Rankin-Selberg L-functions.

Transcription

1

2 Chapter 11 Rankin-Selberg L-functions...I had not recognized in 1966, when I discovered after many months of unsuccessful search a promising definition of automorphic L-function, what a fortunate, although, and this needs to be stressed, unforeseen by me, or for that matter anyone else, blessing it was that it lay in the theory of Eisenstein series. R. P. Langlands Abstract In this chapter we sketch the theory of generic representations and Rankin-Selberg L-functions Paths to the construction of automorphic L-functions Let F be a global field and let π and π be a pair of cuspidal automorphic representations of A GLn \GL n (A F ) and A GLm \GL m (A F ), respectively. An important analytic invariant of this pair is the Rankin-Selberg L-function L(s, π π ). This is a meromorphic function on the complex plane satisfying a functional equation whose poles can be explicitly described in terms of π and π. These L-functions play a crucial role in automorphic representation theory. On the one hand they are used in the statement of the local Langlands correspondence for GL n (and hence, implicitly, for the global Langlands correspondence). This will be discussed in

3 Rankin-Selberg L-functions In this chapter we describe one method by which they can be defined, namely the Rankin-Selberg method. This requires the introduction of the notion of a generic representation. It turns out that all cuspidal representations of GL n (A F ) are generic, but cuspidal representations of other groups need not be generic. A much more detailed treatment of the Rankin-Selberg method is contained in [Cog07], which was our primary reference in the preparation of this chapter. There are other approaches to defining these L-functions. One is via the so-called Langlands-Shahidi method. This method also uses the notion of a generic representation. The idea is that Rankin-Selberg L-functions (and some more general L-functions) occur in the constant term of certain Eisenstein series. This approach was historically important because it suggested both the general definition of a Langlands L-function, discussed in 12.7, and the general formulation of Langlands functoriality. We do not discuss this construction. It is discussed at length in the book [Sha10]. In the special case where π is the trivial representation there is an alternate approach to defining these L-functions due to Godement and Jacquet [GJ72]. It is a direct generalization of Tate s construction of the L-functions of Hecke characters in his famous thesis [Tat67], which in turn is an adelic formulation of the classical work of Hecke Generic characters Let G be a quasi-split group over a global or local field F, let B G be a Borel subgroup, let T B a maximal torus and let N B the unipotent radical of B. We choose a pinning (B, T, {X α } α ) as in Definition 7.2. Let E/F be a Galois extension over which B and T split. Then since T (E) is preserved by Gal(E/F ) the root spaces X α are permuted by Gal(E/F ) and we can choose a set of representatives {β} for the Gal(E/F ) orbits; let O(β) denote the orbit of β. For E-algebras R the exponential map yields a group homomorphism exp : G a (R) G E (R) α O(β) (x α ) α O(β) exp(x α X α ). Its image is a unipotent subgroup whose Lie algebra is spanned by {X α } α O(β). This unipotent subgroup is in fact stable under Gal(E/F ), and hence is the E points of a unipotent subgroup N(β) N that is isomorphic to Res E/F G a.

4 11.3 Generic representations 229 Varying β, we see that we have a subgroup N(β) N, β where the product is over all Gal(E/F )-orbits of simple roots α. We note that inclusion induces an isomorphism N(β) N/N der. β Definition If F is a local field, a character ψ : N(F ) C is called generic if ψ N(β)(F ) is nontrivial for each Gal(E/F )-orbit β. If F is global a character ψ : N(A F ) C trivial on N(F ) is called generic if ψ v is generic for all places v. Example Take G = GL n and let B n GL n be the Borel subgroup of upper triangular matrices. Moreover let N n B n be the unipotent radical. If F is local any character of N n (F ) is of the form N n (F ) C 1 x 12 1 x ψ(m 1 x m n 1 x (n 1)n ) 1 x (n 1)n 1 (11.1) for some m 1,..., m n 1 F, where ψ : F C is a nontrivial character. The character is generic if and only if all of the m i are nonzero. If F is global any character of N n (A F ) trivial on N n (F ) is of the same form, but now the m i must be in F and ψ must be trivial on F. Again, the character is generic if and only if n 1 i=1 m i 0. If all m i = 1 we call this the standard character attached to ψ Generic representations Let G be a quasi-split connected reductive group over a local field F and let ψ : N(F ) C be a character. Let (π, V ) be an admissible representation of G(F ). Definition Assume F is nonarchimedean. A ψ-whittaker functional on V is a continuous linear functional λ : V C such that λ(π(n)ϕ) = ψ(n)λ(ϕ)

5 Rankin-Selberg L-functions for ϕ V and n N(F ). In this nonarchimedean setting, continuous means locally constant. If F is archimedean, we assume that V is a Hilbert space and even that π is unitary. Then for every X U(g) we can define a seminorm X on V sm via ϕ X := π(x)ϕ 2. We can give V sm a locally convex topology via these seminorms, and it then makes sense to speak of continuous linear functionals. It is this notion of continuity we use in the following definition: Definition Assume F is archimedean. A ψ-whittaker functional on V is a continuous linear functional λ : V C such that for ϕ V sm and n N(F ). λ(π(n)ϕ) = ψ(n)λ(ϕ) Assume, as above, that in the archimedean setting (π, V ) is unitary. Assume in addition that ψ is generic. Definition An irreducible admissible representation (π, V ) of G(F ) is ψ-generic if it has a nonzero ψ-whittaker functional on V. The following theorem of Shalika [Sha74] has turned out to be extremely important: Theorem (Shalika) The space of ψ-whittaker functionals on an irreducible admissible representation of G(F ) is at most 1-dimensional. We now define the related notion of a Whittaker function. The space of Whittaker functions on G(F ) is the space of smooth functions W(ψ) := {smooth W : G(F ) C satisfying W (ng) = ψ(n)w (g)}. (11.2) This space admits an action of G(F ): G(F ) W(ψ) W(ψ) (g, W ) (x W (xg)). Let (π, V ) be an irreducible admissible representation π of G(F ). A Whittaker model of π is a nontrivial G(F )-intertwining map V sm W(ψ) (in the nonarchimedian case, V = V sm by assumption). As a corollary of Theorem we have the following (see Exercise 11.2):

6 11.3 Generic representations 231 Corollary A Whittaker model of π exists if and only if π is generic. If π is generic, then it admits a unique Whittaker model. Motivated by Corollary if (π, V ) is generic we let W(π, ψ) (11.3) be the image of any intertwining map V sm explicitly as W(ψ). We write the map V sm W(π, ψ) ϕ W ϕ ψ. By Schur s lemma this map is well-defined up to multiplication by a nonzero complex number. Now assume that F is global, let π be a cuspidal automorphic representation of G(A F ) realized in a subspace V L 2 cusp([g]), and let ϕ V sm. We then define the global ψ-whittaker function W ϕ ψ (g) := ϕ(ng)ψ(n)dn. (11.4) [N] We say that (π, V ) is globally ψ-generic if W ϕ ψ (g) 0 for some ϕ V and g G(A F ) 1. We remark that a priori this notion depends on the realization of π in L 2 cusp([g]) if it does not occur with multiplicity 1. It is not hard to check that if π is globally ψ-generic and π = vπ v then each π v is ψ v -generic (Exercise 11.3). The notion of a generic representation is only useful if we have an interesting supply of generic representations. The following theorem implies that all cuspidal automorphic representations of GL n (A F ) are generic, and moreover they admit an expansion in terms of Whittaker functionals: Theorem Let ϕ L 2 cusp([gl n ]) be a smooth vector in the space of a cuspidal automorphic representation π of A GLn \GL n (A F ). If ψ : N n (A F ) C is a generic character then one has ϕ(g) = γ N n 1(F )\GL n 1(F ) W ϕ ψ (( γ 1 ) g). (11.5) The expression for ϕ in (11.5) is called its Whittaker expansion. It is a generalization (but not the only generalization) of the well-known Fourier expansion of a modular form. We now prove this theorem in the special case n = 2. Consider the function

7 Rankin-Selberg L-functions x ϕ (( 1 x 1 ) g). (11.6) This is a continuous function on the compact abelian group F \A F, and hence admits a Fourier expansion. If we fix a nontrivial character ψ : F \A F C, then all other characters are of the form ψ α (x) := ψ(αx) for α F (see Lemma B.1.2). Thus the Fourier expansion of the function (11.6) is ϕ (( 1 x 1 ) g) ψ(αyx)dy. (11.7) F \A F α F Since ϕ is cuspidal, the α = 0 term vanishes identically. Taking a change of variables x α 1 x for each α F and using the left GL 2 (F )-invariance of ϕ we see that (11.6) is equal to α F W ϕ ψ (( α 1 ) ( 1 x 1 ) g). (11.8) Setting x = 0 we obtain Theorem when n = 2. We will not prove Theorem because we cannot improve on the exposition in [Cog07, 1.1]. The basic idea is to proceed as in the n = 2 case, using abelian Fourier analysis on certain abelian subgroup of N n (A F ). However, since N n (A F ) is no longer abelian for n > 3 one has to combine this with a clever inductive argument using the fact that ϕ is cuspidal. Using Theorem , we obtain the following related theorem of Shalika [Sha74, Theorem 5.5]: Theorem (Multiplicity one for GL n ) An irreducible admissible representation of GL n (A F ) occurs with multiplicity at most one in L 2 cusp([gl n ]). We note that Theorem is false for essentially every reductive group that is not a general linear group. Proof. Let (π 1, V 1 ), (π 2, V 2 ) be two realizations of a given cuspidal automorphic representation (π, V ). Choose equivariant maps L i : V V i. We then obtain Whittaker functionals λ i : V sm C ϕ W Li(ϕ) ψ (I n ), where I n is the identity element. They are nonzero by Theorem By Theorem one therefore has λ 1 = cλ 2 for some c C. Thus L 1 (ϕ)(g) = W L1(ϕ) ψ (( γ 1 ) g) = c γ N n 1(F )\GL n 1(F ) γ N n 1(F )\GL n 1(F ) W L2(ϕ) ψ (( γ 1 ) g)

8 11.4 Formulae for Whittaker functions 233 = L 2 (ϕ)(g). This implies in particular that V 1 and V 2 have nonzero intersection, and hence are equal Formulae for Whittaker functions Let (π, V ) be a globally generic cuspidal representation of the quasi-split group G, realized on a closed subspace V L 2 ([G]). Using Flath s theorem we can choose an isomorphism V sm v V v v W(π v, ψ v ). (11.9) Let ϕ V sm be a pure tensor, i.e. a vector that maps to a vector of the form vϕ v under the first isomorphism. Then, by local uniqueness of Whittaker models upon normalizing the composite isomorphism V sm by multiplying by a suitable nonzero complex number we have W ϕ ψ (g) = v W ϕv ψ v (g). (11.10) Thus to compute the global Whittaker function W ϕ ψ (g) it suffices to compute the local Whittaker functions W ϕv ψ v. When π v and ψ v are unramified this can be accomplished using famous Casselman-Shalika formula [CS80]. For the remainder of this section let F be a nonarchimedean local field and let ψ : F C be a nontrivial unramified character. The Casselman-Shalika formula is valid for any unramified reductive group over F, but to simplify our discussion we will assume that G is a split group. Let B G be the Borel subgroup with split maximal torus T B. We let K G(F ) be a hyperspecial subgroup in good position with respect to (B, T ). We note that by the Iwasawa decomposition one has G(F ) = N(F )µ(ϖ)k (11.11) µ X (T ) where N B is the unipotent radical of B. Using the notation of 7.6, for λ a T C we can then form the induced representation I(λ), where the induction is with respect to the Borel subgroup B. Its unique unramified subquotient is denoted by J(λ). Any unramified representation is equivalent to such a J(λ). The choice of T and B, by duality, give rise to a maximal torus and Borel subgroup T B Ĝ as explained in 7.3. This gives rise to a notion of dominant weight of T in B, say µ X ( T ) = X (T ). By Cartan-Weyl theory, there is a unique isomorphism class V (µ) of irreducible representation of Ĝ attached to µ. Let χ µ be its character.

9 Rankin-Selberg L-functions Theorem (Casselman and Shalika) Any function W in the onedimensional space W(J(λ), ψ) K satisfies W (I) C, I is the identity element. If µ is a dominant weight then W (µ(ϖ)) W (I) = δ 1/2 B (µ(ϖ))χ µ(e λ ). If µ is not a dominant weight then W (µ(ϖ)) = 0. Here one recalls that e λ is discussed in Lemma This is proven in [CS80]; to translate the formula in loc. cit. to the formula above one uses the Weyl character formula. It is instructive to write this formula down more explicitly when G = GL n. In this case it is due to Shintani [Shi76]. We assume that B = B n is the Borel subgroup of upper triangular matrices and T = T n B n is the maximal torus of diagonal matrices. In this case weights can be identified with tuples (k 1,..., k n ) Z n as in (1.9). The cocharacter µ attached to such a tuple is recorded in (1.9). The dominant weights are those with k 1 k n. The associated irreducible representation of GL n is S k1,...,k n (V st ), where V st is the standard representation and S k1,...,k n is the Schur functor attached to the partition k 1,..., k n of n i=1 k n. Let χ k1,...,k n be its character. Corollary (Shintani) If G = GL n and W W(J(λ), ψ) GLn(O F ) is the unique vector satisfying W (I n ) = 1 then ( ) { ϖ k 1 δ 1/2 W. B.. = n (µ(ϖ))χ k1,...,k n (e λ ) if k 1 k n 0 otherwise. ϖ kn 11.5 Local Rankin-Selberg L-functions Let F be a local field and let ψ : F C be a nontrivial character. Our goal in this section is to define local Rankin-Selberg L-functions for generic representations, at least when F is nonarchimedean. The fundamental idea here is that these L-functions are the smallest rational functions in q s that cancel the poles of a family of zeta functions. The definition of this family was originally given in [JPSS83], and the archimedian case was treated definitively in [Jac09]. We refer to these papers for proofs of the unproved statements we make below. Let π, π be irreducible admissible generic representations of GL n (F ) and GL m (F ), respectively. Let ψ : F C be an additive character. Use it to

10 11.5 Local Rankin-Selberg L-functions 235 define a standard generic character of N n (F ) and N m (F ) as in Example We denote these characters again by ψ. Let W W(π, ψ), W W(π, ψ). If m < n let Ψ(s; W, W ) := W ( ) h In m W (h) det(h) s (n m)/2 dh, ( Ψ(s; W, W hx ) ) := W In m 1 dxw (h) det h s (n m)/2 dh, 1 (11.12) where the top integral is over N m (F )\GL m (F ), the outer integral on the bottom is over N m (F )\GL m (F ), and the inner integral on the bottom is over (n m 1) m matrices with entries in F. If m = n, then for each Φ S(F n ) (the Schwartz space) let Ψ(s; W, W, Φ) := W (g)w (g)φ(e n g) det g s dg. (11.13) N n(f )\GL n(f ) Here e n F n is the elementary vector with 0 s in the first n 1 entries and 1 in the last entry. Proposition Assume that F is nonarchimedean. (a) The integrals (11.12) and (11.13) converge for Re(s) sufficiently large. For π and π unitary they converge absolutely for Re(s) 1. For π and π tempered, they converge absolutely for Re(s) > 0. (b) Each integral is a rational function of q s. (c) The C-linear span of the integrals in C[q s, q s ] as W, W (and when m = n the Schwartz function Φ) is a principal ideal I(π, π ). In this nonarchimedean case, one proves that there is a unique polynomial P π,π C[x] satisfying P π,π (0) = 1 such that P π,π (q s ) 1 is a generator of I(π, π ). One sets L(s, π π ) := P π,π (q s ) 1. (11.14) In the archimedean case one actually defines L(s, π π ) using the local Langlands correspondence, which was established very early on in the theory by Langlands himself. We will explain this in more detail in 12.3 below. Right now we note the following important bound: Theorem If π and π are unitary then L(s, π π ) is holomorphic and nozero for Re(s) 1. See [BR94, 2] for a proof. Technically speaking they only state that the L- function is holomorphic at this point, but the argument actually proves that it is nonzero as well.

11 Rankin-Selberg L-functions To discuss the archimedean case we say that a holomorphic function f(s) of s C is bounded in vertical strips if it is bounded in V σ1,σ 2 := {s : σ 1 Re(s) σ 2 } for all σ 1 < σ 2 R. A meromorphic function f(s) is bounded in vertical strips if for each σ 1 < σ 2 there is a polynomial P σ1,σ 2 C[x] such that P σ1,σ 2 (s)f(s) is holomorphic and bounded in vertical strips in the original sense. Theorem Assume F is archimedean. The local Rankin-Selberg integrals (11.12) and (11.13) converge absolutely for Re(s) sufficiently large and admit meromorphic continuations to functions of s that are bounded in vertical strips. Let ( Im wn m) ( w m,n :=, w n := ) GL n (Z) and for functions f on GL n (F ) or GL n (A F ) let ρ(w m,n )f(x) := f(xw m,n ) and f(g) := f(w n g t ). Theorem There is a meromorphic function γ(s, π π, ψ) C(q s ) in the nonarchimedean case such that if m < n we have Ψ(1 s, ρ(w n,m ) W, W ) = ω ( 1) n 1 γ(s, π π, ψ)ψ(s, W, W ) and if m = n we have Ψ(1 s, W, W, Φ) = ω ( 1) n 1 γ(s, π π, ψ)ψ(s; W, W, Φ) for all W W(π, ψ), W W(π, ψ). Here ω is the central quasi-character of π. We warn the reader that the functional equations in [Jac09] are different. The root of this is that the definition of the contragredient W of W in loc. cit. is different. We need one other factor, the local ε-factor: ε(s, π π, ψ) := γ(s, π π, ψ)l(s, π π ) L(1 s, π π. (11.15) ) Proposition In the nonarchimedean case ε(s, π π ) is a function of the form cq fs for some real number f and c C. The ε-factor is a useful arithmetic invariant of the pair (π, π ). Assume that π is trivial. In this case the ideal ϖ f is called the conductor of π. One

12 11.6 The unramified Rankin-Selberg L-functions 237 writes f := f(π), and calls it the exponent of the conductor, or sometimes just the conductor. Let. K 1 (ϖ f(π) ) := g GL n (O F ) : g. (mod ϖf(π) ) Then one has the following (see [Mat13]): Theorem The number f(π) is the smallest nonnegative integer such that the space V of π satisfies V K1(ϖf(π)) 0. Moreover, dim C V K1(ϖf(π)) = The unramified Rankin-Selberg L-functions In this section we assume that F is nonarchimedean and that π and π are irreducible unramified representations of GL n (F ) and GL m (F ), respectively. Assume that ψ : F C is unramified and let W W(π, ψ) and W W(π, ψ) be the unique spherical vectors satisfying W (I n ) = W (I m ) = 1. We assume that π = J(λ) and π = J(λ ), where λ a T nc and λ a T mc. Our aim in this section is to prove the following identity: Theorem For Re(s) sufficiently large if n > m one has If m = n one has ( Ψ(s, W, W, ψ) = det I mn q s e λ e λ ) 1. ( Ψ(s, W, W, 1 O n F ) = det I n 2 q s e λ e λ ) 1. Implicit in the definition of the local Rankin-Selberg integrals is a choice of measure on GL m (F ) and N m (F ) so that we can form a quotient measure on N m (F )\GL m (F ). In order for the equalities above to be valid we must choose the unique Haar measures that assign volume 1 to GL m (O F ) and N m (O F ), respectively. Proof. For each m Z 1 let T + (m) be the set of the tuples µ = (k 1,, k m ) Z m with k 1 k m 0. This can be identified with a subset of the dominant weights of T m or a cocharacter of T m. If n > m let

13 Rankin-Selberg L-functions T + (m) T + (n) be the injection given by extending the tuple k 1,..., k m by zeros. Assume for the moment that n > m. Then by Corollary we have Ψ(s; W, W ) = µ T + (m) = µ T + (m) ( µ(ϖ) W W In m) (µ(ϖ)) det(µ(ϖ)) s (n m)/2 δ 1 B m (µ(ϖ)) χ k1,...,k m (e λ )χ k1,...,k m,0,...,0(e λ )q µ s = det(i mn q s e λ e λ ) 1, where we let µ = k 1 + +k m and, as before, χ k1,...,k m,0,...,0 denotes the character associated to S k1,...,k m,0,...,0. This last identity is known as the Cauchy identity [Bum13, Chapter 38]. Similarly, when n = m we have Ψ(s; W, W, 1 O n F ) = W ( µ(ϖ) ) W (µ(ϖ)) 1 O n F (e n µ(ϖ)) det(µ(ϖ)) s δ 1 B n (µ(ϖ)) µ T + (n) = µ T + (n) χ k1,...,k n (e λ )χ k1,...,kn(e λ )q µ s = det(i n 2 q s e λ e λ ) 1. The calculation above implies that det(i mn q s e λ e λ ) divides L(s, π π ) 1 (as a polynomial in q s )). More is true (see [JPSS83]): Theorem One has L(s, π π ) = det(i mn q s e λ e λ ) Global Rankin-Selberg L-functions Let π and π be cuspidal automorphic representations of A GLn \GL n (A F ) and A GLm \GL m (A F ), respectively. We have previously defined the local Rankin-Selberg L-functions L(s, π v π v). The global Rankin-Selberg L-function is the product L(s, π π ) := v L(s, π v π v). (11.16) If ψ : F \A F C is a nontrivial character we also define

14 11.7 Global Rankin-Selberg L-functions 239 ε(s, π π ) := v ε(s, π v π v, ψ v ). The reason that ψ is not encoded in to the left hand side of this equation is that the right hand side is in fact independent of the choice of ψ. The following theorem collects the basic facts about Rankin-Selberg L- functions: Theorem The Rankin-Selberg L-function admits a meromorphic continuation to the plane, holomorphic except for possible simple poles at s = 0, 1. There are poles at s = 0, 1 if and only if m = n and π = π. One has a functional equation L(s, π π ) = ε(s, π, π )L(1 s, π π ). (11.17) One extremely important consequence of this theorem is the strong multiplicity one theorem: Theorem (Strong multiplicity one) Let π and π be cuspidal automorphic representations of GL n (A F ) and let S be a finite set of places of F. If π S = π S then π = π. We leave the proof as Exercise It is a consequence of Proposition , Theorem and Theorem Despite its name Theorem does not imply the Theorem above. For a beautiful generalization of the strong multiplicity one theorem we refer the reader to the work of Dinakar Ramakrishnan on automorphic analogues of the Chebatarev density theorem. Let ϕ be in the space of π and ϕ in the space in π. We outline the basic idea behind this theorem in the case m = n 1. We will not justify any of the convergence statements made in our outline. For full details we refer the reader to [JPSS83]. In this case we take I(s; ϕ, ϕ ) := ϕ ( h 1 ) ϕ (h) det h s 1/2 dh. (11.18) GL n 1(F )\GL n 1(A F ) Let ϕ(g) := ϕ(g t ) and ϕ (g) := ϕ(g t ). Taking a change of variables h h t we see that I(s; ϕ, ϕ ) = I(1 s, ϕ, ϕ ). (11.19) This simple change of variables is what ultimately powers the proof of theorem Define Ψ(s; W ϕ ψ, W ϕ ψ ) = W ϕ ψ N n 1(A F )\GL n 1(A F ) This is a global Rankin-Selberg integral. ( ( h 1 ) ) W ϕ ψ (h) det h s 1/2 dh.

15 Rankin-Selberg L-functions Theorem For Re(s) sufficiently large I(s; ϕ, ϕ ) = Ψ(s, W ϕ ψ, W ϕ ψ ). In particular, the functional equation (11.19) immediately implies an analogous one for the global Rankin-Selberg integral Ψ(s, W ϕ ψ, W ϕ ψ ). Proof. Replacing ϕ by its Whittaker expansion we have I(s; ϕ, ϕ ) = ϕ ( h 1 ) ϕ (h) det h s 1/2 dh = GL n 1(F )\GL n 1(A F ) GL n 1(F )\GL n 1(A F ) γ W ϕ ψ ( ( γ 1 ) ( h 1 ) ) ϕ (h) det h s 1/2 dh, where the inner sum is over γ N n 1 (F )\GL n 1 (F ). Since ϕ (h) is automorphic on GL n 1 (A F ) and det(γ) = 1 for γ GL n 1 (F ), we may interchange the order of summation and integration and obtain I(s; ϕ, ϕ ) = W ϕ ( ψ ( h 1 ) ) ϕ (h) det h s 1/2 dh. N n 1(F )\GL n 1(A F ) This integral is absolutely convergent for Re(s) 0 which justifies the interchange. Let us first integrate over [N n 1 ]. View N n 1 N n as matrices of the form ( u 1 ). Then for u N n 1 (A F ) one has W ϕ ψ (ug) = ψ(u)w ϕ ψ (g). I(s; ϕ, ϕ ) = = = N n 1(A F )\GL n 1(A F ) W ϕ ψ N n 1(A F )\GL n 1(A F ) W ϕ ψ N n 1(A F )\GL n 1(A F ) = Ψ(s; W ϕ ψ, W ϕ ψ ). [N n 1] W ϕ ψ ( ( u 1 ) ( h 1 ) ) ϕ (uh)du det h s 1/2 dh ( ) h 1 ψ(u))ϕ (uh)du det h s 1/2 dh [Nn 1] ( h 1 ) W ϕ ψ (h) det h s 1/2 dh In the case m = n 1 above Theorem can be deduced from theorems and ; we leave this as Exercise The nongeneric case In the theory above crucial use was made of the fact that cuspidal automorphic representations are globally generic. Indeed, our definition of local L-functions was given in terms of the Whittaker model.

16 11.9 Converse theory 241 We now explain how to handle the general case. Assume first that F is a local field. From 10.6 we know that every pair of admissible irreducible representation π of GL n (F ) and π of GL m (F ) can be written as an isobaric sum π = k i=1π i and π = k j=1π i where the π i and π i are essentially square integrable and hence generic [JS83, 1.2]. We then define, for nontrivial characters ψ : F C, that L(s, π π ) = γ(s, π π, ψ) = ε(s, π π, ψ) = k k L(s, π i π j), i=1 j=1 k k γ(s, π i π j, ψ), i=1 j=1 k k ε(s, π i π j, ψ). i=1 j=1 Of course, one has to check that this is consistent with our earlier definitions. In other words, we must know that if π and π are generic, then this procedure gives the same result as if we used our original definition of the local factors. This can be done. We then adopt the analogous convention in the global case and obtain Rankin-Selberg integrals for isobaric automorphic representations. The analytic properties of these L-functions can be read off from the corresponding ones for cuspidal representations (see Theorem ) Converse theory Let π be an admissible irreducible representation of GL n (A F ). If π is a cuspidal automorphic representation then for all cuspidal automorphic representations π of GL m (A F ) we have seen in Theorem that the L-function L(s, π π ) admits a functional equation and a meromorphic continuation to the plane that is bounded in vertical strips. Converse theory asks if automorphic representations can be characterized by the analytic properties of their L-functions. It has been used to establish cases of Langlands functoriality [CKPSS04]. In addition, it is philosophically important because it says that an L-function that satisfies reasonable assumptions has to come from an automorphic representation, solidifying the tight connection between these two objects.

17 Rankin-Selberg L-functions Theorem Let π be an irreducible admissible representation of GL n (A F ). Assume that for all cuspidal automorphic representations π of GL m with 1 m max(n 2, 1) the L-functions L(s, π π ) and L(s, π π ) have analytic continuations to the plane that are holomorphic and bounded in vertical strips of finite width. Assume moreover that for all π as above L(s, π π ) = ε(s, π π )L(1 s, π π ). Then π is a cuspidal automorphic representation. Here to define ε(s, π π ) we fix a nontrivial character ψ : F \A F C and formally define ε(s, π π ) := v ε(s, π v π v, ψ v ). (11.20) We refer to [CPS99] for a proof of Theorem and for variants. Exercises Let B 1, B 2 GL n be Borel subgroups defined over a nonarchimedean local field F. For 1 i 2 let N i be the unipotent radical of B i and let ψ i be a generic character of N i (F ). Prove that an irreducible admissible representation π of GL n (F ) is ψ 1 -generic if and only if it is ψ 2 -generic Prove Corollary Let G be a split reductive group over a global field F and assume that π is a cuspidal globally ψ-generic representation of V. If π = vπ v prove that each π v is ψ v -generic Prove Theorem when n = Assume that J(λ) is irreducible unitary unramified generic representation of GL n (F ) where λ a T nc. Let ψ : F C be unramified, let W W(π, ψ) and W W(π, ψ) be the unique spherical Whittaker functionals satisfying W (I n ) = W (I n ) = 1. Using the identity det(i2 q s e λ e λ )Ψ(s, W, W, 1 O n F ) = 1 and Proposition prove that every eigenvalue of e λ lies in (q 1/2, q 1/2 ) If π and π are cuspidal automorphic representations of GL n (A F ) such that π S = π S for some finite set of places S of F then π = π Prove Theorem in the case m = n 1 using Theorem and the local functional equations of Theorem

18 References [BR94] Laure Barthel and Dinakar Ramakrishnan. A nonvanishing result for twists of L-functions of GL(n). Duke Math. J., 74(3): , [Bum13] Daniel Bump. Lie groups, volume 225 of Graduate Texts in Mathematics. Springer, New York, second edition, [CKPSS04] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi. Functoriality for the classical groups. Publ. Math. Inst. Hautes Études Sci., (99): , [Cog07] James W. Cogdell. L-functions and converse theorems for GL n. In Automorphic forms and applications, volume 12 of IAS/Park City Math. Ser., pages Amer. Math. Soc., Providence, RI, , 232 [CPS99] J. W. Cogdell and I. I. Piatetski-Shapiro. Converse theorems for GL n. II. J. Reine Angew. Math., 507: , [CS80] W. Casselman and J. Shalika. The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math., 41(2): , , 234 [GJ72] Roger Godement and Hervé Jacquet. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin-New York, [Jac09] Hervé Jacquet. Archimedean Rankin-Selberg integrals. In Automorphic forms and L-functions II. Local aspects, volume 489 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, , 236 [JPSS83] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika. Rankin- Selberg convolutions. Amer. J. Math., 105(2): , , 238, 239 [JS83] Hervé Jacquet and Joseph Shalika. The Whittaker models of induced representations. Pacific J. Math., 109(1): ,

19 244 References [Mat13] Nadir Matringe. Essential Whittaker functions for GL(n). Doc. Math., 18: , [Sha74] J. A. Shalika. The multiplicity one theorem for GL n. Ann. of Math. (2), 100: , , 232 [Sha10] Freydoon Shahidi. Eisenstein series and automorphic L- functions, volume 58 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, [Shi76] Takuro Shintani. On an explicit formula for class-1 Whittaker functions on GL n over P -adic fields. Proc. Japan Acad., 52(4): , [Tat67] J. T. Tate. Fourier analysis in number fields, and Hecke s zetafunctions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages Thompson, Washington, D.C.,

20