Certain L^2-norm and Asymptotic bounds of Whittaker Function for GL(n)
aa r X i v : . [ m a t h . R T ] J u l Certain L -norm and Asymptotic bounds ofWhittaker functions for GL ( n, R ) Hongyu HeDepartment of MathematicsLouisiana State Universityemail: [email protected]
Abstract
Whittaker functions of GL ( n, R ) ([11][12]), are most known for itsrole in the Fourier-Whittaker expansion of cusp forms ( [19] [17]). Theirbehavior in the Siegel set, in large, is well-understood. In this paper,we insert into the literature some potentially useful properties of Whit-taker function over the group GL ( n, R ) and the mirobolic group P n .We proved the square integrabilty of the Whittaker functions with re-spect to certain measures, extending a theorem of Jacquet and Shalika([13]). For principal series representations, we gave various asymptoticbounds of smooth Whittaker functions over the whole group GL ( n, R ).Due to the lack of good terminology, we use whittaker functions to referto K -finite or smooth vectors in the Whittaker model. Let G = GL ( n ) = GL ( n, R ). Let U be the group of unipotent lowertriangular matrices, U be the group of unipotent upper triangular ma-trices, A be the group of positive diagonal matrices, and M be thecentralizer of A in G . Let K be the standard orthogonal group O ( n ).Then we have the Iwasawa decomposition G = KAU .Let ( π, H ) be a irreducible Hilbert representation of G . Let π ∞ bethe Frechet space of smooth vectors in H . Let ( π ∗ ) −∞ be the topolog-ical dual space of π ∞ . This dual space is equipped with the naturalaction of π ∗ . A functional ψ ∈ ( π ∗ ) −∞ is called a Whittaker functionalif π ∗ ( u ) ψ = exp − πi ( X m i u i,i +1 ) ψ, ( ∀ u ∈ U )for some m ∈ R n − with Q m i = 0 ([11] [12]). In [19], J. Shalikashowed that Whittaker functionals, if exist, are unique up to a con-stant for π ∞ . If a Whittaker functional exists, the representation π isoften said to be generic. In [15], Kostant showed that a representationis generic if and only if it has the maximal Gelfand-Kirillov dimension([15]). We then know the classification of generic representations for L ( n ), namely, those irreducible representations induced from the dis-crete series of GL (1) and GL (2) Levi factors.Now fix an m . We can define W h ,m : f ∈ π ∞ → W h, m ( f ) = h π ( g ) f, ψ m i . If m = , the constant vector with entries 1, we write W h f ( g ) = h π ( g ) f, ψ i . Obviously,
W h f ∈ C ∞ ( C × U G ). Now the group G will act from theright. The space { W h f } is known as the Whittaker model of π , intro-duced by Jacquet in his studies of automorphic forms ([12]). Becauseof the Iwasawa decomposition, W h f ( g ) can be uniquely determinedby its restriction on AK . Hence Whittaker model is often regardedas a space of smooth functions on AK . If f is a spherical vector,then W h f ( g ) is uniquely determined by W h f | A . Generally speaking, W h f,m ( g ) is a smooth section of C m × U G → U \ G where C m denote the one dimensional representation defined by thecharacter of U : u → exp 2 πi n − X i =1 u i,i +1 m i . Let P n be the mirabolic subgroup consisting of invertible matrices withlast row (0 , , . . . , , P n is the semidirect product of GL ( n −
1) and R n − . When we refer to GL ( n −
1) as a subgroup of GL ( n ), GL ( n −
1) will always lie in the upper left corner. Consider the restriction
W h f | P n . It is a smooth section of C × U P n → U \ P n , which can be identified with a smooth section of C × U n − GL ( n − → U n − \ GL ( n − GL ( n −
1) and the Eu-clidean measure on R n − , and equip P n with the right invariant mea-sure. Jacquet and Shilika proved the following theorem. Theorem 1.1 ( Jacquet-Shalika [13])
Let ( π, H ) be a generic irre-ducible unitary representation of G . Then W h f | P n ∈ L ( C × U P n ) . The Whittaker model produces a unitary equivalence between ( π | P n , H ) and ( R, L ( C × U P n )) . Here R stand for the right regular representa-tion. By Mackey’s theory, (
R, L ( C × U P n )) is an irreducible unitary rep-resentation of P n , essentially, the unique one with maximal Gelfand-Kirillov dimension ([ ? ] [18]). This implies that ( π | P n , H ) is already rreducible, which was conjectured by Kirillov to be true for all irre-ducible unitary representations of G . In the literature, { W h f | P n } isoften known as the Kirillov model.Here are our main results. Theorem 1.2 (A)
Let π be an irreducible unitary representation of GL ( n ) with a Whittaker model. Let f be a K -finite vector in π ∞ .Write a = diag( a , a , . . . , a n − , in the U AK decomposition of GL ( n − . Then for any l i ∈ N , i ∈ [1 , n − , we have n − Y i =1 ( a i a i +1 ) l i W h f | P n ∈ L ( C × U P n ) . Consequently n − Y i =1 ( a i ) l i W h f | P n ∈ L ( C × U P n ) . Identify L ( C × U P n ) with L ( C × U n − GL ( n − U n − = U ∩ GL ( n − Theorem 1.3 (B)
Let π be an irreducible unitary representation of GL ( n ) with a Whittaker model. Let f be a K -finite vector in π ∞ . Then W h f | GL ( n − ∈ L ( C × U n − GL ( n − , | det | ǫ dg ) for any ǫ ≥ . Indeed, L ( C × U n − GL ( n − , | det | ǫ dg ) provides us a unitary struc-ture of the perturbed representation π ⊗ | det | ǫ | P n . Notice that π ⊗| det | ǫ is never unitary unless ǫ ∈ i R .Theorem A and B suggest that the asymptotic behavior of the Whit-taker function can be similarly understood. The asymptotic behaviorof the Whittaker function W h f ( g ) is well-know for g in the Siegel set.However, wh f ( g ) outside the Siegel set can also be important. In thispaper, we prove the following Theorem 1.4 (C)
Let π ( v, σ ) be the principal series defined over G/M AU .Suppose that ℜ ( v ) < ℜ ( v ) . . . < ℜ ( v n ) . Let ρ = ( n − , n − , . . . , − n − , − n − ) be the half sum of positive rootsfor ( a , u ) . For any f ∈ π ( v, σ ) ∞ , any l i ∈ N , i ∈ [1 , n − , the function a − ρ − v n − Y i =1 ( a i a i +1 ) l i W h f ( g ) is bounded by a constant dependent on f and l . Hence
W h f ( g ) ≤ C f,l a ρ + v n − Y i =1 ( a i +1 a i ) l i . he restriction imposed on v , simply means that v is in the open nega-tive Weyl chamber of a ∗ C . We shall remark that the asymptotic boundwe obtain is the best possible bound. Notice that if σ is trivial and f is spherical, as a i a i +1 → i , the Whittaker function W h f ( a ) will approach c ( v ) a ρ + v with c ( v ) Harish-Chandra’s c function([7], [14]). It can be easily verified that c ( v ) is not zero when v is inthe open negative Weyl chamber. Hence a ρ + v is the best possible ex-ponent. Over the Siegel set with a i a i +1 → ∞ , our theorem implies that W h f ( g ) is fast decaying. More generally, Theorem C gives effectivebounds over g = nak with a in all Weyl chambers.At the boundary where ℜ ( v i ) = ℜ ( v i +1 ) for some i , our theorem isno longer true in general. But a weaker version (with introduction ofcertain small δ , or log-terms) is true. In the general situation where π is induced from discrete series of GL (2) and GL (1), one can embed π into a principal series π ( v, σ ) with v in the closed negative Weylchamber. Similar statement remains true. But it is unlikely that wecan obtain the best results this way. In fact, one would have to usethe leading exponents of π to state Theorem C correctly ([14], [21]).There seems to be a deep connection between the asymptotes of theWhittaker functions and leading exponents of π . We shall not pursuethe asymptotes of the more general Whittaker functions involving thediscrete series of GL (2) in this paper.Finally, we shall remark that Theorem C holds for principal series π ( v, σ ) of all semisimple Lie groups. Theorem 1.5 (D)
Let G be a semisimple Lie group and N AK itsIwasawa decomposition. Let N be the opposite nilpotent group. Let ∆ + ( g , a ) be the positive restricted simple roots. Let π ( v, σ ) be a princi-pal series representation built on G/M AN . Suppose that ℜ ( α i , v ) < for every α i ∈ ∆ + ( g , a ) . Then for any f ∈ π ( v, σ ) ∞ , any l i ∈ N , i ∈ [1 , dim A ] , the function W h f ( g ) is bounded by a constant multiple of a ρ + v + P αi ∈ ∆+( g , a ) − l i α i . Unless otherwise stated, all GL ( n ) in this paper will refer to GL ( n, R ).For n = 2, the Whittaker functions of GL (2) are classical whittakerfunctions and their asymptotes are well-known ([11]). We shall fromnow on assume n ≥ L -norms of Whittaker Model Let G = GL ( n, R ).Fix the invariant measure on GL ( n −
1) = U n − A n − K n − as a − ρ n − ( Y ≤ i Theorem 2.1 Let ( π, H ) be an irreducible unitary representation and ψ m be a Whittaker functional. For any f ∈ H K , W h f,m | A n − ∈ L ( A n − , a − ρ n − daa ) . Proof: Our proof is standard and depend on the fact that the τ -isotypicsubspace H τ is finite dimensional for every τ ∈ ˆ K .Notice that H K = P τ ∈ ˆ K H τ . Without loss of generality, assume that f ∈ H τ and f is in V τ , an irreducible representation of K . Choosean orthonormal basis { e i } in V τ . Then π ( k ) f = P ( π ( k ) f, e i ) e i and W h f,m ( ak ) = (cid:10)X ( π ( k ) f, e i ) π ( a ) e i , ψ m (cid:11) = X ( π ( k ) f, e i ) W h e i ,m ( a ) . Now { ( π ( k ) f, e i ) } are orthogonal to each other in L ( K ), but not nec-essarily in L ( K n − ). However, V τ | K n − is multiplicity free ([5]). Wemay choose the orthonormal basis { E i } from each irreducible sub-representation in V τ | K n − . Now { ( π ( k ) f, E i ) } is an orthogonal set in L ( K n − )5 by Schur’s lemma. We then have Z A n − K n − | W h f,m ( ak ) | dka − ρ n − daa = X k ( π ( k ) f, E i ) k L ( K n − ) k W h E i ,m | A n − k L ( A n − ,a − ρn − daa ) . Hence W h E i ,m | A n − ∈ L ( A n − , a − ρ n − daa ) . It follows that W h f,m | A n − ∈ L ( A n − , a − ρ n − daa ) . (cid:3) Let { X i,j } be the standard basis for the Lie algebra gl ( n ).We have Lemma 2.1 π ∗ ( X i,i +1 ) ψ m = 2 πim i ψ m roof: We have π ∗ ( X i,i +1 ) ψ m ( u ) = ddt | t =0 π ∗ (exp tX i,i +1 ) ψ m = ( ddt | t =0 exp 2 πim i t ) ψ m = 2 πim i ψ m ( u ) . (cid:3) Proof of Theorem A : For every f ∈ π ∞ , we compute W h π ( X i,i +1 ) f,m ( a ) = h π ( a ) π ( X i,i +1 ) f, ψ m i = h π ( a i a i +1 X i,i +1 ) π ( a ) f, ψ m i = a i a i +1 h π ( X i,i +1 ) π ( a ) f, ψ m i = a i a i +1 h π ( a ) f, − π ∗ ( X i,i +1 ) ψ m i = − a i a i +1 πim i h π ( a ) f, ψ m i = − a i a i +1 πim i W h f,m ( a ) (1)Without loss of generality, assume that f ∈ H τ for some τ ∈ ˆ K . Then π ( X i,i +1 ) f will also be K -finite. By Theorem 2.1, a i a i +1 m i W h f,m | A n − ∈ L ( A n − , a − ρ n − daa ) . By induction for any l i ∈ N , n − Y i =1 ( a i a i +1 ) l i W h f | A n − ∈ L ( A n − , a − ρ n − daa ) . Let { e j j ∈ [1 , dim H τ ] } be an orthonormal basis of H τ . The abovestatement is true for every e j . Then n − Y i =1 ( a i a i +1 ) l i W h f ( ak )= n − Y i =1 ( a i a i +1 ) l i h π ( ak ) f, ψ i = n − Y i =1 ( a i a i +1 ) l i h π ( a ) X j ( π ( k ) f, e j ) e j , ψ i = n − Y i =1 ( a i a i +1 ) l i X j ( π ( k ) f, e j ) h π ( a ) e j , ψ i (2)Since each ( π ( k ) f, e j ) ∈ C ∞ ( K n − ) ⊆ L ( K n − ), we have n − Y i =1 ( a i a i +1 ) l i W h f | A n − K n − ∈ L ( A n − K n − , a − ρ n − daa dk ) . ince U \ P n → U n − \ GL ( n − ∼ = A n − K n − , The first statement isTheorem A is proved. We also have a l i i W h f = n − Y j = i ( a j a j +1 ) l i W h f ∈ L ( C × U P n ) ( a n = 1) . The second statement of Theorem A follows immediately. (cid:3) The asymptotes of Whittaker functions over the Siegel set in well-known. Let S ( t ) = Γ U \ U A ( t ) K be the Siegel set with Γ U a lattice in U , t > A ( t ) = { a ∈ A | a i a i +1 ≥ t ( ∀ i ∈ [1 , n − } . Let π be an irreducible smooth representation with a Whittaker modeland f ∈ π ∞ . Then for any l ∈ N n − and uak ∈ S ( t ), | W h f ( uak ) | ≤ C l,f n − Y i =1 ( a i a i +1 ) − l i . (3)Notice that, with respect to the right invariant measure, the Siegel sethas a finite measure. Hence this estimate is inadequate for the discus-sion of W h f ( g ) with g ∈ G .The fast decaying property in Equation 3 is also a special feature ofcusp forms of G . Indeed, in the main lemma of Ch.1 [6], Harish-Chandra proved that if a smooth function of G with zero constantterms along the directions of maximal parabolics containing N A , thenthis function is fast decaying on the Siegel set. Harish Chandra’s in-tention was to apply this lemma to cusp forms. It is not hard to seethat the Whittaker function W h f ( g ) has zero constant terms for allparabolics containing N A . Hence Equation 3 also follows from Harish-Chandra’s lemma. Philosophically, this observation provides us someinitial evidence that many analytic properties of the Whittaker func-tions hold for cusp forms and vice versa.We shall also remark that fast decaying property over Siegel set is of-ten rephrased simply by using the norm k g k − n with g ∈ S ([3]). Thisis no longer good for the purpose of studying W h f ( g ) for arbitrary g .We have to go back to the classical Iwasawa decomposition. Fix a minimal parabolic subgroup M AU , consisting of all invertiblelower triangular matrices. Form the principal series representation ( v, σ ) with g ∈ G acting from the left on smooth functions of theform f ( gmau ) = σ ( m ) − a ρ − v f ( g )where1. π ( v, σ )( g ) f ( x ) = f ( g − x );2. ρ = ( n − , n − , . . . , − n − ) is the half of the positive roots of( u , a );3. v ∈ Hom ( a , C ) = C n is identified with the characters of A and a v = Q ni =1 a v i i ;4. σ = ( σ , σ , . . . , σ n ) is a character of M = { diag( ± , ± , . . . ± } .Each π ( v, σ ) restricted to the center of GL ( n ), produces a centralcharacter diag( a , a , . . . , a ) → n Y i =1 σ i ( sgn( a )) | a | v i . We may speak of representations of central character χ . We use π ( v, σ ) ∞ to denote the linear space of all smooth vector f .The function f is uniquely determined by its value on K and viceversa. Hence the smooth representation π ∞ v,σ can be identified withsmooth sections of the vector bundle K × M C σ . This is often called the (smooth) compact picture, or the compactmodel. Fix an invariant measure on K . We now equip π ( v, σ ) ∞ ∼ = C ∞ ( K × M C σ , K/M ) with the L -norm on K/M . Then π ( v, σ ) be-comes a unitary representation when v is purely imaginary. We use( ∗ , ∗ ) to denote the Hilbert inner product associated with L ( K × M C σ ). In addition, there is a canonical (complex linear) non-degeneratepairing between π ( v, σ ) and π ( − v, σ ∗ ) h f , f i = Z K/M h f ( k ) , f ( k ) i d [ k ] . Here σ ∗ ∼ = σ and h f ( k ) , f ( k ) i = f ( k ) f ( k ) only depends on [ k ] ∈ K/M . It follows that π ( − v, σ ∗ ) can be identified with π ( v, σ ) ∗ , thedual representation of π ( v, σ ).Let k f k denote the L -norm k f k L ( K × M C σ ) . We equip π ∞ ( v, σ ) withthe semi-norms k f k X = k π ( v, σ )( X ) f k with X ∈ U ( g ). Then π ( v, σ ) ∞ becomes a Frechet space. Its dual space, consisting of continuous linearfunctionals, contains π ( − v, σ ∗ ) ∞ as a subspace. It is often denoted by π ( − v, σ ∗ ) −∞ . We retain π ( v, σ )( g )( g ∈ G ) and π ( v, σ )( X )( X ∈ U ( g ))for the group action and Lie algebra action on π ( v, σ ) −∞ . By the Bruhat decomposition, the image of U in G/M AU is open anddense. We consider the pull back i ∗ : π ( v, σ ) ∞ → C ∞ ( U ) . his is essentially the restriction map of f to the group U . It is injec-tive. The group G acts on the image of i ∗ conformally. i ∗ ( π ( v, σ ) ∞ )is often called the noncompact model. Fix the invariant measure, theEuclidean measure on [ u ij ]. If v is purely imaginary, then π ( v, σ )( g )will be unitary operators on L ( U ). Hence we have a unitary repre-sentation ( π ( v, σ ) , L ( U )). Similarly, we have a nondegenerate pairingbetween π ( v, σ ) ∞ and π ( − v, σ ∗ ) ∞ : h f , f i = Z U h f ( u ) , f ( u ) i Y i Lemma 3.1 Suppose that ℜ ( v ) < ℜ ( v ) . . . < ℜ ( v n ) . Let f ∈ π ( v, σ ) ∞ .Then f | K is bounded and f | N ∈ L ( N ) . Indeed for any X ∈ U ( g ), π ( v, σ )( X ) f | K will also be bounded. See forexample Ch. VII. 8,9,10 in [14]. Write( a ∗ C ) − = { v ∈ a ∗ C | ℜ ( v ) < ℜ ( v ) . . . < ℜ ( v n ) } . We call this open negative Weyl chamber in a ∗ C . The closed negativeWeyl chamber cl ( a ∗ C ) − = { v ∈ a ∗ C | ℜ ( v ) ≤ ℜ ( v ) ≤ . . . ≤ ℜ ( v n ) } . Now we take ψ m to be the function exp 2 πi ( P n − i =1 m i u i,i +1 ) when v ∈ ( a ∗ C ) − . Since all vectors in the noncompact model i ∗ ( π ( v, σ ) ∞ ) arein L ( U ), the function exp 2 πi ( P n − i =1 m i u i,i +1 ) defines a continuouslinear functional on π ( v, σ ) ∞ . We then analytically continue ψ m of π ( v, σ ) to all cl ( a ∗ C ) − . Lemma 3.2 We have π ( − v, σ )( a − ) ψ m = a ρ + v ψ Ad ( a ) m where Ad ( a ) m = ( a a m , a a m , . . . , a n − a n m n − ) . For the reason of bookkeeping, we check that π ( − v, σ )( a − ) ψ m ( u ) = ψ m ( au ) = ψ m ( Ad ( a ) ua )= a − ( − ρ − v ) ψ m ( Ad ( a ) u ) = a ρ + v ψ Ad ( a ) m ( u ) . (cid:3) Let f be the spherical vector of π ( ℜ ( v ) , triv ) ∞ with the propertythat f | K ≡ 1. Then f ( u ) ∈ L ( U ) if v ∈ ( a ∗ C ) − by Lemma 3.1. Inaddition, f ( u ) > u ∈ U . Theorem 3.1 Let v ∈ ( a ∗ C ) − and f ∈ π ( v, σ ) ∞ . Then | W h f,m ( ak ) | ≤ C m k f | K k sup a ρ + v . roof: Observe that W h f,m ( ak ) = h π ( v, σ )( k ) f, π ( − v, σ )( a − ) ψ m i = a ρ + v h π ( v, σ )( k ) f, ψ Ad ( a ) m i . Expressed in the noncompact model W h f,m ( ak ) = Z U π ( v, σ )( k ) f ( u ) ψ Ad ( a ) m ( u ) du. Every | π ( v, σ )( k ) f ( u ) | is bounded by k f | K k sup f ( u ) even though f ( u )is the spherical vector of π ( ℜ ( v ) , triv ). Our theorem then follows fromLemma 3.1. (cid:3) Proof of Theorem C : The proof is similar to the proof of Theo-rem A. Let us consider the element in the universal enveloping algebra U ( g ), X l = X l , X l , . . . X l n − n − ,n . Then W h π ( v,σ )( X l ) f,m ( a ) = h π ( v, σ )( X l ) f, π ( − v, σ )( a − ) ψ m i . By Theorem 3.1 and Lemm 2.1, n − Y i =1 ( 2 πm i a i a i +1 ) l i | W h f,m ( a ) | is bounded by k π ( v, σ )( X l ) f | K k sup C m a ρ + v . Similar statement is true for every π ( v, σ )( k ) f : n − Y i =1 ( 2 πm i a i a i +1 ) l i | W h π ( v,σ )( k ) f,m ( a ) | ≤ k π ( v, σ )( X l ) π ( v, σ )( k ) f | K k sup C m a ρ + v . Hence n − Y i =1 ( 2 πm i a i a i +1 ) l i | W h f,m ( ak ) | ≤ C m a ρ + v max k ∈ K ( k π ( v, σ )( X l ) π ( v, σ )( k ) f | K k sup )= C m a ρ + v max k ∈ K ( k π ( v, σ )( Ad ( k − )( X l )) f | K k sup ) . (cid:3) Theorem C can be generalized to cover the case v in the boundary ofthe negative Weyl chamber. But the exact same statement will nothold. Instead, for v ∈ ∂ ( a − C ), we have n − Y i =1 ( a i a i +1 ) l i W h f ( g ) ≤ Ca ρ + v − ǫ , where ǫ ∈ ( a ∗ C ) + and P ni =1 ǫ i = 0. Indeed, we only require that ǫ i > ǫ i +1 if ℜ ( v i ) = ℜ ( v i +1 ). If ℜ ( v i ) < ℜ ( v i +1 ), ǫ i = ǫ i +1 will be llowed. Obviously, these bounds will not be the best bounds. Formany applications, these bounds should be adequate. We shall notpursue this here.We are still left with induced representations from GL (1) and GL (2)-factors. By the theorem of Kostant, representations with Whittakermodel must have the largest Gelfand-Kirillov dimension ([15]). By Vo-gan’s classification of unitary dual of GL ( n ) ([20]), irreducible unitarygeneric representations are of the form π ∼ = Ind GMAN σ ⊗ χ with M A = r Y i =1 GL (2) r Y i =1 GL (1) , (2 r + r = n ) ,σ = r Y i =1 D ( d i + v i , − d i + v i ) ,χ = r Y i =1 C χ i ǫ i . Here d i ∈ Z + , v i ∈ i R , χ i ∈ C , ǫ i = ± 1. The parameter χ satisfies thecondition that the principal series of GL ( r ) induced from χ is unitary. D ( d i + v i , − d i + v i ) is the discrete series with Harish-Chandra param-eter 2 d i and central character k det k v i . The bounds for the Whittakerfunctions of these π is more difficult. At the minimum, We can em-bedded π as a subquotient of a certain principal series π ( v, σ ) suchthat v ∈ cl ( a C ) − . We will be able to obtain bounds on W h f basedon the bounds from the principal series π ( v, σ ). These bounds can befar from optimal. The correct way to write down the bounds is to useleading exponents of π ( v, σ ) ([14]). When σ is a unitary representa-tion of GL (1) factors, the leading exponents are simply a v + ρ . For thegeneral case, the leading exponents are more complex. Hypothetically,the Whittaker functions in this general case should at least share thesame kind of bounds as the spherical principal series π (0 , triv ). Thisshould be adequate for applications in automorphic forms. Let us get back to the setting of Section 2. Let ( π, H ) be an irreducibleunitary representation of G with a Whittaker model and f ∈ H K . Wewould like to give a proof of Theorem C and show that W h f | GL ( n − ∈ L ( U n − \ GL ( n − , | det | s ) ( ∀ s > . Proof of Theorem C: By Theorem A, ∀ t ∈ N , | det | t W h f | GL ( n − = n − Y i =1 ( a i ) t W h f | GL ( n − ∈ L ( C × U n − GL ( n − . bserve that for any s > Z U n − \ GL ( n − | W h f ( g ) | | det( g ) | s d [ g ] ≤ Z U n − \ GL ( n − , | det g | < | W h f ( g ) | d [ g ]+ Z U n − \ GL ( n − , | det g ||≥ | W h f ( g ) | | det( g ) | t d [ g ] < ∞ , where t is an integer greater than s . (cid:3) Now let us perturb the group action of π . We define an action of G on π ∞ : ∀ g ∈ G , π s ( g ) = | det g | s π ( g ) . It is easy to check that ( π s , π ∞ ) is a group representation of G . Itcan never be endowed with a pre-Hilbert structure to make π s unitary.But the following theorem says that π s | P n can be made into a unitaryrepresentation. Indeed, we can perturb the unitary Whittaker modelto obtain a unitary structure for π s | P n . Theorem 4.1 Let s ≥ The map W h : f ∈ π ∞ → W h f ( g ) | P n ∈ L ( C × U P n , | det | s d [ g ]) yields a unitary structure of π s | P n . Here d [ g ] = a − ρ n − daa dk is theright P n -invariant measure on U \ P n . Proof: For any h ∈ P n and f ∈ π ∞ , we compute k W h π s ( h ) f ( p ) k L ( C × U P n , | det | s d [ g ]) = k W h | det h | s π ( h ) f ( P ) k L ( C × U P n , | det | s d [ g ]) = | det h | s k R ( h ) W h f ( p ) k L ( C × U P n , | det | s d [ g ]) = Z | det h | s | W h f ( ph ) | | det p | s d [ p ]= Z | det p | s | W h f ( p ) | d [ p ]= k W h f ( p ) k L ( C × U P n , | det | s d [ g ]) (4)Here R ( h ) stands for the right regular action of h . Hence π s ( h ) pre-serves the Hilbert norm of W h f | P n in L ( C × U P n , | det | s d [ g ]). Infact, π s ( h ) is simply | det h | s R ( h ) in the Kirillov-Whittaker model. Itis a unitary operator of L ( C × U P n , | det | s d [ g ]) . (cid:3) .Write H s = L ( C × U P n , | det | s d [ g ]). Then ( π s | P n , H s ) is a per-turbation of the unitary representation π | P n . 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