On the rationality and holomorphy of Langlands-Shahidi L-functions over function fields
aa r X i v : . [ m a t h . N T ] N ov ON THE RATIONALITY AND HOLOMORPHYOF LANGLANDS-SHAHIDI L -FUNCTIONSOVER FUNCTION FIELDS LUIS ALBERTO LOMEL´I
Abstract.
We prove three main results: all Langlands-Shahidi automorphic L -functions over function fields are rational; after twists by highly ramifiedcharacters our automorphic L -functions become polynomials; and, if π is aglobally generic cuspidal automorphic representation of a split classical groupor a unitary group G n and τ a cuspidal (unitary) automorphic representationof a general linear group, then L ( s, π × τ ) is holomorphic for ℜ ( s ) > s = 1. We also prove the holomorphy andnon-vanishing of automorphic exterior square, symmetric square and Asai L-functions for ℜ ( s ) >
1. Finally, we complete previous results on functorialityfor the classical groups over function fields. introduction
The Langlands-Shahidi method provides a link between the theory of Eisen-stein series and automorphic L -functions for globally generic representations. Themethod was first developed by Shahidi over number fields [36] and is now available inthe case of function fields [31]. We are guided by the analogy that exists in modernNumber Theory between number fields and global function fields. Already presentin Langlands’ work [24], Eisenstein series over number fields have a meromorphiccontinuation. Over function fields, a result of Harder proves that Eisenstein seriesare rational [11]. Based on this, we prove that all Langlands-Shahidi automorphic L -functions over function fields are rational (Theorem 1.2).The connection between Eisenstein series and the Langlands-Shahidi local co-efficient is possible via the Whittaker model of a globally generic representation.Locally, at every unramified place, we have the Casselman-Shalika formula [7]. Atramified places, we find it useful to show in § p -adic groups.In § L -functions and intertwining operators. Weprove a local irreducibility result for induced representations and recall an assump-tion made by Kim concerning normalized intertwining operators. We then, in § L -functions become holomorphic after glob-ally twisting by a highly ramified character. We follow Kim-Shahidi and refine this Mathematics Subject Classification . Primary 11F70, 22E50, 22E55. result in positive characteristic to show that our automorphic L -functions becomeLaurent polynomials (Proposition 4.1).In § p , and viceversa. For example, we make note that Langlands-Shahidi local L -functions, γ -factors and root numbers are respected by the transferfor all split groups. Additionally, we prove in characteristic p , a result that Lapid,Mui´c and Tadi´c proved in characteristic zero concerning the generic unitary dualof classical groups [26].We then focus, starting in §
6, on G n either a split classical group or a quasi-split unitary group. First, we look at the Siegel Levi subgroup M of G n [30]. Inthis case, we show the holomorphy and non-vanishing of exterior square, symmetricsquare and Asai L -functions over function fields for ℜ ( s ) > L -functions for the classical groups and is studied“ab initio” in [30] via the Langlands-Shahidi method in positive characteristic.Furhter information on the location of the poles of these L -functions is knownover number fields, and it might be worhwhile to look into these questions overfunction fields. For example, the approach of Bump and Ginzburg over numberfields [4], continued by Takeda in the twisted case [39], gives precise informationabout the holomorphy of L S ( s, π, Sym ), having possible poles only at s = 1. Usingthe Langlands-Shahidi method, we explore twisted exterior and symmetric square L -functions in [9]. There, we have a Kazhdan transfer from close local fields ofcharacteristic zero to characteristic p .Now, let π be a globally generic cuspidal automorphic representation of a quasi-split classical group G n and let τ a cuspidal (unitary) automorphic representationof GL m (Res GL m in the case G n is a unitary group). Over function fields we canestablish and refine a number fields result of [17, 22] in these cases. More precisely,we prove that L ( s, π × τ ) is holomorphic for ℜ ( s ) > s = 1 (Theorem 6.5).In §
7, we go back and show that our results on functoriality for the split classicalgroups of [28] are valid for any global function field k . Previously, we had madethe assumption char( k ) = 2. In particular, Theorem 9.14 of [ loc. cit. ] (Ramanu-jan Conjecture) and Theorem 4.4(iii) of [29] (Riemann Hypothesis) are now validwithout this restriction. Acknowledgments.
I would like to thank G¨unter Harder for enlightening conver-sations that we held during the time the article was written. I thank Guy Henniartand Freydoon Shahidi for their encouragement to work on this project. I wouldlike to thank W. Casselman for useful discussions concerning the rationality of theLanglands-Shahidi local coefficient. I also thank R. Ganapathy, V. Heiermann, D.Prasad and S. Varma for mathematical communications. I am indebted to theMathematical Sciences Research Institute and the Max-Planck Institute f¨ur Math-ematik for providing excellent working conditions while a Postdoctoral Fellow. Iam grateful to the Instituto de Matem´aticas PUCV, where I have found a greathome institution. Work on this article was supported in part by MSRI under itsNSF Grant, DMS 0932078 and Project VRIEA/PUCV 039.367/2016.
N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 3 Eisenstein series and L -functions over function fields Notation.
Let k be a global function field with field of constants F q and ringof ad`eles A k . At every place v of k , we denote by O v the corresponding ring ofintegers. Similarly, we let p v be the maximal prime ideal with uniformizer ̟ v and q v denotes the cardinality of O v / p v .Let G be a connected quasi-split reductive group defined over k . Fix a borelsubgroup B = TU with maximal torus T and unipotent radical U . Let ∆ denotethe simple roots. We consider only standard parabolic subgroups. There is acorrespondence between subsets θ ⊂ ∆ and standard parabolic subgroup P θ = M θ N θ of G . Recall that if A θ is ( ∩ α ∈ θ ker( α )) , then M θ is the centralizer of A θ in G and N θ is the unipotent radical of P θ . We often drop the subscripts and write P = MN . Given a place v of k and a reductive group H over k , we let H v denotethe group of k v -rational points.Let X ∗ ( M ) k be the set of k -rational characters of M . For every place v of k ,we have the k v -rational characters X ∗ ( M ) v . We let X nr ( M ) k denote the set ofunramified characters of M ( k ). And, similarly for X nr ( M ) v . Let a ∗ M , C = X ∗ ( M ) k ⊗ C . Let ρ P denote half the sum of the positive roots of P . The relation q h s ⊗ χ,H P ( · ) i = | χ ( · ) | sk leads to a surjection(1.1) a ∗ M v , C ։ X nr ( M ) v . at every place v of k . Fix a maximal compact open subgroup K = Q v K v of G ( A k ),as in § I.1.4 of [32]. Then G ( A k ) = P ( A k ) K . For each v of k , H P v ( · ) is trivial on M v ∩ K v . The character q h s ⊗ χ v ,H Pv ( · ) i ∈ X nr ( M ) v , extends to one of G v = G ( k v ).A maximal parabolic P corresponds to a set of the form θ = ∆ − { α } , i.e. P = P θ , for some simple root α . In this case we fix a basis element of the space a ∗ M , C by setting(1.2) ˜ α = h ρ P , α ∨ i − ρ P , where α ∨ is the coroot corresponding to α .Let π = ⊗ ′ π v be cuspidal (unitary) automorphic representation of M ( A k ). Atevery place v of k , we have the induced representationI( s, π v ) = ind G v P v ( π v ⊗ q h s ˜ α,H Pv ( · ) i ) . Then, we globally have the restricted tensor productI( s, π ) = ⊗ ′ I( s, π v ) , with respect to functions f ◦ v,s ∈ I( s, π v ) that are fixed under the action of K v . LUIS ALBERTO LOMEL´I
Eisenstein series.
Let φ : M ( k ) \ M ( A k ) → C be an automorphic form onthe space of a cuspidal (unitary) automorphic representation π of M ( A k ). Then φ extends to an automorphic function Φ : M ( k ) U ( A k ) \ G ( A k ) → C , as in § I.2.17 of[32]. For every s ∈ C , set Φ s = Φ · q h s ˜ α + ρ P ,H P ( · ) i . The function Φ s is a member of the globally induced representation I( s, π ). Weuse the notation of § w = w l w l, M (see also § s, π, ˜ w ) : I( s, π ) → I( s ′ , π ′ ) , defined by M( s, π, ˜ w ) f ( g ) = Z N ′ f ( ˜ w − ng ) dn, for f ∈ I( s, π ). It decomposes into a product of local intertwining operatorsM( s, π, ˜ w ) = ⊗ v A( s, π v , ˜ w ) , which are precisely those appearing in the definition of the Langlands-Shahidi localcoefficient (see § § IV of [32] to the case at hand.
Theorem 1.1 (Harder) . The Eisenstein series E ( s, Φ , g, P ) = X γ ∈ P ( k ) \ G ( k ) Φ s ( γg ) converges absolutely for ℜ ( s ) ≫ and has a meromorphic continuation to a rationalfunction on q − s . Furthermore M( s, π ) = ⊗ v A( s, π v , ˜ w ) is a rational operator in the variable q − s . L -groups. Let Γ be the Galois group over the separable closure k s of k . Thefixed Borel corresponds to the simple roots ∆. This determines a based root datumΨ = ( X ∗ , ∆ , X ∗ , ∆ ∨ ). The dual root datum Ψ ∨ = ( X ∗ , ∆ ∨ , X ∗ , ∆) determinesthe Chevalley group L G ◦ over C . The L -group of G is the semidirect product L G = L G ◦ ⋊ Γ . The based root datum Ψ ∨ fixes a Borel subgroup L B , and we have all standardparabolic subgroups of the form L P = L P ◦ ⋊ Γ. The Levi subgroup of L P is of theform L M = L M ◦ ⋊ Γ, while the unipotent radical is given by L N = L N ◦ . We referto [3] for more details on the semi-direct structure of L G .1.4. Langlands-Shahidi L -functions. Fix a pair of quasi-split reductive groupschemes ( G , M ) such that M is a Levi component of a parabolic subgroup P = MN of G . Let r : L M → End( L n ) be the adjoint representation of L M on the Lie algebra L n of L N . It decomposes into irreducible constituents r = ⊕ m r i =1 r i . We study L -functions that arise from such r i . We refer to [31] for more details. N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 5 Locally, we let L loc ( p ) denote the category of triples ( F, π, ψ ) consisting of: F anon-archimedean local field of characteristic p ; π a generic representation of M ( F );and ψ : F → C × a continuous non-trivial character. We say ( F, π, ψ ) ∈ L loc ( p ) isunramified, if π has an Iwahori fixed vector.Globally, let L glob ( p ) be the category of quadruples ( k, π, ψ, S ) consisting of: k aglobal function field with field of constants F q ; π = ⊗ ′ π v a globally generic cuspidalautomorphic representation of M ( A k ); ψ : k \ A k → C × a non-trivial character; and, S a finite set of places of k such that π v and ψ v are unramified for v / ∈ S .Theorem 6.1 of [31] associates to each triple ( F, π, ψ ) ∈ L loc ( p ) the local factors: γ ( s, π, r i , ψ ) , L ( s, π, r i ) , and ε ( s, π, r i , ψ ) . Globally, we have partial L -functions L S ( s, π, r i ) = Y v / ∈ S L ( s, π v , r i,v ) , for ( k, π, ψ, S ) ∈ L glob ( p ). And, finally, completed global L -functions and ε -factors: L ( s, π, r i ) = Y v L ( s, π v , r i,v ) and ε ( s, π, r i ) = Y v ε ( s, π v , r i,v , ψ v ) . Globally generic representations and rationality.
In the Langlands-Shahidi method, the quadruples we consider ( k, π, ψ, S ) ∈ L glob ( p ) require that π = ⊗ ′ π v be globally generic. In particular, we can assume that π is globally ψ -generic. By definition, there is a cusp form ϕ in the space of π such that W M,ϕ ( m ) = Z U M ( K ) \ U M ( A k ) ϕ ( um ) ψ ( u ) du = 0 . In this case, the Fourier coefficient of the Eisenstein series E ( s, Φ , g ) is given by E ψ ( s, Φ , g, P ) = Z U ( K ) \ U ( A k ) E ( s, Φ , ug ) ψ ( u ) du. The Fourier coefficients of Eisenstein series are also rational functions on q − s [11].We use this to prove the following result. Theorem 1.2.
Let ( k, π, ψ, S ) ∈ L glob ( p ) . Then each L -function L ( s, π, r i ) con-verges absolutely for ℜ ( s ) ≫ and has a meromorphic continuation to a rationalfunction in q − s .Proof. The generic assumption on π , makes it possible to further decompose theFourier coefficients of Eisenstein. From § E ψ ( s, Φ , g, P ) = Y v λ ψ v ( s, π v )(I( s, π v )( g v ) f s,v ) , with f s ∈ I( s, π ), f s,v = f ◦ s,v for all v / ∈ S . Furthermore, we have Corollary 5.2 of[ loc. cit. ]:(1.3) E ψ ( s, Φ , g, P ) = Y v ∈ S λ ψ v ( s, π v )(I( s, π v )( g v ) f s,v ) m r Y i =1 L S (1 + is, π, r i ) − . We explore the details of the local Whittaker functionals λ ψ v ( s, π v )(I( s, π v )( g v ) f s,v )in the next section. In particular, Theorem 2.15 proves that the local Whittakerfunctionals are polynomials in { q sv , q − sv } , where q v = q deg v . Now, the Fourier LUIS ALBERTO LOMEL´I coefficients E ψ ( s, Φ , g, P ) are rational on q − s . Also, recall that each L S ( s, π, r i ) areabsolutely convergent for ℜ ( s ) ≫
0, Theorem 13.2 of [3]. Then, we can concludethat the product m r Y i =1 L S (1 + is, π, r i )extends to a rational function on q − s . The induction step found in § L -function to conclude that each L S (1 + is, π, r i )is rational in the variable q − s . Hence, each L S ( s, π, r i ) is rational. Furthermore,Theorem 6.1 of [ loc. cit. ], gives that locally each L -function L ( s, π v , r i,v ) , for ( k v , π v , ψ v ) ∈ L loc ( p ) , is a rational function on q − sv . Hence, the completed L -function L ( s, π, r i ) = Y v ∈ S L ( s, π v , r i,v ) L S ( s, π, r i )meromorphically continues to a rational function in the variable q − s . (cid:3) On the rationality of local Whittaker models
This section is local in nature and we work over any non-archimedean local field F with residue field F q . We take the opportunity to make available the detailsof the rationality of local Whittaker functionals (Theorem 2.1, and more preciselyTheorem 2.15).2.1. Induced representations and Whittaker models.
We fix a pair of quasi-split reductive groups ( G , M ) and use the local notation of § F, π, ψ ) ∈ L loc ( p ) and denote the space of π by V . For ν ∈ a ∗ M, C , letI( ν, π ) = ind GP ( π ⊗ q h ν,H P ( · ) i ) , where ind denotes normalized unitary parabolic induction, and we denote the spaceof I( ν, π ) by V( ν, π ). We will often abuse notation and identify the representationI( ν, π ) with its space V( ν, π ). Furhtermore, when the parabolic subgroup P of G is clear from context, we simply write Ind( π ) instead of I(0 , π ) or ind GP ( π ).Equation (1.1), is locally given as follows: a ∗ M, C ։ X nr ( M )associates the F -rational character q ν = q h ν,H θ ( · ) i to the element ν ∈ a ∗ M, C . Thekernel of this map is of the form πi log q Λ, for a certain lattice Λ of a ∗ M, C . Thissurjection gives X nr ( M ) the structure of a complex algebraic variety of dimension d = dim R a M . Thus, there are notions of polynomial and rational functions on X nr ( M ) (see § F, π, ψ ) ∈ L loc ( p )has π generic and therefore there is a unique Whittaker functional on V , up tomultiplication by a constant. Let us state a preliminary version of the main resultof this section; it will be stated more precisely as Theorem 2.15. N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 7 Theorem 2.1.
Let ( F, π, ψ ) ∈ L loc ( p ) and let ν ∈ a ∗ M, C . Then the induced repre-sentation I( ν, π ) is also generic. It has a Whittaker functional λ ψ ( ν, π ) which is apolynomial on { q s , q − s } . Given (
F, π, ψ ) ∈ L loc ( p ), we have Whittaker functionals for the representations( π, V ) and (I( ν, π ) , V( ν, π )). Care must be taken when going from one genericcharacter to another. Because of this, we now revisit local Whittaker models.2.2. Local Whittaker models.
We write algebraic group schemes defined over F in bold capital letters, e.g., G , and the corresponding group of rational pointsin regular capital letters, e.g. G = G ( F ). The roots of G with respect to a fixedmaximal split torus T are denoted by Σ, the positive roots by Σ + , which dependon the choice of Borel B . We denote the simple roots by ∆. We write Σ r and Σ + r to be the corresponding sets of reduced roots.For each α ∈ Σ, let N α be the subgroup of U whose Lie algebra is g α + g α .To define a non-degenerate character of U , first write U = Q α ∈ Σ + r N α . The group Q α ∈ Σ + − ∆ N α is a normal subgroup of U and(2.1) U / Y α ∈ Σ + − ∆ N α ∼ = Y α ∈ ∆ N α / N α . If ψ α : N α /N α → C is a smooth character for each α ∈ ∆, then ψ = Q α ∈ ∆ ψ α determines a character of U via the projection map U → U / Q α ∈ Σ + − ∆ N α andthe isomorphism of equation (2.1). A character ψ : U → C defined this way iscalled non-degenerate if each ψ α is non-trivial.Fix ψ , a non-degenerate character of U . Recall that an admissible representa-tion ( ρ, W ) of G is said to be ψ - generic or non-degenerate if there exists a linearfunctional λ ψ : W → C such that λ ψ ( ρ ( u ) w ) = ψ ( u ) λ ψ ( w ) for all u ∈ U ; λ ψ iscalled a Whittaker functional .Having fixed ψ , one can define a non-degenerate character ψ ˜ w of the unipotentradical M ∩ U of M so that ψ and ψ ˜ w are ˜ w - compatible , that is to say(2.2) ψ ˜ w ( u ) = ψ ( ˜ w u ˜ w − ) , u ∈ M ∩ U. Here, ˜ w denotes a representative in N/T of a Weyl group element w . The questionof varying Weyl group element representatives, as well as Haar measures, is taken upin § ψ of U and ψ ˜ w of U M which arecompatible in the sense of (2.2).2.3. Basic facts about twisted Jacquet modules.
The twisted Jacquet moduleof any smooth representation ( σ, V ) of U , is defined to be V ψ,U = V /V ψ ( U ) , where V ψ ( U ) is the span of { σ ( u ) v − ψ ( u ) v | u ∈ U, v ∈ V } . Another characterizationof V ψ ( U ) is as the set of those v ∈ V such that there is a compact open subgroup U ⊂ U with the property that Z U ψ − ( u ) σ ( u ) vdu = 0 . The usual Jacquet module V U is equal to V ,U in this notation. From [7] we havethe following proposition: LUIS ALBERTO LOMEL´I
Proposition 2.2.
Twisted Jacquet modules satisfy the following properties: (i)
The functor V → V ψ,U is exact. (ii) If V ′ is any space on which U acts by ψ then the map V → V ψ,U inducesan isomorphism Hom U ( V, V ′ ) ∼ = Hom C ( V ψ,U , V ′ ) . (iii) Let Ω be the U -morphism I GU ( C ψ ) → C ψ , f f (1) . Then for any smoothcharacter ψ of U , and V a smooth representation of G . Composition with Ω induces an isomorphism Hom G ( V, I GU ( C ψ )) ∼ = Hom C ( V ψ,U , C ) . (iv) If U and U are two subgroups of U , with U = U U , then V ψ ( U ) = V ψ ( U ) + V ψ ( U ) . The Whittaker model of an induced representation.
Fix a parabolicsubgroup P = P θ with Levi M . Also, fix non-degenerate characters ψ and ψ ˜ w which are compatible in the sense of equation (2.2). Let ( F, π, ψ ) ∈ L loc ( p ) be suchthat π is a ψ ˜ w -generic representation of M and denote its Whittaker functionalby λ ψ ˜ w . Let θ = ˜ w θ be the conjugate of θ in ∆. We then have the standardparabolic subgroup P = P θ of G , and we write M = M θ for its Levi and N = N θ for its unipotent radical.Let W be the Weyl group of Σ, and for each α ∈ ∆ let w α be its correspondingreflection. We let W M be the subgroup of W generated by w α , α ∈ θ . Denoteby w l and w l, M the longest elements of W and W M , respectively. Let [ W M \ W ] = (cid:8) w ∈ W | w − θ > (cid:9) , then w = w l w l, M will be the longest element of [ W M \ W ].Weyl group element representatives ˜ w are chosen as in [31].We construct the Whittaker functional λ ψ ( ν, π ) : V( ν, π ) → C of I( ν, π ) from λ ψ ˜ w : V → C . We first define it on a subspace I d P of I = I ( ν, π ), to which we nowturn. Recall that the Bruhat decomposition(2.3) G = a w ∈ [ W M \ W ] P wB gives rise to the Bruhat order among the double cosets
P wB of G . This partialorder is given by P w B ≺ P w B , whenever P w B is contained in the closure of P w B . Here, P w − B is maximal and is the unique open double coset which is densein G . Write d ( w ) = dim( P \ P wB ) and in particular d P = d ( w ) = dim( P \ P w B ).We need a result from Casselman’s notes (section 6 of [5]), which states that thereis a filtration of I by B -stable subspaces0 ⊂ I d P ⊂ · · · ⊂ I ⊂ I, where I n = { f ∈ I | supp( f ) ⊂ [ d ( w ) ≥ n P wB } . Moreover I n /I n +1 = M d ( w )= n I w ,I w = c-Ind BwP w − ∩ B ( w ( π ⊗ q h ν,H P ( · ) i δ / P )) . N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 9 The subspace of importance to us is(2.4) I d P = { f ∈ I ( ν, π ) | supp( f ) ⊂ P w B } , and will also be denoted by I ( ν, π ) d P , since we may want to vary ν .For functions f in I ( ν, π ) d P , let λ ψ ( ν, π ) f = Z N λ ψ w ( f ( w − n )) ψ ( n ) dn. Proposition 2.3.
For each ν ∈ a ∗ M, C , λ ψ ( ν, π ) is a well defined function on I ( ν, π ) d P .Proof. There is an isomorphismI( ν, π ) d P ∼ = c-Ind Bw P w − ∩ B w ( π ⊗ q h ν,H P ( · ) i δ / P ) , where w ( π ⊗ q h ν,H P ( · ) i δ / P )( p ) = π ⊗ q h ν,H P ( · ) i δ / P ( w − pw ) , ∀ p ∈ w P w − . Theisomorphism takes a function f ∈ I ( ν, π ) d P to the function x f ( w − x ), seeProposition 6.3.2 of [5].Since c-Ind is the non-normalized induced representation consisting of functionsof compact support modulo w P w − ∩ B , the function x f ( w − x ) restricted to U has compact support modulo w P w − ∩ U = M ∩ U (notice that w − α < , ∀ α ∈ Σ + − Σ + θ ) and since U = N (cid:0) M ∩ U (cid:1) , the function x f ( w − x ) as a function on N = w P w − \ U has compact support. (cid:3) Let G ∗ be the complement of P w B in G , which is a closed subset of G . Define J = J ( ν, π ) to be the space of locally constant functions f : G ∗ → V such that f ( pg ) = π ( m p ) q h ν + ρ P ,H P ( m p ) i f ( g ), for all p = m p n p ∈ P , g ∈ G ∗ . Restriction givesa morphism of U -spaces from I to J . There is then an exact sequence(2.5) 0 → I d P → I → J → . We have the following:
Lemma 2.4. J ψ,U = 0 .Proof. Because of the fact that J = ⊕ w = w I w and the exactness property of Jacquetmodules, Proposition 2.2, it is enough to show that ( I w ) ψ,U = 0 for w ∈ [ W M \ W ], w = w . This, in turn, is equivalent to showing thatHom N ( I w , C ψ ) ∼ = Hom C (( I w ) ψ,U , C ) = 0 . An element Φ ∈ Hom N ( I w , C ψ ) is in the dual of I w and is an eigenvector for U with eigencharacter ψ , thus Φ ∈ e I w (the contragredient representation of I w ).As a U -space I w = c-Ind UwUw − ∩ U ( w ( π ⊗ q h ν,H P ( · ) i )) , and [5] 2.4.2 says that f I w ∼ = I UwUw − ∩ U ( w ( e σ )) , σ = π ⊗ q h ν,H P ( · ) i . where σ = π ⊗ q h ν,H P ( · ) i . Suppose F : U → e V is the function corresponding to Φ,then on one hand F ( u u ) = e σ ( ˜ w − u ˜ w ) F ( u ) , ∀ u ∈ wU w − ∩ U, u ∈ U, and on the other it is an eigenvector for U so that F ( u ) = ψ ( u ) F (1) , ∀ u ∈ U, thus e σ ( ˜ w − u ˜ w ) F (1) = ψ ( u ) F (1) , ∀ u ∈ wU w − ∩ U. The representation e σ is trivial on N , so in order to get F = 0 all that is needed isto find a u ∈ wN w − ∩ U for which ψ ( u ) = 1 . But for w ∈ [ W M \ W ] , w = w ,there exists an α ∈ ∆ with wα ∈ Σ + − Σ + M . For such an α , choose u ∈ N α with ψ ( u ) = 1, which can be done since ψ is non-degenerate. (cid:3) Given p ∈ P , let m p and n p be the components of p when using the Levidecomposition of P , so that p = m p n p ∈ M N . We also fix a left Haar measure on P . The following is Proposition 1.1 of [28] and its corollary. Proposition 2.5.
For each ν ∈ a ∗ M, C , there is a surjective map P ν : C ∞ c ( G, V ) → I ( ν, π ) , given by P ν ϕ ( g ) = Z P q −h ν,H P ( m p ) i δ / P ( p ) π − ( m p ) ϕ ( pg ) dp . Corollary 2.6.
For each ν ∈ a ∗ M, C : (1) Given a compact open subgroup K of G and a function f ∈ I ( ν, π ) K , thereis a (right) K -invariant function ϕ ∈ C ∞ c ( G, V ) with P ν ϕ = f ; (2) P ν sends C ∞ c ( P w B, V ) onto I ( ν, π ) d P . We now turn towards the rationality of local Whittaker models. In fact, theyare Laurent polynomials. We begin by proving this for I d P . We use the notion of afunction on q ν ∈ X nr ( M ). For this, we fix a basis χ , . . . , χ d of the free Z -module X ( M ), so that ν = s ⊗ χ + · · · + s d ⊗ χ d , s i ∈ C . Then, what is meant by apolynomial (resp. rational) function on q ν is simply a polynomial (resp. rational)function on the variables Z , Z − , . . . , Z d , Z − d , where Z i = q s i . Note that theaffine algebra of C × can be thought as C (cid:2) Z, Z − (cid:3) . When M is maximal, the basisis dictated by ˜ α of equation (1.2). Lemma 2.7.
For any ϕ ∈ C ∞ c ( P w B, V ) , the function λ ψ ( ν, π ) P ν ϕ is a holomor-phic function on ν ∈ a ∗ M, C . In fact, λ ψ ( ν, π ) P ν ϕ is a polynomial in q ν .Proof. If ϕ is such a function, then P ν ϕ ∈ I d P and λ ψ ( ν, π ) P ν ϕ = Z N λ ψ w ( P ν ϕ ( w − n )) ψ ( n ) dn = λ ψ w (cid:18)Z N Z P q −h ν,H P ( m p ) i δ / P ( p ) π − ( m p ) ϕ ( pw − n ) ψ ( n ) dp dn (cid:19) . For y = pw − n ∈ P w − N let Φ ( y ) = Φ ( pw − n ) = π − ( m p ) δ / P ( p ) f ( pw − n ) ψ ( n ) , and extend q h v,H P ( · ) i to P w − N in an obvious way, so that λ ψ ( ν, π ) f = λ ψ w Z P w − N q −h ν,H P ( y ) i Φ ( y ) dy ! . Because ϕ is a locally constant function of compact support, then so is Φ . Hencethis last integral, as with all of the above integrals, can be written as a finite sum.By doing so, it is not hard to see that λ ψ ( ν, π ) P ν ϕ is a polynomial in q ν . (cid:3) N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 11 Remark 2.8.
Assuming Proposition 2.3, the following proposition is derived in § [18] . However, no proof of holomorphicity of the Whittaker model is found inthe literature other than the pricipal series case of [7] . For completion, we extendthese results here to the general case. Proposition 2.9.
For any f ∈ I ( ν, π ) d P , λ ψ ( ν, π )( I ( ν, π )( u ) · f ) = ψ ( u ) λ ψ ( ν, π ) f . For any compact open subgroup U of U , there is a projection operator P ψ,U :I( ν, π ) → I( ν, π ) ψ,U , given by(2.6) P ψ,U f ( g ) = (meas U ) − Z U f ( gu ) ψ ( u ) du . Lemma 2.10.
For any f ∈ I( ν, π ) d P and U a compact open subgroup of U , λ ψ ( ν, π )( P ψ,U f ) = λ ψ ( ν, π ) f Proof.
Notice that λ ψ ( ν, π )( P ψ,U f )= (meas U ) − Z N Z U λ ψ w ( f ( w − nu )) ψ ( u ) ψ ( n ) dudn = (meas U ) − Z U λ ψ ( I ( ν, π )( u ) · f ( w − n )) ψ ( u ) du , then Proposition 2.9 yields the required identity. (cid:3) The next remark follows from the definition, Proposition 2.3 and the computa-tions for Proposition 2.9. It points towards the steps needed to extend the definitionof λ ψ ( ν, π ) from I( ν, π ) d P to all of I( ν, π ). Remark 2.11.
Let f ∈ I( ν, π ) . Suppose we have a compact open subgroup U of U , such that P ψ,U f ∈ I ( ν, π ) d P . Then for any compact open subgroup N of N with N ⊃ U ∩ N Z N λ ψ w ( f ( w − n )) ψ ( n ) dn = Z N λ ψ w ( P ψ,U f ( w − n )) ψ ( n ) dn . Let f ∈ I( ν, π ) and suppose we have a compact open subgroup U of U , suchthat P ψ,U f ∈ I( ν, π ) d P . Under these assumptions we can now define λ ψ ( ν, π ) f = lim N Z N λ ψ w ( P ψ,U f ( w − n )) ψ ( n ) dn = λ ψ ( ν, π ) P ψ,U f , (2.7)where the limit is taken over all compact open subgroups N of N such that N ⊃ N ∩ U . Also, notice that the subgroup U can be replaced by any other compactopen subgroup of U containing U .All that remains is to obtain a compact open subgroup U with the above prop-erties. To study the analytic behavior of λ ψ ( ν, π ), it is important to obtain such agroup independently of ν . Proposition 2.12.
Let K be a compact open subgroup of G . There exists a compactopen subgroup U ⊂ U such that for every ν ∈ a ∗ M, C and every f ∈ I ( ν, π ) K thefunction P ψ,U f lies in I ( ν, π ) d P . Proof.
Let G ∗ be the complement of P w B in G and let J be as in Lemma 2.4.The lemma says that J = J ψ ( U ). Hence, for any f J ∈ J there is a compact opensubgroup U of U with Z U f J ( gu ) ψ ( u ) du = 0 , ∀ g ∈ G ∗ . This gives P ψ,U f J = 0. Since, I ( ν, π ) is admissible, one can enlarge U so that P ψ,U f J = 0 holds for all f J in the image of the finite dimensional I ( ν, π ) K . Becauseof the exact sequence (2.5), for any f ∈ I ( ν, π ) the function P ψ,U f has support in P w B , i.e., P ψ,U f ∈ I ( ν, π ) d P .To choose U independently of ν , let U ⊂ U ⊂ · · · be an exhaustive sequenceof compact open subgroups of U . For each n >
1, let Λ n be the subset of all ν ∈ a ∗ M, C such that P ψ,U n f has support in P w B for all f ∈ I ( ν, π ) K . We havealready shown that a ∗ M, C = ∪ Λ n . By Baire’s lemma, at least one Λ n contains anopen subset of ( a ∗ θ ) C .We show that the complement of such a Λ n is also open, hence Λ n must be allof ( a ∗ θ ) C . Suppose that for some ν ∈ a ∗ M, C there is an f ∈ I ( ν , π ) K with supportoutside of P w B . Then, by Corollary 2.6, there is a ϕ ∈ C ∞ c ( G, V ) such that P ν ϕ = f and ϕ is (right) K -invariant. Then P ν ϕ ∈ I ( ν, π ) K for all ν ∈ a ∗ M, C . Now,there is a g ∗ ∈ G ∗ such that P ν ϕ ( g ∗ ) = 0 (We are using supp f = { g ∈ G | f ( g ) = 0 } ,which can be verified for any f ∈ I ( ν, π )). Now, because ϕ , ν and g ∗ are fixed,it follows from the definition that there is an ε > P ν + ν ε ϕ ( g ∗ ) = 0whenever | ν − ν ε | < ε . (cid:3) Notice that U can be replaced by any other compact subgroup U ∗ of U with U ∗ ⊃ U . Corollary 2.13.
Fix a compact open subgroup K , then there exists a compact opensubgroup U ∗ ⊂ U such that λ ψ ( ν, π ) f = Z U ∗ λ ψ ˜ w ( f ( ˜ w − u )) ψ ( u ) du. for all f ∈ I( ν, π ) K .Proof. Choose U as in the theorem. I( ν, π ) K is finite dimensional and its elementshave compact support modulo P . Thus, there is a compact open subgroup U of U such that P ψ,U f has support in P w U for all f ∈ I( ν, π ) K . Then let U ∗ be acompact open subgroup of U containing U and U to get λ ψ ( ν, π ) f = λ ψ ( ν, π )( P ψ,U ∗ f ) = Z U ∗ λ ψ ˜ w ( f ( ˜ w − n )) ψ ( n ) dn. (cid:3) Remark 2.14.
An alternative way to construct a compact open subgroup with theproperties of U ∗ can be found in Proposition 3.2 of [35] . We note that the letter [6] further comments on several interesting properties of Whittaker functionals thatthe author has found very useful. Theorem 2.15. If π = ( π, V ) is a generic representation of M and ν ∈ a ∗ M, C ,then the induced representation I( ν, π ) is also generic. Explicitly, λ ψ ( ν, π ) will be a N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 13 Whittaker functional of I( ν, π ) . If ϕ ∈ C ∞ c ( G, V ) , then λ ψ ( ν, π ) P ν ϕ is a polynomialfunction on q ν .Proof. λ ψ ( ν, π ) is now defined for any f ∈ I ( ν, π ). It is clear that it is a Whittakerfunctional by writing λ ψ ( ν, π ) f = λ ψ ( ν, π )( P ψ,U ∗ f ) and applying Proposition 2.9to P ψ,U ∗ f ∈ I ( ν, π ) d P .A function ϕ ∈ C ∞ c is K -invariant for some K . The function P ν ϕ lies in I ( ν, π ) K ,for any ν ∈ a ∗ M, C . Let U be as in Proposition 2.12 and write ϕ ν = P ν ϕ , then λ ψ ( ν, π )( ϕ ν ) = λ ψ ( ν, π )( P ψ,U ϕ ν ) . Lemma 2.7 gives the required properties of λ ψ ( ν, π )( ϕ ν ). (cid:3) L -functions and intertwining operators We begin by recalling Shahidi’s tempered L -function conjecture. Then, we ob-tain an irreducibility result for principal series representations (Lemma 3.2). Weconclude the section with a discussion of intertwining operators and an assumptionof Kim in §§ Tempered L -functions and irreducibility of principal series. Shahidi’stempered L -function conjecture was proved in characteristic zero by H. H. Kim, inmost cases [19]. A completely local proof, that is also valid in positive characteristic,is given by V. Heiermann and E. Opdam [12]: Theorem 3.1 (Heiermann-Opdam, Kim) . Let ( F, π, ψ ) ∈ L loc ( p ) be tempered,then each L ( s, π, r i ) is holomorphic on ℜ ( s ) > . We can combine the theory of local L -functions with a result of J.-S. Li [27] toobtain the following: Lemma 3.2.
Let ( F, π, ψ ) ∈ L loc ( p ) be tempered and unramified. Then I( s, π ) isirreducible for ℜ ( s ) > .Proof. We have that π ֒ → Ind( χ ) , with χ an unramified unitary character of T ( F ). From the Satake classification,the character χ corresponds to a complex semisimple conjugacy class in the dualtorus, each Satake parameter having absolute value 1. Let χ s = χ · q h s ˜ α,H P ( · ) i . From the remark following Proposition 4.1 of [22], the function ξ α ( χ s ) defined in § α is either a non-zero constant or a factorappearing in(3.1) m r Y i =1 L (1 + is, ˜ π, r i ) − or m r Y i =1 L (1 − is, π, r i ) − . The result of Li, specifically Theorem 2.2 of [ loc. cit. ], states that I( s, π ) is irreduciblewhen each ξ α ( χ s ) = 0. The local L -functions involved are never zero, and fromTheorem 3.1 we have that the first product Q m r i =1 L (1 + is, ˜ π, r i ) − is non-zero for ℜ ( s ) >
1. We claim that the same is also true for the second product. For this,notice that because π is tempered we can write for each i : L ( s, π, r i ) − = Y j (1 − a i,j q − sv ) , where the parameters a i,j have absolute value 1. Then m r Y i =1 L (1 − is, π, r i ) − = m r Y i =1 Y j (1 − a i,j q − v q isv ) . Each factor in the latter product is non-zero for ℜ ( is ) >
1. In particular, the prod-uct is non-zero for ℜ ( s ) >
1. From Li’s theorem, we must have I( s, π ) irreduciblefor ℜ ( s ) > (cid:3) Intertwining operators.
We have the following connection between the in-tertwining operator and Langlands-Shahidi partial L -functions. Let ( k, π, ψ, S ) ∈ L glob ( p ), then(3.2) M( s, π, ˜ w ) = m r Y i =1 L S ( is, π, r i ) L S (1 + is, π, r i ) O v ∈ S A( s, π v , ˜ w ) . The following Lemma is possible by looking into the spectral theory of Eisensteinseries available over function fields.
Lemma 3.3.
Let ( k, π, ψ, S ) ∈ L glob ( p ) . If ˜ w ( π ) ≇ π , then M( s, π, ˜ w ) and E ( s, Φ , g, P ) are holomorphic for ℜ ( s ) ≥ .Proof. Let Φ be the automorphic functions of § π . Wehave the pseudo-Eisenstein series θ Φ = Z ℜ ( s )= s E ( s, Φ , g, P ) ds. This is first defined for s > h ρ p , α ∨ i by Proposition II.1.6 (iii) and (iv) of [32].Then II.2.1 Th´eor`eme of [ loc. cit. ] gives for self-associate P that h θ Φ , θ Φ i = Z ℜ ( s )= s X ˜ w ∈{ , ˜ w } (cid:10) M( s, π, ˜ w )Φ − ˜ w (¯ s ˜ α ) , Φ s (cid:11) ds, and is zero if P is not self-associate. Now, for P self-associate, the condition˜ w ( π ) ≇ π allows us to shift the imaginary axis of integration by V.3.8 Lemmeof [32] to ℜ ( s ) = 0. However, by IV.1.11 Proposition we have that M ( s, π, ˜ w )is holomorphic for ℜ ( s ) = 0. Thus, we have that M ( s, π, ˜ w ) is holomorphic for ℜ ( s ) >
0. Finally, the poles of Eisenstein series are contained by the constantterms. (cid:3)
An assumption of Kim.
Locally, let (
F, π, ψ ) ∈ L loc ( p ). We have thenormalized intertwining operatorN( s, π, ˜ w ) : I( s, π ) → I( s ′ , π ′ )defined by N( s, π, ˜ w ) = m r Y i =1 ε ( is, ˜ π, r i , ψ ) L (1 + is, ˜ π, r i ) L ( is, ˜ π, r i ) A( s, π, ˜ w ) . Globally, for ( k, π, ψ, S ) ∈ L glob ( p ), we haveN( s, π, ˜ w ) = ⊗ v N( s, π v , ˜ w ) . The following assumption was made by H.H. Kim in his study of local Langlands-Shahidi L -functions and normalized intertwining operators over number fields [16]. N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 15 Having now the Langlands-Shahidi method available in positive characteristic, weare lead at this point to Kim’s assumption.
Assumption 3.4.
Let ( k, π, ψ, S ) ∈ L glob ( p ) . At every place v of k , the normalizedintertwining operator N( s, π v , ˜ w ) is holomorphic and non-zero for ℜ ( s ) ≥ / . It is already known to hold in many cases, including all the quasi-split classicalgroups. See [20] for a more detailed account. We make this assumption for theremainder of the article.
Lemma 3.5.
Let ( k, π, ψ, S ) ∈ L glob ( p ) . If ˜ w ( π ) ≇ π , then m r Y i =1 L ( is, π, r i ) L (1 + is, π, r i ) is holomorphic for ℜ ( s ) ≥ / .Proof. For every place v / ∈ S , we have thatA( s, π v , ˜ w ) f s,v ( e v ) = m r Y i =1 L ( is, π v , r i ) L (1 + is, π v , r i ) f s,v ( e v ) , for ( k v , π v , ψ v ) ∈ L loc ( p ). Globally, the condition ˜ w ( π ) ≇ π allows us to applyLemma 3.3 to ( k, π, ψ, S ) ∈ L glob ( p ). Thus M( s, π, ˜ w ) is holomorphic for ℜ ( s ) ≥ M ( s, π, ˜ w ) f s = m r Y i =1 ε ( is, ˜ π, r i ) − L ( is, ˜ π, r i ) L (1 + is, ˜ π, r i ) ⊗ v N( s, π v , ˜ w ) f s,v , where f s = ⊗ v f s,v ∈ I( s, π ). Now, with an application of Assumption 3.4, we canprove the Lemma. (cid:3) Global twists by characters
We now extend a number fields result of Kim-Shahidi to the case of positivecharacteristic. In particular, Proposition 2.1 of [21] shows that, up to suitableglobal twists, Langlands-Shahidi L -functions become holomorphic. In the case offunction fields, Harder’s rationality result allows us to prove the stronger propertyof L -functions becoming polynomials after suitable twists.Let ξ ∈ X ∗ ( M ) be the rational character defined by ξ ( m ) = det (Ad( m ) | n ) , where n is the Lie algebra of N . At every place v of k , we obtain a rational character ξ v ∈ X ∗ ( M ) v . Given a gr¨ossencharakter χ = ⊗ χ v : k × \ A k → C × , we obtain thecharacter χ · ξ = ⊗ χ v · ξ v of M ( A k ). Let ( k, π, ψ, S ) ∈ L glob ( p ). For n ∈ Z ≥ , form the automorphic repre-sentation π n,χ = π ⊗ ( χ · ξ n ) . Proposition 4.1.
Let ( k, π, ψ, S ) ∈ L glob ( p ) and fix a place v ∈ S . Then thereexist non-negative integers n and f v such that for every gr¨ossencharakter χ = ⊗ χ v with cond( χ v ) ≥ f v , we have that L ( s, π n,χ , r i ) , ≤ i ≤ m r , is a polynomial function on { q s , q − s } .Proof. In order to apply Lemmas 3.3 and 3.5, we first need the existence of aninteger n such that ˜ w ( π n,χ ) ≇ π n,χ , for suitable χ . For this, we follow Shahidi[37] to impose the right condition on the conductor of χ v . Write P = P θ , with θ = ∆ − { α } , so that M = M θ and N = N θ . Let A θ be the split torus of M θ andlet A θ = A θ ( k v ), the group of k v -rational points. Let A θ = ˜ w ( A θ ) A − θ = (cid:8) a ∈ A θ | ˜ w ( a ) = a − (cid:9) . Then Lemma 2 of [ loc. cit. ] gives ξ | A θ = 1 . In fact, we have that ξ : A θ → G m is onto. Given an integer n , we choose ℓ v ∈ Z ≥ such that 1 + p ℓ v v ⊂ ξ n ( A θ )and ℓ v is minimal with this property. Let ω v be the central character of π v . Take(4.1) f v ≥ max { ℓ v , cond( ω v ) } , which depends on n . Then(4.2) ˜ w ( ω v ⊗ ( χ v · ξ n )) ≇ ω v ⊗ ( χ v · ξ n ) , for cond( χ v ) ≥ f v .Let ( G i , M i ) be as in Proposition 6.4 of [31], where G i ֒ → G and we havethe corresponding parabolic P i = M i N i of G i . Let ξ i ∈ X ∗ ( M i ) be the rationalcharacter ξ i ( m ) = det (Ad( m ) | n i ) , where n i is the Lie algebra of N i . There are then integers n , . . . , n m r , such thatupon restriction to M i we have ξ n i i = ξ n and χ · ξ n i i = χ · ξ n . With this choice of n = n , we choose f v as in (4.1) at the place v ∈ S . Then, for any gr¨ossencharakter χ : k × \ A × k → C × with cond( χ v ) ≥ f v . Equation (4.2) at v ensures that˜ w ( π n,χ ) ≇ π n,χ . Now, Lemma 3.3 together with (1.3) give that m r Y i =1 L S (1 + is, π n,χ , r i )is holomorphic and non-zero for ℜ ( s ) ≥
0. The induction step found in § L -function and conclude that each L S ( s, π n,χ , r i )is holomorphic and non-zero on ℜ ( s ) ≥
1. Furthermore, with the aid of Lemma 3.5we conclude that each L S ( s, π n,χ , r i ) is holomorphic on ℜ ( s ) ≥ / ℜ ( s ) ≤ /
2. The automorphic L -functions L ( s, π n,χ , r i ) being now entire, inaddition to being rational by Theorem 1.2, must be polynomials on { q − s , q s } . (cid:3) The following corollary is obtained from the proof of the Proposition.
Corollary 4.2.
Let ( k, π, ψ, S ) ∈ L glob ( p ) be such that ˜ w ( π ) ≇ π . Then L ( s, π, r i ) , ≤ i ≤ m r , is a polynomial on { q − s , q s } . N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 17 Transfer and the generic unitary dual of classical groups
We recall the results of Ganapathy on the Kazhdan transfer between close localfields and the Langlands-Shahidi local coefficient [8]. We use this to transfer a resultof Lapid, Mui´c and Tadi´c on unitary generic representations of the classical groups[26]. In particular, for the split classical groups, Ganapathy and Varma show thatlocal L-functions and γ -factors are compatible with the Kazhdan transfer [10]; weobserve that this holds in more generality (Theorem 5.2). An alternative approachto the results of this section would be to use the techniques of [1, 2].5.1. Kazhdan transfer.
Let F and F ′ be non-archimedean local fields that are m -close in the sense of Kazhdan [15], i.e, there is an isomorphism of rings O F / p mF ∼ = O F ′ / p mF ′ . Let G be a split connected reductive group scheme defined over Z . Let I denote the standard Iwahori subgroup scheme of G and let I m = Ker {I ( O F ) → I ( O F / p mF ) } . We use the results of Ganapathy, who establishes a refined Kazhdan tranfer for theHecke algebra H ( G, I m ) in [8].Let R ( G ) be the category of smooth complex representations of G . Let R m ( G )be the subcategory of representations ( π, V ) of G such that π ( G )( V I m ) = V . If H ( G, I m )-mod is the category of H ( G, I m )-modules, Proposition 3.16 of [8] showsthat R m ( G ) is closed under sub-quotients and gives a functor R m ( G ) → H ( G, I m )-mod , which is an equivalence of categories.Ganapathy constructs a Kazhdan isomorphism H ( G, I m ) −→ H ( G ′ , I ′ m )which also gives a bijection Isomorphism classes ofirreducible admissible π of G with π I m = 0 ζ m −−→ Isomorphism classes ofirreducible admissible π ′ of G ′ with π I ′ m = 0 . We use the notation of [10], to denote π ∼ m π ′ whenever we are in the abovesituation. A property that is preserved under the transfer is the Moy-Prasad depth[34], depth( π ) = depth( π ′ ). We refer to [8] for more details, and also §
13 of [10]for a summary of results.
Remark 5.1.
Kazhdan [15] , produces an isomorphism of Hecke algebras usingprincipal congruence subgroups K m of G , for m large enough, instead of the Iwahorifiltration. Since I m ⊂ K m , we have that H ( G, K m ) ⊂ H ( G, I m ) . Corollary 3.15of [8] , proves that ζ m | H ( G, K m ) is the Kazhdan isomorphism between H ( G, K m ) and H ( G, K ′ m ) for appropriate m . Parabolic induction and transfer.
Fix a pair ( G , M ) of split reductivegroup schemes such that M is a Levi component of a parabolic subgroup P of G .We have the transfer of the previous section for M , so that given m -close local fields F and F ′ , the map ζ m gives a transfer σ ∼ m σ ′ between admissible representationsof M and M ′ . The results of § § κ m for the correspondingparabolically induced representations. The Kazhdan transfer is compatible withparabolic induction. For this, suppose that we have π I m = 0, then Ind( π ) of G I GP π of M κ , π ′ of M ′ Ind( π ′ ) of G ′ I G ′ P ′ ζ m +3 Suppose we are given two characters ψ : F → C × and ψ ′ : F ′ → C × . We write ψ ∼ m ψ ′ if cond( ψ ) = cond( ψ ′ ) = t and ψ | p t − mF / p tF ∼ = ψ ′ | p t − mF ′ / p tF ′ . An important result is the following: Let π be a generic representation of G n and τ a smooth irreducible representation of GL r ( F ). Let m ≥ π ), depth( τ ) < m . There exists an integer l , depending on n, r, m , such thatfor each F ′ that is l -close to F , we have(5.1) γ ( s, π × τ, ψ ) = γ ( s, π ′ × τ ′ , ψ ′ ) , if π ∼ l π ′ , τ ∼ l τ ′ and ψ ∼ l ψ ′ . This is Proposition 13.5.3 of [10] for the splitclassical groups. For a general split reductive group scheme, we observe that wehave: Theorem 5.2.
Let ( F ′ , π ′ , ψ ′ ) ∈ L loc ( p ) be a generic representation of M ′ = M ( F ′ ) . Let m ≥ be such that depth( π ′ ) < m . There exists an integer l , dependingon m , such that for every non-archimedean local field F that is l -close to F ′ we have L ( s, π, r i ) = L ( s, π ′ , r i ) ε ( s, π, r i , ψ ) = ε ( s, π ′ , r i , ψ ′ ) , if π ∼ l π ′ and ψ ∼ l ψ ′ , ≤ i ≤ m r .Proof. First assume that(5.2) γ ( s, π, r i , ψ ) = γ ( s, π ′ , r i , ψ ′ )is true when π ∼ l π ′ are tempered representations. Then, the construction of L -functions and ε -factors of [36] and [31], builds the general case from temperedrepresentations aided by Langlands classification. The theorem then follows, sincethe Kazdan transfer preserves temperedness, is compatible with parabolic inductionand twists by unramified characters (see § F ′ , π ′ , ψ ′ ) ∈ L loc ( p ). This is done by aninduction argument on the number of irreducible constituents of the adjoint actionof L M on L n , namely r = ⊕ m r i =1 r i . If r = r is irreducible, we have that γ ( s, π ′ , r , ψ ′ ) = C ψ ′ ( s, π ′ , ˜ w ′ ) . If m r >
1, we can use Lemma 4.4 of [31]. Then for each i , 1 ≤ i ≤ m r , there exists apair ( G i , M i ), such that the adjoint action r ′ of L M i on L n i has r ′ = ⊕ m r ′ j =1 r ′ j , with m r ′ < m r and r i = r ′ . We continue in this way until we find a pair ( G i k , M i k ), suchthat r i = r ( k )1 and the adjoint action after k steps r ( k ) = r ( k )1 is irreducible. Fromthe way the construction is made, there is a triple ( F ′ , π ′ ( k ) , ψ ′ ) ∈ L ( G i k , M i k , p ),for which we have that γ ( s, π ′ , r i , ψ ′ ) = C ψ ′ ( s, π ′ ( k ) , ˜ w ′ ) . N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 19 Finally, thanks to Theorem 5.5 of [8], we have that there is a sufficiently large l , sothat for every non-archimedean local field F , if we let π ∼ l π ′ and ψ ∼ l ψ ′ , then C ψ ( s, π ( k ) , ˜ w ) = C ψ ′ ( s, π ′ ( k ) , ˜ w ′ ) . From this, equation (5.2) follows. (cid:3)
Generic unitary dual.
Assume F has characteristic zero. Let π be a (uni-tary) generic representation of G n = G n ( F ). From [26] we see that π is a con-stituent of an induced representation of the form(5.3) Ind( δ ν β ⊗ · · · ⊗ δ d ν β ⊗ ρ ν α ⊗ · · · ⊗ ρ ′ e ν α e ⊗ π ) . The representations δ i and ρ j are unitary discrete series; the Langlands parametersare of the form 1 > β > · · · β d > / > α > · · · > α e >
0; and, π is temperedgeneric of a classical group G n of the same kind as G n . In fact, except for thecase of G n = SO n , the set { β , . . . , β d } is empty and the Langlands parametersare bounded by 1 / Theorem 5.3.
Let F ′ be a non-Archimedean local field of characteristic p andlet π ′ be a (unitary) generic representation of G ′ n = G n ( F ′ ) . Then π ′ ∼ l π is aconstituent of an induced representation of the form (5.4) Ind( δ ′ ν β ⊗ · · · ⊗ δ ′ d ν β ⊗ ρ ′ ν α ⊗ · · · ⊗ ρ ′ e ν α e ⊗ π ′ ) , with the notation of (5.3) .Proof. Let us show that unitarity is preserved under the Kazhdan transfer. Let F be or characteristic 0 and such that it is l -close to F ′ . Take l sufficiently large, andlet π be a representation of G , such that π ′ ∼ l π and depth( π ′ ) = depth( π ) < m .This is possible by the results of § loc. cit. ], where l depends on m .Since ( π, V ′ ) ′ is unitary, there is a positive definite Hermitian form h ′ on V ′ .For every positive integer r > l , we obtain a positive definite Hermitian form h ′ r on V ′I r , defined by h ′ r ( v ′ , w ′ ) = h ′ ( v ′ , w ′ ) for v ′ , w ′ ∈ V ′I r . We can transport thestructure, using the Kazhdan isomorphism ζ r to V I r . We thus obtain a positivedefinite hermitian form h r on V I r . Let t > r > l , then at every stage we havecompatibility V I r ∪ V I t ζ t V ′I ′ t V ′I ′ r ζ r ∪ for the Hermitian forms. We can then define a hermitian form h on V . Let v, w ∈ V \ { } , then there exists an r > l such that v, w ∈ V I r . Then h ( v, w ) = h r ( v, w ) >
0. Hence the form is positive definite and π is unitary.Genericity is respected by the Kazhdan transfer, this is the content of § π of G is then unitary generic. Since we are now incharacteristic 0, we have the Lapid-Mui´c-Tadic result, hence π is a constituent ofan induced representation of the form given by equation (5.3). With the notationfor this setting for π , we can take π ′ ∼ l π , δ ′ i ∼ l δ i , ρ ′ j ∼ l ρ j , for 1 < i < d ,1 < j < e , all with depth less than m and l large enough.The Kazhdan transfer behaves well under parabolic induction, see § by unramified characters. Thus the Lapid-Mui´c-Tadi´c presentation for π transfersto π ′ , and we obtain the desired form. (cid:3) On the holomorphy of L -functions for the classical groups Let G n be either SO n +1 , Sp n , SO n or a quasi-split unitary group U N (with N = 2 n + 1 or 2 n ). We leave the study of GSpin N groups, in addition to non-split quasi-split SO ∗ n and GSpin ∗ n , for another occasion. Our groups are definedover a global function field k . In the case of a quasi-split unitary group, thereis a degree-2 extension defining the hermitian form. To include this case in auniform way, we make the following convention: let K = k if G n is split, and K/k denotes a separable quadratic extension if G n is a quasi-split unitary group. Thegroup Res GL m denotes the usual general linear group GL m in the split case and isobtained via restriction of scalars in the non-split quasi-split case.6.1. Exterior square, symmetric square and Asai L -functions. We let M be the Siegel Levi subgroup of G n as in § v of k may be split. In this casewe set K v = k v × k v . In particular, we let L loc ( p, G , M ) be the class of triples( K v /k v , π v , ψ v ) consisting of: a non-archimedean local field k v ; K v = k v if G n issplit; K v a degree-2 finite ´etale algebra over k v if G n is a quasi-split unitary group;an irreducible generic representation π of M ( k v ); and, a non-trivial continuouscharacter ψ v of k v . In our setting, the triples ( K v /k v , π v , ψ v ) always arise from thecorresponding global objects. More precisely, we use the notation L glob ( p, G , M )to denote the corresponding global class of quadruples ( K/k, π, ψ, S ).In these cases, L -functions and related local factors arise from representations ofGL n . Let ρ n denote the standard representation of GL n ( C ). The adjoint action iseither irreducible, and we write r = r in the following cases(6.1) r = Sym ρ n if G = SO n +1 ∧ ρ n if G = SO n r A if G = U n , or reducible, when we have r = r ⊕ r as follows(6.2) r = (cid:26) ρ n ⊕ ∧ ρ n if G = Sp n ( ρ n ⊗ ρ ) ⊕ ( r A ⊗ η K/k ) if G = U n +1 . We refer to [30] for any unexplained notation concerning the Siegel Levi case.All of these cases share the advantage that the Ramanujan conjecture holds forGL n over function fields thanks to the work of L. Lafforgue [23]. We can actuallyprove a general theorem for any pair ( G , M ) under the assumption that π satisfiesthe Ramanujan conjecture, which we now recall. Conjecture 6.1.
Let π be a globally generic cuspidal automorphic representationof a quasi-split connected reductive group H . Then each π v is tempered. Remark 6.2.
In Section 6.2 below, we only make use of Corollary 6.4, wherethe assumption of the next proposition holds. We do not assume the Ramanujanconjecture for the classical groups to prove Theorem 6.5.
Proposition 6.3.
Let ( K/k, π, ψ, S ) ∈ L glob ( p, G , M ) , with M a maximal Levisubgroup of a quasi-split connected reductive group G . Assume that π satisfies the N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 21 Ramanujan conjecture. Then, for each i , ≤ i ≤ m r , the automorphic L -function L ( s, π, r i ) is holomorphic for ℜ ( s ) > .Proof. Let (
K/k, π, ψ, S ) ∈ L glob ( p, G , M ). From Theorem 3.1, we know that atplaces v ∈ S each of the L -functions L ( s, π v , r i,v ) is holomorphic for ℜ ( s ) >
0. Webegin with the known observation that local components of residual automorphicrepresentations are unitary representations. Furthermore, if the local representationI( s, π v ) is irreducible, then it cannot be unitary. From Lemma 3.2, we conclude thatthe global intertwining operator M( s, π, ˜ w ) must be holomorphic for ℜ ( s ) > m r Y i =1 L S ( is, π, r i ) L S (1 + is, π, r i )is holomorphic on ℜ ( s ) >
1. Since the poles of Eisenstein series are contained inthe constant terms, with an application of equation (1.3) we can further concludethat m r Y i =1 L S (1 + is, π, r i ) − is holomorphic for ℜ ( s ) >
1. Now, the induction step found in § L -function and prove that each L S ( s, π, r i )is holomorphic for ℜ ( s ) > (cid:3) Corollary 6.4.
Let ( K/k, τ, ψ, S ) ∈ L glob ( p, G , M ) , with M the Siegel Levi sub-group of a quasi-split classical group. Then, for each i , ≤ i ≤ m r , the automorphic L -function L ( s, τ, r i ) is holomorphic and non vanishing for ℜ ( s ) > .Proof. The assumptions of Proposition 6.3 are satisfied as noted in Remark 6.2,which shows the holomorphy result. To prove non vanishing, we look at all possi-bilities for r i given by equations (6.1) and (6.2). The L -functions corresponding to ρ n and ρ n ⊗ ρ are understood and known to satisfy the properties of the corollary[13, 14]. The L -functions L ( s, τ, r ) for r = Sym , ∧ , r A or r A ⊗ η K/k satisfy thefollowing relationships L ( s, τ × τ ) = L ( s, τ, Sym ) L ( s, τ, ∧ ) L ( s, τ × τ θ ) = L ( s, τ, r A ) L ( s, τ, r A ⊗ η K/k ) . Each L -function L ( s, τ, r ) appearing on the right hand sides of the previous twoequations is holomorphic for ℜ ( s ) >
1. The Rankin-Selberg product L -functionson the left hand sides are non vanishing for ℜ ( s ) > L ( s, τ, r ) inturn must be non vanishing for ℜ ( s ) > (cid:3) Rankin-Selberg L -functions for the classical groups. We begin by adapt-ing the notation to the classical groups. Let L glob ( p, G n , Res GL m ) be the classconsisting of quintuples ( K/k, π, τ, ψ, S ): k a global function field of characteris-tic p ; K = k if G n is split and K/k a separable quadratic extension if G n is aquasi-split unitary group; π = ⊗ ′ π v a globally generic cuspidal automorphic rep-resentation of G n ( A k ); τ = ⊗ ′ τ v a cuspidal (unitary) automorphic representationof Res GL m ( A k ) = GL m ( A K ); ψ : k \ A k → C × a global additive character; and S a finite set of places of k such that π v and ψ v are unramified for v / ∈ S . In the case of a separable quadratic extension, we can think of the additive character ψ K : K \ A K → C × obtained via the trace. Theorem 6.5.
Let ( K/k, π, τ, ψ, S ) ∈ L glob ( p, G n , Res GL m ) . Then L ( s, π × τ ) isholomorphic for ℜ ( s ) > and has at most a simple pole at s = 1 .Proof. Thanks to the work of L. Lafforgue [23], each local component τ v of thecuspidal unitary τ arises from an induced representation of the formInd( τ ,v ⊗ · · · ⊗ τ f,v ) , with each τ l,v tempered. Also, the classification of generic unitary representationsof G n ( k v ) gives that every π v is a constituent of(6.3) Ind( δ ,v ν β ⊗ · · · ⊗ δ d,v ν β d ⊗ δ ′ ,v ν α ⊗ · · · ⊗ δ ′ e,v ν α e ⊗ π ,v ) . Here, the notation is that of (5.3) and Theorem 5.3 for the split classical groups.For the quasi-split unitary groups see Remark 6.6.Then, the multiplicativity property of Langlands-Shahidi L -functions gives L ( s, π v × τ v ) = L ( s, π v, × τ v ) d Y i =1 e Y j =1 f Y l =1 L ( s + β i , δ i,v × τ l,v ) L ( s + α j , δ ′ j,v × τ l,v ) × d Y i =1 e Y j =1 f Y l =1 L ( s − β i , δ i,v × τ l,v ) L ( s − α j , δ ′ j,v × τ l,v ) . From Theorem 3.1, each of the L -functions appearing in the right hand side isholomorphic for ℜ ( s ) large enough. This carries through to the left hand side andwe can conclude that L ( s, π v × τ v ) is holomorphic for ℜ ( s ) > β . In particular, for ℜ ( s ) > σ = τ ⊗ ˜ π , so that ( K/k, σ, ψ, S ) ∈ L glob ( p, G , M ). Where G is a classical group of rank l = m + n of the same type as G n and M = Res GL m × G n is a maximal Levi subgroup. As observed in the number fields case by H. H. Kimin [17], and from [32, 33] over function fields, if the global intertwining operatorM( s, Φ , g, P ) has a pole at s , then a subquotient of I( s , σ ) would belong to theresidual spectrum and we would have that almost every I( s , σ v ) is unitary.However, to obtain a contradiction, we claim for ℜ ( s ) > s, σ v ) cannot be unitary for at least one v / ∈ S (we can actually show this for all v / ∈ S ). For this, we begin by fixing a place v / ∈ S , which we assume remains inertif we are in the case of a non-split quasi-split classical group (see Remark 6.6). Inthese cases we now apply equation (6.3) for the group G l . If I( s , σ v ) were unitarythen, up to rearrangement if necessary, it would be of the form(6.4)Ind( χ ,v ν s ⊗ · · ·⊗ χ m,v ν s ⊗ µ ,v ν β ⊗ · · ·⊗ µ d,v ν β d ⊗ µ ′ ,v ν α ⊗ · · ·⊗ µ ′ e,v ν α e ⊗ µ ,v ) , where we now have unramified unitary characters µ ,v , µ i,v and µ ′ j,v ; the character µ ,v is taken to be trivial unless we are in the odd unitary group case. The Langlandsparameters are of the form 1 > β > · · · β d > / > α > · · · > α e >
0. Theclassification tells us that this cannot be the case if ℜ ( s ) >
1. Hence, the globalintertwining operator M( s, Φ , g, P ) must be holomorphic for ℜ ( s ) > m r Y i =1 L S ( is, σ, r i ) L S (1 + is, σ, r i ) N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 23 is then holomorphic on ℜ ( s ) >
1. For the classical groups we precisely have that m r = 2 and(6.5) m r Y i =1 L S ( s, σ, r i ) = L S ( s, σ, r ) L S (2 s, τ, r ) , where r = ρ n ⊗ ˜ ρ m and L S ( s, σ, r ) = L S ( s, τ, r ) has r either an exterior square,symmetric square or Asai L -function. The induction step is given by the Siegel Levicase of Corollary 6.4. We can then cancel the second L -functions and conclude thatthe quotient(6.6) L S ( s, π × τ ) L S (1 + s, π × τ )is holomorphic for ℜ ( s ) > L S (1 + s, π × τ ) − is holomorphic for ℜ ( s ) >
1. In thisway, we can cancel the now non-zero denominator in equation (6.6) to concludethat L S ( s, π × τ ) is holomorphic on ℜ ( s ) >
1. We have also noted that L ( s, π v × τ v )is holomorphic for ℜ ( s ) > v ∈ S . We then obtain the requiredholomorphy property for the completed L -function L ( s, π × τ ).Now, we have that the poles of the global intertwining operator M ( s, σ, ˜ w ) areall simple [32]. Then, by equation (3.2), the quotient L S ( s, π × τ ) L S (2 s, τ, r ) L S (1 + s, ˜ π × ˜ τ ) L S (1 + 2 s, ˜ τ , r )has at most a simple pole at s = 1. From Corollary 6.4, L S (2 , τ, r ) and L S (3 , ˜ τ , r )are different from zero. Thus, L ( s, π × τ ) has at most a simple pole at s = 1. (cid:3) Remark 6.6.
The case of unitary groups can also be established by choosing asplit place v , and using the known result for GL n . This is because the argumentonly requires one place v where I( s, σ v ) cannot be unitary for ℜ ( s ) > . Eachrepresentation π w i , i = 1 or , w i | v is known to have Langlands parameters ≤ t i,d i ≤ · · · ≤ t i, < / . On functoriality for the classical groups
Throughout this section, we restrict ourselves to the split case. Theorems 7.1 and7.2 were established in [28, 29] under the assumption char( k ) = 2; we now removethis restriction. The corresponding results for the quasi-split unitary groups areproved in [31]. Theorem 7.1.
Let G n be a split classical group defined over a function field k . (i) (Ramanujan Conjecture). If π = ⊗ ′ π v is a globally generic cuspidal auto-morphic representation of G n ( A k ) , then each local component π v is tem-pered.Let ( k, π, τ, ψ ) ∈ L glob ( p, G n , G m ) , with G m either GL m or a split classical groupof rank m . Then (ii) (Rationality). L ( s, π × τ ) converges absolutely on a right half plane and hasa meromorphic continuation to a rational function on q − s . (iii) (Functional Equation). L ( s, π × τ ) = ε ( s, π × τ ) L (1 − s, ˜ π × ˜ τ ) . (iv) (Riemann Hypothesis). The zeros of L ( s, π × τ ) are contained in the line ℜ ( s ) = 1 / . The proof of the Ramanujan conjecture is essentially that of [28]. And the proofsof Properties (ii)–(iv) are those appearing in [29]. However, they are completedwith the results of this article. Basically, we now prove Theorem 7.2 below in acharacteristic free way for any global function field.Let us summarize the results of [28] on the globally generic functorial lift for theclassical groups. Let G n be a split classical group of rank n defined over a globalfunction field k . The functorial lift of [28] takes globally generic cuspidal auto-morphic representations π of G n ( A k ) to automorphic representations of H N ( A k ),where H N is chosen by the following table. G n L G n ֒ → L H N H N SO n +1 Sp n ( C ) × W k ֒ → GL n ( C ) × W k GL n Sp n SO n +1 ( C ) × W k ֒ → GL n +1 ( C ) × W k GL n +1 SO n SO n ( C ) × W k ֒ → GL n ( C ) × W k GL n Table 1
Theorem 7.2.
Let G n be a split classical group. Let π be a globally generic cuspidalrepresentation of G n ( A k ) ; n ≥ if G n = SO n . Then, π has a functorial lift to anautomorphic representation Π of H N ( A k ) . It has trivial central character and canbe expressed as an isobaric sum Π = Π ⊞ · · · ⊞ Π d , where each Π i is a unitary self-dual cuspidal automorphic representation of GL N i ( A k ) such that Π i ≇ Π j for i = j . Furthermore, Π v is the local Langlands functorial liftof π v at every place v of k . That is, for every ( k v , π v , τ v , ψ v ) ∈ L loc ( p, G n , GL m ) there is equality of local factors γ ( s, Π v × τ v , ψ v ) = γ ( s, π v × τ v , ψ v ) L ( s, Π v × τ v ) = L ( s, π v × τ v ) ε ( s, Π v × τ v , ψ v ) = ε ( s, π v × τ v , ψ v ) . Proof.
Let Π ′ be the automorphic representation of GL N ( A k ) obtained from π viathe weak functorial lift of Theorem 8.5 of [28]. This lift has the property thatΠ ′ v is the unramified lift of π v for all v / ∈ S of § loc. cit. ]. From theclassification of automorphic forms for GL N [14], it is possible to find a globallygeneric representation Π of GL N ( A k ) such thatΠ v ∼ = Π ′ v for almost all places v of k . It is the unique generic subquotient of an globallyinduced representation of the form Ind(Π ′ ⊗ · · · ⊗ Π ′ d ) with each Π ′ i a cuspidalrepresentation of GL N i ( A k ). We write this as an isobaric sumΠ = Π ′ ⊞ · · · ⊞ Π ′ d , The representation Π is unitary, what we need to show is that each Π ′ i is unitary. Forthis, write Π ′ i = | det( · ) | t i Π i with each Π i unitary and t d ≥ · · · ≥ t , t ≤
0. BecauseΠ is unitary, we cannot have t >
0. Consider the quadruple ( k, π, Π , ψ, S ) ∈ N RATIONALITY AND HOLOMORPHY OF LANGLANDS-SHAHIDI L -FUNCTIONS 25 L glob ( p, G n , GL N ), then L ( s, π × ˜Π ) = L ( s, Π × ˜Π )= Y i L ( s + t i , Π i × ˜Π ) . From Theorem 6.5 we have that L ( s, π × ˜Π ) is holomorphic for Re( s ) >
1. However, L ( s + t , Π × ˜Π ) has a simple pole at s = 1 − t ≥
1. This pole carries throughto a pole of L ( s, π × ˜Π ), which gives a contradiction unless t = 0. A recursiveargument shows that all the t i ’s must be zero.Also, every time we have Π i ∼ = Π j , we add to the multiplicity of the pole at s = 1of the L -function(7.1) L ( s, π × ˜Π i ) = Y j L ( s, Π j × ˜Π i ) . From Theorem 6.5, L ( s, π × ˜Π j ) has at most a simple pole at s = 1. Hence, wemust have Π i ≇ Π j , for i = j .To show that each Π i is self-dual, we argue by contradiction and assume Π i ≇ ˜Π i .Notice that for ( k, π, Π i , ψ, S ) ∈ L glob ( p, G n , GL N i ), the representation σ = ˜Π i ⊗ ˜ τ of M ( A k ) satisfies w ( σ ) ≇ σ . From Corollary 4.2, the L -function L ( s, π × ˜Π i ) hasno poles. Now, we have that a pole appears on the right hand side of equation (7.1)from the L -function L ( s, Π i × ˜Π i ). This is a contradiction, unless Π i is self-dual.We now follow the proof of Proposition 9.5 of [31] to obtain that Π v is theunique local Langlands functorial lift of π v at every place v of k . A crucial stepis the stability property of γ -factors under twists by highly ramified characters,Theorem 6.10 of [28], which is available in characteristic two. Lemmas 9.2 and 9.3of [31] indicate how to obtain the desired equality of local factors. (cid:3) References [1] A.-M. Aubert, P. Baum, R. Plymen and M. Solleveld,
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