Effective equidistribution for some unipotent flows in $PSL(2, \mathbb{R})^k$ mod cocompact, irreducible lattice

Let $d \geq 2$, and let $\Gamma \subset PSL(2, \mathbb{R})^d$ be an irreducible, cocompact lattice. We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $\Gamma \backslash PSL(2, \mathbb{R})^d$.


INTRODUCTION
There has been greater interest recently in making Ratner's theorems effective. Green-Tao proved all Diophantine nilflows on any nilmanifold become equidistributed at polynomial speed, see [5]. Flaminio-Forni proved rather sharp estimates on the speed of equidistribution for a class of higher step nilmanifolds.
We will now describe the setting for our equidistribution result. Let k ≥ 2 and let Γ ⊂ PSL(2, R) k be an irreducible, cocompact lattice, where PSL(2, R) k is the direct product of k copies of PSL(2, R). Let M = Γ\ PSL(2, R) k .
The vector fields on M are elements of the Lie algebra sl(2, R) k , the direct product of k copies of the Lie algebra sl(2, R). For each 1 ≤ i ≤ k, define the vector fields {U i , V i } in sl(2, R) k by U i := (0, . . . , 0, U, 0, . . . , 0) , where U and V are elements of sl(2, R) that occur in the i th position of the k-tuple, and they are given by is the complete set of generators for unipotent flows on M whose eigenspace for the zero eigenvalue has codimension 1.
The space of diagonal elements in sl(2, R) k is k-dimensional. For each 1 ≤ i ≤ k, define the diagonal element occurs in the i th position and generates the geodesic flow on PSL(2, R).
For i ∈ {1, 2, . . . , k}, We will study the rate of equidistribution of the unipotent flow {φ i t } t∈R on M given by Our approach is via unitary representations and invariant distributions. The main ingredients for the proof are the work of Flaminio-Forni on the equidistribution of horocycle flows on quotients of PSL(2, R) by a lattice (see [3]) and a result of Kelmer-Sarnak on the spectral gap for irreducible, cocompact lattices Γ ⊂ PSL(2, R) k (see [6]).
Let L 2 (M ) be the separable Hilbert space of complex-valued square-integrable functions on M . Let C ∞ (M ) be the space of smooth functions on M and let E ′ (M ) = (C ∞ (M )) ′ be its distributional dual space.
Any element of the Lie algebra sl(2, R) k acts on E ′ (M ) via the right regular representation. The Laplacian operator △ is a second-order, elliptic element in the enveloping algebra of sl(2, R) k . It is an essentially self-adjoint differential operator on L 2 (M ) and is given by The Sobolev space of order s ∈ R + is the maximal domain W s (M ) of the inner product f, g s : The center of the enveloping algebra of sl(2, R) k contains the second-order differential operator . A result by Kelmer-Sarnak shows i has a spectral gap on L 2 (M ) in the sense that the positive eigenvalues of i are bounded below by some µ 0 > 0 (see Theorem 2 and Section 1.3 of [6]).
Our estimates are given in terms of Sobolev norms involving a finite number of derivatives. In what follows, we let s > 3k/2 + 1, and fix The representation dπ is related to the derived representation of the right regular representation by the following simple lemma.
Differentiating at t = 0 gives the lemma. Hence, Then with respect to a positive Stieltjes measure the unitary representation π has the following direct integral decomposition where there Casimir element i acts as the constant µ on each unitary representation space H µ,s , and dβ(µ) is a positive Stieltjes measure. Each representation space H µ,s is a direct sum of an at most countable number of irreducible unitary representation spaces.
By irreducibility and (2), the vector fields U i , X i and V i are decomposable into the irreducible representations of π in the sense that where H s µ ⊂ H µ,s inherits the inner product from W s (M ). There is a decomposition where • for µ = 0, the space I 0 is spanned by the PSL(2, R)-invariant volume; For T ≥ 1, let log + T := max{1, log T }.
Additionally, we have the following lower bound. For every D ∈ B + , there is a constant C s (D) > 0 such that for all sufficiently large T ≥ 1,  Then for all f ∈ Ann s (M ), there exists a unique g ∈ W r (M ) (up to additive constants) such that Moreover, there is a constant C r,s := C r,s (Γ) > 0 such that Proof. First say r is an integer. By Theorem 1.2 of [3], for any f ∈ Ann s (M ) there is a g ∈ W r (M ) such that and observe that Because U i commutes with △ 0 , we have Then for any ǫ > 0, (4) gives By Lemma 6.3 of [8], for all α, β ∈ Z ≥0 , there is a constant C α+β > 0 such that By interpolation, the same holds for all α, β ∈ R ≥0 , see [7]. Hence, Letting ǫ < s − 2n − 1, and combining (5), (6) and (7), we have This finishes the proof of Theorem 2.1 in the case when r ∈ Z ≥0 . The general case for r ≥ 0 follows by interpolation.

PROOF OF THEOREM 1.2
For all (x, T ) ∈ M × R + , write γ x,T as We may project γ x,T onto a basis of invariant distributions described in Theorem 1.1. Then It follows from Lemma 5.2 of [3] that for some constant C s := C s (Γ) > 0, the quantity We prove Theorem 1.2 by controlling each of the terms in (9). We begin by estimating R γ −s .
There is a constant C s := C s (Γ) > 0 such that for all T ≥ 1, Proof. Let f ∈ Ann s (M ). Then by Theorem 2.1, for any 3k 2 < r < s − 1, there is a constant C r,s := C r,s (Γ) > 0 and a function g ∈ W s (M ) satisfying U i g = f and g r ≤ C r,s f s . Then as in Lemma 5.5 of [3], The dependence of C r,s on r can be removed by taking r = s/2 + (3k − 2)/4.
Proof. The unipotent arc {φ i t } T t=0 contracts under right multiplication by the map e αX i , for α > 0. Setting α = log T , we get e − log T X i γ = γ xe log T X i ,1 .
By (10) and the Sobolev embedding theorem, the projection of γ xe log T X i ,1 onto the component with continuous spectrum satisfies [3] gives a constant C s > 0 such that exp(tX i ) exp(C(γ xe log T X i ,1 ) −s ≤ C s (1 + |t|)e −t/2 .
Then the lemma would now be immediate except that the orthogonal splitting I s (M ) ⊕ I s (M ) ⊥ is not preserved under the geodesic flow. However, by iteratively applying the geodesic map e X i to the arc γ xe log T X i ,1 and using Lemma 3.1 to control any additional contribution to the remainder distribution, Lemma 3.2 follows. See Section 5.3 of [3] for details. Lower bounds can be obtained by an argument in [3] involving the L 2 version of the Gottschalk-Hedlund Lemma. See Lemma 5.7, Lemma 5.8, Lemma 5.9 and Lemma 5.13 of this paper for details.

ACKNOWLEDGEMENTS
The author would like to thank Giovanni Forni for helpful discussions and encouragement.