Temperedness at ramified places for globally generic exceptional groups

1 Introduction

Let X be a smooth projective geometrically connected curve over \F_q, let F=\F_q(X), and let G/F be a split connected adjoint absolutely simple group. Fix a split Borel subgroup B=TN, and write \A_F for the adeles of F. A cuspidal automorphic representation \pi of G(\A_F) is globally generic if some vector has a nonzero Whittaker coefficient

for a character \psi nontrivial in every simple-root coordinate. The generalized Ramanujan conjecture predicts tempered local components for such representations. Here temperedness means that the unitary local representation is weakly contained in the regular representation. This is the generic temperedness prediction within the broader framework of Arthur parameters [1, 25].

We prove the exceptional-group case, including the places where the representation is ramified. The global input is the unramified Ramanujan theorem of the companion paper [21, Corollary 1.2].

The distinction between spherical and ramified components is substantial. An unramified parameter is determined by its Satake class, whereas at a ramified place a Weil–Deligne parameter also contains a nilpotent monodromy operator. The local-global theorem available for general reductive groups identifies only the semisimple Weil parameter. The argument below supplies the additional monodromy comparison needed for temperedness.

1.1 Context and the local problem

Over function fields, Drinfeld proved Ramanujan for \GL_2 [10], and Laurent Lafforgue proved it for \GL_n [16, Théorème VI.10(i)]. For other reductive groups, genericity selects the part of the cuspidal spectrum for which temperedness is expected. The Langlands–Shahidi method relates this condition to local coefficients and automorphic L-functions. Lomelí developed this method over function fields and proved generic Ramanujan for split classical groups and quasi-split unitary groups [18, 19, 20]. In particular, the later split classical results include characteristic 2 [20, Theorem 7.1(i)].

Vincent Lafforgue’s excursion operators attach semisimple global parameters to cuspidal automorphic representations of general reductive groups [17]. Genestier–Lafforgue construct semisimple local parameters and prove local-global compatibility at every place [14]. Gan–Harris–Sawin, with an appendix by Beuzart-Plessis, show that the local parameter of a tempered representation admits an essentially tempered completion [11, Theorem 1.2 and Corollary 1.3]. These results make it possible to compare the global and representation-theoretic sources of monodromy without assuming a full local correspondence.

On the unramified side, Sawin–Templier prove temperedness under a monomial geometric supercuspidal hypothesis and a cyclic base-change hypothesis [24]. Ciubotaru–Harris obtain temperedness at unramified places under hypotheses including a generic unramified place and a tempered place [3]. The companion result used here gives the unramified conclusion for split adjoint absolutely simple groups from a generic unramified component alone [21, Corollary 1.2]. The present paper proves the local implication that extends this conclusion to the places of level.

Two established principles guide that implication. First, genericity is related to regularity at 1 of the adjoint L-factor, as formulated in the Gross–Prasad/Rallis conjecture [15, Conjecture 2.6]; see, for example, the criterion proved by Gan–Ichino under local-correspondence and local-factor hypotheses [12, Appendix B, Proposition B.1]. Second, adjoint regularity is equivalent to openness of the monodromy orbit over a fixed semisimple parameter [4, Proposition 3.5] and [6, Proposition 6.10]. We give the short Lie-algebra proof of the implication needed here. The broader principle that purity strongly constrains a completion of a semisimple Weil representation also appears in Taylor–Yoshida [26, Section 1].

Our task is to obtain the required adjoint regularity for arbitrary generic inducing data using the semisimple correspondence alone. We compare the total local coefficient with a product of Weil–Deligne L-factor ratios. Multiplicativity, rank-one Tate factors, the globalization theorem of Gan–Lomelí, and the crude functional equation of Lomelí provide the comparison [13, 19]. An elementary cancellation shows that these ratios do not depend on monodromy up to nonvanishing monomials. This is exactly the amount of local-factor compatibility required by the argument. Genestier–Lafforgue also construct local \gamma-factors from semisimple parameters and characterize them through global functional equations [14, Section 7].

1.2 The proof and its reusable local criterion

The proof has a local part and a global source of purity. At a fixed place, let r be the semisimple Weil parameter of a generic local representation. A Weil–Deligne completion is a nilpotent operator N satisfying \Ad(r(w))N=|w|N. The local result is Theorem 5.2: if a completion has adjoint representation pure of weight zero, then the representation is tempered. Purity here prescribes Frobenius eigenvalue sizes along monodromy chains, with weights symmetric about zero. This criterion applies to every split adjoint absolutely simple group over a local function field.

To see the mechanism, suppose the representation is not tempered. Its Langlands datum consists of a tempered representation \sigma of a proper Levi subgroup and a strictly positive real exponent \nu. A tempered completion for \sigma, twisted by \nu, gives a completion (r,N_M). The long intertwiner has generic image, so its scalar on Whittaker coinvariants is nonzero. The local-coefficient comparison then makes the adjoint L-factor of (r,N_M) regular at 1. A completion (r,N_*) with pure adjoint has the same regularity. Both operators therefore lie in the unique open orbit of the centralizer of r, and are conjugate. Figure 1 displays this comparison.

Purity now passes to the positive dual nilradical, a direct summand of the adjoint Weil–Deligne representation. A pure weight-zero representation has Frobenius determinant of absolute value 1. The positive exponent \nu makes that same determinant strictly smaller than 1. This contradiction proves the local criterion.

Globally, the companion theorem makes every unramified component tempered. The adjoint local system attached to an occurring excursion parameter is consequently pointwise pure of weight zero. Deligne’s theorem on local monodromy for curves [9, Théorème 1.8.4] supplies a pure adjoint completion at each place. Semisimple local-global compatibility identifies its Weil part with the parameter of \pi_v, and the local criterion applies.

Section 2 fixes conventions and states the parameter inputs. Section 3 proves the monodromy and purity lemmas. Section 4 establishes the local-coefficient comparison, including its behavior under specialization. Section 5 proves the local criterion, and Section 6 applies it to the global representation.

2 Local conventions and parameter theorems

We record the precise parameter theorems used below and explain how their semisimple Weil parameters relate to monodromy. The distinction is essential: the global theorem and the tempered-completion theorem will produce two potentially different monodromy operators over the same Weil parameter.

2.1 Groups, induction, and Whittaker characters

Let k be a nonarchimedean local field of positive characteristic, with residue field of cardinality Q. Fix a split adjoint absolutely simple group G over k, a split maximal torus T, and a Borel subgroup B=TU. In the local arguments, H denotes either G or a standard Levi subgroup of G; a standard parabolic of H is written P=MR. Dual groups have compatible positive systems, so positive coroots of H are positive roots of \widehat H. Put \widehat{\mathfrak n}=\Lie(\widehat R). All parabolic induction is normalized and is denoted I_P^H.

For a split group, a Whittaker character is a smooth unitary character of its maximal unipotent subgroup that is nontrivial on every simple root group. A smooth representation is generic if it admits a nonzero equivariant functional for such a character. We use uniqueness of Whittaker functionals for irreducible generic representations, heredity under normalized induction, and exactness of twisted unipotent coinvariants. These statements, and the local-coefficient constructions based on them, are valid over local function fields; see [19, Sections 1–2] and [7, Section 2]. Exactness is also [2, Proposition 1.9(a)]. For the foundational uniqueness and heredity results, see also [23, Théorèmes 2 and 4].

The following elementary observation ensures that Whittaker data can be chosen compatibly throughout the argument, even in small characteristic.

Write \mathfrak a_{M,\C}^*=X^*(M)\otimes_{\mathbb Z}\C. For z\in\mathfrak a_{M,\C}^*, the notation \sigma_z denotes the unramified twist in which s\chi acts as |\chi|^s. Via duality, X^*(M) is the cocharacter lattice of Z(\widehat M)^\circ; a central weight b on a representation of \widehat M therefore pairs with z, giving b(z). In particular, the weights on \widehat{\mathfrak n} satisfy b(\nu)>0 when \nu is in the open positive chamber for P.

2.2 Weil–Deligne parameters and local factors

Fix \ell\ne\operatorname{char}(k) and an isomorphism \iota:\overline{\Q}_\ell\simeq\C. All absolute values of coefficients below are taken after \iota. Choose the square roots in normalized induction and in the parameter theorems to correspond to positive real square roots. We use geometric Frobenius \Fr, so |\Fr|=Q^{-1}. Local class field theory sends a uniformizer to geometric Frobenius, and Satake parameters are normalized accordingly. Thus a twist by |\chi|^s on representations gives the same |\cdot|^s-twist on parameters.

A Frobenius-semisimple Weil–Deligne parameter for H is a pair (r,N), where r:W_k\longrightarrow\widehat H(\C) has finite inertia image and semisimple Frobenius, and N\in\Lie(\widehat H) is nilpotent with

For an algebraic representation of \widehat H on V, use the same symbols for the induced operators and set

This is a reciprocal polynomial in Q^{-s}, with constant term 1; in particular, it has no zeros. We often write L(s,a\circ(r,N)) when the algebraic representation a needs to be specified.

A tempered L-parameter is a homomorphism \phi:W_k\times\SL_2(\C)\longrightarrow\widehat H(\C), algebraic on \SL_2, whose Weil image is bounded. Its associated pair is

Thus the Weil part r of a tempered completion need not itself be bounded. This familiar distinction is the reason to retain monodromy.

We write f\doteq g for equality up to a nonzero constant times a monomial in the exponential unramified-twist coordinates. Finite covers of twist tori are allowed. In additive coordinates, the omitted factor is a constant times the exponential of a linear form, hence is holomorphic and nowhere zero. In one variable it has the form cQ^{-as}.

2.3 The parameter inputs

We use the following forms of the global and local parameter theorems. They specify the precise information carried from representations to parameters; no compatibility of monodromy is included.

The global statement is [17, Théorème 0.1]; the local statements, including representations without an integrality condition, are [14, Théorème 0.1 and Remarque 0.2]. For compatibility with central characters, apply the functoriality for homomorphisms with normal image to the inclusion of the central split torus.

Here is how to interpret the coefficient and occurrence assertions for complex representations. A finite-order central character allows a cocompact central lattice in its kernel. Fixed-level cusp spaces modulo that lattice are finite-dimensional and commute with coefficient extension. Project an irreducible Hecke module of level invariants to any excursion summand where its projection is nonzero; that projection is injective, so it supplies the required occurrence. Uniqueness of the excursion parameter of \Pi is unnecessary. For the adjoint group no central lattice is needed.

Locally, after transport by \iota, irreducibles may be realized over a finite extension of \Q_\ell. Indeed a fixed-compact-open Hecke algebra over \Q_\ell is of finite type [5, Theorem 1.1], so its finite-dimensional simple module over \overline{\Q}_\ell descends to such an extension. The corresponding irreducible representation also descends: form induction from this Hecke module and quotient by the largest subrepresentation with zero invariants under that compact open subgroup. The latter subrepresentation is characterized by vanishing of every averaged translate, a condition compatible with scalar extension. This permits use of the finite-extension formulations of the parameter theorems.

2.4 The Weil part of a global restriction

The local monodromy theorem turns a restriction of \Sigma_\Pi into a Weil–Deligne pair. The next observation verifies that its Frobenius-semisimple Weil part is exactly the parameter in Theorem 2.2(iii), also as a dual-group-valued parameter.

Thus a global restriction and a tempered local completion can have the same r while retaining distinct operators N. The next section gives two ways to work with this distinction: an L-factor ratio that forgets N, and a regularity condition that determines N up to conjugacy.

3 Adjoint regularity and monodromy

The semisimple Weil parameter does not determine individual local L-factors, since those factors also involve monodromy. We first show that the ratio occurring in a local functional equation is nevertheless determined up to a unit. We then explain how purity controls local L-factors and determinants. Finally, we prove that regularity of the adjoint L-factor at 1 determines the monodromy orbit. These statements will allow us to compare two monodromy operators once their semisimple Weil parameters have been identified, without assuming compatibility of the monodromy operators themselves.

3.1 A ratio independent of monodromy

3.2 Consequences of purity

Recall that the monodromy filtration centered at zero of a nilpotent operator N on V is the unique finite increasing filtration M_\bullet V, indexed by the integers, satisfying

Here M_jV=0 for sufficiently negative j and M_jV=V for sufficiently positive j. On a Jordan chain v,Nv,\ldots,N^dv, the successive vectors have degrees d,d-2,\ldots,-d. This description constructs the filtration and shows that it commutes with direct sums. Its uniqueness also shows that the Weil action preserves it, since conjugation by r(w) multiplies N by the nonzero scalar |w|.

We say that (r,N) is pure of weight zero, with respect to the fixed complex realization of the coefficients, if every Frobenius eigenvalue on \operatorname{Gr}^{M}_jV has absolute value Q^{j/2}. When N=0, this says that all eigenvalues of r(\mathrm{Fr}) have absolute value 1. Nonzero monodromy allows eigenvalues of different absolute values, arranged in symmetric Jordan chains.

For example, take trivial inertia on V=\mathbb C e_1\oplus \mathbb C e_2 and set

The degrees of e_1,e_2 are -1,1, so this representation is pure of weight zero although the image of r is unbounded. It comes from the L-parameter trivial on W_k and standard on \mathrm{SL}_2(\mathbb C). Replacing N by 0 destroys purity and introduces a pole at s=1/2 in the local L-factor. This is why boundedness of the Weil part and purity of a completed Weil–Deligne representation must be distinguished.

3.3 Adjoint regularity determines the orbit

The next proposition is the implication from adjoint regularity to an open monodromy orbit established in [4, Proposition 3.5] and [6, Proposition 6.10]. We give the elementary argument in the semisimple case needed here. The Killing form identifies the obstruction to an orbit being open with the space responsible for a pole of the adjoint L-factor at 1.

4 Local coefficients and semisimple parameters

The comparison in this section expresses the divisor of a Shahidi local coefficient in terms of the semisimple local parameter. It applies to arbitrary generic inducing representations, including ramified supercuspidals. We use multiplicativity to reduce to supercuspidal data, and then isolate the prescribed place in a global functional equation. Only the product of local factors occurring in a local coefficient is needed. Lemma 3.1 allows this product to be compared without identifying monodromy operators.

Let H be a split adjoint group under consideration, or a standard Levi subgroup of such a group, over the local function field k. Fix a split maximal torus T, a Borel subgroup B=TU, and a nondegenerate character \psi of U(k). Write P=MR for a proper standard parabolic subgroup of H. All induction is normalized, and we write

Here \sigma is an irreducible generic representation of M(k) and z\in\mathfrak a_{M,\mathbb C}^*=X^*(M)\otimes_{\mathbb Z}\mathbb C. Whittaker characters and Weyl representatives are chosen compatibly. Lemma 2.1 gives a single torus orbit of nondegenerate characters for these groups, so such a choice is possible for every generic \sigma.

Let w_H and w_M denote the longest Weyl elements of H and M, and put w=w_Hw_M. Let P' be the standard parabolic with Levi wMw^{-1}. The standard intertwining operator

is defined by its usual unipotent integral and meromorphic continuation. In particular, A_H has no normalization by local L-functions. We use the convention

for the local coefficient. We also use this definition for a Weyl element carrying the simple roots of its source Levi into the ambient simple roots. These are the intertwiners that occur in multiplicativity.

Let \widehat P=\widehat M\widehat R be the corresponding standard parabolic in \widehat H. Decompose its unipotent Lie algebra under the connected center of \widehat M:

The index b runs over the weights that occur, r_b is the representation of \widehat M on V_b, and b(z) is the pairing with the central cocharacter specified by the twist z.

We first explain how local coefficients behave under specialization and induction in stages. We then prove the rank-one and principal-series cases of the proposition. Globalization supplies the remaining supercuspidal case. The local-coefficient constructions and the crude functional equation are those of Lomelí [19, §§1–2 and Theorem 4.3]; the globalization method is that of Gan–Lomelí [13, Theorem 1.1 and §5]. For the related construction of \gamma-factors from semisimple parameters, see also [14, Section 7].

4.1 Whittaker lines in families

For a smooth representation V of H(k), write V_{U,\psi} for its twisted coinvariants. When V=I_P^H(\sigma_z) with \sigma generic, these coinvariants are one-dimensional. The scalar of the standard intertwining operator on this line is the inverse local coefficient. To use this description at a reducibility point, one must choose generators that remain nonzero after specialization.

Choose a finite cover of the torus of unramified twists on which the characters in use are algebraic, and let \mathcal A be its Laurent polynomial coordinate ring. The family \sigma_z is a smooth \mathcal A[M(k)]-module, and its induced family is a smooth \mathcal A[H(k)]-module. Meromorphic operators are obtained by extending scalars to the fraction field of \mathcal A.

4.2 Elementary intertwiners and reduction of the inducing data

We record the root decomposition that governs both sides of (4.3). For a Weyl element v, set

For a standard Levi L, write \Delta_L for its simple roots.

The standard intertwining integrals factor according to Lemma 4.3. First one uses Fubini’s theorem in a convergence chamber, and then meromorphic continuation. At each step, normalized induction in stages identifies the elementary operator with the maximal-parabolic operator in H_j. Taking Whittaker scalars and using Lemma 4.2 therefore gives multiplicativity of local coefficients, up to the allowed units. This is the multiplicativity of [19, Proposition 2.3].

We explain more precisely how it reduces the inducing representation. Suppose for the moment that P is maximal in H, with omitted simple root \alpha. Let \widetilde\alpha\in\mathfrak a_{M,\mathbb R}^* be proportional to the character giving the determinant on \operatorname{Lie}(R) and normalized by

Its pairing with every simple coroot of M is zero. The decomposition (4.2) is then conventionally written

where \widehat{\mathfrak h}_{\beta^\vee} is the corresponding root space in \operatorname{Lie}(\widehat H) and i ranges over the positive integers that occur. Along z=s\widetilde\alpha, the arguments in (4.3) are is and 1-is.

Every other twist direction is a sum of this relative direction and a direction in X^*(H)\otimes\mathbb C. A twist from H twists both sides of the intertwiner and changes its Whittaker scalar by at most a unit. On the dual side it acts through Z(\widehat H)^\circ and acts trivially on \widehat{\mathfrak n}. Thus it suffices to prove the maximal-parabolic comparison on the relative line.

By the subrepresentation theorem, \sigma embeds in normalized induction in M from a supercuspidal representation \sigma_0 of a standard Levi M_0\subseteq M. Exactness and heredity of Whittaker coinvariants show that \sigma_0 is generic. They also show that the embedding induces an isomorphism on Whittaker lines, since both lines are one-dimensional. The same holds after every unramified twist.

The element w=w_Hw_M carries the simple roots of M_0 into those of H. In the induced realization from M_0, its intertwining integral still uses exactly the roots outside M turned negative by w: w preserves the positive system of M in its target Levi. Thus induction in stages identifies the intertwiner on \sigma with the restriction of this intertwiner on the smaller inducing data. The isomorphisms of Whittaker lines just established identify their local coefficients, up to units.

Apply Lemma 4.3 with L=M_0, and write M_j=w_{<j}M_0w_{<j}^{-1} for the successive Levi subgroups. At step j, the inducing representation is the transport of \sigma_0 to M_j, and the twist is w_{<j}(s\widetilde\alpha). Its relative coordinate in H_j is c_js, where

Indeed, w_{<j}^{-1}\alpha_j^\vee belongs to the original inversion set of coroots, all of which are positive and outside M. Pairing with \widetilde\alpha is strictly positive on this set. Each elementary coefficient consequently has a genuinely varying argument; the reduction does not restrict a meromorphic coefficient to an identically singular locus.

The same factorization decomposes the dual representations. Transport the nilradical Lie algebra for M_j\subset H_j back by w_{<j}^{-1}. It is invariant under \widehat M_0, because the original algebra is invariant under \widehat M_j. The coroot partition (4.4) identifies their direct sum with \widehat{\mathfrak n} restricted to \widehat M_0. A step summand of relative degree i_j has twist exponent c_ji_js. To see this, decompose w_{<j}\widetilde\alpha into c_j\widetilde\alpha_j and a character direction from H_j; the latter pairs to zero with every coroot in H_j.

Compatibility of semisimple parameters with normalized induction identifies \rho_\sigma with the parameter obtained from \rho_{\sigma_0}. Taking monodromy zero, the preceding decomposition therefore makes the right side of (4.3) the product of the right sides for the elementary maximal parabolics. We have proved that comparison for maximal parabolics with supercuspidal inducing data implies comparison for maximal parabolics with arbitrary generic data.

4.3 The principal-series comparison

The rank-one case supplies the local comparisons needed away from the place prescribed in globalization. If a maximal proper Levi M is a torus, the derived root system of H has rank one. The local coefficient is the Tate gamma factor for the character obtained by pulling \sigma back along the coroot. Tate’s formula and local class field theory give

This calculation permits an arbitrary smooth inducing character; see [19, §1.3]. It also applies to the isogeny types occurring here. Pulling the integral back through the root homomorphism from \operatorname{SL}_2 identifies the root groups and the unipotent integration, and the inducing character pulls back through the stated coroot. Extra central tori have no effect on this calculation.

An irreducible generic constituent of a principal series can be embedded in a principal series, with a possibly Weyl-conjugate inducing character, by the subrepresentation and supercuspidal support theorems. The preceding elementary factorization with M_0=T reduces its comparison to (4.6). Thus the maximal-parabolic case of (4.3) is already proved whenever the inducing representation is a generic principal-series constituent. In particular it is available for ramified principal series; an Iwahori-fixed hypothesis is not needed.

4.4 Globalization of supercuspidal data

We now prove the remaining maximal-parabolic comparison. Let \sigma be a generic supercuspidal representation of M(k). We may first twist it so that its central character has finite order. Indeed, Z(M) is a split torus, a smooth character of its maximal compact subgroup has finite image, and restriction

has finite cokernel. Unramified twists can therefore adjust the finitely many uniformizer values to roots of unity. Twisting back shifts the relative parameter s; the remaining twist direction is from H and has the behavior already described. Twist compatibility of semisimple parameters gives the identical shift on the proposed Galois expression.

Write k=k_0((t)), and put K=k_0(t). Let v_0 be the rational place t=0, so K_{v_0}=k. Use the constant split models of H and M over K. The finite-order central character just obtained extends to a finite-order automorphic character of Z(M)(\mathbb A_K). Here is an explicit extension, to check the central-character hypothesis of globalization.

For one split central factor, the idele-class group of K fits into the split exact sequence

The degree-zero description follows from the triviality of \operatorname{Pic}^0(\mathbb P^1), and the uniformizer idele at v_0 splits the degree map. Prescribe the given unit character at v_0, cancel its restriction to k_0^\times using a character of the residue field at infinity, and take trivial characters at the remaining unit groups. This defines a finite-order character of the left group in (4.7). Give the degree generator the prescribed finite-order value of the local character on t. The resulting automorphic character has the desired restriction at v_0. Repeat for every split central factor.

Choose a nontrivial global additive character and the resulting generic character of the Borel unipotent of M. Its local character at v_0 is in the torus orbit for which \sigma is generic. The globalization theorem of Gan–Lomelí [13, Theorem 1.1] now produces a globally generic cuspidal representation \Pi of M(\mathbb A_K) such that

and whose central character is the finite-order character just constructed. Their theorem preserves the prescribed unipotent period; for the Borel unipotent and a nondegenerate character this is precisely global genericity. It imposes no restriction on the depth of \sigma or on the positive characteristic.

Choose a global parameter \Sigma occurring for \Pi, and enlarge a finite set S of places containing v_0 so that normalized Satake matching and all unramified conditions for the crude functional equation hold outside S. For the maximal parabolic under consideration, Lomelí’s crude functional equation is

In [19, Theorem 4.3], the denominator is written using the contragredient \widetilde\Pi and r_i. At an unramified place, contragredience inverts the Satake class up to the Levi Weyl group, so those Euler factors are exactly the factors for \Pi and r_i^\vee displayed here.

The same partial Euler products are the partial L-functions of r_i\circ\Sigma. Write \Phi_x for the local Weil–Deligne parameter obtained from \Sigma at x. The functional equation for a lisse sheaf on a curve, with ramified factors defined by inertia invariants, gives

This is the usual curve functional equation [8, §§9–10]; it follows from the trace formula and duality for middle extensions, and does not require purity. To check the direction of the local ratio, write the complete functional equation as L(s,V)=\varepsilon(s,V)L(1-s,V^\vee). Removing the Euler factors in S puts L_x(1-s,V^\vee)/L_x(s,V) on the right. The global epsilon factor is a nonzero constant times a monomial. After the substitutions s\mapsto is, its contribution has exactly the form suppressed by \doteq.

The \ell-adic functional equation is an identity of rational functions and is transported through the fixed coefficient identification. Its partial L-functions agree with the automorphic ones because their Euler factors agree outside S. The ramified factors are those of \Phi_x: the invariant stalk of the lisse sheaf is the inertia-invariant kernel of its monodromy, and Frobenius semisimplification does not change its determinant.

We can now isolate v_0. For each x\in S\setminus\{v_0\}, the representation \Pi_x is generic and is a principal-series constituent. The principal-series comparison identifies its factor in (4.8) with its factor in (4.9), up to a unit. Local-global compatibility, together with Lemma 2.4, identifies the Weil part of the Frobenius-semisimplified pair \Phi_x with \rho_{\Pi_x}. Lemma 3.1 then permits us to discard its monodromy in comparing the local ratios. Cancellation leaves

At v_0 the same compatibility and Lemma 3.1 replace \Phi_{v_0} by (\rho_\sigma,0), or by any specified completion (\rho_\sigma,N_\sigma). This proves (4.3) for maximal parabolics with supercuspidal data. The reduction above proves it for every generic inducing representation of a maximal Levi.

4.5 Completion of the comparison for an arbitrary parabolic

5 A local criterion for temperedness

We now apply the local-coefficient comparison to a generic Langlands quotient. It first supplies adjoint regularity for a completion built from the tempered inducing representation. A completion of the same semisimple parameter with pure adjoint will then force the inducing exponent to vanish.

The criterion separates the local issue from the global source of purity. In particular it requires neither a full local Langlands correspondence nor an identification of the two monodromy operators in advance.

6 From unramified Ramanujan to every place

We apply Theorem 5.2 to a global parameter. The only input specific to the global Ramanujan problem is the following result from the companion manuscript [21, Corollary 1.2].