As I promised in this previous post here, today we’re going to compute some explicit constants in Ratner’s paper on the rate of mixing of geodesic flows by simply bookkeeping them while trying to follow the article. In particular, we will borrow the notations from it (and, a fortiori, they will be the same as in this post here).
By the way, let me say that the decision of bookkeeping constants instead of trying more elegant methods is partly motivated by a conversation I had with Livio Flaminio (in the occasion of a summer school at Bedlewo, Poland) where we saw that, even though one disposes of (classical) constructions of the principal, complementary and discrete series, it seems that direct computation with them only give useful information on explicit constants in the case of the principal and discrete series (and in these cases the computation is actually quite elegant). Thus, since our main motivation for this post comes from the complementary series, I decided to stick to the non-elegant straightforward computation presented below the fold.
Let be a non-trivial irreducible unitary -representation in a complex separable Hilbert space . Denoting by , , we define, for each ,
Then, one has . Furthermore, by irreducibility of , we have that or . In this way, one can construct an orthonormal basis of such that if and only if .
Denote by , where is the (positive) diagonal 1-paramter subgroup of . In the sequel, we will be interested in the decay properties of as . To perform this study, we will follow M. Ratner by making a series of preparations.
As it is shown in Lemma 2.1 of Ratner’s paper, satisfies the following ODE
Furthermore, by the discussions after equation (2.12) of Ratner’s paper and by equation (2.13) of this paper, one has and . Hence,
for every , we obtain that the constant appearing in equation (2.14) of Ratner’s paper is
with as above.
Similarly, , so that is bounded by the quantity for every . On the other hand, this last quantity is bounded by
Because and (since ), we see that
In other words, the constant appearing in equation (2.14) of Ratner’s paper is
with as above.
Next, we observe that the constant in equation (2.16) of Ratner’s paper is slightly different from the previous constant in equation (2.14) of this paper. Indeed, by denoting by and the roots of the characteristic equation of the ODE satisfied by , the fact that implies that
because . However, the constant in equation (2.16) of Ratner’s paper remains the same of equation (2.14) of this article:
Concluding our series of preparations, we recall the definitions of the following two functions
introduced after equation (2.18) of Ratner’s paper. These functions appear naturally in our context because the ODE verified by can be rewritten as where is the differentiation operator (with respect to ). Thus, since , we have and, hence,
where is a constant. In particular, we can write
if , and
if , where . Moreover, by using these equations, and the fact that as (a consequence of the non-triviality of , that is, it has no invariant vectors), we can deduce that
in equations (2.19) and (2.20) of Ratner’s paper.
After these preparations, we are ready to pass to the next section, where we will render more explicit the constants appearing in Lemma 2.2 of Ratner’s paper about the speed of decay of the matrix coefficients as .
2. Decay of matrix coefficients of irreducible unitary -representations
By following closely the proof of Lemma 2.2 of Ratner’s paper, we will show the following explicit variant of it:
- if ;
- if ;
- if ;
Remark 1 In Ratner’s article, the function is slightly different from the one above (when ): indeed, in this paper, if , and if . In particular, this allows to gain over the factor of (in front of the exponential functions , ) when is not close to at the cost of permitting larger constants. However, since we had in mind the idea of getting uniform constants regardless of and the factor of does not seem very substantial, we decided to neglect this issue and to stick to the function as defined in Lemma 1 above.
Proof: We begin with the case , i.e., . In this case, from (7), we know that
Since and (because ), we conclude that
and, a fortiori,
Define . By integration by parts, ,
and , where
It follows that
where . That is, we can take in the equation (2.22) of Ratner’s paper. Also, since , we get
with , that is, this constant works in equation (2.24) of Ratner’s paper. Finally, by integrating by parts,
On the other hand, since , one has . By combining these facts, we see that
Thus, using these estimates to control and the estimate (12) above to control , we obtain
where and .
The second step in the analysis of the regime is the control of the quantities and . Since , and , , we can estimate the first quantity as follows:
Here, we used that (so that ) and whenever . Similarly, we can estimate the second quantity as follows:
where and . Inserting these estimates above into (8), we deduce that
so that we can take and in the equation (2.28) of Ratner’s paper.
Next, we observe that when and imply and
In next (and final) section, we will apply Lemma 1 to derive explicit variants of Theorems 1 and 3 of Ratner’s paper. To do so, we need some notation. We denote by , . Given an unitary -representation , we denote by the set of vectors such that is . Finally, if the map is , we denote by
the Lie derivative of along the direction of of the infinitesimal generator of the rotation group .
In particular, in the case of an irreducible unitary -representation , since when , we have that
for every .
3. Two applications of Lemma 1
where , and are as in Lemma 1.
Proof: Following the proof of Theorem 1 of Ratner’s paper (at page 283), we write
with , (and , ) as in page 276 of this paper. We have
by Lemma 1.
On the other hand, since for all , we know that
The desired result follows.
In the sequel, denotes the Casimir operator associated to (see this post).
If , let and . Assume that when . Let with . Then, for all ,
Closing this post, we note that the arguments in Ratner’s paper and the previous variants of Ratner’s theorems have nice consequences to the study of rates of mixing of the geodesic flow on hyperbolic surfaces. More precisely, we consider the regular representation of on where is a hyperbolic surface of finite area (i.e., is a lattice of ). Then, by noticing that the lift to the unit tangent bundle of of a function is constant along the orbits of , one has that the Lie derivative of such lifts vanish. Therefore, by direct application of Theorem 3, one gets:
Remark 2 In Appendix A of this paper here, it was observed that the this corollary can be used to show that certain cyclic covers of the hyperbolic surface of degree have complementary series. Here, of course, the lower bound depends on the fact that the constant is very explicit (for instance, since , we have ).