EXPLICIT GEODESIC FLOW-INVARIANT DISTRIBUTIONS USING SL2(R)-REPRESENTATION LADDERS

An explicit construction of a geodesic flow-invariant distribution lying in the discrete series of weight 2k isotopic component is found, using techniques from representation theory of SL2(ℝ). It is found that the distribution represents an AC measure on the unit tangent bundle of the hyperbolic plane minus an explicit singular set. Finally, via an averaging argument, a geodesic flow-invariant distribution on a closed hyperbolic surface is obtained.


Introduction
Flow-invariant quantities play a major role in dynamics, namely, in answering ergodic questions, and are related, via the cohomological equation, to the problem of describing time changes for flows (see, e.g., [3,4,5,10] and the references therein).
In particular, it is well known that geodesic flow-invariant quantities appear quite naturally in quantum chaos: geodesic flow is a model for the time evolution of a classical mechanical system; while the time evolution of the quantum mechanical system is given in terms of the eigenfunctions of a certain selfadjoint operator.The transition between a classical mechanical system and the quantum version of the same system involves a method commonly known as quantization.The inverse process, that of going from a quantum mechanical system to a classical system, involves a limit which the physicists would like to claim as unique.(This is known as the correspondence principle or the semiclassical limit.)A central problem in physics since the late twenties has been trying to understand this transition.One of the main stumbling blocks seems to stem from the fact that there are many possible quantizations for the same classical system and it is not clear which of these should be the "correct one," furthermore the question of the uniqueness of the limit is also relevant.A mathematical approach to the problem can be stated as the unique quantum ergodicity question, where one is interested in finding the weak* limit points of microlocal lifts of eigenfunctions of the hyperbolic Laplacian 1300 Explicit geodesic flow-invariant distribution [2,8,13,16,18,23,24,25].It is well known that such limits must be geodesic flowinvariant measures on the unit (co-)tangent bundle of the Riemann surface [19], however, it is unknown which invariant measures actually occur.
On this matter, Zelditch, Šnirel'man, and Colin de Verdière [2,18,23] independently showed that almost all limits are Liouville measures for a compact hyperbolic surface.There is evidence supporting that this limit is unique for M = PSL 2 (Z)\PSL 2 (R) (see [13,16]), while on the other hand, Jakobson [8] showed that for the flat tori, the possible limits are of the form ν = φ(x)d Vol, where φ(x) is a trigonometric polynomial satisfying a rigid geometric condition.
As a further step in trying to understand the possible measures that could arise from such a limit, we undertake the study of geodesic flow-invariant distributions lying completely in the discrete series of weight 2k isotopic component of SL 2 (R) by explicitly constructing them and examining some of their properties.This is done using a "ladder" construction on the (square root of the) unit tangent bundle of the hyperbolic plane, by methods based on the representation theory of SL 2 (R).
The background material is presented in Section 2, while the actual construction is carried out in Section 3 and can be summarized as follows: essentially, a distribution on T * 1 H that lies completely in the discrete series unitary representation is constructed, then it is made geodesic flow-invariant by requiring it to be in the kernel of the operator corresponding to geodesic flow.The first requirement is met by using a "ladder" construction in which a suitable holomorphic form of a given weight 2k is raised to weights 2k + n, n a positive integer, and these are all summed with a yet unspecified coefficient to form an infinite series.The requirement that this series be geodesic flow-invariant characterizes completely the coefficients (up to a constant) and further analysis shows that this is in fact a distribution of order 0 with singularities of logarithmic type or milder (Theorem 3.8).Furthermore, the singular set has a simple geometrical description (Figures 2.1 and 2.2).Finally, a geodesic flow-invariant distribution is obtained on a closed hyperbolic surface by the usual procedure of averaging over the group, this result is presented as Theorem 3.12.

Background
As mentioned in the introduction, we will be using the representation theory of SL 2 (R), hence we introduce some terminology that may not be common to all readers.

Notation and terminology. Let
The action of G on H is by fractional linear transformations.We adopt the usual convention that z = x + iy, so the action of a b c d ∈ G is given by a b c d : z → (az + b)/(cz + d).Notice that this action factors through the center, hence it is an action of PSL 2 (R) = SL 2 (R)/ ± Id.
Let µ g (z) = cz + d; it is called the automorphy factor and satisfies the cocycle relation µ g g (z) = µ g (gz) • µ g (z) for all g ,g ∈ G.Moreover, d(gz)/dz = µ g (z) −2  Since K fixes i ∈ H, then there is a natural identification between SL 2 (R)/K and H given by gK → gi = (ai + b)/(ci + d).In fact we can identify SL 2 (R) with T * 1 H the square root (the double cover) of the unit cotangent bundle via the map g → (gi,(µ . This can also be seen by recalling [12] that an element g ∈ SL 2 (R) has the unique Iwasawa decomposition hence the rule for z = x + iy ∈ H and θ the argument for the root cotangent vector, provides the required equivalence between SL 2 (R) and T * 1 H.
1302 Explicit geodesic flow-invariant distribution Next, we define an automorphic form on SL 2 (R) for the group Γ with weight l ∈ C as a function f l : SL 2 (R) → C satisfying (i) (left Γ-invariance) f l (γg) = f l (g) for all γ ∈ Γ and g ∈ G; (ii) (right action by K) f l (gk) = µ k (i) −l f l (g) for all k ∈ K. Notice that µ k (i) = e iθ since k = cos θ sin θ − sin θ cos θ ; (iii) f l satisfies an SL 2 (R)-invariant differential equation.With this definition in mind, it is easy to recover the more common definitions of an automorphic form as a function on H.The key lies in the following procedure that takes functions defined on H and lifts them to functions on SL 2 (R) and vice versa: where z = gi for some g ∈ G.It is interesting to note that we can rewrite f l (g) = f (z)y l/2 e ilθ , and that f l transforms on the right according to the character cos θ sinθ − sin θ cos θ → e ikθ of K.With the procedure described above, we have then that an automorphic form on H for the group Γ with weight l ∈ C is a function f : (ii) f is the solution to an SL 2 (R)-invariant differential equation.Note that further regularity and growth properties follow from the last condition in each of the two definitions.
As stated, this definition of automorphic form is very wide; by allowing for different growth conditions, we can change the number of automorphic forms that are allowed; the more strict the condition, the less forms satisfying it.

Some representation theory of SL 2 (R)
. Let M = Γ\H, where Γ is a cofinite group of isometries of H. Then the Lie Algebra of M is locally identified with p, the space of symmetric matrices with trace zero, contained in sl 2 (R), which in turn enables us to use the techniques of representation theory.In accordance with Bargmann's classification theorem (see, e.g., [12,Theorem IV.6.8]),there are four different types of unitary representations of SL 2 (R): discrete series, mock discrete series, principal series, and complementary series.All of these are infinitesimally isomorphic to some explicit subspaces of functions on SL 2 (R).The explicitness of these subspaces is what will enable us to work with them [11,12].
Consider the Iwasawa decomposition of SL In particular, let ψ n ∈ H(s) be the function such that ψ n (r(θ)) = e inθ .Let π s be the representation of SL 2 (R) on H(s) by right translation.By Bargmann's classification theorem, the discrete series representation of weight m corresponds to ±s = m − 1, where m is an integer greater than or equal to 2. Consider the following two subspaces of H(m − 1): (2.4) Alvaro Alvarez-Parrilla 1303 The subspace 2k) .Recall that this is just a realization of the discrete series unitary representation.

The construction
Let V be the space of functions on SL 2 (R) such that f |K ∈ L 2 (K) and that are also geodesic flow-invariant.Then SL 2 (R) acts on V by right translation.If π is an irreducible representation of SL 2 (R) on V , then V (π) = Image(ϕ), with ϕ ∈ Hom(π,V ), is the π-isotopic component of V and consists of "copies" of the π irreducible subspaces.
Denote by H(2k − 1)(π 2k ) the discrete series of weight 2k isotopic component in V , and by H(2k − 1)(π 2k ) the corresponding distributions.Consider the formal sum T = n f n , with f n ∈ H(2k − 1)(π 2k ) for all n.By abusing notation, we define T as a distribution on SL 2 (R) associated to the formal sum.We will say that the distribution T lies completely in It is now possible to specifically state the problem addressed in this paper.
Main question.Suppose we have a geodesic flow-invariant distribution lying completely in the discrete series of weight 2k isotopic component H(2k − 1)(π 2k ).What can we say about the distribution?

The plan of action.
In order to answer the above question, an explicit construction of a geodesic flow-invariant distribution is carried out.The idea behind the construction is as follows: start with a suitable holomorphic form on a hyperbolic manifold M and then via a "ladder" construction obtain the distribution lying in the discrete series of weight 2k.Since M is a hyperbolic quotient, work on H by requiring the objects to be automorphic forms of weight 2k.The use of relative Poincaré series allows the construction of such automorphic forms; recall that the Petersson series, Θ k,γn , are automorphic forms of weight 2k associated to primitive hyperbolic elements γ n ∈ Γ (for further details, see [9]).Θ k,γn is constructed by summing over the group the function . By conjugating the group with an appropriate element, it is possible to use a primitive hyperbolic element γ 0 ∈ Γ whose axis is the imaginary axis (in the upper half-plane model of H).In this case, Q γ0 (z) = (a −1 − a)z.Thus the construction is carried out using Θ k,γ0 .
Since the representation theory of SL 2 (R) is to be used, we use the lifting procedure, outlined in (2.3), to lift forms on H to functions on SL 2 (R), as well as the identification of SL 2 (R) with T * 1 H the square root (double cover) of the unit cotangent bundle of H.This allows us to establish a correspondence between tensors on H and functions on SL 2 (R): let f (z)dz k be a symmetric k-tensor on H, consider first the balanced tensor f (z)y k dz k/2 d z−k/2 (recall that the hyperbolic metric is ds = y −1 dz 1/2 d z−1/2 ).Then associate, under the identification of SL 2 (R) with T * 1 H, the weight 2k function on SL 2 (R) given by Ψ k (z,θ) = f (z)y k e i2kθ .The function Ψ k (z,θ) on SL 2 (R) is called the lift of f .Under this procedure, the function Q −k γ0 (z) is lifted (up to a constant) to the function g k,γ0 (z,θ) = y k e i2θk /z k .The intention then is to construct a "ladder" in H 2k out of the lift of the Petersson series Θ k,γ0 , and use the techniques of representation theory of SL 2 (R) to make the "ladder" geodesic flow-invariant.
Instead of summing over the group immediately (and obtaining the lift of Θ k,γ0 (z)), we opt to first obtain its "ladder" in H 2k .
In order to construct the "ladder", we will need to compute the action of the following elements of the Lie algebra of SL 2 (R): , and W = 0 1 −1 0 .The infinitesimal representation dπ s acts on elements of H(s) via the Lie derivative, hence abusing notation and denoting by E + , E − , and W the Lie derivatives of E + , E − , and W, respectively, we have these are the "raising," "lowering," and "weight" operators, respectively, (since they raise, lower, and pick out the weights of the ψ n ).Notice that one can think of the operator E + as generalizing the complex exterior differential ∂ which maps forms of type dz k to forms of type dz k+1 , also note that formally we have E − = E + , and since we are interested in calculating ladders in H 2k = H (2k) ⊕ H (−2k) , then it follows by the above remark and the following standard lemma (whose proof can be found in, say [11,12]), that it is only necessary to calculate in H (2k) .
Lemma 3.1 (discrete series ladder).Let u ∈ H(s).Suppose that (1) With this in mind, the construction should proceed as follows.
(1) We will start with the γ 0 -invariant function g k,γ0 , and proceed to construct its ladder (2) Next we will determine the coefficients a m which make the ladder geodesic flowinvariant (Section 3.3).
(3) Finally sum over the group to obtain the geodesic flow-invariant distribution on M lying only in H 2k (Section 3.4).

Recursion relations for some polynomials.
The following subsection puts together some facts about some polynomials that will be fundamental for the rest of the construction.
Alvaro Alvarez-Parrilla 1305 Let k ∈ Z + be fixed.Consider the polynomials The polynomials defined by (3.3) satisfy the following recursion relation: Proof.The proof is a simple induction argument on n.
Lemma 3.3 (generating function for polynomials).The polynomials defined by (3.3) have generating function We have the following lemma.
Lemma 3.4.The polynomials p n (x) = (1 − x) 2k−1 q n (x) satisfy (i) p n (x) is a polynomial in x of degree n + 2k − 1, (ii) p n (x) has exactly 2k terms, that is, (iii) the coefficients b n, j , for Proof.From (3.6), and since deg(p n In the proof of statement (ii), we will drop the index corresponding to n in the coefficients of the polynomial p n (x), that is, we will write b j instead of b n, j (since n is fixed).
In order to prove this statement, we first need a recursion relation for the polynomials p n , we obtain this from (3.4).From the definition of p n (x), we have that 1306 Explicit geodesic flow-invariant distribution so substituting in (3.4), we obtain the desired recursion relation for the polynomials p n : We now proceed by induction on n.For n = 0, the result follows by the binomial theorem as follows: Next we consider n = m + 1.By induction hypothesis, hence substituting in (3.9) and collecting terms of the same degree, (3.12) So, indeed p n (x) has exactly 2k terms.Finally to prove statement (iii), we know that p n (x) = (1 − x) 2k−1 q n (x), hence by (3.3) and by the binomial theorem, we have Alvaro Alvarez-Parrilla 1307 and since then indeed b n,m is a polynomial in n of degree ≤ k − 1.

Explicit construction on SL 2 (R)
. Let γ 0 ∈ Γ be a hyperbolic element whose axis is the imaginary axis on H. Consider the function on SL 2 (R) (recall that this is the lift of Q −k γ0 (z)), and its ladder Notice that, as required by Lemma 3.1, Wg k,γ0 = i2k g k,γ0 and E − g k,γ0 = 0.
To find the coefficients a m , we will make use of the fact that we require t k,γ0 to be geodesic flow-invariant.The operator generating geodesic flow is given in terms of E + and E − by G = (E + + E − )/2.Hence we require that Gt k,γ0 = 0, or equivalently, n≥0 This will enable us to determine explicitly the coefficients that make the ladder (3.17) geodesic flow-invariant.
Remark 3.5.It is to be noted that since (E + + E − )/2 = 1 0 0 −1 , then the requirement that the ladder (3.17) is to be geodesic flow-invariant can be stated as saying that it must be A-invariant (where A < SL 2 (R) is the subgroup of diagonal matrices).

Lemma 3.6 (geodesic flow-invariant coefficients). The coefficients that make the ladder (3.17) geodesic flow-invariant are given by
for m ∈ Z.In other words, with the above choice of coefficients (formal) A-invariance of the ladder (3.17) exists (up to the choice of a 0 , see proof).
Proof.The proof of the lemma proceeds as follows.
Since E + and E − are the raising and lowering operators, they satisfy the commutation relation (3.20) 1308 Explicit geodesic flow-invariant distribution where W = ∂/∂θ corresponds to the element 0 1 −1 0 and "picks out the weight" of the function on which it acts (so Wg k,γ0 = i2k g k,γ0 ).
Thus from (3.18), we obtain the following recursion relation: From this and the fact that E − g k,γ0 = 0, since g k,γ0 is a holomorphic form, we have that a 0 is a free coefficient.Without loss of generality, we let a 0 = 1.
with the convention that E 0 + ≡ 1.But since g k,γ0 is a holomorphic form, then E − g k,γ0 = 0, and since W "picks out" the weight, WE m + g k,γ0 = 2i(k + m)g k,γ0 , then Hence substituting in (3.21), we obtain that is, Since a 0 = 1, we obtain that for n = 2m, Alvaro Alvarez-Parrilla 1309 On the other hand for n odd, we have from (3.21) that a 1 E − E + g k,γ0 = 0, and by the same arguments as above, we see that hence a 1 = 0, and by (3.25), a n = 0 for n odd.This proves the lemma.
Remark 3.7.Notice that in the proof of the above lemma, we have used the facts that g k,γ0 is an eigenvector of the Casimir operator ω and that ω commutes with E + , E − , and W. Furthermore, the determination of the coefficients (and hence of the ladder (3.17)) is unique up to the choice of the coefficient a 0 .
It will now be convenient to introduce the change of variables α = z/z = e −i2ψ and β = e i2θ , where z = x + iy = re iψ , and θ is the fiber variable for T * 1 H. Recall that since y > 0, then 0 < ψ < π, and θ represents the "direction" in the unit cotangent bundle, so 0 ≤ θ < π (see Figure 2.1).
In these new variables, we have An easy calculation shows that where q n (α) is a polynomial in α of degree n satisfying the following recursion relation: By Lemma 3.2, we have that Hence, where , (3.33) 1310 Explicit geodesic flow-invariant distribution so (3.17) becomes We can now state the following theorem.
Theorem 3.8 (geodesic flow-invariant ladder on SL 2 (R)).The formal sum on SL 2 (R) given explicitly by is a distribution of order 0, with a pole of order k − 1 at α = 1, and singularities of logarithmic type or milder; otherwise.Furthermore, it is invariant under the right action of geodesic flow and the left action of γ 0 , and lies completely in H(2k − 1)(π 2k ), the discrete series of weight 2k isotopic component.
Proof.Notice that by construction, the formal sum given by (3.34) (alternatively by (3.35)) lies in H(2k − 1)(π 2k ) and it is invariant under geodesic flow and the left action of γ 0 .
In order to prove that the formal sum is in fact a distribution as stated in the theorem, we show that it represents a holomorphic function on each of the variables α and β on S 1 × S 1 except on an explicit singular set k,γ0 (see below).
Hence, this lemma shows that the series (3.34) defines a holomorphic function on Alvaro Alvarez-Parrilla 1311 Now, we notice that in the case k = 1, we have C 2m,1 = 1/(2m + 1), so the expression (3.34) for t k,γ0 reduces to so in particular the series defines a distribution of order 0 on S 1 × S 1 − 1,γ0 , where 1,γ0 = {(α, β) : For the cases of k > 1, we argue as follows.The general term of the series in question is where p n (x) = (1 − x) 2k−1 q n (x).By Lemma 3.4(ii), we obtain that (3.43) where the coefficients b 2m, j and b 2m,2m+k+ j are polynomials in 2m of degree k − 1.On the other hand, C −1 2m,k is a polynomial in 2m of degree k, and in fact for large m, the products b 2m, j C 2m,k and b 2m,2m+k+ j C 2m,k behave like 1/(2m + r), with r ≥ 1 being an odd positive integer.Hence we obtain 2k power series, on each of which we may again apply Fatou's theorem to obtain convergence of t k,γ0 on its regular points on S 1 × S 1 .Thus we see that t k,γ0 has a pole of order k − 1 arising from the factor (1 − α) 1−k and singularities of logarithmic type or milder arising from the 2k terms each of which has logarithmic or milder behavior.Remark 3.10.Notice that from Remark 3.7 and the fact that the discrete series representations occur with multiplicity, any other irreducible subrepresentation is a multiple of this explicit one (obtained by changing the coefficient a 0 alluded to in Remark 3.7).
or equivalently in terms of Arctanh, and so forth.From these, it is clear that the same description of the singular set that was found for k = 1,2 works as well for k ≥ 3.

Explicit construction of an
(3.51) Proof.Notice first that for γ ∈ SL 2 (R), a formal calculation shows that Moreover, observe that every geodesic flow-invariant distribution on the discrete series of weight 2k is a linear combination of T Re k,γ for different primitive geodesics [γ], this follows directly from the fact that the space S 2k (Γ) of Γ-automorphic forms of weight 2k is generated by {Θ k,γ : γ ∈ Ᏻ}, where Ᏻ is a finite set of primitive geodesics on M (see [9]).
Remark 3.13.Recall that the choice of holomorphic form of weight 2k, with which the geodesic flow-invariant distribution on H is constructed, is chosen such that the automorphic geodesic flow-invariant distribution is in fact the ladder of raising of the Petersson 1314 Explicit geodesic flow-invariant distribution series of weight 2k.This is of interest since there is a large volume of work on the Petersson series but no one has (to the extent of my knowledge) examined the natural extension of studying the ladder of a Petersson series.
The study of other properties of T k,γ0 , as well as extending the method presented in this paper to the other unitary representations given by Bargmann's classification, is underway, and will be presented elsewhere.

automorphic geodesic flow-invariant distribution.
Because the Riemann surfaces in question are quotients of H, it is natural to consider relative Poincaré series of the geodesic flow-invariant distributions: notice that t k,γ0 is not Γ-automorphic, but by summing over the group, one formally obtains the relative Poincaré series T k,γ0 (z,θ) = mentioned before, the lift to functions on SL 2 (R) of the Petersson series Θ k,γ0 .Theorem 3.12 (weight 2k discrete series).Assume that k > 1.Let T Re k,γ0 = 2Re T k,γ0 .The discrete series component of weight 2k of a geodesic flow-invariant distribution on a closed hyperbolic surface M is given by a linear combination of T Re k,γ for different primitive geodesics [γ].Furthermore, T k,γ0 is a distribution of order ε with singularities of mild type, and it satisfies