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.

**1. Preliminaries **

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

where

and

Furthermore, by the discussions after equation (2.12) of Ratner’s paper and by equation (2.13) of this paper, one has and . Hence,

and

Since

for every , we obtain that the constant appearing in equation (2.14) of Ratner’s paper is

i.e.,

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

i.e.,

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:

because .

Concluding our series of preparations, we recall the definitions of the following two functions

and

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

and

Finally, from the estimates (2), (4) above, and the facts and , we can estimate:

and

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:

where

- if ;
- if ;
- if ;

and

Remark 1In 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, by definition, , we can apply (2), (4) above to obtain

On the other hand, using that , , the equations (5), (6), (9), (10) above, and the fact that in the present case, we get

Since and (because ), we conclude that

where

and

Next, we notice that, when , by (8), and (11), (12) above,

If , we have that , and , so that (9), (10) and the equations (5), (6) and (15) above imply

and, *a fortiori*,

where

and

If , we have that , so that , , and . Putting this into (9), (10), and the equations (5), (6), and (15) above, we get

and

where

and

Now we pass to the case . In this regime, is not bounded, so that we can’t control by using (11). So, we follow the arguments in page 281 of Ratner’s paper: we recall that

and

Define . By integration by parts, ,

and , where

and

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

where .

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

where

and

Finally, we consider the case . We begin by estimating and : using (2), (4) above and , , we obtain

and

Thus,

so that we can take and in the equation (2.28) of Ratner’s paper.

Next, we observe that when and imply and

Therefore, from the previous discussion and , it follows that

where

and

At this stage, from (14), (16), (17), (18), (19) above, we see that the proof of the desired lemma is complete.

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**

Theorem 2Let be a non-trivial irreducible unitary representation of in and let . Let and . Then, for all ,

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

so that

by Lemma 1.

On the other hand, since for all , we know that

and

The desired result follows.

In the sequel, denotes the Casimir operator associated to (see this post).

Theorem 3Let be an unitary representation of having no non-zero invariant vectors in . Write the spectrum of the Casimir operator and

If , let and . Assume that when . Let with . Then, for all ,

where ,

*Proof:* This is an immediate consequence of Theorem 2 and the arguments from pages 285–286 of Ratner’s paper.

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:

Corollary 4Let be a lattice of and let be the regular representation of on where . Given with , it holds

where

is the hyperbolic Laplacian on , is its first eigenvalue, is the size of the spectral gap (if ), and the constants are given by the equations (1), (3).

Remark 2In 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 ).

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