Certain L2-norms on automorphic representations of SL(2)
aa r X i v : . [ m a t h . R T ] A ug Certain L -norms on Automorphic Representations of SL (2 , R ) Hongyu He ∗ Department of MathematicsLouisiana State Universityemail: [email protected]
Abstract
Let Γ be a non-uniform lattice in SL (2 , R ). In this paper, we study various L -norms of auto-morphic representations of SL (2 , R ). We will bound these norms with intrinsic norms defined on therepresentation. Comparison of these norms will help us understand the growth of L -functions in asystematic way ( [9]). Let Γ be a non-uniform lattice in SL (2 , R ). By an automorphic representation of SL (2 , R ), wemean a finitely generated admissible representation of SL (2 , R ), consisting of Γ-invariant functionson SL (2 , R ) ([5]). Among all automorphic representations, L automorphic representations, i.e., sub-representations of L ( G/ Γ), are of fundamental importance. Since L automorphic representationsare unitary and completely reducible, we assume L automorphic representations to be irreducible.By Langlands theory, L automorphic representations come from either the residues of Eisensteinseries or the cuspidal automorphic representations. Throughout this paper, we shall mostly focus onirreducible cuspidal representations, even though our results also apply to unitary Eisenstein serieswith vanishing constant term near a cusp.Let G = SL (2 , R ) and π be an irreducible admissible representation of G . We say an automorphicrepresentation is of type π if the automorphic representation is infinitesimally equivalent to π . Inparticular, we write L ( G/ Γ) π for the sum of all L -automorphic representations of type π . It iswell-known that L ( G/ Γ) π is of finite multiplicity ([5]). The main purpose of this paper is to studyvarious L -norms of the automorphic forms at the representation level. In the literature, automor-phic forms, the K -finite vectors in an automorphic representation, are the main focus of interests.Our main focus here is the L -norms of automorphic forms, in comparison with (intrinsic) normsin the representation. We hope to gain some understanding of various L -norms of automorphicrepresentation as a whole, without references to automorphic forms. We believe this may lead to abetter understanding of the Fourier coefficients and L -functions.Our estimates of L -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 ∗ Key word: Authomorphic forms, automorphic representation over R , SL (2), Iwasawa decomposition, Fourier coeffi-cients, K -invariant norm, principal series, cusp forms, complementary series onstant terms of automorphic forms. We give estimates of various L norms of the restriction ofautomorphic representation to AN and the Siegel set. The KN A decomposition, on the other hand,seems to be a potentially useful tool to study the L -function associated with the automorphic rep-resentation. In this paper, we give various estimates on the L -norm of automorphic representationrestricted to Ω A , with Ω a compact domain in KN .Our view point and setup are very similar to those of Harish-Chandra ([5]). The group action will befrom the left and the standard cusp will be at zero instead of ∞ . Working in the general frameworkof harmonic analysis on semisimple Lie groups, Harish-Chandra gave a very detailed account of thetheory of cusp forms and Eisenstein series, mainly due to Selberg, Gelfand and Piatetsky-Shapiro,and Langlands. Our goal here is quite limited: we only treat the group G = SL (2 , R ) and we studyvarious L -norms of automorphic representations of type π . Most of our results are stated in terms ofautomorphic distribution ([1] [16] [15]). The reason is simple. There are two types of norms involved,one for the automorphic forms, and one for the representation. Using automorphic distributions,automorphic forms can be viewed as matrix coefficients of K -finite vectors and a fixed automorphicdistribution. This allows us to compare norms of automorphic forms and norms of the representation.These results will shed lights on the growth of the Rankin-Selberg L -functions ([9]).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 -norm on the fundamental domain is k f k L ( G/ Γ) = Z F | f ( kan ) | a daa dndk. We have
Theorem 1.1
Let π = P ( u, ± ) be a unitary representation in the principal series. Let H be acuspidal representation in L ( G/ Γ) π . Then for any ǫ > , there exists a C ǫ > such that Z F | f ( kan ) | a ǫ daa dndk ≤ C ǫ k f k L ( G/ Γ) , ( ∀ f ∈ H ) . For any ǫ < , there exists a C ǫ > such that Z F | f ( kan ) | a ǫ daa dndk ≤ C ǫ ||| f ||| ǫ − u , ( ∀ f ∈ H ∞ ) . Here u = ℜ ( u ) and the norm ||| f ||| ǫ − u is defined on H ∞ ( Def. 4.5) , smooth vectors in therepresentation in H . Our theorem essentially says that every f ∈ L ( G/ Γ) π is also in L ( F , a ǫ daa dndk ) for every ǫ > L ( F , a daa dndk ) ⊇ H → L ( F , a ǫ daa dndk )is bounded for every ǫ > L ( F , a daa dndk ) → L ( F , a ǫ daa 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. e 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 > D n (see Cor. 3.2). They are proved by studying the L -norms of Fourier coefficients of the auto-morphic distribution, defined in Schmid ([16] and Bernstein-Resnikov ([1]). Similar theorem can beestablished for automorphic representations of GL ( n, R ) ([8]). We shall mention one problem worthyof further investigation. Problem : Let G be a semisimple Lie group, Γ an arithmetic lattice and S a Siegel domain. Findthe best exponents α such that i : L ( G/ Γ) π → L ( S, a α daa dndk )is bounded. Here G = KAN is the Iwasawa decomposition.If α is ”bigger” than 2 ρ , the sum of positive roots, i is automatically bounded. The problem isto find the ”smallest” α such that i is bounded. We shall remark that cusp forms will remain to bein L ( S, a α daa dndk ) for any α since they are fast decaying in the Siegel set. Hence our problem isabout cuspidal representations, rather than cusp forms.The second main result is an L -estimates of f on Ω A 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 acompact domain in KN . Let ǫ > . Then there exists a positive constant C depending on ǫ, H and Ω such that k f k L (Ω A,a ǫ d aa dtdk ) ≤ C ||| f ||| − ǫ ( f ∈ H ∞ ) . We shall remark that in the
KN A decomposition, the invariant measure is given by dkdn daa . Hence,the L -norm here is a perturbation of the canonical L -norm. In addition, Ω A has infinite measure.The perturbation is needed because our theorem fails at ǫ = 0. At ǫ = 0, the norm ||| f ||| − ǫ is theoriginal Hilbert norm k f k of the cuspidal representation. There is no chance that k f k L (Ω A,a ǫ d aa dtdk ) can remain bounded for all f .Throughout our paper, the Haar measure on A will be daa . We use c or C as symbolic constants and c ǫ,u to indicate the dependence on ǫ and u . L -norm of Γ -invariant functions Let G = SL (2 , R ). Let N = { n t = (cid:18) t (cid:19) : t ∈ R } ,K = { k θ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) : θ ∈ [0 , π ) } A = { (cid:18) a a − (cid:19) : a ∈ R + } , and w = (cid:18) − (cid:19) ∈ K . We call w the Weyl element. Let Γ be a discrete subgroup of G suchthat Γ ∩ N is nontrivial. Without loss of generality assume thatΓ ∩ N = N p = { n t : t ∈ p Z } ith p ∈ N + .Let M = {± I } ⊆ K . Fix P = M AN , the minimal parabolic subgroup. Then the identitycomponent P = AN . Fix daa dt as the left invariant measure on an t ∈ P and dT daa as the rightinvariant measure on N T a ∈ P . We shall keep the notion that an t = N T a . Then T = a t, t = a − T, daa dT = a daa dt, daa dt = a − daa dT. 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 dt daa dk = dT daa dk. Let N T = { n T : 0 ≤ T ≤ T } if T > N T = { n T : 0 ≥ T ≥ T } if T <
0. Let X T = KN T A equipped with the canonical measure dkdT daa . Let ǫ ∈ R . For f ∈ C ( G/ Γ) or moregenerally L loc ( G/ Γ), we would like to estimate k f k T ,ǫ = k f k L ( X T ,a ǫ daa dkdT ) . Here L loc ( G/ Γ) is the space of locally square integrable function on G/ Γ.Let a ∈ R + . Let A + a = { a ≥ a } and A − a = { < a ≤ a } . By abusing notation, we simplyuse a ∈ R + as an element in A . Write X ( T , a ) ± = KN T A ± a , P ( T , a ) ± = N T A ± a . Write k f k L ( X ( T ,a ) ± ,a ǫ daa dT dk ) as k f k T ,a ± ,ǫ . k f k T ,a − ,ǫ Without loss of generality, assume T >
0. Observe that P ( T , a ) − = { ≤ T ≤ T , < a ≤ a } = { < a ≤ a , ≤ t ≤ a − T } . We have
Proposition 2.1
Let f ∈ L loc ( P ) such that f ( xN p ) = f ( x ) for a fixed period p ∈ Z . Then for any ǫ ∈ R , Z a a ǫ ⌊ T p a ⌋ Z p | f ( an t ) | dt daa ≤ k f k L ( P ( T ,a ) − ,a ǫ daa dT ) ≤ Z a a ǫ ( ⌊ T p a ⌋ +1) Z p | f ( an t ) | dt daa . Proof: We have k f k L ( P ( T ,a ) − ,a ǫ daa dT ) = Z a Z T a ǫ k f ( n T a ) k dT daa = Z a Z a − T a ǫ k f ( an t ) k dt daa ≥ Z a a ǫ ⌊ T a p ⌋ ( Z p k f ( an t ) k dt ) daa (1) ere ⌊∗⌋ is the floor function. The other direction is similar. (cid:3) For T negative, we have a similar statement. Combining these two cases, we have Theorem 2.1
Assume that f ∈ L loc ( G ) and f ( xN p ) = f ( x ) for a fixed period p . Let a > and ǫ ∈ R . Then Z K Z a a ǫ ⌊ | T | p a ⌋ Z p | f ( kan t ) | dt daa dk ≤ k f k T ,a − ,ǫ ≤ Z K Z a a ǫ ( ⌊ | T | p a ⌋ +1) Z p | f ( kan t ) | dt daa dk. k f k T ,a +1 ,ǫ To estimate k f k T ,a +1 ,ǫ , we must utilize the Weyl group element w . We assume that | f ( xw ) | = | f ( x ) | ( ∀ x ∈ G ) . Let a ∈ [ a , ∞ ). By the Iwasawa decomposition n T aw = k ( T, a ) n T ′ a ′ , a ′ = √ T + 1 a , T ′ = − T and k ( T, a ) ∈ K . This defines a coordinate transform from ( T, a ) to ( T ′ , a ′ ). Let ( P ( T , a ) + ) ′ bethe coordinate transform of P ( T , a ) + w in terms of ( T ′ , a ′ ) coordinates. We have( P ( T , a ) + ) ′ = {− T ≤ T ′ ≤ , < a ′ ≤ p ( T ′ ) + 1 a } . It is easy to see that P ( − T , a ) − ⊆ ( P ( T , a ) + ) ′ ⊆ P ( − T , p T + 1 a ) − , and KP ( − T , a ) − ⊆ KP ( T , a ) + w ⊆ KP ( − T , p T + 1 a ) − . Observe that a ǫ daa dT = ( p ( T ′ ) + 1) ǫ ( a ′ ) − ǫ dT ′ da ′ a ′ and | f ( kn T a ) | = | f ( kn T aw ) | = | f ( kk ( T, a ) n T ′ a ′ ) | . We obtain
Proposition 2.2
Let f ∈ L loc ( G ) , a > and ǫ ∈ R . Suppose that f ( xN p ) = f ( x ) and | f ( xw ) | = | f ( x ) | . Then k f k − T , ( a ) − , − ǫ ≤ k f k T ,a +1 ,ǫ ≤ ( q T + 1) ǫ k f k − T , ( √ T
21 +1 a ) − , − ǫ ( ǫ ≥ q T + 1) ǫ k f k − T , ( a ) − , − ǫ ≤ k f k T ,a +1 ,ǫ ≤ k f k − T , ( √ T
21 +1 a ) − , − ǫ ( ǫ ≤ .4 Estimates of k f k T ,ǫ Choose a = 1. We have k f k − T , (1) − , − ǫ ≤ k f k T , + ,ǫ ≤ ( q T + 1) ǫ k f k − T , ( √ T +1) − , − ǫ ( ǫ ≥ q T + 1) ǫ k f k − T , (1) − , − ǫ ≤ k f k T , + ,ǫ ≤ k f k − T , ( √ T +1) − , − ǫ ( ǫ ≤ Theorem 2.2
Let f be a locally square integrable function on SL (2 , R ) such that f ( xN p ) = f ( x ) and | f ( xw ) | = | f ( x ) | . If ǫ > , then Z K Z ( a ǫ + a − ǫ ) ⌊ T p a ⌋ Z p | f ( kan t ) | dt daa dk ≤ k f k T ,ǫ ≤ Z K Z a ǫ ( ⌊ T p a ⌋ + 1) Z p | f ( kan t ) | dt daa dk + ( q T + 1) ǫ Z K Z √ T +10 a − ǫ ( ⌊ T p a ⌋ + 1) Z p | f ( kan t ) | dt daa dk. (2) If ǫ ≤ , then Z K Z ( a ǫ + ( q T + 1) ǫ a − ǫ ) ⌊ T p a ⌋ Z p | f ( kan t ) | dt daa dk ≤ k f k T ,ǫ ≤ Z K Z a ǫ ( ⌊ T p a ⌋ + 1) Z p | f ( kan t ) | dt daa dk + Z K Z √ T +10 a − ǫ ( ⌊ T p a ⌋ + 1) Z p | f ( kan t ) | dt daa dk. (3)If p = 1 and T = 1, we have k f k T ,ǫ ≤ C ǫ Z K Z √ ( a ǫ + a − ǫ ) Z | f ( kan t ) | dt daa dk. Notice that for 0 < a ≤ √ ⌊ a ⌋ +1 ≤ a . Hence, we have bounded the norm of f on X T . Generally,we have Theorem 2.3
Suppose that f is a locally square integrable function on SL (2 , R ) such that f ( xN p ) = f ( x ) and | f ( xw ) | = | f ( x ) | . Let ǫ ∈ R . Then there exists a positive constant c T ,ǫ, p such that k f k T ,ǫ ≤ c T ,ǫ, p Z K Z √ T ( a ǫ + a − ǫ ) Z p | f ( kan t ) | dt daa dk. Proof: We choose a positive constant c such that ⌊ T p a ⌋ + 1 ≤ c T p a ( ∀ < a ≤ q T ) . Then let c T ,ǫ, p = c max(2 , p T ) ǫ ). (cid:3) Observe that the right hand side of our inequality involves an integral over a Siegel set. How-ever the measure on this Siegel set can be larger than the invariant measure a dk daa dt . What wehave achieved is a bound of k f k T ,ǫ by an integral on a Siegel set. In the next section, we shall giveestimation of the norms of f on A − a N/N p and on KA − a N/N p . Matrix Coefficients and Analysis on P /N p Now we shall focus on L automorphic representations of type π where π is a principal series repre-sentation. According to Langlands, L automorphic representations come from either the residue ofEisenstein series or cuspidal automorphic forms. In either cases, the restrictions of L automorphicrepresentations fail to be L on P /N p , when P /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 -integrability of f | P with f ∈ L ( G/ Γ) π with respect to themeasure a ǫ daa dt . We will consequently discuss the L -norm on a Siegel subset. We conduct ourdiscussion in terms of matrix coefficients with respect to periodical distributions with no constantterm. More precisely, the function f | P will be regarded as the matrix coefficient of v ∈ H π and aperiodical distribution in ( H ∗ ) −∞ . Our view is similar to Schmid and Bernstein-Reznikov ([16] [1]). SL (2 , R ) Principal series representations of G can be easily constructed using homogeneous distributions on R . See for example [3] [11]. In this section, we shall focus on the smooth vectors and the spaceof distributions associated with them. Let ( π u, ± , P ( u, ± )) be the unitarized principal series rep-resentation with trivial and nontrivial central character. P ( u, ± ) includes unitary principal series P ( u, ± ) (with u ∈ i R ) and complementary series P ( u, +) (with u ∈ ( − , ∪ (0 , P (0 , − ). In addition P ( u, ± ) ∼ = P ( − u, ± ).Consider the noncompact picture. We have for any g = (cid:18) a bc d (cid:19) , f ∈ P ( u, ± ) ∞ , π u, ± ( g ) f ( x ) = χ ± ( a − cx ) | a − cx | − − u f ( dx − ba − cx ) . Here χ − ( x ) is the sign character on R − { } and χ + ( x ) is the trivial character. In particular, we have π u, ± (cid:18) a a − (cid:19) f ( x ) = | a | − − u f ( a − x ) , ( a ∈ R + ); π u, ± (cid:18) b (cid:19) f ( x ) = f ( x − b ); π u, ± ( w ) f ( x ) = χ ± ( − x ) | x | − − u f ( − x ); π u, ± (cid:18) cos θ − sin θ sin θ cos θ (cid:19) f ( x ) = χ ± (cos θ − x sin θ ) | cos θ − x sin θ | − − u f ( x cos θ + sin θ cos θ − x sin θ ) . There is a G -invariant pairing between P ( u, ± ) ∞ and P ( − u, ± ) ∞ . This allows us to write the dualspace 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 sat-isfying certain conditions at infinity. According to [ ? ] [16] [14], every L automorphic form of type π can be written as matrix coefficientsof an automorphic distribution and a vector in the unitary representation π . Equivalently, in our etting, there exists a distribution τ ∈ P ( u, ± ) −∞ such that the automorphic forms of type π can bewritten as linear combinations of f m ( g ) = h π u, ± ( g ) τ, v m i , with v m ( x ) = (1 + x ) − − u ( xi − xi ) m . 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 τ = ∗ X n ∈ p − Z ,n =0 b n exp 2 πixn, Here p is a positive integer and P ∗ denote the weak summation ([6]). We call such τ a periodicaldistribution without constant term.Let τ ∈ P ( u, ± ) −∞ be a periodic distribution without constant term. We compute the matrixcoefficient formally: h π u, ± ( an t ) τ, v i = h X n ∈ p − Z ,n =0 a − − u b n exp 2 πi ( a − x − t ) n, v ( x ) i = a − − u ∗ X n ∈ p − Z ,n =0 Z b n exp(2 πia − xn ) exp( − πitn ) v ( x ) dx = a − − u ∗ X n ∈ p − Z ,n =0 ( F v )( − na − ) b n exp( − πitn ) . (4)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 v ∈ P ( − u, ± ) ∞ with ℜ ( − u ) > − Lemma 3.1
Let u = u + iu with u < and τ = ∗ X n ∈ p − Z ,n =0 b n exp 2 πixn ∈ P ( u, ± ) −∞ . For v ∈ P ( − u, ± ) ∞ , we have h π u, ± ( an t ) τ, v i = a − − u ∗ X n ∈ p − Z ,n =0 ( F v )( − na − ) b n exp( − πitn ) Z p |h π u, ± ( an t ) τ, v i| dt = p X n ∈ p − Z a − − u | b n | |F v ( − na − ) | . Proof: Suppose ℜ ( v ) <
1. The functions in P ( − u, ± ) ∞ are smooth functions of the form (1 + x ) − − u φ ( xi − xi ) with φ an odd or even smooth function on the unit circle. They are slowly growingfunction. Their Fourier transforms exist. Since the derivatives v ( n ) are of this form and they areintegrable , we see that F v ( ξ ) will decay faster than any polynomial at ∞ . The weak sum in Equation(4) becomes a convergent sum. Our lemma is proved. (cid:3) We shall make a few remarks here. Since v ∈ P ( − u, ± ) ∞ and τ ∈ P ( u, ± ) −∞ , the matrix coef-ficient h π u, ± ( an t ) τ, v i is automatically smooth. Our lemma simply provided a Fourier expansion, hich is generally known as the Fourier-Whittaker expansion over the whole group G . The restric-tion that u < u ≥ 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 compositionfactors. Hence, automorphic representations that are discrete series, can be treated by consideringthe reducible P ( − u, ± ). We shall refer readers to Schmid’s paper [ ? ] for details. When P ( − u, ± ) isirreducible, F v ( ξ ) is a fast decaying continuous function off from zero. Our lemma is still valid inthis case. However,if u > F v ( ξ ) will fail to be a locally integrable function near zero and need tobe regularized to be a Schwartz distribution. From now on, without further mentioning, wewill restrict our scope to u < . In this situation, h exp 2 πixn, v i shall be interpreted as − πin h exp 2 πinx, dvdx i . If P ( u, ± ) is unitary, then ℜ ( u ) ∈ ( − , π is a discrete series representation, then π can beembedded into a principal series representation P ( − u, ± ) with u <
1. Hence our assumption isadequate for the discussion of L automorphic representations. L -norms on P /N p Let us first study the L norms of f ( g ) = h π u, ± ( g ) τ, v i on P /N p . τ and v are given in Lemma 3.1.Now we compute Z a Z p | f ( an t ) | dta ǫ daa = p Z a a ǫ X n ∈ p − Z ,n =0 a − − u | b n | |F v ( − na − ) | daa = p Z ∞ a − a − ǫ X n ∈ p − Z ,n =0 a u | b n | |F v ( − na ) | daa = p X n ∈ p − Z ,n =0 Z ∞ a − a − ǫ +1+ u | b n | |F v ( − na ) | daa = p X n ∈ p − Z ,n> X ± Z ∞ na a − ǫ +1+ u n ǫ − − u | b ± n | |F v ( ∓ a ) | daa = p X ± Z ∞ a p a − ǫ + u |F v ( ∓ a ) | X p ≤ n ≤ aa ,n ∈ p − Z n ǫ − − u | b ± n | da (5)We summarize this in the following proposition. Proposition 3.1
Let u = u + iu with u < . Let v ∈ P ( − u, ± ) ∞ and τ ∈ P ( u, ± ) −∞ : τ = ∗ X n ∈ p − Z ,n =0 b n exp(2 πinx ) . Let f ( an t ) = h π u, ± ( an t ) τ, v i . Then f ( an t ) is a smooth function on P and Z a Z p | f ( an t ) | dta ǫ daa = p X ± Z ∞ a p a − ǫ + u |F v ( ∓ a ) | X p ≤ n ≤ aa ,n ∈ p − Z n ǫ − − u | b ± n | da. n particular, Z ∞ Z p | f ( an t ) | dta ǫ daa = p X ± X p ≤ n,n ∈ p − Z n ǫ − − u | b ± n | Z ∞ a − ǫ + u |F v ( ∓ a ) | da. (6)Proof: Since f ( g ) is a smooth function on G , f ( an t ) is a smooth function on P . Both equations holdwithout any assumptions on convergence. Hence both sides of the equations converge or diverge atthe same time. (cid:3) b n We can now provide some estimates of certain sum of Fourier coefficients. These estimates are moreor less known for automorphic forms ([1] [16] [15] [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 ) = h π u, ± ( an t ) τ, v i is bounded on P . 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 ) | ≤ C µ,f a µ for some µ > , i. e., f ( an t ) decays faster than a µ near the cusp , thenwe have for each ǫ ∈ ( − µ, , X n> ,n ∈ p − Z n ǫ − − u | b ± n | < ∞ .
2. For each ǫ > , there exists a C ǫ,τ > such that k X n = p ,n ∈ p − Z n ǫ − − u | b ± n | < C ǫ,τ k ǫ ( k > . Let me make a remark about the ± or ∓ signs. If F v ( a ) is nonvanishing on R − , then b ± n should beread as b + n ; if F v ( a ) is nonvanishing on R + , then b ± n should be read as b − n . The proof should beread in the same way.Proof: Fix f ( an t ) = h π u, ± ( an t ) τ, v i bounded on P by C f . Suppose that F v ( a ) is nonvanishingon R − or R + .1. Suppose that | f ( an t ) | ≤ C µ,f a µ for µ >
0. For − µ < ǫ <
0, the left hand side of Equation(6) converges. Since F v ( a ) is nonvanishing on R ∓ , R ∞ a − ǫ + u |F v ( ∓ a ) | da >
0. Then the sum P p ≤ n n ǫ − − u | b ± n | becomes a factor and must remain bounded by a constant depending on f and ǫ . . Let ǫ > δ > a > δ p . By Prop. 3.1 we have( X p ≤ n ≤ a δ,n ∈ p − Z n ǫ − − u | b ± n | ) Z ∞ δ a − ǫ + u |F v ( ∓ a ) | da ≤ Z ∞ δ a − ǫ + u |F v ( ∓ a ) | ( X p ≤ n ≤ a a,n ∈ p − Z n ǫ − − u | b ± n | ) da ≤ Z ∞ a p a − ǫ + u ( X p ≤ n ≤ aa ,n ∈ p − Z n ǫ − − u | b ± n | ) |F v ( ∓ a ) | da ≤ p − Z a Z p | f ( an t ) | dta ǫ daa ≤ C f a ǫ ǫ (7)Now fix a δ > R ∞ δ a − ǫ + u |F v ( ∓ a ) | da is positive. It follows that there exists C ǫ,f > a = kδ , X p ≤ n ≤ k,n ∈ p − Z n ǫ − − u | b ± n | < C ′ f a ǫ ǫ = 2 C ′ f k ǫ δ − ǫ ǫ − = C ǫ,f,δ k ǫ . Notice that δ depends on v , therefore also on f . We can write c ǫ,f,δ as c f,δ / (cid:3) If τ is a cuspidal automorphic distribution in a unitary principal series or complementary seriesrepresentation, then all automorphic forms f ( g ) will be bounded and rapidly decaying near the cuspat zero. In this situation, the estimates in Theorem 3.1 were well-known ( [16] [1]). The first estimatecan 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) automorphicrepresentation 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 − N or − p − N . Ourestimates of Fourier coefficients also follow similarly upon applying the intertwining operator. Thedetails of how to treat the discrete series representations can be found in [16] [15]. L -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.1. If ǫ < and P n =0 ,n ∈ p − Z | n | ǫ − − u | b n | < ∞ , then there exists positive constant C ǫ,τ such that Z a Z p | f ( an t ) | dta ǫ daa ≤ C ǫ,τ X ± Z ∞ a p a − ǫ + u kF v ( ± a ) | da. In particular, Z ∞ Z p | f ( an t ) | dta ǫ daa ≤ C ǫ,τ X ± Z ∞ a − ǫ + u kF v ( ± a ) | da.
2. If ǫ > and P | n |≤ k,n ∈ p − Z | n | ǫ − − u | b n | < C ǫ,τ k ǫ for any k > , then Z a Z p | f ( an t ) | dta ǫ daa ≤ C ǫ,τ a ǫ p X ± Z ∞ a p a u kF v ( ± a ) | da. e shall remark that this theorem holds even P ( u, ± ) is not unitary.Combining Theorems 3.1 and 3.2, we have Corollary 3.1 ( ǫ > ) Under the same assumption as Prop. 3.1, suppose for some φ ∈ P ( − u, ± ) ∞ the function f ( an t ) = h π u, ± ( an t ) τ, φ i is bounded on P and F φ ( a ) is nonvanishing on both R + and R − . Then for any ǫ > and v ∈ P ( − u, ± ) ∞ , we have Z a Z p |h π u, ± ( an t ) τ, v i| dta ǫ daa ≤ C ǫ,τ a ǫ Z | a |≥ a p | a | u kF v ( a ) | da. (8) In particular, if P ( u, ± ) is unitary, we have Z a Z p |h π u, ± ( an t ) τ, v i| dta ǫ daa ≤ C ǫ,τ a ǫ k v k P ( − u, ± ) , Z K Z a Z p |h π u, ± ( kan t ) τ, v i| dta ǫ daa dk ≤ C ǫ,τ a ǫ k v k P ( − u, ± ) for every v ∈ P ( − u, ± ) . Proof: We only need to prove the second statement. If u = 0, i.e., P ( u, ± ) is a unitary principalseries, then Z | a |≥ a | a | u kF v ( a ) | da ≤ kF v ( x ) k L ( R ) = k v k P ( u, ± ) . If P ( − u, +) is a complementary series representation, then the unitary Hilbert norm k v k P ( − u, ± ) isgiven by exactly the square root of Z | x | u kF v ( x ) | dx, up to a normalizing factor depending on u . Hence we have Z a Z p | f ( an t ) | dta ǫ daa ≤ C ǫ,τ a ǫ k v k P ( − u, ± ) . Observe that h π u, ± ( kan t ) τ, v i = h π u, ± ( an t ) τ, π − u, ± ( k − ) v i and k π ( − u, ± )( k − ) v k P ( − u, ± ) = k v k P ( − u, ± ) . The inequalities in the second statement hold for v ∈ P ( − u, ± ) ∞ . Therefore, they must also hold for v ∈ P ( − u, ± ). (cid:3) .Notice that Inequality (8) is true for all ℜ ( u ) <
1, in particular for u with P ( − u, ± ) reducible.Hence it applies to discrete series representation D n . In addition, the norm on the right hand sideof Inequality (8) is bounded by C ǫ,τ a ǫ Z a ∈ R | a | u kF v ( a ) | da By the Kirillov model, this integral is a constant multiple of the unitary norm k v k D n ([10]). We have Corollary 3.2 (discrete series case)
Let D n be a discrete series representation. Let τ be a peri-odic distribution in D −∞ n with period p . Suppose that for some φ ∈ D ∞− n , the function h D n ( an t ) τ, φ i is bounded on P . Then for any ǫ > and v ∈ D ∞− n , Z a Z p |h D n ( an t ) τ, v i| dta ǫ daa ≤ C ǫ,τ a ǫ k v k D − n , K Z a Z p |h D n ( kan t ) τ, v i| dta ǫ daa dk ≤ C ǫ,τ a ǫ k v k 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 < ) Let P ( u, ± ) be a unitary representation. Under the assumptions of Prop.3.1, suppose that ǫ < and P n =0 | n | ǫ − − u | b n | < ∞ . Then there exists C ǫ,τ > such that Z K Z a Z p |h π u, ± ( kan t ) τ, v i| dta ǫ daa dk ≤ C ǫ,τ Z | x | > a p | x | − ǫ + u kF ( π − u, ± ( k ) v )( x ) | dx. In particular, Z K Z Z p |h π u, ± ( kan t ) τ, v i| dta ǫ daa dk ≤ C ǫ,τ Z K Z ∞−∞ | x | − ǫ + u kF ( π − u, ± ( k ) v )( x ) | dxdk ; Both inequalities hold for those v ∈ P ( − u, ± ) with which the right hand sides converge. In the case of automorphic forms, our L norms are estimated over a Siegel subset, but with themeasure a ǫ daa dkdt , while the Siegel set is often equipped with the measure a daa dkdt . The bounds wehave are certain norms on the representation. This allows us to treat everything at the representationlevel. If ǫ >
0, the bounds come from the Hilbert norm of the automorphic representation. We havenothing to improve on. If ǫ < ||| v ||| ǫ − u = Z K Z | x | − ǫ + u kF ( π − u, ± ( k ) v )( x ) | dxdk in more details. Our goal is to bound ||| v ||| ǫ − u by a more tangible norm. A natural choice is a normcoming 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 areindeed fast decaying at ∞ , but singular at zero. For any bounded smooth function φ with locallysquare integrable Fourier transform, let us define k φ k C u = Z | x | − u kF ( φ )( x ) | dx, ( ∀ u ∈ ( − , C u , upto a normalizing factor. The standard norm k ∗ k u for the complementary series is oftenconstructed using the standard intertwining operator A u ([11]). Our norm k ∗ k C u differs from the k ∗ k u by a normalizing factor. The standard norm k ∗ k u has a pole at u = 0. The norm k ∗ k C u doesnot. Hence k ∗ k C u is potentially easier to use. In this section, we will first review the basic theoryof complementary series. Then we will use k · k C u to bound the norm |||·||| u . Our main references are[11] [3]. .1 Intertwining operator and complementary series The standard intertwining operator A u : P ( u, +) ∞ → P ( − u, +) ∞ is well-defined for ℜ u > C . In the noncompact picture, A u ( f )( x ) = Z f ( y ) | x − y | − u dy. Let h∗ , ∗i be the complex linear G -invariant pairing P ( u, +) × P ( − u, +) → C . For any φ, ψ ∈ P ( u, +), we define h φ, ψ i u = h A u ( φ ) , ψ i . This is a G -invariant bilinear form on P ( u, +) ∞ . When u is real and 0 < u < φ, ψ ) u = h A u ( φ ) , ψ i u yields an G -invariant inner product on P ( u, +) ∞ . Its completion is often called a complementaryseries 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 v ( u )2 m = (1 + x ) − u ( 1 + xi − xi ) m ( m ∈ Z ) . The intertwining operator A u maps v ( u )2 m to c ( u )2 m v − u m . The constant c ( u )2 m = ( − m − u π Γ( u )Γ( u +12 + m )Γ( u +12 − m ) = 2 − u Γ( u )Γ( m + − u +12 ) sin( u +12 π )Γ( u +12 + m ) . See [3]. We make two observations here. First, the formula above in fact uniquely determined theanalytic continuation of the intertwining operator A u . Secondly, for u / ∈ Z + 1,Γ( − u +12 + m )Γ( u +12 + m ) ∼ c u m − u ( m → ∞ ) . We have
Lemma 4.1
For a fixed u ∈ ( − , or u ∈ (0 , , there exist positive constants c u , c ′ u such that c ′ u (1 + | m | ) − u ≤ ( v ( u ) m , v ( u ) m ) u ≤ c u (1 + | m | ) − u ( m ∈ Z )) . The intertwining operator A u has a pole at u = 0. Hence we must exclude u = 0 from our estimates. ( ∗ , ∗ ) u Recall that for u ∈ (0 , φ, ψ ) u = Z Z φ ( x ) ψ ( y ) | x − y | − u dxdy ( φ, ψ ∈ P ( u, +) ∞ ) , and ( φ, ψ ) C u = Z | ξ | − u F ( φ )( ξ ) F ( ψ )( ξ ) dξ. y Fourier inversion formula, we have( φ, ψ ) C u = G ( u )( φ, ψ ) u , where R | ξ | − u exp − πixξdξ = G ( u ) | x | − u . This is true for u ∈ (0 ,
1) and can be analyticallycontinued to u ∈ ( − , G ( u ) can be expressed in terms of Γ-functions andpossesses a zero at u = 0 ([14]). Hence we have k v ( u ) m k C u = G ( u ) k v ( u ) m k u for u ∈ ( − , Theorem 4.1
For u ∈ ( − , , there exist positive constants q u , q ′ u depending continuously on u such that q = q ′ = 1 and q ′ u (1 + m ) − u ≤ k v ( u )2 m k C u ≤ q u (1 + m ) − u . P ( iλ, +) case Fix v ∈ P ( iλ, +) ∞ with λ ∈ R . Recall that we are interested in the norm ||| v ||| u = Z K Z | x | − u kF ( π iλ, + ( k ) v )( x ) | dxdk = Z K k π iλ, + ( k ) v k C u dk ( u < . Clearly, this norm is K -invariant. Hence we will need to estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( iλ )2 m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = k v ( iλ )2 m k C u . Theorem 4.2
Let u ∈ ( − , . Then there exists a positive constant c u such that ∀ m ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( iλ )2 m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ≤ c u (1 + | m | − u ) . Proof: Observe that v ( iλ )2 m ( x ) = (1 + x ) − iλ + u v ( u )2 m . Under the compact picture of P ( u, +), v ( iλ )2 m becomes | sin θ | iλ − u exp 2 miθ, (cot θ = x ) . The function | sin θ | iλ − u has period π and L derivative. Hence its Fourier series expansion X k ∈ Z a k exp 2 kiθ, satisfy that | a k | ≤ h u (1 + k ) − . We obtain v ( iλ )2 m = X k ∈ Z a k v ( u )2 m +2 k . It follows that k v ( iλ )2 m k C u = X k ∈ Z | a k | k v ( u )2 m +2 k k C u ≤ h u q u X k ∈ Z (1 + ( m + k ) ) − u k + 1 ≤ h u q u X k ∈ Z (1 + 2 m ) − u (1 + 2 k ) − u k + 1 , which will be bounded by a multiple of (1 + m ) − u . (cid:3) For u ∈ ( − ,
0) the map v ( x ) ∈ P ( iλ, +) ∞ → (1 + x ) iλ − u v ( x ) ∈ P ( u, +) ∞ reserves the K action and maps v ( iλ )2 m to v ( u )2 m . By Theorem 4.1 and 4.2 there is a constant c u suchthat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( iλ )2 m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ≤ c u k v ( u )2 m k C u We have
Theorem 4.3
For u ∈ ( − , and λ ∈ R , there exists a positive constant c u such that ||| v ( x ) ||| u ≤ c u k (1 + x ) iλ − u v ( x ) k C u ( ∀ v ( x ) ∈ P ( iλ, +) ∞ ) . Under the assumption of Cor 3.3, applying Theorem 4.3, h π u, ± ( kan t ) τ, v i will be in L ( G/N p , a ǫ daa dtdk )as long as k (1 + x ) iλ − ǫ v ( x ) k C ǫ is bounded with ǫ ∈ ( − , P ( iλ, − ) case Let u ∈ ( − ,
0) and λ ∈ R . The K -types in P ( iλ, − ) are v ( iλ )2 m +1 ( x ) = (1 + x ) − iλ + u ( 1 + xi − xi ) m + ( m ∈ Z ) . Here x = cot θ ( θ ∈ (0 , π )) and ( xi − xi ) m + = exp i (2 m + 1) θ is well-defined. Our goal is to estimate k v ( iλ )2 m +1 ( x ) k C u . We still have v ( iλ )2 m +1 ( x ) = v ( u )2 m ( 1 + xi − xi ) (1 + x ) u − iλ . In the compact picture of P ( u, +), v ( iλ )2 m +1 ( x ) becomes sgn(sin θ ) | sin θ | − u + iλ exp(2 m + 1) iθ . Noticethat this function has period π and take the same value as | sin θ | − u + iλ exp(2 m + 1) iθ when θ ∈ [0 , π ].Observe that sgn(sin( θ + π )) | sin( θ + π )) | − u + iλ = − sgn(sin θ ) | sin θ | − u + iλ . Let sgn(sin θ ) | sin θ | − u + iλ = P k ∈ Z b k − exp(2 k − iθ be its Fourier expansion. Again, the function sgn(sin θ ) | sin θ | − u + iλ has L -derivative. Hence | b k − | ≤ c u | k − | . By a similar argument as P ( iλ, +) case we have Theorem 4.4
Let u ∈ ( − , . Then there exists a positive constant c u such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( iλ )2 m +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ≤ c u (1 + | m | − u ) . Theorem 4.5
For u ∈ ( − , and λ ∈ R , there exists a positive constant c u such that ||| v ( x ) ||| u ≤ c u k (1 + x ) iλ − u ( 1 + xi − xi ) v ( x ) k C u ( v ( x ) ∈ P ( iλ, − ) ∞ ) . Proof: Consider the map I : P ( iλ, − ) ∞ → P ( u, +) −∞ defined by I ( v )( x ) = (1 + x ) iλ − u ( 1 + xi − xi ) v ( x ) .I maps the orthogonal basis { v ( iλ )2 m − : m ∈ Z } of |||∗||| to orthogonal basis { v ( u )2 m : m ∈ Z } of thecomplementary series C u . In addition, one can easily check that I is bounded. Our theorem thenfollows. Contrary to the spherical case, the operator I is no longer K -invariant. (cid:3) .5 Bounds by the complementary norm: P ( u, +) case Let u ∈ ( − , P ( u, +) is the complementary series C u . For µ < v ∈ P ( u, +) ∞ , we areinterested in ||| v ( x ) ||| u + µ = Z K Z R | x | − u − µ |F ( π u, + ( k ) v )( x ) | dxdk. For our purpose, we will assume that u + µ > − Theorem 4.6
Let u ∈ ( − , and µ ∈ ( − − u, . Then there exists a positive constant c µ,u suchthat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( u )2 m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + µ ≤ c µ,u (1 + | m | ) − u − µ . If u + µ ≤
0, our proof is similar to the proof of Theorem 4.2. If 0 < u + µ <
1, the proof will bedifferent. We will be a little sketchy.Proof: We have v ( u )2 m = v ( u + µ )2 m (1 + x ) µ . Under the compact picture, v ( u )2 m = v ( u + µ )2 m | sin θ | − µ . Let P k ∈ Z a k exp 2 kiθ be the Fourier expansion of | sin θ | − µ . Since | sin θ | − µ has L -derivative, we musthave | a k | ≤ h µ (1 + k ) − for a positive constant h µ . We obtain v ( u )2 m = X k ∈ Z a k v ( u + µ )2 m +2 k . Notice u + µ > −
1. If u + µ ≤
0, by Theorem 4.1, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( u )2 m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + µ = k v ( u )2 m k C u + µ = X k ∈ Z | a k | k v ( u + µ )2 m +2 k k C u + µ ≤ h µ q u + µ X k ∈ Z (1 + ( m + k ) ) − u + µ k + 1 ≤ h µ q u + µ X k ∈ Z (1 + 2 m ) − u + µ (1 + 2 k ) − u + µ k + 1 , which will be bounded by a multiple of (1 + m ) − u + µ .If u + µ > m = 0, we have X k ∈ Z (1 + ( m + k ) ) − u + µ k + 1 = X | k | > | m | (1 + ( m + k ) ) − u + µ k + 1 + X | k |≤ | m | (1 + ( m + k ) ) − u + µ k + 1 . The first sum is bounded by P | k | > | m | k +1 ≤ c | m | − ≤ c | m | − u − µ , since u + µ <
1. The second sumis bounded by c ′ | m | − u − µ . We see that k v ( u )2 m k C u + µ ≤ c u,µ (1 + | m | ) − u − µ . (cid:3) By essentially the same proof as Theorem 4.3, we have
Theorem 4.7
For u ∈ ( − , and µ ∈ ( − − u, , there exists a positive constant c u,µ such that ||| v ( x ) ||| u + µ ≤ c u,µ k (1 + x ) − µ v ( x ) k C u + µ ( v ( x ) ∈ P ( u, +) ∞ ) . 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 ofcusps, z , z , . . . , z l . Write G/ Γ as the union of Siegel sets S , S , . . . S l with a compact set C ([2]).Since Γ action is on the right, our standard Siegel set will be near 0, not ∞ . Let dg = a da dt dk bethe invariant measure of G under the KAN decomposition. Over each Siegel set S i , the invariantmeasure 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 ( G/ Γ) be a cuspidal auto-morphic representation of type P ( − u, ± ) . Given any K -invariant measure ν on G/ Γ such that ν isbounded by dg on C and bounded by a ǫi da i a i dt i dk on S i , there exists a constant C depending on ν (hence on ǫ ) and H such that1. If ǫ > , then k f k L ( G/ Γ ,dν ) ≤ C k f k L ( G/ Γ ,dg ) , ( f ∈ H );
2. If ǫ < , then for any f ∈ H ∞ ∼ = P ( u, ± ) ∞ k f k L ( G/ Γ ,dν ) will be bounded by a multiple of thecomplementary norm ||| f ||| ǫ − u 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 ) = h π u, ± ( kan t ) σ, v i . Then for any h ∈ G , the left action L ( h ) f ( g ) = f ( h − g ) = h π u, ± ( h − g ) σ, v i = h π u, ± ( g ) σ, π − u, ± ( h ) v i . We see that the left action on f ( kan t ) is equivalent to the action of P ( − u, ± ) on v . Fix H ⊆ L ( G/ Γ),a cuspidal automorphic representation of type P ( − u, ± ). By [16] [1], there exists a Γ-invariant distri-bution τ ∈ P ( u, ± ) −∞ such that all smooth vectors in H ∞ can be written as h π u, ± ( g ) τ, v i 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 ofHarish-Chandra, this amounts to choose a cuspidal pair ( P, A ). By Cor. 3.1, for each cusp z i , wecan choose a Siegel set S i and find a constant C i such that kh π u, ± ( g ) τ, v ik L ( S i ,a ǫi daiai dt i dk ) ≤ c i k v k P ( − u, ± ) = c ′ i kh π u, ± ( g ) τ, v ik L ( G/ Γ) . Obviously, for the compact set C , kh π u, ± ( g ) τ, v ik L ( C ,dg ) ≤ kh π u, ± ( g ) τ, v ik L ( G/ Γ) . Hence, our first inequality follows.Fix ǫ <
0. By Cor. 3.3, kh π u, ± ( g ) τ, v ik L ( S i ,dν ) ≤ C ||| v ||| ǫ − u defined for each cusp z i . In thecases of P ( − iλ, +), By Theorem 4.3, the norm ||| v ||| ǫ ≤ C i k (1 + x ) iλ − ǫ v ( x ) k C ǫ . Observe that the map from P ( − iλ, +) ∞ to P ( ǫ , +) ∞ defined by v ( x ) → (1 + x ) iλ − ǫ v ( x ) s K -invariant and the k ∗ k C ǫ is independent of the choices of the unipotent subgroup N . Hence k (1 + x ) iλ − ǫ v ( x ) k C ǫ remains the same for different choices of cusps. Over C , we have kh π iλ, + ( g ) τ, v ik L ( C ,dg ) ≤ kh π iλ, + ( g ) τ, v ik L ( G/ Γ) = C k v k P ( − iλ, +) ≤ C k (1 + x ) iλ − ǫ v ( x ) k C ǫ . We obtain kh π iλ, + ( g ) τ, v ik L ( G/ Γ ,dν ) ≤ C k (1 + x ) iλ − ǫ v ( x ) k C ǫ . The complementary series case P ( u, +) is similar. The nonspherical unitary principal series P ( iλ, − )is more delicate. Essentially, norms ||| v ||| ǫ with respect to different N i will be mutually bounded.Hence we still have kh π iλ, + ( g ) τ, v ik L ( G/ Γ ,dν ) ≤ C k (1 + x ) iλ − ǫ ( 1 + xi − xi ) v ( x ) k C ǫ . (cid:3) Ω A The
KAN decomposition fits naturally in the theory of Fourier-Whittaker coefficients of automorphicforms. It is used by number theorists to conduct analysis on automorphic forms, often over a Siegelset. However to understand the L -function of automorphic representation, in particular, the growthof L-function, the natural choice seems to be the KN A decomposition. Both
KAN and
KN A orig-inated in the Iwasawa decomposition and are closely related to Cartan decomposition. The analysisbased on these decomposition seems to be of different flavor and have different implications. The G -invariant measure with respect to KAN decomposition is a daa dndk or a − daa dadndk dependingon 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 representedby a zeta integral over M A ∼ = GL (1). Hence it is desirable to have an estimate of the L -norm ofautomorphic forms over Ω A , where Ω a compact set with finite measure in KN . Theorem 5.2
Let Γ be a nonuniform lattice in SL (2) . Suppose that w ∈ Γ and N p ⊆ Γ . Let H bea cuspidal automorphic representation of G of type P ( iλ, ± ) . Then there exists a positive constant C depending on ǫ, H and T such that k f k T ,ǫ ≤ C ||| f ||| − | ǫ | ( f ∈ H ∞ ) . Proof: By Theorem 2.3, k f k T ,ǫ ≤ c T ,ǫ, p Z K Z √ T ( a ǫ + a − ǫ ) Z p | f ( kan t ) | dt daa dk ≤ C T ,ǫ, p ((1 + T ) ǫ + 1) Z K Z √ T a −| ǫ | Z p | f ( kan t ) | dt daa dk. Since H is cuspidal, there are K -finite functions in H that are bounded and rapidly decaying nearthe cusp 0. Again, we write f ( g ) ∈ H as matrix coefficient h π iλ, ± ( g ) τ, v i for some v ∈ P ( − iλ, ± )and τ ∈ P (ı λ, ± ) −∞ . Obviously, τ will have no constant term in Fourier expansion. Its Fouriercoefficients have the convergence specified in Theorem 3.1. By Cor 3.1 3.3, there exists C ǫ, H ,T > k f k T ,ǫ ≤ C ǫ, H ,T ||| f ||| − | ǫ | ( f ∈ H ∞ ) . Our theorem then follows. (cid:3) orollary 5.1 Let Γ be a nonuniform lattice in SL (2 , R ) . Suppose that w ∈ Γ and N p ⊆ Γ . Let H be a cuspidal automorphic representation of G of type P ( iλ, ± ) . Let Ω be a compact 2 dimensionaldomain in KN . Let ǫ ∈ R . Then there exists a positive constant C depending on ǫ, H and Ω suchthat k f k L (Ω A,a ǫ d aa dtdk ) ≤ C ||| f ||| − | ǫ | ( f ∈ H ∞ ) . Proof: Obviously, any compact set Ω in KN is contained in some KN T . Hence Ω A ⊆ X T . Thenour assertion follows from the previous theorem. (cid:3) We shall remark that our results also apply to cuspidal automorphic representations of type P ( − u, +)with u ∈ ( − , ||| f ||| − u − | ǫ | as in Theorem 5.2. We shall remark that the Theorem 5.2 remains to be true if1. w ∈ Γ and N p ⊆ Γ;2. the Fourier coefficient b n of τ satisfies the conditions that b = 0 and P | n | − ǫ − − u | b n | < ∞ for ǫ > 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 h π iλ, ± ( g ) τ, v i with τ ∈ P ( iλ, ± ) −∞ . Let ǫ > and suppose τ = P ∗ n ∈ p − Z ,n =0 b n exp 2 πixn with P | n |≤ k | n | − ǫ − | b n | < ∞ . Then kh π iλ, ± ( g ) τ, v ik T ,ǫ ≤ C ||| v ||| − ǫ ( v ∈ H ∞ ) . 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 Eisensteinseries of type P ( iλ, ± ) and ǫ ∈ R . Suppose that V has zero constant term with respect to N . Then k f k T ,ǫ ≤ C ||| f ||| − | ǫ | ( f ∈ V ) . Proof: The Fourier coefficients of Eisenstein series for congruence subgroups are computable ([4]). Itcan be checked that P | n | − ǫ − | b n | < ∞ for ǫ > (cid:3) References [1] Bernstein, J and Reznikov, A. Sobolev norms of automorphic functionals and Fourier coefficientsof cusp forms. C. R. Acad. Sci. Paris Sr. I Math. 327 (1998), no. 2, 111-116.[2] A. Borel
Automorphic forms on SL (2) Cambridge Tracts in Mathematics, 130. Cambridge Uni-versity Press, Cambridge, 1997.[3] W. Casselman Admissible Representations of SL Automorphic Forms and L-functions for the Group GL ( n, R ), Cambridge UniversityPress, Cambridge 2006.[5] Harish-Chandra Automorphic Forms on Semisimple Lie Groups , Notes by J. G. M. Mars, LNM62, Springer-Verlag, 1968.
6] H. He, Generalized matrix coefficients for infinite dimensional unitary representations. J. Ra-manujan Math. Soc. 29 (2014), no. 3, 253-272.[7] H. He, “Representations of ax + b group and Dirichlet Series,”preprint, 2019.[8] H. He “Certain L -norms on authomorphic representations of GL ( n, R ), (in process).[9] H. He “Growth of L -funtions, ”, preprint, 2020.[10] H. Jacquet, J. Shalika, “On Euler Products and the Classification of Automorphic Representa-tions I ”, Amer. Jour. Math 103 (Vol 3), 499-558, 1981.[11] A. Knapp Representation theory of semisimple Groups
Princeton University Press 2002.[12] S. Lang
SL(2) . GTM 105, Springer-Verlag, New York, 1985.[13] R. Langlands
On the functional equations satisfied by Eisenstein series , Lecture Notes in Math.,Springer-Verlag Berlin,Heidelberg, New York, 1976.[14] S. D. Miller and W. Schmid, Automorphic distributions, L-functions, and Voronoi summationfor GL(3). Ann. of Math. (2) 164 (2006), no. 2, 423-488.[15] S. D. Miller and W. Schmid, The Highly Oscillatory Behaviorof Automorphic Distributions for SL (2), Letters in Mathematical Physics, volume 69, (2004), 265-286.[16] W. Schmid, Automorphic distributions for SL(2,R). Conference Moshe Flato 1999, Vol. I (Dijon),345-387, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000.(2), Letters in Mathematical Physics, volume 69, (2004), 265-286.[16] W. Schmid, Automorphic distributions for SL(2,R). Conference Moshe Flato 1999, Vol. I (Dijon),345-387, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000.