WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2)

Transcription

1 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 ALEXANDRU A. POPA Abstract. In this note, we present a complete theory of Whittaker newforms for local representations π of GL(2, which are functions in the Whittaker model of π whose Mellin transform equals the L-function of the representation. We connect this problem with that of finding test vectors for toric linear forms on the representation space of π. This method allows us to treat in an unified manner the real and complex fields, thus identifying Whittaker newforms in all cases, including those previously unknown. 1. Introduction A Whittaker newform for a representation π of G = GL(2 over a local field F is a function in the Whittaker model whose Mellin transform equals the L-function of π. The existence and properties of such Whittaker functions have been known for quite some time, at least in the nonarchimedean and real cases (e.g. [Ge75], Proposition 6.17, but the proofs are not readily available in the literature. In this paper, we fill the void by collecting the facts that were previously known, with proofs, together with some new results in the complex archimedean case. Our approach follows a suggestion of Benedict H. Gross [Gr1, 11-12], and is inspired by a connection between this problem, and that of determining test vectors for toric linear forms on the representation space of π. The motivation for the present paper comes from the local theory of Rankin- Selberg L-functions, where the Whittaker newforms play a central role. See the work of S. W. Zhang [Zh1], and a forthcoming paper of the author [Po4] for applications. In S. W. Zhang s paper, the facts proved here are stated only in the nonarchimedean and some real cases. The connection with test vectors for linear forms allows us to treat the real and complex fields in a unified manner, thus identifying Whittaker newforms in the complex case as well. To state the problem more precisely, let F be a local field, and let π be an admissible, irreducible, infinite dimensional representation of G(F. It is well known that such a representation is generic, that is it admits a Whittaker model, whose definition we recall. Let ψ be a nontrivial additive character of F. Then there is a unique space W (π, ψ of locally constant functions W on G(F (Schwartz functions if F is archimedean satisfying: W (( 1 x g = ψ(xw (g for all x F, and such that the right regular representation of G(F on W (π, ψ is isomorphic to π. 1

2 2 ALEXANDRU A. POPA For any W W (π, ψ, define the Mellin transform: ( x (1 Ψ W (s, g = g x s 1/2 d x F W where d x is the invariant measure on F such that the set of units in F has measure 1 in the nonarchimedean case, and the multiplicative Lebesgue measure on R or C if F = R or F = C respectively. The integral converges if the real part of s is large enough, and it can be meromorphically continued to the complex plane. The L-function L(s, π is defined as the greatest common denominator of all Ψ W (s, g, appropriately normalized. The choice of exponent s 1/2 in formula (1 is made so that the Mellin transform has a functional equation for s 1 s. Our goal is to identify explicit vectors W π W (π, ψ, which we call (following [Zh1] Whittaker newforms, such that: (2 Ψ Wπ (s, e = L(s, π, where e is the unit matrix in G(F. Such Whittaker functions are of course not unique, but they become unique if we further require that they are invariant under a certain compact congruence subgroup of G(F in the nonarchimedean case, or have a given weight in the archimedean case. We shall see that these invariance properties arise naturally from a connection between this problem, and the problem of finding test vectors for certain toric linear forms on the representation space of π. Regarding the terminology, we remark that the use of the term newform here is compatible with its use in the classical theory of modular forms over the upper half plane. Indeed, if f is a cuspidal newform over the rational numbers Q, then there is an automorphic form φ f on GL(2, A, with A the adeles of Q, constructed from f in the standard manner (e.g. as in [Ge75, 3]. The Whittaker coefficient of φ f decomposes as a product of local Whittaker functions, and the local coefficients can be shown to be exactly the Whittaker newforms attached to the local representations of GL(2, which are the local factors of the global automorphic representation associated with f. See [Ge75, 6] for a more detailed discussion of the connection between the classical Hecke theory and Whittaker newforms, or [Zh1] for an overview. The paper is organized as follows. Following a suggestion of B. H. Gross, in section 2 we interpret the Mellin transform as a linear form on the space W (π, ψ, on which the diagonal torus acts by a certain character. In the nonarchimedean case, the results of [GP91] provide a one dimensional subspace of W (π, ψ on which any such toric newform is not zero, namely the space of functions invariant under a compact congruence subgroup of G(F of level equal to the conductor of π. By using the action of certain elements in the Hecke algebra, we show in section 3 that the identity (2 holds, for an appropriately normalized vector W π in this subspace. The result is known, and we only include the proof here for the sake of completeness. In the archimedean case, the connection with test vectors for linear forms provides a one dimensional candidate space for the Whittaker newform. In section 4, we show that the identity (2 holds, for a function W π belonging to this one dimensional subspace. The proof uses the differential equations satisfied by Whittaker functions in the archimedean case to relate their values on the diagonal torus with values of Bessel functions. The facts needed about Bessel functions are recalled in the Appendix.

3 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL( Test vectors for linear forms Let F be a local field, and let π : G(F Aut V be an admissible, irreducible, infinite dimensional representation with central character ω. If L is a separable quadratic extension of F, we denote by T the algebraic group over F with T (F GL 1 (L. Let χ T : T (F Aut C be a character whose restriction to F equals ω 1. Consider the representation π χ T of the group H = G(F T (F ; this is an admissible, irreducible representation, in which the diagonally embedded subgroup F acts trivially. Viewing L as a two dimensional vector space over F, we can embed the torus T (F inside G(F = Aut F (L by regarding elements in T (F L as acting on L by left multiplication. Then T (F embeds diagonally in H, and we denote the image of this embedding by. An object that has received attention in the literature is the space of linear forms on V C invariant under, that is linear forms on V on which T acts by χ 1 T. A theorem of Waldspurger and Tunnel shows that the space Hom T (π χ T, C of such forms is at most one dimensional, and the question arises of finding natural test vectors on which such a form is nonzero. This problem has been treated in the nonarchimedean case by B. H. Gross and D. Prasad in [GP91]. In this paper we are interested in the case that L = F + F is the split algebra, so T (F is a split torus embedded in G(F. In this case the results of Waldspurger and Tunner imply that the space Hom T (π χ T, C is exactly one dimensional. Test vectors for the forms belonging to this space can be obtained as follows. If F is nonarchimedean and χ is unramified, the results of [GP91] give a line on which any nonzero form m Hom T (π χ T, C does not vanish, namely the one dimensional subspace of V fixed by an order R of M 2 (F of reduced discriminant ϖf CO F, which contains T (O F, where C is the conductor of π (see Section 3 for the definitions. If F is archimedean, let K be the maximal compact subgroup of G(F = Aut F (L fixing the positive definite quadratic form associated with the split extension L/k. Let W be the minimal K-type of π that contains vectors on which the compact torus K T (F acts by χ 1 T (see Section 4 for the definitions. B.H. Gross suggested in [Gr1] that the space of such vectors is one dimensional, and that it provides the desired test vectors for forms in Hom T (π χ T, C. We prove the first claim in Section 4, while we show that the second claim is true in the cases that are relevant to the problem of determining Whittaker newforms for π. The connection between the test vector problem and that of finding Whittaker newforms arises in the case that T (F is the diagonal torus inside G(F, which we assume herafter. Then the character χ T equals ω 1 when restricted to the center Z(F of G(F, hence there exists a character χ of F such that: a (3 χ T = χ(ad 1 ω 1 (d. d Let l : V C be the Whittaker functional, the unique linear form on which the group of unipotent matrices acts by the character ψ. If χ T, χ are related by (3, consider the form m χ obtained by averaging : (4 m χ (v = F l (( t v χ(td t.

4 4 ALEXANDRU A. POPA If this integral converges, it is easy to check that m χ (tv = χ T (t 1 m χ (v, for t = a T (F, d that is m χ Hom T (π χ T, C. Taking χ to be the character χ s (t := t s 1/2 F, the integral m χ (v becomes the Mellin transform denoted by Ψ Wv (s, e before, where W v W (π, ψ is given by W v (g := l(gv for v V. We therefore expect that Whittaker test vectors for m χ (for χ = χ s have their Mellin transform equal to a nonzero multiple of L(s, π, and we show that the multiple is 1 if the test vector is appropriately normalized. In the nonarchimedean case, the results of [GP91] mentioned above imply that the linear form m χ (for χ = χ s does not vanish on the one dimensional subspace of V fixed by the congruence subgroup K 1 (ϖf C, where C is the conductor of π (see Section 3 for the definitions. We show that if v π is an appropriately normalized vector in this subspace, then the corresponding Whittaker function W π W (π, ψ satisfies the identity: (5 Ψ Wπ (s, e = m χ (v π = L(s, π. In the archimedean case, the representation π, restricted to the standard maximal compact subgroup K of G, decomposes into a sum of finite dimensional representations, called K-types. Inside each K-type W we consider the subspace of vectors W T. We prove in section 4 that W T is at most one dimensional, and when it is trivial, the linear form m χ vanishes on W. The Whittaker newform is then found to reside in W T, for the smallest possible K-type W for which W T is one dimensional (again when χ = χ s. on which the compact torus T K acts by the character χ 1 T 3. Nonarchimedean case Keeping the notations of the last section, assume now that F is a nonarchimedean field with uniformizer ϖ F, and residue field of order q = ϖ F 1. We denote by O F, U F the ring of integers and the group of units of F. Assume for simplicity that the fixed character ψ used to define the Whittaker model W (π, ψ is unramified, that is it is trivial on O F, but not on ϖ 1 F O F. The integral (1 converges for Re(s large enough, and it can be continued to a rational function of q s with at most two poles. The L-function L(s, π is normalized such that it is of the form P (q s 1 where P is a polynomial of degree two or less with P ( = 1. The conductor of π is the smallest integer C such that one of the following equivalent conditions are satisfied: There is a nonzero function W W (π, ψ which is invariant under { } K 1 (ϖf C a b := GL c d 2 (O F : c ϖ CF O F, d 1 + ϖ CF O F ; There ( is a nonzero function W W (π, ψ such that W (gk = W (gω(d for all a b k = K c d (ϖf C, where ω is the central character of π, and { } K (ϖf C a b := GL c d 2 (O F : c ϖ CF O F.

5 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 5 The existence of such an integer follows immediately from the fact that π is admissible. We then have the following: Proposition 1 (Ca73. Let C be the conductor of π as defined above. The space of K 1 (ϖf C -invariant functions in W (π, ψ is one dimensional. Moreover, the results of [GP91] imply, via the connection with linear forms from the previous section, that the Mellin transform of a nonzero Whittaker function in the space of the Proposition is not zero. Therefore we are led to define the Whittaker newform W π attached to π, to be the unique function in the one dimensional space of the proposition which takes value 1 at the unit element of GL 2 (F (from the proof of the proposition in [Ca73] it follows that the functions in the one dimensional subspace above do not vanish at the identity matrix e. Proposition 2. The Whittaker newform W π satisfies: Ψ Wπ (s, e = L(s, π. PROOF. Let C be the conductor of π. We first observe that the values: ( a (6 W π, only depend on a, and are if a > 1. Indeed, for any α O F, β O F, the right invariance of W π under K (ϖ C implies ( ( ( a a α β 1 aβ aα W π = W π = W π = = ψ(aβw π (( aα. If β is small enough, it follows that the values (6 depend only on a ; if a > 1, take β such that ψ(aβ 1 (this is possible since the maximal ideal on which ψ is nontrivial is O F to conclude that the values (6 are. Using these properties of W π in formula (1, we obtain: (7 Ψ Wπ (s, e = k= W π ( ϖ k F ϖ F k(s 1/2. ϖ k Therefore we need to compute the sequence w k = W F π for k. If π is unramified the values w k are computed in [Go7, Theorem 1.11] and the claim is easily verified. We assume therefore that π is ramified, that is C >. We adapt the method used to treat the unramified case in [Go7] (see also [Bu97], that is we will find a recursion relation for w k using the action of a certain Hecke operator. To simplify notation, denote by K the compact ( subgroup K 1 (ϖ C. ϖ Let χ be the characteristic function of the set K K. It is an element in the Hecke algebra H K of K-biinvariant functions on G(F. Since the space W (π, ψ K of fixed H K vectors is one dimensional and invariant under H K, it follows that there is a constant α F such that: (8 π(χw π (g = αw π (g for all g G(F.

6 6 ALEXANDRU A. POPA On the other hand, using the definition of the Hecke algebra action we have: ( ϖ k (9 π(χw π = K ϖ ϖ k W π g dg. K Now we use the following disjoint decomposition, valid for C > : ϖ ϖ b K K = K, b (mod ϖ together with the fact that ( ( ϖ k ϖ b W π g = ψ(ϖ k ϖ k+1 bw π g. Since W π is right invariant under K, it follows that, up to a nonzero constant which depends on the measure normalization, the right hand side of the equation (9 equals w k+1. Combining this with the equality (8, we obtain: ϖ k+1 ϖ k W π = βw π = β k+1, for some constant β F. Introducing these values in equation (7 we obtain: 1 Ψ Wπ (s, e = 1 βq 1/2 q. s But we know a priori that the L-function L(s, W π is the inverse of a linear polynomial in q s with constant term 1 (linear because π is ramified, such that Ψ Wπ (s, e/l(s, π is holomorphic for all s. It follows that the only possibility is Ψ Wπ (s, e = L(s, π, which proves the proposition. Using this proposition, one can easily compute the values of the newform W π on the diagonal torus: Corollary 1. Let α 1, α 2 C be such that L(s, π = 1,2(1 α i ϖ F s 1, that is the Satake parameters of π in case π is unramified. Let W π be the Whittaker newform. Then we have: { a if a > 1, W π = a 1/2 k+l=ν(a αk 1α2 l otherwise, where we make the convention that = 1, in case one or both of α 1, α 2 is. PROOF. We have already seen in the previous proof that the desired values only depend on a and are if a > 1. The previous proposition and the identity (7 then yield: 2 L(s, π = (1 α i ϖ F s 1 ϖ k = W F π ϖ F k(s 1/2. i=1 Expanding the left hand side as a power series in ϖ F s and equating coefficients yields the desired formula. k=

7 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL( Archimedean case Consider now the case F is R or C. We fix the characters ψ(x = e 2πix of R and ψ(z = e 2πi(z+ z of C. Denote by K the standard maximal compact subgroup of G = GL 2 (F, so K = O 2 (R or K = U 2 (C in the real or complex case. Let π be an admissible irreducible representation of G(F with central character ω, that is a (g, K-module, where g is the complexification of the Lie algebra of GL 2 (F. 1 As representation space for π we take the Whittaker model W (π, ψ, consisting of smooth, K-finite functions on G(F. The notion of level in the nonarchimedean case is replaced by that of weight in the archimedean case. The representation π restricted to K decomposes into a direct sum of finite dimensional nontrivial subspaces, which we call K-types: π K = W n W n+2 W n+4..., indexed by the infinite set of nonnegative integers {n+2k} k Z. The smallest one, n, is called the weight of π. For F = R, the K-type W m is the span of the vectors in the representation space of π having weight ±m under the action of SO 2 (R. The vector v is said to have weight m if: π(k θ v = e imθ cos θ sin θ v for all k θ = SO sin θ cos θ 2 (R. The space W m is 1-dimensional for m = and 2-dimensional for m >. If π is a principal series representation with central character µ(t = t r sgn(t n with n {, 1}, then the weight of π is n; if π is a discrete series representation σ(µ 1, µ 2 with µ 1 µ 1 2 (t = tp sgnt for some integer p >, then the weight of π is p + 1. For F = C, the K-type W m is the (m + 1-dimensional subspace of π on which SU 2 (C acts by its unique irreducible (m + 1-dimensional representation ρ m. We recall that the representation ρ m can be realized on the space V m of degree m homogenous polynomials in two variables: on which SU 2 (C acts by: ϕ(x, y = m/2 k= m/2 a k x m/2+k y m/2 k, (1 ρ m (gϕ(x, y = ϕ[(x, yg]. If π is the principal series representation π(µ 1, µ 2, with µ 1 µ 1 2 (z = zp z q such that p q is an integer, then the weight of π is p q. Recall that all irreducible, admissible representations of G(C are isomorphic to a principal series representation. To state the formula for the L-function L(s, π, recall first the formula for the L-functions attached to characters of R and C, via Tate s theory for GL 1. In terms of the gamma factors ( G 1 (s = π s s 2 Γ, G 2 (s = 2(2π s Γ(s = G 1 (sg 1 (s + 1, 2 we have, if µ(t = t r R sgnm (t is a character of R, with m {, 1}: L R (s, µ = G 1 (s + r + m; 1 See [Bu97, p.2] for a definition of (g, K-modules.

8 8 ALEXANDRU A. POPA while if µ(z = z p z q is a complex character with p q Z, then: L C (s, µ = G 2 (s + max(p, q, where by the maximum of two complex numbers a, b such that a b R, we mean a + max(, b a. If F = R and π is a principal series representation π(µ 1, µ 2 of G(R, we have: L(s, π = L R (s, µ 1 L R (s, µ 2. On the other hand, if π is a discrete series representation σ(µ 1, µ 2, then π can be associated via the local Jacquet-Langlands correspondence to a character λ of C, and we have L(s, π = L C (s, λ. More precisely, one can assume without loss of generality that µ 1 = s1, µ 2 = s2 sgn m2, with s 1 s 2 = S a positive integer, m 2 {, 1}, and S m 2 odd. Then π comes from the complex character λ(z = z s1 z s 2 (or from its complex conjugate, and hence: L(s, π = L C (s, λ = G 2 (s + s 1. If F = C, then all infinite dimensional, irreducible, admissible representations of GL 2 (C belong to the principal series. If π is a principal series representation π(µ 1, µ 2 with µ 1, µ 2 characters of C, then: L(s, π = L C (s, µ 1 L C (s, µ 2. Assume now F is either R or C, and recall that the representation π of G(F has central character ω. As in Section 2, let χ T be a character of the diagonal torus T whose restriction to Z(F equals ω 1, and let χ be the corresponding character of F given by Eqn. (3. Towards our goal of finding a Whittaker vector W π whose Mellin transform equals the L-function, we first identify inside each K-type of π a natural test space for the nonvanishing of any linear form m Hom T (π χ T, C. Definition 1. If W is a K-type of π, let W T W be the (possibly zero subspace on which the compact torus T c := T K acts by χ 1 T. Explicitly, for F = R: (11 W T = { v W : π(ɛv = χ( 1 1 v }, 1 where ɛ = T c, while for F = C: (12 W T = { v W : π[t(a]v = ω(a 1 χ(a 2 v, a S 1}, a where S 1 = {a C : a = 1}, and t(a = T ā c for a S 1. The following proposition shows the connection between the space W T and the vanishing of linear forms on which T acts by χ 1 T. Proposition 3. (i If W is an arbitrary K-type of π, then the space W T is at most one dimensional. (ii If W is a K-type of π such that dim W T =, then m(v = for every linear form m Hom T (π χ T, C and for every v W. PROOF. Case 1: F = R. If W is one dimensional (i.e. π has weight and W is the minimal K-type, then it is obvious that W T is or 1 dimensional, depending on the eigenvalue of ɛ on W, hence dim W T 1 [see Eqn. (11]. The second part is obvious in this case.

9 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 9 On the other hand, if W is two dimensional, the action of ɛ on W decomposes into two eigenspaces with eigenvalues ±1, hence dim W T = 1. There is nothing to prove for part (ii in this case. Case 2: F = C. Let W be the (unique (n + 1-dimensional subspace of π on which SU 2 (C acts by its irreducible representation ρ n. Let T 1 = T SU 2 (C be the diagonal subgroup of SU 2 (C. Part (i follows from the fact that the restriction of ρ n to T 1 decomposes completely into a direct sum of one dimensional representations, each appearing with multiplicity one. If the character ω 1 χ 2 restricted to T 1 S 1 is not among them, then W T = ; otherwise, dim W T = 1. For part (ii, notice that if v W and if m Hom T (π χ T, C, then by definition we have: m[π(t(av] = ω 1 (aχ 2 (am(v if a = 1. This shows that m(v = if T c acts on v by a character different from ω 1 χ 2 (see Eqn. (12. If W T =, we have observed above that W is spanned by such vectors v, hence m vanishes on W. The previous proposition suggests that the space W T, when nonzero, provides test vectors on which linear forms m Hom T (π χ T, C do not vanish. To show that this is indeed the case, one can use the specific form m χ Hom T (π χ T, C, defined on Whittaker functions by averaging as in (4. Note that its values on functions W W T are given by: (13 m χ (W = W t χ(td t, which follows from equations (11 and (12. The diagonal values of Whittaker functions of a given type can be computed in favorable circumstances in terms of Bessel functions, as in the proof of the next proposition, thus proving that m χ (W does not vanish. However, in this paper we are mostly interested in identifying Whittaker newforms, hence we restrict ourselves to the case χ(t = χ s (t := t s 1/2 F from now on, where t R = t sgnt and t C = t t. The linear form m χ (W becomes the Mellin transform Ψ W (s, e for this choice of χ. Note that for each K-type W, the space W T does not depend on s for this choice of χ [see (11, (12], thus justifying the notation. The following proposition implies that, at least for values of s for which L(s, π does not have a zero or pole, the forms in Hom T (π χ T, C do not vanish on the line W T, for the minimal K-type W for which W T {}. A similar result can be proved by the same methods when χ is arbitrary, but we do not pursue the matter further. Proposition 4. In both the real and the complex case, let W be the minimal K-type in the Whittaker model W (π, ψ such that dim W T = 1. Then m χ (W = Ψ W (s, e = L(s, π for some W W T, and for Re(s large enough. The proof occupies the last section. For convenience, let us describe explicitly the minimal K-type of the proposition.

10 1 ALEXANDRU A. POPA F = R: Let π be a representation of G(R of weight n. The minimal K-type W for which W T is W n, unless π is the weight representation π( r 1 sgn, r 2 sgn, when it is W 2. F = C : Let π be a principal series representation π(µ 1, µ 2 of G(C, where µ i (z = z pi z qi are characters of C, with p i, q i C such that p i q i Z. If n = p 1 q 1 (p 2 q 2 is the weight of π, and m = q 1 p 1 +q 2 p 2 is the exponent of ω 1 (a for a S 1, then the minimal K-type W for which W T is W N, with N = max(n, m. In the case F = R, it will follow from the proof that the Whittaker newform W π can be taken to be the function of minimal weight under the action of SO 2 (R (appropriately normalized, unless π is the weight zero representation π( r1 sgn, r2 sgn, in which case W π can be taken of weight two. The values of the Whittaker newform on the diagonal torus can also be extracted from the proof, and they are used in [Po4] and [Zh1] to compute the Rankin-Selberg convolution associated to a pair of representations of G(R. 5. Proof of Proposition 4 First we sketch the proof. For W W (π, ψ and t R, let f W (t be the function of one variable: t (14 f W (t = W 1/2 sgnt t 1/2. Using the lowering and raising operators in the complexification of the Lie algebra of GL 2 (F, we will show that if W W T, where W is the minimal K-type of the proposition, then f W satisfies a second order differential equation satisfied also by the function t a J u (4πt for some a, u C, where J u is the Bessel function described in the appendix (this holds in all cases, except for the discrete series in the real case, when f W has a simpler form. Since f W is of rapid decay at infinity, and the Bessel equation has only one solution of rapid decay, it follows that f W (t is of the form t a J u (4πt (up to a constant. The identity then follows from equation (13, using the integral formula (A.3. The details are supplied below. The real case. We consider four cases, depending on the representation π. Case 1: π is the discrete series representation σ( s 1, s 2 sgn m of weight k = S +1, where S = s 1 s 2 is a positive integer, m {, 1}, and S m odd. Let W k, W k W k be Whittaker function of weight k, k, normalized such that π(ɛw k = W k, and let f k, f k be the corresponding function of one variable given by Eq. (14. Then the Whittaker function W + = W k + W k belongs to W T k. Since the lowering operator annihilates W k, the function f k satisfies the following differential equations (see [Go7], p. 2.2: 2tf k(t + (4πt kf k (t =, and a similar equation is satisfied by f k. Since the Whittaker functions are of moderate growth at infinity, the only solutions are (up to a constant, which is chosen a posteriori so that the conclusion holds: { 2t k 2 e 2πt if t, f k ( t = f k (t = if t <.

11 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 11 The integral (13 for W = W + W T, can now be easily computed, using the integral definition of the Γ function: Ψ W+ (s, e = 2Ψ Wk (s, e = 4 x s 1 +s 2 +k 2 e 2πx x s 1/2 d x = 2G 2 (s + s 1, which is exactly 2L(s, π. Hence the Whittaker newform can be taken to be W k. Case 2: π is the weight principal series π( r1, r2. Let r = r 1 r 2. Let W W be a weight Whittaker function, and let f (t be the function corresponding to W by Eq. (14. Since π(ɛw = W, it follows that W W T, and that f is an even function of t. Translating the action of the raising and lowering operators on W in terms of the function f, we obtain the following differential equation: 2 1 r f 2 (t + 4t 2 4π 2 f (t =. This equation cannot be solved explicitly, the solutions being classical Whittaker functions; however, if we set J(2πt = f (tt 1/2 for t >, we notice that J satisfies (after the change of variables y = 2πt: (1 + r2 (15 J (y + J (y y 4y 2 J(y =. This is the equation satisfied by the classical Bessel function J r/2 defined in the appendix. In fact, since the equation (15 admits a unique solution of moderate growth at infinity (up to a constant, it follows that we can take J(t = 2J r/2 (t for t >. Hence the integral (13 becomes for W = W W T : Ψ W (s, e = 4 x r 1 +r J r/2 (2πxx s 1/2 d x = L(s, pi, where the last equality follows from formula (A.3. Case 3: π is the weight 1 principal series representation π( r1, r2 sgn. Let r = r 1 r 2 and let W 1, W 1 be two nonzero Whittaker functions of weight 1, -1 respectively, normalized such that LW 1 = rw 1, where L is the lowering operator in g. Then π(ɛw 1 = W 1, hence the Whittaker function W + := W 1 + W 1 belongs to W1 T. The corresponding functions f 1, f 1 given be Eq. (14 satisfy the differential equations (see [Go7] p. 2.19, or [JL7], the proof of lemma : 3 rf 1 (t = 2tf 1(t (4πt + 1f 1 (t rf 1 (t = 2tf 1(t + (4πt 1f 1 (t. Not that if r = these equations can be solved explicitly, yielding: { t 1/2 e 2πt if t >, f 1 (t = if t <. The claim is then easy to check as in Case 1. If r one proceeds as follows. 2 This is the equation (78 in [Go7] p. 2.2, where it is stated inexactly. 3 We warn the reader that there is a factor of 2π missing in [JL7], where u should be replaced by 2πu ; it is corrected in [Go7].

12 12 ALEXANDRU A. POPA By adding and subtracting the two equations above, one checks that the function f(t = f 1 (t + f 1 (t, which corresponds to the Whittaker function W + via (14, satisfies (for t > : f (t f (t t ( (r t 2 + 4π 2 f(t = Moreover, the function f is even since f 1 ( t = f 1 (t (which follows from the corresponding relation between the Whittaker functions. For t >, define the function J(2πt = t 1 f(t. Writing the previous equation in terms of J, we obtain (after the change of variables y = 2πt: J (y + J (y (r 1 2 y 4y J(y = This is the equation satisfied by the Bessel function J (r 1/2, and we conclude as before that we can take J(t = 2J (r 1/2 (t for t >. Hence the Eq. (13, written for W = W + W1 T becomes Ψ W+ (s, e = 4 x r 1 +r J (r 1/2 (2πxx s 1/2 d x = L(s, π, where for the second equality we have used formula (A.3. Finally observe that the function f, corresponding to W = W 1 W 1 via (14, is odd as a function of t hence the integral Ψ W (s, e vanishes. Therefore: Ψ 2W1 (s, e = Ψ W+ (s, e + Ψ W (s, e = L(s, π, which proves that the Whittaker newform can be taken to be the weight one function 2W 1. Case 4: π is the weight principal series π( r1 sgn, r2 sgn. Let r = r 1 r 2. Let W 2, W, W 2 W 2 be Whittaker vectors of weights 2,, 2 respectively, normalized such that L 2 W 2 = (r 1LW = (r 2 1W 2, where L g is the lowering operator. Then it is shown in [Go7], p. 2.7, Eq. (21, that π(ɛw 2 = W 2, hence W := W 2 W 2 W2 T. Let f 2, f, f 2 be the functions corresponding to W 2, W, W 2 by (14. The action of the raising and lowering operators translates into the following system of differential equations [JL7]: (r + 1f 2 (t = 2tf (t 4πtf (t (r + 1f 2 (t = 2tf (t + 4πtf (t f (t [4π 2 + (r 2 1/4t 2 ]f (t = Subtracting the first two equations we obtain (r = r 1 r 2 1 because π is a principal series representation: f 2 (t f 2 (t = 8πtf (t/(r + 1, while the third equation has as solution f (t = t 1/2 J r/2 (2πt (compare with equation (A.4 in the appendix. Equation (13 becomes, for W = W W2 T : Ψ W (s, e = t (r 1+r 2 /2 t 3/2 J r/2 (2πtt s 1/2 d t = L(s, π

13 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 13 where for the second equality we have used formula (A.3. As in Case 3, the integral Ψ W+ (s, e vanishes for W + = W 2 + W 2. Therefore: Ψ 2W2 (s, e = Ψ W (s, e + Ψ W+ (s, e = L(s, π, which proves that the Whittaker newform can be taken to be the weight two function 2W 2. The complex case. Since all irreducible, admissible representations of GL 2 (C are principal series, we can assume that π is of the type π(µ 1, µ 2. Here µ i (z = z pi z q i are characters of C, with p i, q i C such that p i q i Z. Let µ(z := µ 1 µ 1 2 (z = zp z q, and assume without loss of generality that p q (since π(µ 1, µ 2 π(µ 2, µ 1. Then π has weight n = p q, and the restriction of π to SU 2 (C decomposes as follows: (16 π SU2(C ρ n ρ n+2 ρ n+4, where ρ N is the unique (N + 1-dimensional irreducible representation of SU 2 (C, which acts by right translation on the space V N of homogenous polynomials in two variables of degree N. Next we review the theory of differential equations satisfied by certain Whittaker functions in W (π, ψ, following [JL7]. Fix an integer N n, having the same parity as n, so that ρ N appears in the decomposition (16, and fix an SU 2 (C intertwining map: (17 i N : V N W (π, ψ. For k between N/2 and N/2, let W k W (π, ψ be the image of the monomial x N/2+k y N/2 k. That is W k is in the image of i N and transforms as follows under T c : W k [gt(a] = a 2k W k (g for a = 1. It is clear that W k is then completely determined by the function: t 1/2 (18 f k (t := W k t 1/2 for t >. Using the action of certain elements in the center of the universal enveloping algebra of GL 2 (C, it is shown in [JL7] that the functions f k (t satisfy the following differential equations 4 for N/2 k N/2: (19 f k (t (1 2k f k t (2 f k (t (1 + 2k f k t (16π 2 + p2 (1 k 2 t 2 (16π 2 + q2 (1 + k 2 t 2 f k (t = f k (t = = 8πi(N/2 + k f k 1 t = 8πi(N/2 k f k+1. t 4 There is a factor of 4π missing in [JL7]. To correct the equations there, u has to be replaced by 4πu

14 14 ALEXANDRU A. POPA Taking k = N/2 in equation (19 and k = N/2 in equation (2 and comparing with equation (A.4 in the appendix we see immediately that f N/2, f N/2 are related to Bessel functions as follows (up to a constant which we choose to be one: (21 f N/2 (t = t 1+N/2 J p (4πt, f N/2 (t = t 1+N/2 J q (4πt. Moreover, if N = n (the minimal K-type one can compute all the functions f k (t, n/2 k n/2, in terms of Bessel functions (which does not seem easily feasible for higher N. With our assumption that n = p q, one can show proceeding recursively that: (22 f k (t = t n/2+1 J q+n/2 k (4πt, for k = n/2, n/2 1,, n/2. Coming back to the proof of the proposition, denote by m the integer such that ω 1 (a = a m if a = 1, that is m = q 1 p 1 + q 2 p 2. As pointed out following the statement of the proposition, the minimal K-type W such that W T is isomorphic to ρ N, where By equation (13, for W W T we have: (23 Ψ W (s, e = N = max(n, m. R > W t t 2s 1 d t. To compute this integral, we consider two cases: Case 1: N = m > n, that is (p 1 q 1 (p 2 q 2 >. Since we have assumed p q, we must have p i > q i, hence m = N < and: L(s, π = G 2 (s + p 1 G 2 (s + p 2. On the other hand, the space W T is spanned by W π = W N/2, (the image of y N under the intertwining map i N of equation (17, hence by equation (21: t W π = ω(t 1/2 t 1/2 W π t 1/2 = t p1+p2+1 J p1 p 2 (4πt. Plugging this expression back into the formula (23 and using formula (A.3, we obtain the desired identity Ψ(W π, s = L(s, π (provided W π is appropriately normalized. Case 2: N = n m, that is (p 1 q 1 (p 2 q 2. Since we have assumed p q, we have also p 1 q 1, q 2 p 2, hence: L(s, π = G 2 (s + p 1 G 2 (s + q 2. In this case, the space W T is spanned by W π = W m/2, hence by equation (22 we have (taking into account the relations among the p i, q i : t W π = ω(t 1/2 t 1/2 W π t 1/2 = t p 1+q 2 +1 J p1 q 2 (4πt. As before, it follows that Ψ(W π, s = L(s, π.

15 WHITTAKER NEWFORMS FOR LOCAL REPRESENTATIONS OF GL(2 15 Appendix on Bessel Functions For each complex number u, the Bessel function J u is a solution of the following differential equation: (A.1 J u (y + J u(y (1 + u2 y y 2 J u (y = for y >. It can be shown that (up to a constant this equation admits a unique solution of moderate growth at infinity. If normalized appropriately, this solution satisfies the following identities: (A.2 (A.3 e y(t+t 1 t u d t = 2J u (2y, s + u J u (yy s d y = 2 s 2 Γ Γ 2 ( s u where y > in the first equation, and Re s > Re u in the second. We also need the equation satisfied by the functions G(y = y a J u (y, where a C. It can be easily seen that: (A.4 G (y + (1 2a G (y y (1 + u2 a 2 References y 2 2, G(y =. [Bu97] D. Bump: Automorphic forms and representations, Cambridge Studies in Advanced Mathematics 55 (1997 [Ca73] W. Casselman: On some results of Atkin and Lehner, Math. Ann. 21 (1973, pp [Ge75] S. Gelbart: Automorphic forms on adele groups, Annals of math. studies no. 83, Princeton University Press (1975 [Go7] R. Godement: Notes on Jacquet-Langlands theory, Inst. for Advanced Study, Princeton (197 [Gr1] B.H. Gross: Heegner points and representation theory, MSRI 21, preprint [GP91] B.H. Gross, D. Prasad: Test vectors for linear forms, Math. Ann. 291 (1991, pp [JL7] H. Jacquet, R. Langlands: Automorphic forms on GL 2, Lect. Notes in Math. 114, Springer- Verlag (197 [Po4] A. Popa: Central values of Rankin L-series over real quadratic fields, preprint, 24 [Zh1] S. W. Zhang: Gross-Zagier formula for GL 2, Assian J. Math 5 (21,