Certain L2-norms on automorphic representations of SL(2)

Let $\Gamma$ be a non-uniform lattice in $SL(2, \mathbb R)$. In this paper, we study various $L^2$-norms of automorphic representations of $SL(2, \mathbb R)$. We bound these norms with intrinsic norms defined on the representation. Comparison of these norms will help us understand the growth of $L$-functions in a systematic way.


Introduction
Let Γ be a non-uniform lattice in SL (2, R).By an automorphic representation of SL(2, R), we mean a finitely generated admissible representation of SL(2, R), consisting of Γ-invariant functions on SL(2, R) ( [5]).Among all automorphic representations, L 2 automorphic representations, i.e., subrepresentations of L 2 (G/Γ), are of fundamental importance.Since L 2 automorphic representations are unitary and completely reducible, we assume L 2 automorphic representations to be irreducible.By Langlands theory, L 2 automorphic representations come from either the residues of Eisenstein series or the cuspidal automorphic representations.Throughout this paper, we shall mostly focus on irreducible cuspidal representations, even though our results also apply to unitary Eisenstein series with vanishing constant term near a cusp.
Let G = SL(2, R) and π be an irreducible admissible representation of G.We say an automorphic representation is of type π if the automorphic representation is infinitesimally equivalent to π.In particular, we write L 2 (G/Γ) π for the sum of all L 2 -automorphic representations of type π.It is well-known that L 2 (G/Γ) π is of finite multiplicity ( [5]).The main purpose of this paper is to study various L 2 -norms of the automorphic forms at the representation level.In the literature, automorphic forms, the K-finite vectors in an automorphic representation, are the main focus of interests.Our main focus here is the L 2 -norms of automorphic forms, in comparison with (intrinsic) norms in the representation.We hope to gain some understanding of various L 2 -norms of automorphic representation as a whole, without references to automorphic forms.We believe this may lead to a better understanding of the Fourier coefficients and L-functions.
Our estimates of L 2 -norms essentially involve two decompositions, the Iwasawa decomposition KAN , and its variant KN A. The KAN decomposition is utilized mainly to define Fourier coefficients and To state our results in a simpler form, let Γ = SL(2, Z).Fix the usual Iwasawa decomposition G = KAN with N the unipotent upper triangular matrices.Let F be the fundamental domain of G/Γ contained in a Siegel set.Recall that the L 2 -norm on the fundamental domain is We have Theorem 1.1 Let π = P(u, ±) be a unitary representation in the principal series (see Section 3.1 for the definition).Let H be a cuspidal representation in L 2 (G/Γ) π .Then for any ǫ > 0, there exists a C ǫ > 0 such that For any ǫ < 0, there exists a C ǫ > 0 such that Here u 0 = ℜ(u) and the norm |||f ||| ǫ 2 −u0 is defined on H ∞ , smooth vectors in the representation in H( see Eq. 3.7 for the definition of |||f |||).
Our theorem essentially says that every f ∈ L 2 (G/Γ) π is also in L 2 (F , a ǫ da a dndk) for every ǫ > 0. In other words, the natural injection is bounded for every ǫ > 0 even though the natural map L 2 (F , a 2 da a dndk) → L 2 (F , a ǫ da a dndk) is not bounded unless ǫ ≥ 2. In terms of the parameter ǫ, there is a natural barrier at ǫ = 0, namely, as ǫ → 0, the norms of these bounded operators go to infinity.
We shall remark that our estimates are true for all nonuniform lattices of any finite covering of SL(2, R) (see Theorem 5.1).In addition, the first bound with ǫ > 0 also holds for discrete series D n (see Cor. 3.2).They are proved by studying the L 2 -norms of Fourier coefficients of the automorphic distribution, defined in Schmid ( [15]) and Bernstein-Resnikov ( [1]).For the general linear group GL(n, R), similar results should hold.The following problem is worthy of further investigation.
Problem: Let G be a semisimple Lie group, Γ an arithmetic lattice and S a Siegel domain.Find the best exponents α such that is bounded.Here G = KAN is the Iwasawa decomposition.
Notice that if α = 2ρ, the sum of positive roots of gl(n), the measure on the right hand side is the invariant measure of G restricted to S. In this case, i is automatically bounded.This shows that if α is "bigger" than 2ρ, i is also bounded.The problem is to find the "smallest" α such that i is bounded.We shall remark that cusp forms will remain to be in L 2 (S, a α da a dndk) for any α since they are fast decaying on the Siegel set.Hence our problem is about cuspidal representations, rather than cusp forms.
The second main result is an L 2 -estimates of f on ΩA where Ω is a compact domain in G/A.Theorem 1.2 Let Γ be a nonuniform lattice in SL(2, R).Suppose that the Weyl element w ∈ Γ and Γ ∩ N = {I}.Let H be a cuspidal automorphic representation of G of type P(iλ, ±).Let Ω be a compact domain in KN .Let ǫ > 0. Then there exists a positive constant C depending on ǫ, H and Ω such that See Eq. 3.7 for the definition of |||f |||.
We shall remark that in the KN A decomposition, the invariant measure is given by dkdn da a .Hence, the L 2 -norm here is a perturbation of the canonical L 2 -norm.In addition, ΩA has infinite measure.The perturbation is needed because our theorem fails at ǫ = 0.At ǫ = 0, the norm |||f ||| − ǫ 2 is the original Hilbert norm f of the cuspidal representation.There is no chance that f L 2 (ΩA, d a a dtdk) can remain bounded for all f ∈ H.
Throughout our paper, the Haar measure on A will be da a .We use c or C as symbolic constants and c ǫ,u to indicate the dependence on ǫ and u. and w = 0 1 −1 0 ∈ K.We call w the Weyl element.Let Γ be a discrete subgroup of G such that Γ ∩ N is nontrivial.Without loss of generality assume that Let M = {±I} ⊆ K. Fix P = M AN , the minimal parabolic subgroup.Then the identity component P 0 = AN .Fix da a dt as the left invariant measure on an t ∈ P 0 and dT da a as the right invariant measure on N T a ∈ P 0 .We shall keep the notion that an t = N T a. Then Fix dk = dθ as the invariant measure on K.We write g = k θ an t for the KAN decomposition and g = k θ n T a for the KN A decomposition.Fix the standard invariant measure dg = a 2 dt da a dk = dT da a dk.
Here L 2 loc (G/Γ) is the space of locally square integrable function on G/Γ.
By abusing notation, we simply use a ∈ R + as an element in A. Write Write f L 2 (X(T1,a1) ± ,a ǫ da a dT dk) as f T1,a ± 1 ,ǫ .

Estimates on
Without loss of generality, assume T 1 > 0. Observe that We have Proof: We have Here ⌊ * ⌋ is the floor function.The other direction is similar.
For T 1 negative, we have a similar statement.Combining these two cases, we have

Estimate on
To estimate f T1,a + 1 ,ǫ , we must utilize the Weyl group element w.We assume that Let a ∈ [a 1 , ∞).By the Iwasawa decomposition and k(T, a) ∈ K.This defines a coordinate transform from (T, a) to (T ′ , a ′ ).Let (P (T 1 , a 1 ) + ) ′ be the coordinate transform of P (T 1 , a 1 ) + w in terms of (T ′ , a ′ ) coordinates.We have It is easy to see that Observe that We obtain Choose a 1 = 1.We have Combined with Theorem 2.1, we have (2.2) If p = 1 and T 1 = 1, we have Hence, we have bounded the norm of f on X T1 .Generally, we have Then there exists a positive constant c T1,ǫ,p such that Proof: We choose a positive constant c such that Observe that the right hand side of our inequality involves an integral over a Siegel set.However the measure on this Siegel set can be larger than the invariant measure a 2 dk da a dt.What we have achieved is a bound of f T1,ǫ by an integral on a Siegel set.In the next section, we shall give estimation of the norms of f on A − a1 N/N p and on KA − a1 N/N p .
3 Matrix Coefficients and Analysis on P 0 /N p Now we shall focus on L 2 automorphic representations of type π where π is a principal series representation.According to Langlands, L 2 automorphic representations come from either the residue of Eisenstein series or cuspidal automorphic forms.In either cases, the restrictions of L 2 automorphic representations fail to be L 2 on P 0 /N p , when P 0 /N p is equipped with the left invariant measure.However if we perturb the invariant measure correctly, automorphic forms will be square integrable.
In this section, we will discuss the L 2 -integrability of f | P0 with f ∈ L 2 (G/Γ) π with respect to the measure a ǫ da a dt.We will consequently discuss the L 2 -norm on a Siegel subset.We conduct our discussion in terms of matrix coefficients with respect to periodical distributions with no constant term.More precisely, the function f | P0 will be regarded as the matrix coefficient of v ∈ H π and a periodical distribution in (H * ) −∞ .Our view is similar to Schmid and Bernstein-Reznikov ( [15] [1]).
Consider the noncompact picture ( [10]).The noncompact picture is essentially the restriction of Here χ − (x) is the sign character on R − {0} and χ + (x) is the trivial character.In particular, we have There is a G-invariant pairing between P(u, ±) ∞ and P(−u, ±) ∞ .This allows us to write the dual space of P(u, ±) ∞ as P(−u, ±) −∞ .
Unless otherwise stated, P(u, ±) will refer to the noncompact picture.The space P(u, ±) ∞ will then be a subspace of infinitely differentiable functions on N ∼ = R satisfying certain conditions at infinity.

Matrix coefficients with respect to periodical distribution with zero constant term
According to [1] [15] [13], every L 2 automorphic form of type π can be written as matrix coefficients of an automorphic distribution and a vector in the unitary representation π.Equivalently, in our setting, there exists a distribution τ ∈ P(u, ±) −∞ such that the automorphic forms of type π can be written as linear combinations of . For P(u, +), the weight m can only be an even integer.For P(u, −), the weight m must be an odd integer.If τ is cuspidal, τ has a Fourier expansion Here p is a positive integer and * denote the weak summation ( [6]).We call such τ a periodical distribution without constant term.
Let τ ∈ P(u, ±) −∞ be a periodic distribution without constant term.We compute the matrix coefficient formally: Here F is the Fourier transform, and v is in a suitable subspace of P(−u, ±) −∞ .The formula above, also known as the Fourier-Whittaker expansion in a more general context, is valid for with φ an odd or even smooth function on the unit circle.They are slowly decreasing functions.Their Fourier transforms exist.Since the derivatives v (n) are of this form and they are integrable , we see that F v(ξ) will decay faster than any polynomial at ∞.The weak sum in Equation (3.1) becomes a convergent sum.Our lemma is proved.
We shall make a few remarks here.Since v ∈ P(−u, ±) ∞ and τ ∈ P(u, ±) −∞ , the matrix coefficient π u,± (an t )τ, v is automatically smooth.Our lemma simply provided a Fourier expansion, which is generally known as the Fourier-Whittaker expansion over the whole group G.The restriction that u 0 < 1 is somewhat unsatisfactory.When u 0 ≥ 1, F v(ξ) may fail to be a function even for v smooth.This happens when P(−u, ±) is reducible and discrete series will appear as composition factors.Hence, automorphic representations that are discrete series, can be treated by considering the reducible P(−u, ±).We shall refer readers to Schmid's paper [15] for details.When P(−u, ±) is irreducible, F v(ξ) is a fast decaying continuous function off from zero.Our lemma is still valid in this case.However,if u 0 > 1, F v(ξ) will fail to be a locally integrable function near zero and need to be regularized to be a Schwartz distribution.
From now on, without further mentioning, we will restrict our scope to u 0 < 1.We do not lose any generalities here.If P(u, ±) is unitary, then ℜ(u) ∈ (−1, 1).If π is a discrete series representation, then π can be embedded into a principal series representation P(−u, ±) with u < 1. Hence our assumption is adequate for the discussion of L 2 automorphic representations.When ℜ(u) < 1 and v ∈ P(−u, ±) ∞ , exp 2πixn, v shall be interpreted as exp 2πinx, dv dx .
3.3 L 2 -norms on P 0 /N p Let us first study the L 2 norms of f (g) = π u,± (g)τ, v on P 0 /N p .τ and v are given in Lemma 3.1.Now we compute We summarize this in the following proposition.
Let f (an t ) = π u,± (an t )τ, v .Then f (an t ) is a smooth function on P 0 and Proof: Since f (g) is a smooth function on G, f (an t ) is a smooth function on P 0 .Both equations hold without any assumptions on convergence.Hence both sides of the equations converge or diverge at the same time.

Estimates of Fourier coefficients b n
We can now provide some estimates of certain sum of Fourier coefficients.These estimates are more or less known for automorphic forms ([1] [15] [14] [4]).Our setting is more general.
Theorem 3.1 Under the same assumption as Prop.3.1, suppose that there exists a v ∈ P(−u, ±) ∞ such that f (an t ) = π u,± (an t )τ, v is bounded on P 0 .Suppose that F v(a) is nonvanishing on R − or R + .Then we have the following estimates about the Fourier coefficients b n .
1.If |f (an t )| 2 ≤ C µ,f a µ for some µ > 0, i. e., f (an t ) decays faster than a µ near the cusp 0, then we have for each ǫ ∈ (−µ, 0), 2. For each ǫ > 0, there exists a C ǫ,τ > 0 such that Let me make a remark about the ± or ∓ signs.If F v(a) is nonvanishing on R − , then b ±n should be read as b +n ; if F v(a) is nonvanishing on R + , then b ±n should be read as b −n .The proof should be read in the same way.
Proof: Fix f (an t ) = π u,± (an t )τ, v bounded on P 0 by C f .Suppose that F v(a) is nonvanishing on R − or R + .
1. Suppose that |f (an t )| 2 ≤ C µ,f a µ for µ > 0. For −µ < ǫ < 0, the left hand side of Equation becomes a factor and must remain bounded by a constant depending on f and ǫ.
If τ is a cuspidal automorphic distribution in a unitary principal series or complementary series representation, then all automorphic forms f (g) will be bounded and rapidly decaying near the cusp at zero.In this situation, the estimates in Theorem 3.1 were well-known ( [15] [1]).The first estimate can also be obtained by observing that the Rankin-Selberg L(f × f, s) has a pole at s = 1 for suitable f and the coefficients of the Dirichlet series are all nonnegative ( [4]).If the (cuspidal) automorphic representation is a discrete series representation, the automorphic distribution τ can be embedded in P(u, ±) −∞ for a suitable u and will have its Fourier coefficients supported on p −1 N or −p −1 N. Our estimates of Fourier coefficients also follow similarly upon applying the intertwining operator.The details of how to treat the discrete series representations can be found in [15] [14].

L 2 -norms of Bounded Periodical Matrix coefficients
By considering the converse of Theorem 3.1, the equations in Prop.3.1 also imply the following.
Theorem 3.2 Under the same assumption as Proposition 3.1, we have the following estimates.
In particular, We shall remark that this theorem holds even P(u, ±) is not unitary.
Hence it applies to discrete series representation D n .In addition, the norm on the right hand side of Inequality (3.6) is bounded by By the Kirillov model, this integral is a constant multiple of the unitary norm v Dn ([9]).We have Corollary 3.2 (discrete series case) Let D n be a discrete series representation.Let τ be a periodic distribution in D −∞ n with period p. Suppose that for some φ ∈ D ∞ −n , the function D n (an t )τ, φ is bounded on P 0 .Then for any ǫ > 0 and v ∈ D ∞ −n , for every v ∈ D ∞ −n and therefore v ∈ D −n .Here D −n is the dual of D n .
Notice that Theorem 3.2 holds for each π −u,± (k)v.We obtain Corollary 3.3 (ǫ < 0) Let P(u, ±) be a unitary representation.Under the assumptions of Prop.
3.1, suppose that ǫ < 0 and n =0 |n| In particular, Both inequalities hold for those v ∈ P(−u, ±) with which the right hand sides converge.
In the case of automorphic forms, our L 2 norms are estimated over a Siegel subset, but with the measure a ǫ da a dkdt, while the Siegel set is often equipped with the measure a 2 da a dkdt.The bounds we have are certain norms on the representation.This allows us to treat everything at the representation level.If ǫ > 0, the bounds come from the Hilbert norm of the automorphic representation.We have nothing to improve on.If ǫ < 0 , we will need to further study the norm in more details.Our goal is to bound |||v||| ǫ 2 −u0 by a more tangible norm.A natural choice is a norm coming from the complementary series construction.

K-invariant Norms and complementary series
Let ℜ(u) > −1.Recall that the smooth vectors in the noncompact picture of unitarizable P(u, ±) are bounded smooth functions on R with integrable Fourier transform.The Fourier transforms are indeed fast decaying at ∞, but singular at zero.For any bounded smooth function φ with locally square integrable Fourier transform, let us define whenever such an integral converges.This norm is indeed the unitary norm of the complementary series C u , upto a normalizing factor.The standard norm * u for the complementary series is often constructed using the standard intertwining operator A u ([10]).Our norm * Cu differs from the * u by a normalizing factor.The standard norm * u has a pole at u = 0.The norm * Cu does not.Hence * Cu is potentially easier to use.In this section, we will first review the basic theory of complementary series.Then we will use • Cu to bound the norm |||•||| u .Our main references are [10] [3].

Intertwining operator and complementary series
The standard intertwining operator A u : P (u, +) ∞ → P (−u, +) ∞ is well-defined for ℜu > 0 and has meromorphic continuation on C. In the noncompact picture, Let * , * be the complex linear G-invariant pairing For any φ, ψ ∈ P(u, +), we define This is a G-invariant bilinear form on P(u, +) ∞ .When u is real and 0 < u < 1, (φ, ψ) u = A u (φ), ψ u yields an G-invariant inner product on P(u, +) ∞ .Its completion is often called a complementary series representation of G, which is irreducible and unitary.
In the noncompact picture, the standard basis for the K-types of P(u, +) is given by The intertwining operator A u maps v .
See [3].We make two observations here.First, the formula above in fact uniquely determined the analytic continuation of the intertwining operator A u .Secondly, for u / ∈ 2Z + 1, We have Lemma 4.1 For a fixed u ∈ (−1, 0) or u ∈ (0, 1), there exist positive constants c u , c ′ u such that The intertwining operator A u has a pole at u = 0. Hence we must exclude u = 0 from our estimates.

Bounds by the complementary norm: P(iλ, +) case
Fix v ∈ P(iλ, +) ∞ with λ ∈ R. Recall that we are interested in the norm Clearly, this norm is K-invariant.Hence we will need to estimate v Proof: Observe that Under the compact picture of P(u, +), v 2m becomes | sin θ| iλ−u exp 2miθ, (cot θ = x).
The function | sin θ| iλ−u has period π and L 1 derivative.Hence its Fourier series expansion k∈Z a 2k exp 2kiθ, for some positive constant h u .We obtain It follows that which will be bounded by a multiple of (1 + m 2 ) − u 2 .
If u + µ > 0 and m = 0, we have k∈Z The first sum is bounded by By essentially the same proof as Theorem 4.3, we have Theorem 4.7 For u ∈ (−1, 1) and µ ∈ (−1 − u, 0), there exists a positive constant c u,µ such that

K-invariant Norms over G/Γ
Let Γ be a nonuniform lattice in SL(2, R).Then G/Γ has a finite volume and a finite number of cusps, z 1 , z 2 , . . ., z l .Write G/Γ as the union of Siegel sets S 1 , S 2 , . . .S l with a compact set C 0 ( [2]).Since Γ action is on the right, our standard Siegel set will be near 0, not ∞.Let dg = a da dt dk be the invariant measure of G under the KAN decomposition.Over each Siegel set S i , the invariant measure can be written as dg = a i da i dt i dk.
Theorem 5.1 Let Γ be a nonuniform lattice in SL(2, R).Let H ⊆ L 2 (G/Γ) be a cuspidal automorphic representation of type P(−u, ±).Given any K-invariant measure ν on G/Γ such that ν is bounded by dg on C 0 and bounded by a ǫ i dai ai dt i dk on S i , there exists a constant C depending on ν (hence on ǫ) and H such that and |||f ||| ǫ 2 −u0 will be bounded the complementary norm given in Theorems 4.3 4.5 4.7.We shall remark that our theorem can be generalized to all nonuniform lattice of a finite covering of SL(2, R).
Proof: Let v ∈ P(−u, ±) ∞ and σ ∈ P(u, ±) −∞ .Let f (kan t ) = π u,± (kan t )σ, v .Then for any h ∈ G, the left action We see that the left action on f (kan t ) is equivalent to the action of P(−u, ±) on v. Fix H ⊆ L 2 (G/Γ), a cuspidal automorphic representation of type P(−u, ±).By [15] [1], there exists a Γ-invariant distribution τ ∈ P(u, ±) −∞ such that all smooth vectors in H ∞ can be written as π u,± (g)τ, v for some v ∈ P(−u, ±) ∞ .Fix ǫ > 0. For each cusp z i , we can use the action of k i so that k i z i = 0.In the language of Harish-Chandra, this amounts to choose a cuspidal pair (P, A).By Cor.3.1, for each cusp z i , we can choose a Siegel set S i and find a constant C i such that Obviously, for the compact set C 0 , Hence, our first inequality follows.
Fix ǫ < 0. By Cor.3.3, π u,± (g)τ, v L 2 (Si,dν) ≤ C|||v||| ǫ 2 −u0 defined for each cusp z i .In the cases of P(−iλ, +), By Theorem 4.3, the norm Observe that the map from P(−iλ, +) is K-invariant and the * C ǫ 2 is independent of the choices of the unipotent subgroup N .Hence remains the same for different choices of cusps.Over C 0 , we have We obtain The complementary series case P(u, +) is similar.The nonspherical unitary principal series P(iλ, −) is more delicate.Essentially, norms |||v||| ǫ 2 with respect to different N i will be mutually bounded.Hence we still have

Bounds with respect to ΩA
The KAN decomposition fits naturally in the theory of Fourier-Whittaker coefficients of automorphic forms.It is used by number theorists to conduct analysis on automorphic forms, often over a Siegel set.However to understand the L-function of automorphic representation, in particular, the growth of L-function, the natural choice seems to be the KN A decomposition.Both KAN and KN A originated in the Iwasawa decomposition and are closely related to Cartan decomposition.The analysis based on these decomposition seems to be of different flavor and have different implications.The G-invariant measure with respect to KAN decomposition is a 2 da a dndk or a −2 da a dadndk depending on the choices of N .The G-invariant measure with respect to KN A decomposition is simply dk dn da.
Recall that L-function for a cuspidal automorphic representation of SL(2, R) can be represented by a zeta integral over M A ∼ = GL(1).Hence it is desirable to have an estimate of the L 2 -norm of automorphic forms over ΩA, where Ω a compact set with finite measure in KN .Since H is cuspidal, the K-finite functions in H are bounded and rapidly decaying near the cusp 0. Again, we write f (g) ∈ H as matrix coefficient π iλ,± (g)τ, v for some v ∈ P(−iλ, ±) and τ ∈ P(ıλ, ±) −∞ .Obviously, τ will have no constant term in Fourier expansion.Its Fourier coefficients have the convergence specified in Theorem 3.1.By Cor 3.1 3.3, there exists C ǫ,H,T1 > 0 such that Our theorem then follows.Proof: Obviously, any compact set Ω in KN is contained in some KN T1 .Hence ΩA ⊆ X T1 .Then our assertion follows from the previous theorem.

Applications to Unitary Eisenstein series
We shall remark that the Theorem 5.2 remains to be true if The following proposition follows directly from Theorem 3.2.Proposition 5.1 Let Γ be a discrete subgroup of SL(2) such that w ∈ Γ and N p ⊆ Γ.Let V be an automorphic representation of type P(iλ, ±).In addition, we can assume V is given by π iλ,± (g)τ, v with τ ∈ P(iλ, ±) −∞ .Let ǫ > 0 and suppose τ = * If Γ is a congruence subgroup containing w and the unitary Eisenstein series is cuspidal at 0 and ∞, we have Corollary 5.2 Let Γ be a congruent subgroup of SL(2, R) such that w ∈ Γ.Let V be an Eisenstein series of type P(iλ, ±) and ǫ ∈ R. Suppose that V has zero constant term with respect to N .Then Proof: The Fourier coefficients of Eisenstein series for congruence subgroups are computable ( [4]).It can be checked that |n| − ǫ 2 −1 |b n | 2 < ∞ for ǫ > 0.
Theorem 4.1 and 4.2 there is a constant c u such that v