E_{1}^{−}}{ir_{1}} )ϕ_{2k+2,2t}⟩ + O(∥ f ∥_{∞}Nr_{1}^{−1})

= ⟨\frac{E_{1}^{+}}{ir_{1}} (fϕ_{2l−2,2m}), \frac{E_{1}^{−}}{ir_{1}} ϕ_{2k+2,2t}⟩ + O (∥ E_{1}^{+}f ∥_{∞} r_{1}^{−1})

+ O (∥ f ∥_{∞} Nr_{1}^{−1})

= ⟨fϕ_{2l−2,2m}, \frac{E_{1}^{+}E_{1}^{−}}{r_{1}^{2}} ϕ_{2k+2,2t}⟩ + O_{f}(Nr_{1}^{−1}).

But

E_{1}^{+} E_{1}^{−}ϕ_{2k+2,2t} = −[r_{1}^{2} + O(N^{2})]ϕ_{2k+2,2t}

and (∗∗) is established. Iterating the above calculation at most N_{0} times gives

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ = ⟨fϕ_{2l−2k,2m−2t}, ϕ_{00}⟩ + O_{f}(N max(r_{1}^{−1}, r_{2}^{−1}))

and so

⟨fΦ_{N}, Φ_{N}⟩ = ∑_{l,m}^{N} \frac{(N − |l| + 1)(N − |m| + 1)}{(2N + 1)^{2}}⟨fϕ_{2l,2m}, ϕ_{0,0}⟩+O_{f}(N max(r_{1}^{−1}, r_{2}^{−1})).

Since for |l|, |m| ≤ N_{0},

\frac{(N − |l| + 1)(N − |m| + 1)}{(2N + 1)^{2}} = 1 + O_{f}(N^{−1})

and using the K-finiteness of f, we see that

⟨fΦ_{N}, Φ_{N}⟩ = ∑_{l,m}⟨fEϕ_{2l,2m}, ϕ_{00}⟩ + O_{f}(N^{−1}) + O_{f}(N max(r_{1}^{−1}, r_{2}^{−1})).

If for a dense subset of C(Γ\G) consisting only of K-finite functions lim_{r_{1},r_{2}→∞} I_{ϕ}(f) exists, then this limit is given by

I(f) := lim_{r_{1},r_{2}→∞} I_{ϕ}(f) = lim_{N→∞} lim_{r_{1},r_{2}→∞}⟨fΦ_{N}, Φ_{N}⟩.

Since if f ≥ 0 ⟨fΦ_{N}, Φ_{N}⟩ ≥ 0, we see that there is a measure μ s.t. I(f) = ∫ fdμ. This measure is a probability measure since I_{ϕ}(1) = 1 for all elements of our sequence. Therefore, we have the following:

Corollary 3.2. 1. There is a measure μ_{ϕ} on Γ\G (namely |Φ_{N}|^{2}d vol_{Γ\G} where N is min(r_{1}^{1/2}, r_{2}^{1/2})) so that

|I_{ϕ}(f) − ∫ f dμ_{ϕ}| = O_{f}(max(r_{1}^{−1/2}, r_{2}^{−1/2})).

2. Any weak^{∗} limit of the I_{ϕ}(·) is a probability measure on Γ\G.

In particular the above corollary allows us to lift a weak^{∗} limit of the |ϕ|^{2}d vol to a measure μ on Γ\G, which, as we will see in the next section, has nice invariance properties.

4. Invariance Under the Cartan Flow.

Let

A = \{((\begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix}) , (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix})) : t, s > 0\} ,

A_{i} = A ∩ G_{i} for i = 1, 2.

Theorem 4.1. Any μ obtained as in Corollary 3.2 is invariant under A.

Theorem 4.1 is equivalent to having ∫ (H_{i}f)dμ = 0 for i = 1, 2, f ∈ C^{∞}(Γ\G) since H_{i} is the generator for the corresponding one parameter flow obtained by multiplication on the right by elements of A_{i}. This, in turn, is equivalent to I_{ϕ}(H_{i}f) → 0 as r_{1}, r_{2} → ∞, but we have in fact the following exact statement, which for the 2 dimensional case was found by Zelditch. Since to prove Theorem 3 it is enough to consider f which are K-finite, we consider only such functions in the proposition below.

Proposition 4.2. For j = 1, 2

I_{ϕ}(2ir_{j}H_{j} + H_{j}^{2} + 4X_{j}^{2})f) = 0;

In particular, I_{ϕ}(H_{j}f) = O_{f}(r_{j}^{−1})

Proof. We assume through the proof that j = 1, and use E^{+}, H, … to denote E_{1}^{+}, H_{1}, etc. The proof of this proposition is a simple calculation.

Write, formally, Φ_{∞} = ∑_{n,m=−∞}^{∞} ϕ_{2n,2m}. This sum can be given an

explicit meaning, but in fact in the following arguments can be replaced by a sufficiently big finite sum at the cost of additional indexation. The following identities are easily verified:

I_{ϕ}(f) = ⟨fΦ_{∞}, ϕ_{0,0}⟩
⟨(W f)Φ_{∞}, ϕ_{0,0}⟩ = −⟨f(W Φ_{∞}), ϕ_{0,0}⟩
E^{+}Φ_{∞} = (ir_{1} − iW − 1)Φ_{∞}
E^{−}Φ_{∞} = (ir_{1} + iW − 1)Φ_{∞}
E^{−}E^{+}ϕ_{∞} = (−r_{1}^{2} − (iW − 1)^{2})Φ_{∞}

We further recall that E^{−} + E^{+} = 2H, where H is the generator of the 1-parameter Cartan action. Thus (with 2H = E^{+} + E^{−}, 2iV = E^{+} − E^{−}, and X = \frac{W +V}{2} )

−(r_{1}^{2} + 1)I_{ϕ}(f) = ⟨fΦ_{∞}, E^{−}E^{+}ϕ_{0,0}⟩ = ⟨E^{−}E^{+}fΦ_{∞}, ϕ_{0,0}⟩ =

= ⟨f(E^{−}E^{+}Φ_{∞})+(E^{+}f)(E^{−}Φ_{∞})+(E^{−}f)(E^{+}Φ_{∞})+(E^{−}E^{+}f)Φ_{∞}, ϕ_{0,0}⟩ =

= −(r_{1}^{2} + 1)I_{ϕ}(f) + 2ir_{1}I_{ϕ}(Hf)+
⟨[(W^{2} − 2iW + E^{−}E^{+} − 2H − 2W V)f] Φ_{∞}, ϕ_{0,0}⟩ =

= −(r_{1}^{2} + 1)I_{ϕ}(f) + 2ir_{1}I_{ϕ}(Hf) + I_{ϕ}((H^{2} + (W + V)^{2})f)

and so

I_{ϕ}(Hf) = \frac{i}{2r_{1}} I_{ϕ}((H^{2} + 4X^{2})f)

5. measures of invariant tori

While it is possible to define and discuss the Hecke operators without invoking the more modern adelic language, using this language greatly simplifies the analysis of these operators, particularly when the class number of the quaternion algebra we are considering is greater than one. While this assumption is quite natural for Γ\ℍ , it seems overly restrictive for Γ\ℍ × ℍ . To avoid having to deal with groups with class number greater than one, we shall deviate from [RS] by working with the group SL_{2}(ℝ) (and products of such) instead of P GL_{2}(ℝ). Our treatment here is quite general and will work for any irreducible congruence lattice in the product of any number of SL_{2}(ℝ) (the case of SL_{2}(ℝ) included).

We start by defining the manifolds on which we show no scarring occurs. We start from a quaternion algebra over a number field L, i.e. an algebra H(L) whose elements are

α = X_{0} + X_{1}ω + X_{2}Ω + X_{3}ωΩ

where

ω^{2} = a; Ω^{2} = b, ωΩ = −Ωω (5.1) with a, bϵO_{L} ≡ O. The trace tr(α) and norm n(α) for α ∈ H(L) are defined in the usual way.

We require further that L is totally real (i.e. that all the embeddings L → ℂ take only real values), and let V_{∞}^{r}, V_{∞}^{u} and V_{f} denotes the infinite places ν of L where a H is ramified, i.e. H(L_{ν}) is a division ring, the infinite places at which H does not ramify, hence

H(L_{ν}) ≅ M_{2}(ℝ) (5.2)

and the finite places of L respectively. Recall that V_{∞}^{r} ∪ V_{∞}^{u} can be identified with the embeddings L → ℝ, and V_{f} with the prime ideas in L. Let G be the algebraic group over L of norm 1 quaternions in H. We now take n = |V_{∞}^{u}|

\begin{aligned}G_{∞}^{u} & = ∏_{ν∈V_{∞}^{u}} G(L_{ν}) \\ G_{∞}^{r} & = ∏_{ν∈V_{∞}^{r}} G(L_{ν}) \\ G_{f} & = ∏_{ν∈V_{f}} G(L_{ν}) \\ X & = G(L)\G(𝔸)/(G_{∞}^{r} × G̃) \end{aligned} (5.3)

and G̃ to be a compact open subgroup of G_{f}.

Since G is simply connected (as an algebraic group), it follows from the Strong Approximization Theorem [RP, Chapter 7] that X is connected and is of the form

Γ\G_{∞}^{u} ≅ Γ\SL_{2}(ℝ)^{h}

where

Γ = G(L) ∩ [G_{∞}^{u} × G_{∞}^{r} × G̃]

where we identify G(L) with both its image in G(𝔸) and G_{∞}^{u}, and similarly for Γ < G(L).

We remark that for any γ ∈ Γ < G(L) (considered as an element of H(L)),

tr(γ), n(γ) ∈ O_{L}

We fix some isomorphism map G_{∞}^{u} ≅ SL_{2}(ℝ)^{n}, and as in other sections of this note, take A to denote the positive diagonal subgroup in SL_{2}(ℝ)^{n}, and take K to be the maximal compact

SO(2, ℝ)^{n} ≅ K < G_{∞}^{u}.

We set M = X/K, a quotient manifold of ℍ^{n}. Similarly to the n = 2 case, the algebra of invariant differential operators on M is spanned by the Laplacian acting on each coordinate of ℍ^{n}, which we denote by Δ_{1}, …, Δ_{n}.

There is a standard way to attach to all but finitely many prime ideals ν ∈ V_{f} a Hecke operator; one way to do this is as follows: Equation (5.1) can be used to get a concrete model for G as a subvariety of GL_{4}; by the definition of the adelic topology, G̃ can be presented as

G̃ = G̃_{0} × ∏_{ν∈V_{f} \S} G(O_{L_{ν}})

with G̃_{0} an open compact subset of ∏_{ν∈S} G(L_{ν}). We may further assume

that H does not ramify for all ν ∈ V_{f} \ S, and that we can identify G(L_{ν}) with SL_{2}(L_{ν}) and G(O_{L_{ν}}) with SL_{2}(O_{L_{ν}}). Let t_{ν} ∈ O_{ν} be a generator for νO_{ν} in O_{ν}. Set a_{ν} ∈ G(𝔸) by

[a_{ν}]_{ν′} = \begin{cases}e & for ν′ ≠ ν; \\ (\begin{matrix}t_{ν} \\ & t_{ν}^{−1} \end{matrix}) & for ν′ = ν, \end{cases}

and let π be the projection G(K)^{S(𝔸)} → X. For any x ∈ X , the set π(π^{−1}(x)a_{ν}^{k}) contains exactly [O : ν]^{2k−1}([O : ν] − 1) points (and does not depend on the specific isomorphism G(L_{ν}) ≅ SL_{2}(L_{ν}) we use). We will denote these points by T_{ν^{2}k}(x) , and this also defines an operator on L^{2}(X) by

T_{ν^{2k}} : f(x) ↦ ∑_{y∈T_{ν^{2k}} (γ)} f(g)

For our purposes, we will need to give a slightly more explicit description of the Hecke points: take x̃ ∈ G_{∞}^{u} so that x = Γx̃. A point y ∈ Γ_{ν^{2}k}(x) if and only if there is some h ∈ G(L) ↩ ∏_{ν′∈V_{f}} G(L_{ν})

h ∈ G̃a_{ν}^{k}G̃ ∩ G(L)

such that y = Γhx̃. As an element in the quaternion algebra, it is clear that

n(h) = 1;
tr(h) ∈ ν^{−k},

since if tr(h′) ∉ O_{L_{ν}} the image of the sequence h′^{n} in G(L_{ν}) is unbounded, and G̃ is compact.

Note that an immediate consequence of this definition is that Hecke operators corresponding to different prime ideals commute. Another consequence is that the Hecke operators are self adjoint, or what amounts to the same

x ∈ T_{ν^{2}}(y)iffy ∈ T_{ν^{2}}(x)

A very similar construction allows us to define the Hecke operators also on L^{2}(M) (which we will also denote in the same way).

Theorem 5.1. Let M, X, be as above. Let ϕ_{i} be a sequence of joints eigenfunctions of Δ_{1}, Δ_{2}, …, Δ_{n} and all Hecke operators T_{ν^{2}}. Let μ be a weak^{∗} limit to the micro local lifts of the measures |ϕ_{i}|^{2}d vol. Then for any x̃ ∈ X whose A-orbit x̃A is an invariant tori

μ(x̃A) = 0. (5.4)

The proof of this theorem rests on the following lemma, as well as the well-known fact that there are no eigenfunctions of the Laplician on a regular tree with finite L^{2}-norm.

Lemma 5.2. Let x̃ ∈ X be a point such that x̃A is an invariant torus. Then there are infinitely many ν ∈ V_{f} for which for some open e ∈ B ⊂ A the toral pieces

{x̃B, T_{ν^{2}}(x̃)B, T_{ν^{4}}(x̃)B, … }

are all disjoint.

Proof. Assume that x̃A is an invariant tori with x̃ = Γx. Then there is a γ ∈ Γ such that

γx ∈ xA (5.5)

(in fact, the set of such γ is an n-dimensional commutative subgroup of G(L) ). We can identify γ ∈ Γ < G(L) with an element in the quaternion algebra H(L). Then L(γ) < H(L) is isomorphic to a number L′, [L′ : L] = 2, with γ mapped by the isomorphisms to α ∈ O_{L′}.

For every ν ∈ V_{∞}^{r} the image of γ in G(L_{ν}) has (as an element in a compact group) only eigenvalues on the units circle. Note that the eigenvalues of this image are exactly the two possible images of α by embeddings τ of L′ coinciding with ν on L, each appearing with multiplicity two; in particular τ is complex. On the other hand, by (5.5), for every ν ∈ V_{∞}^{u} the eigenvalues of γ in G(L_{ν}) (which are exactly the eigenvalues of the 2-by-2 matrix γ is sent to by the isomorphisms G(L_{ν}) ≅ SL_{2}(ℝ) are in ℝ. Using the fact that the image of the natural embedding of L′ into the product of all its infinite valuations is discrete, we see that there is a η > 0 such that for any α′ ∈ O_{L′}, for at least one embedding τ of L′ to ℝ or ℂ

|log |τ(α)|| > 2η.

Let B be defined by

B = B^{−1} = \{(( \begin{matrix}e^{s_{1}} & 0 \\ 0 & e^{−s_{1}} \end{matrix} ) , …, ( \begin{matrix}e^{s_{n}} & 0 \\ 0 & e^{−s_{n}} \end{matrix} )) : ∀i |s_{i}| < \frac{η}{2}\}

From now to the end of the proof, we fix ν ∈ V_{f} \ S, i.e. a prime ideal in L, that in addition remains inert in L′. It follows immediately from the Cebotarev Density Theorem (See [C], p. 159) that there are infinitely many such prime ideals in L (indeed, one can probably give an easier direct proof for this fact along the lines of the argument in [RS]).

Assume now that for some Γy ≠ Γy′ ∈ ⋃_{k=0}^{∞} T_{ν^{2k}} (x̃),

yB ∩ y′B ≠ ∅. (5.6)

Note that while L′ and B depends on x, they remain fixed if we replace x with any other point in G(L)x. This is because Γ ∩ xAx^{−1} and Γ ∩ hxAx^{−1}h^{−1} are comeasurable for any H ∈ G(L). Thus we may assume x = y′ and define k by Γy ∈ T_{ν^{2k}}(x̃), where as before x̃ = Γx.

From the explicit description of the Hecke points, we know that y′ can be presented as hx with h ∈ G(L) < H(L) satisfying tr(h) ∈ ν^{−k} and (like any element in G(L)) n(h) = 1. Since

hx ∈ xB^{2} ⊂ xA

it follows from (5.5) and the fact that A is a commutative group that [γ, h] = 0. This implies that h is in the sub algebra L[γ] ≅ L′ of H(L) (otherwise L(γ, h) would be an extension of L that properly contains L′, and hence H(L) would contain an element whose minimal polynomial over L is of degree ≥ 3 – which is of course impossible). From the minimal polynomial of h,

h^{2} − tr(h)h + n(h) = 0

we know that h ∈ ν^{−k}O_{L′} and that n_{L′/L}(h) = 1. Since ν is inert, νO_{L′} ◁ O_{L′} is a prime ideal and h̄ ∈ ν^{−k}O_{L′}. The only way to reconcile these conditions is if h ∈ O_{L′}^{∗}. From the same reasoning we applied to γ, for any embedding τ : L′ → ℂ extending an embedding from V_{∞}^{r}, we have that |τ(h)| = 1. Thus unless h = 1, which is clearly ridiculous, hx cannot be in xB^{2}, by definition of B.

Next we cite the following basic theorem about harmonic analysis on trees. Let T_{d} be a d+1-regular tree, and define S_{T_{d}}: L^{2}(T_{d}) → L^{2}(T_{d}) by replacing the value of f ∈ L^{2}(T_{d}) at any node of the tree with the sum of the value of f on all of its neighbors. This operator is essentially the Laplacian on T_{d}, and is well-defined for any locally bounded function on T_{d}. For more background on harmonic analysis on trees see [FTN].

Theorem 5.3. S_{T_{d}} has no discrete spectrum (i.e. has no L^{2}-normalized eigenfunctions).

The following corollary is an obvious consequence of the above theorem, using standard compactness arguments. While we do not need this, it can be seen that R can be taken to be O(ϵ^{−1}) independently of the degree of the tree T_{d}.

Corollary 5.4. Fix o ∈ T_{d}. For any ϵ > 0 there is a R such that if f ∈ ℝ_{d}^{T} is an eigenfunction of S_{T_{d}} (note we do not assume f ∈ L^{2}(T_{d})),

|f(0)|^{2} ≤ ϵ ∑_{{y∈T_{d}, \atop 1≤d(x,y)≤R}} |f(y)|^{2}

Proof of Theorem 5.1. The first thing we need to understand is how the assumption that ϕ_{i} are Hecke eigenfunctions is manifested in the properties of μ, the weak^{∗} limit of their micro local lifts.

Our starting point is Theorem 3.1. Note that purely from notational reasons this theorem is stated for joint eigenfunctions of the algebra of invariant differential operators only for the case of n = 2, i.e. G_{∞}^{u} ≅ SL_{2}(ℝ) × SL_{2}(ℝ) – the proof works essentially as written also for the general case. In order not to introduce new and contradictory notations we shall assume that n = 2 also in this proof; but this assumption is in no way essential, and the proof carries out verbatim also for the general case, e.g. the much studied case of n = 1.

A key player in this theorem is the function Φ_{N}^{(j)} (the dependence on j in Theorem 3.1 is implicit)

Φ_{N}^{(j)} = \frac{1}{2N + 1} ∑_{a,b=−N}^{N}(φ_{j})_{2a,2b}

for the general case, we sum over n indices instead of 2, and normalize accordingly. If the ϕ_{j} are Hecke eigenfunctions on M, then Φ_{N}^{(j)} are Hecke eigenfunctions on X; furthermore, by Theorem 3.1 for any f ∈ C^{∞}(X)

∫ fdμ = lim_{N→∞} lim_{j→∞} ∫ f|Φ_{N}^{(j)}|^{2}dvol_{X}. (5.7)

Suppose now that x̃A = ΓxA is an invariant measure, and let ν ∈ V_{f} and B ⊂ A be as in Lemma 5.2. Take ϵ > 0 arbitrary, R depending on ϵ as in Corollary 5.4. Take B′ ⊂ G_{∞}^{u} to be an open set containing B such that

{ỹB^{1} : ỹ ∈ Y_{x,R}}

with

Y_{x,R} = ∪_{i=1}^{R} T_{ν^{2i}}(x̃)

are all disjoint. For any ỹ ∈ X, we can view

{ỹ} ∪ ∪_{i=1}^{∞} T_{ν^{2i}}(ỹ)

as a regular tree T_{d} of degree d = [ν : O_{L}]([ν : O_{L}] + 1), whose edges are (ỹ′, ỹ′′) with ỹ′ ∈ T_{ν^{2}}(ỹ′′). The Hecke operator T_{ν^{2}} is identical to S_{T_{d}} on every such Hecke-tree. Thus we see that for any N and b ∈ B′

|Φ_{N}^{(j)}(x̃b)|^{2} ≤ ϵ ∑_{ỹ∈Y_{x,R}} |Φ_{N}^{(j)}(ỹb)|^{2}

hence

\begin{aligned}∫_{x̃B^{1}} |Φ_{N}^{(j)}|^{2}dvol_{X} & ≤ ϵ ∑_{ỹ∈Y_{x},R} ∫_{ỹB^{1}} |Φ_{N}^{(j)}|^{2}dvol_{X} \\ & ≤ ϵ ∫_{X} |Φ_{N}^{(j)}|^{2}dvol_{X} = ϵ \end{aligned}

and

μ(x̃B) = 0.

Appendix A. Relation to the standard micro local lift

In this appendix we omit the proofs of the statements since they are not needed for any of the main results of this note.

Let ϕ_{i} be eigenfunctions of Δ_{1} and Δ_{2}, with eigenvalues tending to −∞, and let μ′ be a weak^{∗} limit of |ϕ_{i}|^{2}d vol. We wish to relate the standard micro local lift μ′′ of μ′ to our lift μ. Recall that μ′′ is a measure on S^{∗}M (where M is Γ\ℍ × ℍ), invariant under the geodesic flow. On S^{∗}M (equipped with the natural volume measure) the geodesic flow is not ergodic, with each ergodic component being the induced volume on a smooth submanifold of this space. Consider first S^{∗}(ℍ × ℍ), of which S^{∗}M is a quotient. The points of this space are of the form (x_{1}, x_{2}, ξ_{1}, ξ_{2}), where x_{j} ∈ ℍ, and ξ_{j} ∈ T_{x_{j}}^{∗}(ℍ) satisfying ∥ξ_{1}∥^{2} + ∥ξ_{2}∥^{2} = 1. The ratio ∥ξ_{1}∥/∥ξ_{2}∥ is well defined on the quotient S^{∗}M and is invariant under the geodesic flow. This gives us an invariant fibration of S^{∗}M where all but two fibers are of codimension 1 and can be identified naturally with Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ). The exceptional fibers are identified with Γ\P SL_{2}(ℝ) × ℍ and Γ\ℍ × P SL_{2}(ℝ). Under this identification the geodesic flow becomes the right action by a oneparameter subgroup of the Cartan group A (on each fiber a different one-parameter subgroup). It follows from the Howe-Moore Theorem [Zi] that with respect to the natural volume form on each fiber the geodesic flow is ergodic.

The micro local lift is defined whenever the ϕ_{i} are eigenfunctions of the Laplacian. Using the additional information that the ϕ_{i} are eigenfunctions of Δ_{1} and Δ_{2}, one easily sees that μ′′ is supported on one of the fibers in the above fibration of S^{∗}M. The fiber on which μ′′ is supported is determined by the limit ρ of λ_{1,i}/λ_{2,i}. The exceptional fibers correspond to ρ = ∞ and ρ = 0 respectively. Both of these facts follow easily by evaluating the integrals of ∥ξ_{1}∥^{n}.

The identification of the regular fibers with Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ) sends μ′′ to our lift μ. However, from the standard construction, one can only get invariance under a one-parameter flow, whereas we get invariance under a two-parameter group.

If ρ = 0 the standard micro local lift will give us a measure on Γ\ℍ× P SL_{2}(ℝ); but as long as both λ_{1,i} and λ_{2,i} tend to −∞ our lift gives us a measure on Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ). This measure is again invariant under a two-parameter group; its projection to Γ\ℍ×P SL_{2}(ℝ) can be identified with the standard micro local lift. The case ρ = ∞ is similar.

Appendix B. Conjecture 1.4 for G = P SL_{2}(ℝ) × P SL_{2}(ℝ)

In this appendix we show that for uniform lattices Γ, Conjecture 1.4 does indeed imply that any A ergodic invariant measure on Γ\P SL_{2}(ℝ)× P SL_{2}(ℝ) is either Haar measure or Haar measure on a compact A orbit. We consider first the case Γ is co-compact. It is easy to classify the closed connected groups A < F < G such that xF is closed. The Lie algebra of F is invariant under Ad A ; therefore, it is a union of eigenspaces for Ad A. Let F′ be the Lie group corresponding to the algebra generated by all eigenspaces on which Ad A does not act trivially, or equivalently, F′ is the group generated by all unipotent elements of F. One such eigenspace is

\{((\begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}))\};

but if

\{((\begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}))\} ∈ Lie F

then also

\{((\begin{matrix}0 & 0 \\ ∗ & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}))\} ∈ Lie F.

To see this note first that \overline{xF′} ⊂ xF is by Ratner’s theorem [R] homogeneous, i.e. there is a closed connected subgroup H > F′ so that \overline{xF′} = xH. Furthermore, xH is of finite volume (according to the induced metric on this submanifold).

Unless the above claim is true, the action of A does not preserves the volume of xH , and in fact since the dependence of vol(exp(a)xH) on a ∈ Lie A is linear, vol(exp(a)xH) can be made to go to 0 which is a contradiction to the compactness of Γ\P SL_{2}(ℝ) × P SL_{2}(ℝ) . It only remains to eliminate the possibility that

F = \{((\begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix}) , (\begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix}))\} .

The unipotent radical of H is equal to the radical of H (see [Sh, §3] or the appendix in [LW]), and so F′ = H. But this, in particular, this means that |F′ ∩ x^{−1}Γx| = ∞, which shows that the projection of the irreducible lattice x^{−1}Γx from P SL_{2}(ℝ) × P SL_{2}(ℝ) to P SL_{2}(ℝ) is neither injective nor is the image finite, which is a contradiction to Margulis Normal Subgroup Theorem [Zi].

Acknowledgements: This work has been strongly influenced by Peter Sarnak, who suggested that the rigidity cojectures in ergodic theory might be linked to the Quantum Unique Ergodicity conjecture and provided consistent encouragement. The author was introduced to the topic of Quantum Chaos from discussions with Zeev Rudnick. While writing section §5, I greatly benefited from discussions with Armand Borel, Alex Lubutzki, Hee Oh, Peter Sarnak and Kannan Soundararajan. I would also like to thank Shahar Mozes, George Tomanov and Barak Weiss for stimulating discussions. Part of this note has been written during my stay at the Newton Institute in the summer of 2000.

To all of them, and all others who helped me during this work I am deeply grateful.

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