The local Langlands correspondence for inner forms of $SL_n$

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $SL_n (F)$ enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of $SL_n (F)$. An analogous result is shown in the archimedean case. To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of $GL_n (F)$, when the fields are close enough compared to the depth of the representations.


Background
Let F be a local field and let D be a division algebra with centre F , of dimension d 2 ≥ 1 over F . Then G = GL m (D) is the group of F -rational points of an inner form of GL md . We will say simply that G is an inner form of GL n (F ), where n = md. It is endowed with a reduced norm map Nrd: GL m (D) → F × . The group G := ker(Nrd : G → F × ) is an inner form of SL n (F ). (The split case D = F is allowed here.) In this paper, we will complete the local Langlands correspondence for G .
We sketch how it goes and which part of it is new. For any reductive group over a local field, say H, let Irr(H) denote the set of (isomorphism classes of) irreducible admissi-ble H-representations. Let (H) be the collection of (equivalence classes of) Langlands parameters for H, as defined in [13].
The local Langlands correspondence (LLC) for GL n (F ) was established in the important papers [26,28,37,38,44,52]. Together with the Jacquet-Langlands correspondence [7,19,43], this provides the LLC for inner forms G = GL m (D) of GL n (F ), see [4,29]. Recall that (GL m (D)) (GL n (F )) if GL m (D) is not split. For these groups, every L-packet φ (G) is a singleton and the LLC is a canonical bijective map rec D,m : Irr(GL m (D)) → (GL m (D)). (1) The LLC for inner forms of SL n (F ) is derived from the above, in the sense that every L-packet for G consists of the irreducible constituents of Res G G ( φ (G)). Of course these L-packets have more than one element in general. To parametrize the members of φ (G ), one must enhance the Langlands parameter φ with an irreducible representation of a suitable component group. This idea originated for unipotent representations of p-adic reductive groups in [40, § 1.5]. For SL n (F ), φ is a map from the Weil-Deligne group of F to PGL n (C) and a correct choice is the group of components of the centralizer of φ in PGL n (C). Indeed, using this component group the enhanced Langlands correspondence for SL n (F ) was already established by Gelbart and Knapp [22, § 4]-at that time still assuming that it could be done for GL n (F ).
In general, more subtle component groups S φ are needed, see [3,32,48]. Our enhanced L-parameters will be pairs (φ , ρ) consisting of a Langlands parameter φ for G , enhanced with a ρ ∈ Irr(S φ ). We consider such enhanced L-parameters for G modulo the equivalence relation coming from the natural PGL n (C)-action. When two L-parameters φ 1 and φ 2 are conjugate, there is a canonical bijection Irr(S φ 1 ) → Irr(S φ 2 ), coming from conjugation by any g ∈ PGL n (C) with g −1 φ 1 g = φ 2 . With this in mind, it makes sense to speak about Irr(S φ ) for φ ∈ (G ). The LLC for G should be an injective map which satisfies several natural properties. The map will almost never be surjective, but for every φ which is relevant for G the image should contain at least one pair (φ , ρ). The image should consist of all such pairs, which satisfy an additional relevance condition on ρ. This form of the LLC was proven for "GL n -generic" representations of G in [29], under the assumption that the underlying local field has characteristic zero. A remarkable aspect of Langlands' conjectures [48] is that it is better to consider not just one reductive group at a time, but all inner forms of a given group simultaneously. Inner forms share the same Langlands dual group, so in (2) the right-hand side is the same for all inner forms H of the given group. The hope is that one can turn (2) into a bijection by defining a suitable equivalence relation on the set of inner forms and taking the corresponding union of the sets Irr(H) on the left-hand side. Such a statement was proven for unipotent representations of simple p-adic groups in [41].
Let us make this explicit for inner forms of GL n (F ), respectively, SL n (F ). We define the equivalence classes of such inner forms to be in bijection with the isomorphism classes of central simple F -algebras of dimension n 2 via M m (D) → GL m (D), respectively, M m (D) → GL m (D) der . This equivalence relation can also be motivated with Galois cohomology, see Sect. 2.
As Langlands dual group, we take GL n (C), respectively, PGL n (C). Langlands parameters for GL n (F ) (respectively, SL n (F )) take values in the dual group, and they must be considered up to conjugation. The group with which we want to conjugate Langlands parameters should be a central extension of the adjoint group PGL n (C), but apart from that there is some choice. It does not matter for the equivalence classes of Langlands parameters, but it is important for the component group of centralizers that we will obtain. The interpretation of inner forms via Galois cohomology entails [3] that we must consider the conjugation action of the simply connected group SL n (C) on the dual groups and on the collections of Langlands parameters for GL n (F ) or SL n (F ).
For any Langlands parameter φ for SL n (F ), we define the groups Notice that the centralizers are taken in SL n (C) and not in the Langlands dual group PGL n (C), where the image of φ lies. It is worth noting that our group C(φ ) coincides with the group S + φ defined by Kaletha in [32] with Z taken to be equal to the centre of SL n (F ).
More often one encounters the component group It related to (3) by the short exact sequence Given a Langlands parameter φ for GL n (F ), we can define C(φ), S φ and Z φ by the same formulas as in (3). Again C(φ) coincides with the group that Kaletha considers. It is easily seen that Z GL n (C) (im φ) is connected, so S φ ∼ = Z φ . The usual component group S φ is always trivial for GL n (F ). Hence, S φ (resp. S φ ) has more irreducible representations than S φ (resp. S φ ). Via the Langlands correspondence, the additional ones are associated with irreducible representations of non-split inner forms of GL n (F ) (resp. SL n (F )). For example, consider a Langlands parameter φ for GL 2 (F ) which is elliptic, that is, whose image is not contained in any torus of GL 2 (C). Then S φ = Z(SL 2 (C)) ∼ = {±1}. The pair (φ, triv S φ ) parametrizes an essentially square-integrable representation of GL 2 (F ) and (φ, sgn S φ ) parametrizes an irreducible representation of the non-split inner form D × , where D denotes a noncommutative division algebra of dimension 4 over F .
For general linear groups over local fields, we prove a result which was already known to experts, but which we could not find in the literature: There is a canonical bijection between: • pairs (G, π) with π ∈ Irr(G) and G an inner form of GL n (F ), considered up to equivalence; • pairs (φ, ρ) with φ ∈ (GL n (F )) and ρ ∈ Irr(S φ ).
For these Langlands parameters, S φ = Z φ and a character of Z φ determines an inner form of GL n (F ) via the Kottwitz isomorphism [36]. In contrast to the usual LLC, our packets for general linear groups need not be singletons. To be precise, the packet φ contains the unique representation rec −1 D,m (φ) of G = GL m (D) if φ is relevant for G, and no G-representations otherwise.
A similar result holds for special linear groups, but with a few modifications. Firstly, one loses canonicity, because in general there is no natural way to parametrize the members of an L-packet φ (G ) if there are more than one. Already for tempered representations of SL 2 (F ), the enhanced LLC is not canonical, see [6,Example 11.3]. It is possible to determine a unique parametrization of Irr(SL n (F )) by fixing a Whittaker datum and of Irr(G ) by adding more data [32], but this involves non-canonical choices.
Secondly, the quaternion algebra H turns out to occupy an exceptional position. Our local Langlands correspondence for inner forms of the special linear group over a local field F can be stated as follows:
(a) The group G determines ρ Z φ and conversely.
(b) The correspondence satisfies the desired properties from [13, § 10.3], with respect to restriction from inner forms of GL n (F ), temperedness and essential square integrability of representations.
In the archimedean case, the classification of Irr(SL m (D)) is well known, at least for D = H. The main value of our result lies in the strong analogy with the non-archimedean case. The reason for the lack of bijectivity for the special linear groups over the quaternions is easily identified. Namely, the reduced norm map for H satisfies Nrd(H × ) = R >0 , whereas for all other local division algebras D with centre F the reduced norm map is surjective, that is, Nrd(D × ) = F × . Of course, there are various ad hoc ways to restore the bijectivity in Theorem 1.2, for example by decreeing that SL m (H) appears twice among the equivalence classes of inner forms of SL 2m (R). This can be achieved in a natural way with strong inner forms, as in [1]. But one may also argue that for SL m (H) one would actually be better off without any component groups.
For p-adic fields F , the above theorem can be derived rather quickly from the work of Hiraga and Saito [29].
By far the most difficult case of Theorem 1.2 is that where the local field F has positive characteristic. The paper [29] does not apply in this case, and it seems hard to generalize the techniques from [29] to fields of positive characteristic.
Our solution is to use the method of close fields to reduce it to the p-adic case. Let F be a local field of characteristic p, o F its ring of integers and p F the maximal ideal of o F . There exist finite extensions F of Q p which are l-close to F , which means that o F /p l F is isomorphic to the corresponding ring for F . Let D be a division algebra with centre F , such that D and D have the same Hasse invariant. Let K r be the standard congruence subgroup of level r ∈ N in GL m (o D ), and let Irr(G, K r ) be the set of irreducible representations of G = GL m (D) with nonzero K r -invariant vectors. Define K r ⊂ GL m ( D) and Irr(GL m ( D), K r ) in the same way.
For l sufficiently large compared to r, the method of close fields provides a bijection which preserves almost all the available structure [7]. But this is not enough for Theorem 1.2; we also need to relate to the local Langlands correspondence. The l-closeness of F and F implies that the quotient of the Weil group of F by its l-th ramification subgroup is isomorphic to the analogous object for F [18]. This yields a natural bijection between Langlands parameters that are trivial on the respective l-th ramification groups. We show that: Theorem 1.3 (see Theorems 6.1 and 6.2) Suppose that F and F are l-close and that l is sufficiently large compared to r. Then the maps (1), (4) and (5) form a commutative diagram In the special case D = F and D = F, this holds for all l > 2 n−1 r.
The special case was also proven by Ganapathy [20,21], but without an explicit lower bound on l. Theorem 1.3 says that the method of close fields essentially preserves Langlands parameters. The proof runs via the only accessible characterization of the LLC for general linear groups: by means of -and γ -factors of pairs of representations [27].
To apply Henniart's characterization with maximal effect, we establish a result with independent value. Given a Langlands parameter φ, we let d(φ) ∈ R ≥0 be the smallest number such that φ / ∈ d(φ) (GL n (F )). That is, the smallest number such that φ is nontrivial on the d(φ)-th ramification group of the Weil group of F with respect to the upper numbering. For a supercuspidal representation π of GL n (F ), let d(π) be its normalized level, as in [14].

Proposition 1.4 (see Proposition 4.2)
The local Langlands correspondence for supercuspidal representations of GL n (F ) preserves depths, in the sense that d(π) = d(rec F,n (π)).

Inner forms of GL n (F)
Let F be a local field and let D be a division algebra with centre F , of dimension dim F (D) = d 2 . The F -group GL m (D) is an inner form of GL md (F ), and conversely every inner form of GL n (F ) is isomorphic to such a group.
In the archimedean case, there are only three possible division algebras: R, C and H. The group GL m (H) is an inner form of GL 2m (R), and (up to isomorphism) that already accounts for all the inner forms of the groups GL n (R) and GL n (C). One can parametrize these inner forms with characters of order at most two of Z(SL n (C)), such that GL n (F ) is associated with the trivial character and GL m (H) corresponds to the character of order two of Z(SL 2m (C)).
Until further notice we assume that F is non-archimedean. Let us make our equivalence relation on the set of inner forms of GL n (F ) precise. Following [29], we consider inner twists of GL n , that is, pairs (G, ξ ) where G is an inner form of GL n and ξ : GL n → G is an isomorphism defined overF , such that ξ −1 • σ • ξ • σ −1 is an inner automorphism of GL n for every σ in the absolute Galois group of F . Equivalence classes of inner twists are classified by the first Galois cohomology group of F with values in the adjoint group of GL n , H 1 (F, PGL n ). For a more explicit interpretation, we note that H 1 (F, PGL n ) also parametrizes the isomorphism classes of central simple F -algebras of dimension n 2 . In other words, we have defined that our equivalence classes of inner forms of GL n (F ) are in bijection with the isomorphism classes of central simple F -algebras of dimension n 2 via M m (D) → GL m (D). By [36,Proposition 6.4], there exists a natural bijection In fact, such a map exists on general grounds [36, § 6.5], in characteristic 0 it is bijective by [33,Satz 2] and in positive characteristic by [47]. Clearly, the map is bijective. The composition of these two maps can also be interpreted in terms of classical number theory. For M m (D) with md = n, the Hasse invariant h(D) (in the sense of Brauer theory) is a primitive d-th root of unity. The element of H 1 (F, PGL n ) associated with M m (D) has the same image h(D) in {z ∈ C × : z n = 1}. In particular, 1 ∈ C × is associated with M n (F ) and the primitive n-th roots of unity correspond to division algebras of dimension n 2 over their centre F . We warn the reader that our equivalence relation on the set of inner forms of GL n (F ) is rougher than isomorphism. Namely, if h(D ) = h(D) −1 , then M m (D ) is isomorphic to the opposite algebra of M m (D) and is a group isomorphism. Since x → x −T is the only nontrivial outer automorphism of GL n , all isomorphisms between groups GL m (D) arise in this way (up to inner automorphisms).
Furthermore, there is a standard presentation of the division algebras D, which is especially useful in relation to the method of close fields. Let L be the unique unramified extension of F of degree d, and let χ be the character of Gal(L/F ) ∼ = Z/dZ, which sends the Frobenius automorphism to h(D). If F is a uniformizer of F , then D is isomorphic to the cyclic algebra [L/F, χ, F ], see Definition IX.4.6 and Corollary XII.2.3 of [50]. We will call a group of the form a standard inner form of GL n (F ). The local Langlands correspondence for G = GL m (D) has been known to experts for considerable time, although it did not appear in the literature until recently [4,29].
We need to understand it well for our later arguments, so we recall its construction. It generalizes and relies on the LLC for general linear groups: rec F,n : Irr(GL n (F )) → (GL n (F )).
The latter was proven for supercuspidal representations in [26,28,38] and extended from there to Irr(GL n (F )) in [52].
As G is an inner form of GL n (F ),Ǧ = GL n (C) and the action of Gal(F /F ) on GL n (C) determined by G is by inner automorphisms. Therefore, we may take as Langlands dual group L G =Ǧ = GL n (C).
Let us recall the notion of relevance of Langlands parameters for non-split groups. Let φ ∈ (GL n (F )), and letM ⊂ GL n (C) be a Levi subgroup that contains im(φ) and is minimal for this property. As for all Levi subgroups, for some integers n i with k i=1 n i = n. Then φ is relevant for G if and only ifM corresponds to a Levi subgroup M ⊂ G. This is equivalent to m i := n i /d being an integer for all i. Moreover, in that case By definition, (G) consists of the G-relevant elements of (GL n (F )). Consider any φ ∈ (G). Conjugating by a suitable element ofǦ, we can achieve that Levi subgroups of GL n (C) and GL m (D), respectively; ) and im(φ i ) not contained in any proper Levi subgroup of GL n i (C); • φ i and φ j are not equivalent if i = j.
Then rec −1 F,n i (φ i ) ∈ Irr(GL n i (F )) is essentially square-integrable. Recall that the Jacquet-Langlands correspondence [7,19,43] is a natural bijection JL : Irr essL 2 (GL m (D)) → Irr essL 2 (GL n (F )) between essentially square-integrable irreducible representations of G = GL m (D) and GL n (F ). It gives We remark that ω is square-integrable modulo centre if and only all rec −1 F,n i (φ i ) are so, because this property is preserved by the Jacquet-Langlands correspondence. The Zelevinsky classification for Irr(GL n i (F )) [52] (which is used for rec F,n i ) shows that, in the given circumstances, this is equivalent to φ i being bounded. Thus, ω is square-integrable modulo centre if and only φ is bounded.
The assignment φ → (M, ω) sets up a bijection It is known from [19, Theorem B.2.d] and [8] that for inner forms of GL n (F ) normalized parabolic induction sends irreducible square-integrable (modulo centre) representations to irreducible tempered representations and that every irreducible tempered representation can be obtained in that way. This shows that the tempered part of Irr(G) is naturally in bijection with the tempered part (i.e. ω tempered) of the right-hand side of (12). Now we analyse the non-tempered part of Irr(G). Let M 1 be a Levi subgroup of G containing M such that ω is square-integrable modulo the centre of M 1 , and such that M 1 is maximal for this property. Let P 1 be a parabolic subgroup of M 1 with Levi factor M. Then ω 1 = I M 1 P 1 (ω) is irreducible and independent of P 1 , while by the aforementioned results the restriction of ω 1 to the derived group of M 1 is tempered. Furthermore, the absolute value of the character of ω 1 on Z(M 1 ) is regular in the sense that no root of (G, Z(M 1 )) annihilates it. Hence, there exists a unique parabolic subgroup P 2 of G with Levi factor M 1 , such that (P 2 , ω 1 ) satisfies the hypothesis of the Langlands classification [34,37]. That result says that I G P 2 (ω 1 ) has a unique irreducible quotient L(P 2 , ω 1 ) and that every irreducible G-representation can be obtained in this way, from data that are unique up to G-conjugation. This provides a canonical bijection between Irr(G) and the righthand side of (12).
To summarize the above constructions, let P be a parabolic subgroup of G with Levi factor M, such that PM 1 = P 2 . By the transitivity of parabolic induction, I G P (ω) has a unique irreducible quotient, say L(P, ω), and it is isomorphic to L(P 2 , ω 1 ). The composite map is the local Langlands correspondence for GL m (D). Since it is bijective, all the L-packets is tempered if and only if ω is square-integrable modulo centre, which by the above is equivalent to boundedness of φ ∈ (G).
We denote the inverse of (13) by Because both the LLC for Irr essL 2 (GL n i (F )), the Jacquet-Langlands correspondence and I G P respect tensoring with unramified characters, rec D,m (L(P, ω ⊗ χ)) and rec D,m (L(P, ω)) differ only by the unramified Langlands parameter for M which corresponds to χ.
In the archimedean case, Langlands [37] himself established the correspondence between the irreducible admissible representations of GL m (D) and Langlands parameters. The paper [37] applies to all real reductive groups, but it completes the classification only if parabolic induction of tempered representations of Levi subgroups preserves irreducibility. That is the case for GL n (C) by the Borel-Weil theorem and for GL n (R) and GL m (H) by [11, § 12].
The above method to go from essentially square-integrable to irreducible admissible representations is essentially the same over all local fields and stems from [37]. There also exists a Jacquet-Langlands correspondence over local archimedean fields [19,Appendix D]. Actually it is very simple, the only nontrivial cases are GL 2 (R) and H. Therefore, it is justified to say that (11)- (14) hold in the archimedean case.
With the S-groups from [3], we can build a more subtle version of (14). Since Z GL n (C) (φ) is connected, Let χ G ∈ Irr Z(SL n (C)) be the character associated with G via (7) or (6).
Proof This can be derived with [2, Corollary 2.2] and [29, Lemma 9.1]. However, we prefer a more elementary proof.
Replacing φ by a suitable GL n (C)-conjugate L-parameter, we may assume that a Levi subgroup minimally containing φ is As explained in (10), φ is relevant for GL m (D) if and only d divides every n j .
The determinant of a typical element e z 1 I n 1 , . . . , e z k I n k is exp(n 1 z 1 +· · ·+n k z k ). Hence, the Lie algebra of Z SL n (C) (im φ) is determined by the equation n 1 z 1 + · · · + n k z k = 0 and . , e z k I n k |z i ∈ C, n 1 z 1 + · · · + n k z k = 0} .
For any integer l, we have e z 1 I n 1 , . . . , e z k I n k = e 2π il/n I n if and only if z j 2π i ∈ Z + l n for all j. This lies in C(φ) • if and only if there are integers l j such that k j=1 n j (l/n + l j ) = l + k j=1 n j l j is zero. That is only possible if l is a multiple of the greatest common divisor g of the n j .
Reconsider GL m (D) as above. As discussed after (7), the character χ GL m (D) has order d. In particular, its kernel consists of the d-th powers in Z(SL n (C)). Now we can conclude with some equivalences: We regard e (inn GL n (F )) := (φ, ρ) : φ ∈ (GL n (F )), ρ ∈ Irr(S φ ) as the collection of enhanced Langlands parameters for all inner forms of GL n (F ). With this set, we can establish the local Langlands correspondence for all such inner forms simultaneously. To make it bijective, we must choose one group in each equivalence class of inner forms of GL n (F ). In the archimedean case it suffices to say that we use the quaternions, and in the non-archimedean case we take the standard inner forms (8).

Theorem 2.2 Let F be a local field. There exists a canonical bijection
It extends the local Langlands correspondence for GL n (F ), which can be recovered by considering only pairs (φ, triv S φ ).
Proof The elements of e (inn GL n (F )) with a fixed φ ∈ (GL n (F )) are First, we consider the non-archimedean case. By Lemma 2.1 and (7), (16) is in bijection with the equivalence classes of inner forms of GL n (F ) for which φ is relevant. In each such equivalence class, there is a unique standard inner form Hence, e (inn GL n (F )) contains precisely one element for every pair (G, φ) with G a standard inner form of GL n (F ) and φ a Langlands parameter, which is relevant for G. Now we apply the LLC for G (13) and define that the map in the theorem sends (φ, χ) to ,m is a canonical bijection, so is the thus obtained map.
In the archimedean case, the above argument does not suffice, because some characters of Z(SL n (C)) do not parametrize an inner form of GL n (F ). We proceed by direct calculation, inspired by [37, § 3].
Suppose that F = C. Then W F = C × and im(φ) is just a real torus in GL n (C). Hence, Z GL n (C) (φ) is a Levi subgroup of GL n (C) and C(φ) = Z SL n (C) (φ) is the corresponding Levi subgroup of SL n (C). All Levi subgroups of SL n (C) are connected, so S φ = C(φ)/C(φ) • = 1. Consequently, e (inn GL n (C)) = (GL n (C)), and the theorem for F = C reduces to the Langlands correspondence for GL n (C). Now we take F = R. Recall that its Weil group is defined as Let M be a Levi subgroup of GL n (C) which contains the image of φ and is minimal for this property. Then φ(C × ) is contained in a unique maximal torus T of M. By replacing φ by a conjugate Langlands parameter, we can achieve that is standard and that T is the torus of diagonal matrices. Then the projection of φ(W R ) on each factor GL i (C) of M has a centralizer in GL i (C), which does not contain any torus larger than Z(GL i (C)). On the other hand, φ(τ ) normalizes Z M (φ(C × )) = T , so Z GL n (C) (φ) = Z T (φ(τ )). It follows that n i = 0 for i ≥ 2. The projection of φ(τ ) on each factor GL 2 (C) of M is either 0 1 −1 0 or 0 −1 1 0 . Hence, Z GL n (C) (φ) contains the torus Together with (15) this shows that S φ = 1 if n is odd or if n is even and φ is not relevant for GL n/2 (H). Now suppose n 1 = 0. Then n = 2n 2 , φ is relevant for GL n 2 (H) and T φ = (z j I 2 ) n 2 j=1 : z j ∈ C × . We see that T φ ∩ SL n (C) has two components, determined by whether n 2 j=1 z j equals 1 or −1. Write φ = n 2 j=1 φ j with φ j ∈ (GL 2 (R)). We may assume that φ is normalized such that, whenever φ j is GL 2 (C)-conjugate to φ j , actually φ j = φ j = φ k for all k between j and j . Then Z M n (C) (φ) is isomorphic to a standard Levi subalgebra A of M n 2 (C), via the ring homomorphism Thus, we checked that for every φ ∈ (GL n (R)), Irr(S φ ) parametrizes the equivalence classes of inner forms G of GL n (R) for which φ is relevant. To conclude, we apply the LLC for G.

Inner forms of SL n (F)
As in the previous section, let D be a division algebra over dimension d 2 over its centre F , with reduced norm Nrd: D → F . We write Notice that it equals the derived group of GL m (D). It is an inner form of SL md (F ), and every inner form of SL n (F ) is isomorphic to such a group. We use the same equivalence relation and parametrization for inner forms of SL n (F ) as for GL n (F ), as described by (7) and (6).
As Langlands dual group of G = GL m (D) , we take In particular, every Langlands parameter for G = GL m (D) gives rise to one for G . In line with [13, § 10], the L-packets for G are derived from those for G in the following way. It is known [49] that every φ ∈ (G ) lifts to a φ ∈ (G). The L-packet φ (G) from (13) consists of a single G-representation, which we will denote by the same symbol. Its restriction to G depends only on φ , because a different lift φ of φ would produce φ (G), which only differs from φ (G) by a character of the form We call the restriction of φ (G) to G π φ (G) . In general it is reducible, and with it one associates the L-packet φ (G ) := π ∈ Irr(G ) : π is a constituent of π φ (G) .
The goal of this section is an analogue of Theorem 2.2. First we note that every irreducible G -representation (say π) is a member of an L-packet φ (G ), because it appears in a G-representation (for example in Ind G G π). Thus, the main problem is the parametrization of the L-packets. Such a parametrization of φ (G ) was given in [29] in terms of S-groups, at least when F has characteristic zero and φ (G) is "GL n -generic". After recalling this method, we will generalize it. Put Notice that every element of X G ( φ (G)) is a character, which by Schur's lemma is trivial on Z(G). Since G/G Z(G) is an abelian group and all its elements have order dividing n, the same goes for X G ( φ (G)). Moreover, X G ( φ (G)) is finite, as we will see in (21). On general grounds [29,Lemma 2.4], there exists a 2-cocycle κ φ such that More explicitly, by [29,Lemma 2.4] there are elements which together form a basis of End G ( φ (G)). The map γ → I γ is in general not multiplicative, but it is multiplicative up to scalars, and that accounts for the cocycle κ φ . By [29, Corollary 2.10], the decomposition of π φ (G) as a representation of G × X G ( φ (G)) is Let us compare C(φ ) with C(φ) = Z SL n (C) (φ). Since the kernel of the projection GL n (C) → PGL n (C) is central and φ is the projection of φ, the subgroups C(φ) ⊂ C(φ ) of SL n (C) have the same Lie algebra. Consequently, We also note that Assume for the moment that D H, so Nrd: D → F is surjective by [50, Proposition X.2.6]. Letγ : W F → C × ∼ = Z(GL n (C)) correspond to γ ∈ Irr(F × ) ∼ = Irr(G/G ) via local class field theory. By the LLC for G, φ is GL n (C)-conjugate to φγ for all γ ∈ X G ( φ (G)). As (φγ ) = φ , φ and φγ are in fact conjugate by an element of C(φ ) ⊂ SL n (C). This gives an isomorphism showing in particular that the left-hand side is abelian. Since C(φ )/C(φ) is the component group of the centralizer of the subset im(φ ) of the algebraic group PGL n (C), the groups in (21) are finite. Thus, we obtain a central extension of finite groups The algebra (17) can be described with the idempotent such that for any s ∈ S φ the subspaces CsZ φ on both sides correspond. Moreover, any two such isomorphisms differ only by a character of S φ /Z φ .
Proof First we determine the difference of two such isomorphisms. Their composition is an algebra automorphism of C[S φ /Z φ , κ φ ], which preserves each of the subspaces CsZ φ . The multiplication rules in this algebra show that the automorphism is the C-linear extension of sZ φ → λ(s)sZ φ for a character λ of S φ /Z φ . Now we suppose that char(F ) = 0 and that the representation φ (G) is tempered. In the archimedean case, the cocycle κ φ is trivial by [29, Lemma 3.1 and p. 69]. In the nonarchimedean case, the theorem is a reformulation of [29,Lemma 12.5]. We remark that this is a deep result, and its proof makes use of endoscopic transfer and global arguments.
Next we consider the case where char(F ) = 0 and we have a possibly unbounded Langlands parameter φ ∈ (G ), with a lift φ ∈ (G). Let Y be a connected set of unramified twists φ χ of φ, such that C(φ χ ) = C(φ) and C(φ χ ) = C(φ ) for all φ χ ∈ Y . It is easily seen that we can always arrange that Y contains bounded Langlands parameters, confer [5,Proposition 3.2]. The reason is that for any element (here the image of a Frobenius element of W F under φ) of a torus in a complex reductive group, there is an element of the maximal compact subtorus which has the same centralizer.
The construction of the intertwining operators from (18) is similar to that for R-groups. It determines the 2-cocycle κ φ by The I γ can be chosen independently of χ ∈ X nr (M), so the κ φ χ do not depend on χ. For φ χ tempered, we already have the required algebra isomorphisms, and now they extend by constancy to all φ χ ∈ Y . This concludes the proof in the case char(F ) = 0.
The proof of the case char(F ) > 0 requires more techniques; we dedicate Sects. 4-6 to it.
For a character χ of Z φ or of Z(SL n (C)), we write We will use this with the characters χ G = χ G from Lemma 2.1. We still assume that D H. As shown in [29, Corollary 2.10], the isomorphism (17) and Theorem 3.2 imply that defines an irreducible G -representation for every ρ ∈ Irr(S φ , χ G ). More precisely, (24) determines a bijection where [π] denotes the isomorphism class of π. In general π(φ , ρ) is not canonical, it depends on the choice of an algebra isomorphism as in Theorem 3.2. Hence, the map ρ → π(φ , ρ) is canonical up to an action of on Irr e χ G C[S φ ] . Via (19) and Theorem 3.2, this corresponds to an action of Irr X G ( φ (G)) on φ (G), which can be described explicitly. Since . This means that every c ∈ Irr X G ( φ (G)) determines a coset g c H in G. Now the formula defines the action of Irr X G ( φ (G)) on φ (G ). In other words, the representation is irreducible, the action of G on φ (G ) is in fact transitive, which means that with suitable choices one can arrange that π(φ , ρ) is any element of this L-packet.
For D = H, some modifications must be made. In that case, G = G Z(G), so Res G G preserves irreducibility of representations and X G ( φ (G)) = 1. Moreover, G/G ∼ = R × >0 R × , which causes (21) and (22) to be invalid for D = H. However, (23) still makes sense, so we define As mentioned before, Hiraga and Saito [29] have established the local Langlands correspondence for irreducible "GL n -generic" representations of inner forms of SL n (F ), where F is a local field of characteristic zero. We will generalize this on the one hand to local fields F of arbitrary characteristic and on the other hand to all irreducible admissible representations. We will do so for all inner forms of SL n (F ) simultaneously, to obtain an analogue of Theorem 2.2.
Notice that the restriction of ρ to Z φ ∼ = Z(SL n (C))/Z(SL n (C)) ∩ C(φ ) • determines an inner form G ρ of GL n (F ) (up to equivalence) via (7) and Lemma 2.1. Its derived group G ρ is the inner form of SL n (F ) associated with ρ. We note that the actions of PGL n (C) on the various e (G ) combine to an action on e (inn SL n (F )). With the collection of equivalence classes e (inn SL n (F )), we can formulate the local Langlands correspondence for all such inner forms simultaneously.
First we consider the non-archimedean case. As for GL n (F ), we fix one group in every equivalence class of inner forms. We choose the groups GL m ([L/F, χ, F ]) with [L/F, χ, F ] as in (8) and call these the standard inner forms of SL n (F ).
Proof Let φ ∈ (SL n (F )) and lift it to φ ∈ (GL n (F )). Then By this and Lemma 2.1, the set of standard inner forms of SL n (F ) for which φ is relevant is in natural bijection with Hence, the collection of (φ , ρ) ∈ e (inn SL n (F )) with φ fixed is The description of G ρ in part (a) is a reformulation of (7) and our choice of standard inner forms of SL n (F ). By (25), the map of the theorem provides a bijection between (28) and the set of pairs (G , π) where G is a standard inner form of SL n (F ) for which φ is relevant and π ∈ Irr(G ) is a constituent of φ (G). Now we see from Lemma 3.1 and Theorem 2.2 that the map of the theorem is bijective. Part (b) is a consequence of (17) and Theorem 3.2, see [29, Corollary 2.10]. Parts (c) and (d) follow from the analogous statements for inner forms of GL n (F ) (which were discussed after (13)) in combination with [46, Proposition 2.7].
Let us formulate an archimedean analogue of Theorem 3.3, that is, for the groups SL n (C), SL n (R) and SL m (H). In view (27), we cannot expect a bijection, and part (b) has to be adjusted.

Theorem 3.4 Let F be R or C. There exists a canonical surjection
with the following properties: (a) The preimage of Irr(SL n (F )) consists of the (φ , ρ) with Z φ ⊂ ker ρ, and the map is injective on this domain. The preimage of Irr(SL n/2 (H)) consists of the (φ , ρ) such that ρ is not trivial on Z φ , and the map is two-to-one on this domain. (b) Suppose that φ is relevant for G = SL m (D) and lifts to φ ∈ (G). Then the restriction of φ (G) to G is irreducible if D = C or D = H and is isomorphic to ρ∈Irr(S φ /Z φ ) π(φ , ρ) ⊗ ρ in case D = R.

(c) π(φ , ρ) is essentially square-integrable if and only if φ (W F ) is not contained in any proper parabolic subgroup of PGL n (C)). (d) π(φ , ρ) is tempered if and only if φ is bounded.
Proof Theorem 2.2 and the start of the proof of Theorem 3.3 show that (28) is also valid in the archimedean case. To see that the map thus obtained is canonical, we will of course use that the LLC for GL m (D) is so. For SL n (F ), the intertwining operators admit a canonical normalization in terms of Whittaker functionals [29, pp. 17 and 69], so the definition (24) of π(φ , ρ) can be made canonical. For SL m (H), the definition (27) clearly leaves no room for arbitrary choices.
Part (a) and part (b) for D = R follow as in the non-archimedean case, except that for D = H the preimage of π(φ , ρ) is in bijection with Irr(S φ , e H × ). To prove part (b) for D = C and D = H, it suffices to remark that Res G G preserves irreducibility, as G = G Z(G). The proof of parts (c) and (d) carries over from Theorem 3.3.
It remains to check that the map is two-to-one on Irr(SL m (H)). For this, we have to compute Consider φ ∈ (SL m (H)) with two lifts φ, φ ∈ (GL m (H)) that are conjugate under GL 2m (C). The restriction of φ −1 φ to C × ⊂ W R is a group homomorphism c : C × → Z(GL 2m (C)). Clearly φ and φ can only be conjugate if c = 1, so φ can only differ from φ on τ ∈ W R . Since Recall the standard form of φ exhibited in the proof of Theorem 2.2, with image in the Levi subgroup GL 2 (C) m of GL 2m (C). It shows that the Langlands parameter φ determined by φ (τ ) = −φ(τ ) is always conjugate to φ, for example by the element diag(1, −1, 1, . . . , −1) ∈ GL 2m (C). Therefore, (29) has precisely two elements. Now e H × C[S φ ] is a two-dimensional semisimple C-algebra, so it is isomorphic to C ⊕ C. We conclude that Irr(S φ , e H × ) has two elements, for every φ ∈ (SL m (H)).

Characterization of the LLC for some representations of GL n (F)
In this section, F is any local non-archimedean field. It is known from [27] that generic representations of GL n (F ) can be characterized in terms of γ -factors of pairs, where the other part of the pair is a representation of a smaller general linear group. We will establish a more precise version for irreducible representations that have nonzero vectors fixed under a specific compact open subgroup. Let F s be a separable closure of F , and let Gal(F s /F ) l be the l-th ramification group of Gal(F s /F ), with respect to the upper numbering. We define It is known that the set of l's at which Gal(F s /F ) l jumps consists of rational numbers and is discrete [45,Chap. IV,§ 3]. In particular, there exists a unique rational number d(φ) such that φ / ∈ d(φ) (GL n (F )) and φ ∈ l (GL n (F )) for any l > d(φ).
We will say that φ ∈ (GL n (F )) is elliptic if its image is not contained in any proper Levi subgroup of GL n (C). Recall the Swan conductor swan(φ) from [25, § 2].

(31)
Proof Recall that the filtration of Gal(F s /F ) with the lower numbering only has jumps at integer values of the index. Let c(φ) denote the smallest integer such that Let a(φ) denote the Artin conductor of φ. Because φ is assumed to be elliptic, the restriction of φ to W F is irreducible. The equality Then it follows from the definition of c(φ) that d(φ) is the largest rational number such that Let m, m be integers satisfying m > m ≥ m/2 . There is a canonical isomorphism given by x → x − 1. This leads to an isomorphism from p −1 P −m /p −1 P −m to the Pontryagin dual of U m +1 (A)/U m+1 (A), explicitly given by We recall from [16, (1.5)] that a stratum is a quadruple [A, m, m , β] consisting of a hereditary o F -order A in M n (F ), integers m > m ≥ 0, and an element β ∈ M n (F ) with A-valuation ν A (β) ≥ −m. A stratum of the form [A, m, m − 1, β] is called fundamental [16, (2.3)] if the coset β + p −1 P 1−m does not contain a nilpotent element of M n (F ). We remark that the formulation in [14] is slightly different because the notion of a fundamental stratum there allows m to be 0.
Fix an irreducible supercuspidal representation π of GL n (F ). According to [14,Theorem 2], there exists a hereditary order A in M n (F ) such that either where (m, A) ranges over all pairs consisting of an integer m ≥ 0 and a hereditary o F -order in M n (F ) such that π contains the trivial character of U m+1 (A).
The following result was claimed in [51, Theorem 2.3.6.4]. Although Yu did not provide a proof, he indicated that an argument along similar lines as ours is possible.
Recall that f (π) is an integer, called the conductor of π. It follows from (33) and (34) that In the case when π is an unramified representation of F × , the inertia subgroup I F is contained in ker φ, with φ = rec F,1 (π). Hence, (31) implies that a(φ) = 0. On the other hand, π is trivial on o × F , and a fortiori trivial on 1 From now on, we will assume that we are not in the above special case, that is, we assume that n = 1 or that π is ramified. Let A be a principal o F -order in M n (F ) such that e(A) = n/ gcd(n, f (π)), and let K(A) denote the normalizer in GL n (F ) of A. By [14,Theorem 3], the restriction of π to K(A) contains a nondegenerate (in the sense of [14, (1.21)]) representation of K(A), and we have [14, (3.7)] where d( ) ≥ 0 is the least integer such that  (35) and (36) that for every nondegenerate irreducible representation ρ of K(A) which occurs in the restriction of π to K(A). It follows from the definition (32) of d(π), that for every nondegenerate irreducible representation ρ of K(A) which occurs in the restriction of π to K(A). We will check that (38) is actually an equality. The case where d(π) = 0 is easy, so we only consider d(π) > 0.
Let A be any hereditary o F -order A in M n (F ), and define m A (π) to be the least nonnegative integer m such that the restriction of π to U m+1 (A ) contains the trivial character. Then choose A so that m A (π)/e(A ) is minimal, and let [A , m A (π), m A (π) − 1, β] be a stratum occurring in π. By [14,Theorem 2 ], this is a fundamental stratum. By [14, (3.4)], we may assume that the integers e(A ) and m A (π) are relatively prime. Hence, we may apply [14, (3.13)]. We find that A is principal and that every irreducible representation of K(A ) which occurs in the restriction of π to K(A ), and such that the restriction of to U m A (π ) (A ) contains ψ β , is nondegenerate. In particular, we have d( ) = m A (π).
It remains to check that the principal order A satisfies e(A ) = n/ gcd(n, f (π)).
Hence, we get that is, Hence, we have Since e(A ) and d(ρ ) = m A (π) are relatively prime, we deduce that e(A ) divides n . Thus, we have e(A ) = n , which means that (39) holds. We conclude that (38) is indeed an equality, which together with (37) shows that d( ) = d(π).
As congruence subgroups are the main examples of groups like U m (A) above, they have a link with depths. This can be made precise. Let K 0 = GL n (o F ) be the standard maximal compact subgroup of GL n (F ) and define, for r ∈ Z >0 : We denote the set of irreducible smooth GL n (F )-representations that are generated by their K r -invariant vectors by Irr(GL n (F ), K r ). To indicate the ambient group GL n (F ), we will sometimes denote K r by K r,n .

Lemma 4.3
For π ∈ Irr(GL n (F )) and r ∈ Z >0 , the following are equivalent: Proof For this result, it is convenient to use the equivalent definition of depth provided by Moy and Prasad [42]. In their notation, the group K r is P o,(r−1)+ , where o denotes the origin in the standard apartment of the Bruhat-Tits building of GL n (F ). From the definition in [42, § 3.4], we read off that any π ∈ Irr(GL n (F ), K r ) has depth ≤ r − 1.
Conversely, suppose that d(π) ≤ r − 1. Then π has nonzero vectors fixed by the group P x,(r−1)+ , where x is some point of the Bruhat-Tits building. Since we may move x within its GL n (F )-orbit and there is only one orbit of vertices, we may assume that x lies in the star of o. As r − 1 ∈ Z ≥0 , there is an inclusion P x,(r−1)+ ⊃ P o,(r−1)+ = K r , so π has nonzero vectors fixed by K r .
Let us recall some basic properties of generic representations, from [31, Section 2]. Let ψ : F → C × be a character which is trivial on o F but not on −1 F o F . We note that ψ is unitary because F /o F is a union of finite subgroups. Let U = U n be the standard unipotent subgroup of GL n (F ), consisting of upper triangular matrices. We need a character θ of U , which does not vanish on any of the root subgroups associated with simple roots. Any choice is equally good, and it is common to take Let (π, V ) ∈ Irr(GL n (F )). One calls π generic if there exists a nonzero linear form λ on V such that Such a linear form is called a Whittaker functional, and the space of those has dimension 1 (if they exist). Let W (π, θ ) be the space of all functions W : G → C of the form Then W (π, θ ) is stable under right translations, and the representation thus obtained is isomorphic to π via v ↔ W v . Most irreducible representations of GL n (F ), and in particular all the supercuspidal ones, are generic [23].
We consider one irreducible generic representation π of GL n (F ) and another one, π , of GL n−1 (F ). For W ∈ W (π, θ ) and W ∈ W (π , θ ) one defines the integral where μ denotes the quotient of Haar measures on GL n−1 (F ) and on U n−1 . This integral is known to converge absolutely when Re(s) is large [31,Theorem 2.7.i]. The contragredient representationsπ andπ are also generic. We defineW ∈ W (π, θ ) by where g −T is the transpose inverse of g and w n is the permutation matrix with ones on the diagonal from the lower left to the upper right corner. We denote the central character of π by ω π . With these notations the L-functions, -factors and γ -factors of the pair (π, π ) are related by see [31,Theorem 2.7.iii]. We regard these equations as definitions of the -and γ -factors.
Suppose that det φ corresponds to the central character of π via local class field theory and that holds in one of the following cases: (a) for n = n − 1 and every generic π ∈ Irr(GL n (F ), K 2r−1,n ); (b) for every n such that 1 ≤ n < n, and for every supercuspidal representation π in Irr(GL n (F ), K 2r−1,n ).
Proof (b) By Proposition 4.2 d(rec F,n (π)) ≤ r − 1. By Lemma 4.3, the assumption applies to every supercuspidal π ∈ Irr(GL n (F )) of depth ≤ 2r − 2. The point is that with π i supercuspidal. The multiplicativity of γ -factors also gives the equality Hence, to establish (43) it suffices to show that whenever σ is a supercuspidal. In the case d(σ ) > 2r − 2, this is the content of [21,Theorem 7.4]. We note that this result uses both the assumption on the central character of π and Proposition 4.2.
By the assumption of the theorem, the right-hand side equals This finishes the proof of (45) and of (43). Now we can apply [27, Théorème 1.1], which says that π ∼ = rec −1 F,n (φ).

The method of close fields
Kazhdan's method of close fields [18,35] has proven useful to generalize results that are known for groups over p-adic fields to groups over local fields of positive characteristic. It was worked out for inner forms of GL n (F ) by Badulescu [7]. Let F and F be two local non-archimedean fields, which we think of as being similar in a way that will be made precise below. Let G = GL m (D) be a standard inner form of GL n (F ) and let G = GL m ( D) be the standard inner form of GL n ( F ) with the same Hasse invariant as G.
In this section, an object with a tilde will always be the counterpart over F of an object (without tilde) over F , and a superscript means the subgroup of elements with reduced norm 1. Then G = G der is an inner form of SL n ( F ) with the same Hasse invariant as G and Let o D be the ring of integers of D, D a uniformizer and p D = D o D its unique maximal ideal. The explicit multiplication rules in D [50,Proposition IX.4.11] show that we may assume that a power of D equals F , a uniformizer of F .
Generalizing the notation for GL n (F ), let K 0 = GL m (o D ) be the standard maximal compact subgroup of G and define, for r ∈ Z >0 : . We denote the category of smooth G-representations that are generated by their K rinvariant vectors by Mod(G, K r ). Let H(G, K r ) be the convolution algebra of compactly supported K r -biinvariant functions G → C. According to [12,Corollaire 3.9] Mod(G, K r ) → Mod(H(G, K r )), is an equivalence of categories. The same holds for ( G, K r ).
From now on, we suppose that F and F are l-close for some l ≥ r, that is, as rings. As remarked in [18], for every local field of characteristic p > 0 and every l ∈ N there exists a finite extension of Q p which is l-close to F . Notice that (47) With [7,Théorème 2.4], (47) also gives rise to a ring isomorphism which in turn induces a group isomorphism We note that, whenever r ≤ r ≤ l, one choose λ r such that it induces λ r . Then refines GL m (λ r ). Recall that the Cartan decomposition for G says that K 0 \G/K 0 can be represented by : a 1 ≤ · · · ≤ a m . Clearly A + is canonically in bijection with the analogous set A + of representatives for K 0 \ G/ K 0 (which of course depends on the choice of a uniformizer D ). Since K r \G/K r can be identified with K r \K 0 × A + × K 0 /K r , that and GL m (λ r ) determine a bijection ζ r : K r \G/K r → K r \ G/ K r .
Most of the next result can be found in [7,10].
Theorem 5.1 Suppose that F and F are sufficiently close, in the sense that the l in (47) is large. Then the map 1 K r gK r → 1 ζ r (K r gK r ) extends to a C-algebra isomorphism This induces an equivalence of categories such that: (a) ζ G r respects twists by unramified characters and its effect on central characters is that of (48). Proof The existence of the isomorphism ζ G r is [7,Théorème 2.13]. The equivalence of categories follows from that and (46). (a) Let G 1 be the subgroup of G generated by all compact subgroups of G, that is, the intersection of the kernels of all unramified characters of G. Since K r and K r are compact, ζ r restricts to a bijection Moreover, because A + → A + respects the group multiplication whenever it is defined, the induced bijection G/G 1 → G/ G 1 is in fact a group isomorphism. Hence, ζ r induces an isomorphism which clearly satisfies, for π ∈ Mod(G, K r ) and χ ∈ X nr (G): The central characters can be dealt with similarly. The characters of Z(G) appearing in Mod(G, K r ) are those of Now we note that ζ G r and (48) have the same restriction to the above group. (b) By [7, Théorème 2.17], ζ G r preserves cuspidality and square integrability modulo centre. Combining the latter with part (a), we find that it also preserves essential square integrability. A variation on the proof of [7,Théorème 2.17.b] shows that temperedness is preserved as well. Alternatively, one can note that every irreducible tempered representation in Mod(G, K r ) is obtained with parabolic induction from a square-integrable modulo centre representation in Mod(M, M ∩ K r ), and then apply part (c). (c) This property, and its analogue for Jacquet restriction, are proven in [10,Proposition 3.15]. We prefer a more direct argument. The constructions in [7, § 2] apply equally well to (M, K r ∩ M), so ζ r induces an algebra isomorphism ζ M r and an equivalence of categories ζ M r . By [17,Corollary 7.12] the parabolic subgroup P determines an injective algebra homomorphism commutes [17,Corollary 8.4]. The construction of t P in [17, § 7] depends only on properties that are preserved by ζ G r (and its counterparts for other groups), so commutes. Now we combine (53) with (52) for G and G. (d) The contragredient of a G-representation (π, V ) is the representationπ, on the smooth part of the dual vector space of V , defined byπ (g)(λ) = λ • π(g −1 ). Similarly, the contragredient of a H(G, K r )-module W is W * with the action where the involution on H(G, K r ) is given by f * (g) = f (g −1 ). Furthermore, (V * ) K r ∼ = (V K r ) * , so the equivalence of categories (46) commutes with the formation of contragredients. The map ζ G r does so because ζ G r commutes with the involution *. (e) For the γ -factors, see [7,Théorème 2.19].
Consider the L-function of a supercuspidal σ ∈ Irr(G, K r ). By [24, Propositions 4.4 and 5.11] L(s, σ ) = 1 unless m = 1 and σ = χ•Nrd with χ : F × → C × unramified. This property is preserved by ζ G r , so L s, ζ G r (σ ) = 1 if the condition is fulfilled. In the remaining case The proof of part (a) shows that ζ G Thus, ζ G r preserve the L-functions of supercuspidal representations. By [30, § 3], the L-functions of general π ∈ Irr(G, K r ) are determined by the L-functions of supercuspidal representations of Levi subgroups of G, in combination with parabolic induction and twisting with unramified characters. In view of parts (a),(c) and the above, this implies that ζ G r always preserves L-functions. Now the relation (s, π, ψ) = γ (s, π, ψ) L(s, π) L(1 − s,π ) and part (d) show that ζ G r preserves -factors.
For r ≤ r ≤ l, Mod(G, K r ) is a subcategory of Mod(G, K r ) and it follows from (50) that In [9], Badulescu showed that Theorem 5.1 has an analogue for G and G , which can easily be deduced from Theorem 5. In view of (55) and the isomorphism o F /p r Hence, it induces an algebra isomorphism As preparation for the next section, we will show that in certain special cases the functors ζ G r preserve the L-functions, -factors and γ -factors of pairs of representations, as defined in [31].
Suppose that F is l-close to F and that ψ :

Theorem 5.3
Assume that F and F are l-close for some l > r and that ψ is l-close to ψ. Let π ∈ Irr(GL n (F ), K r,n ) be supercuspidal and let π ∈ Irr(GL n−1 (F ), K r,n−1 ) be generic. Then Remark It will follow from Theorem 6.1 that the above remains valid with any natural number instead of n − 1 (except that the L-functions need not equal 1).
After the first version of this paper was put on the arXiv, the authors were informed that a similar result was proved in [20,Theorem 2.3.10]. See also [21,Theorem 7.6]. Our proof differs from Ganapathy's and yields a better bound on l, namely l > r compared to l ≥ n 2 r + 4.
Proof Since π andπ are supercuspidal, whereas π andπ are representations of a general linear group of lower rank, [31,Theorem 8.1] assures that all the L-functions appearing here are 1. By (42) this implies that the relevant γ -factors are equal to the -factors of the same pairs. Hence, it suffices to prove the claim for the -factors. We note that by Theorem 5.1 is constant on sets of the form U n−1 gK r,n−1 . Since the subgroup K r,n−1 is stable under the automorphism g → g −T , the functionsW andW are also right K r,n−1 -invariant. Both transform under left translations by U n−1 as θ , so GL n−1 (F ) → C : g →W g 0 0 1 W (g) defines a function U n−1 \GL n−1 (F )/K r,n−1 → C. Since det(K r,n−1 ) ⊂ o × F and det(U n−1 ) = 1, the function | det | F can also be regarded as a map U n−1 \GL n−1 (F )/K r,n−1 → C. Now the idea is to transfer these functions to objects over F by means of the Iwasawa decomposition as in [39, § 3], and to show that neither side of (41) changes.
Let A F ⊂ GL n (F ) be the group of diagonal matrices all whose entries are powers of F . The Iwasawa decomposition states that GL n (F ) = a∈A F U n aK 0,n .
This, the canonical bijection A F → A F : a → a and the isomorphism GL n (λ r ) from (49) combine to a bijection ζ r : U n \GL n (F )/K r,n → U n \GL n ( F )/ K r,n , U n ak K r,n → U n aGL n (λ r )(k) K r,n .
Because ψ is l-close to ψ we may apply [39, Lemme 3.2.1], which says that there is a unique linear bijection ρ n : W (π, θ ) K r,n → W ζ GL n (F ) r (π), θ K r,n which transforms the restriction of functions to A F K 0,n according to ζ r . We will use (58) and (59) also with n − 1 instead of n. Put W = ρ n (W ) and W = ρ n−1 (W ). As (58) commutes with g → g −T , W = ρ n (W ) andˇ W = ρ n−1 (W ).
It follows immediately from the definition of ζ r that | det(ζ r (g))| F = | det(g)| F .
For the computation of (s, W, W ), we may normalize the measure μ such that every double coset U n−1 \U n−1 gK r,n−1 has volume 1, and similarly for the measure on U n−1 \GL n−1 ( F ). The equalities (61)  The previous two equalities and (56) prove that all terms in (41), expect possibly the -factors, have the same values as the corresponding terms defined over F . To establish the desired equality of -factors, it remains to check that (s, W, W ) is nonzero for a suitable choice of right K r,n−1 -invariant functions W and W .
Take v as above, but nonzero. Then W = W v is nonzero because V ∼ = W (π , θ ). Choose g 0 ∈ GL n−1 (F ) with W (g 0 ) = 0 and define H : GL n−1 (F ) → C by H(g) = W (g) if g ∈ U n−1 g 0 K r,n−1 and H(g) = 0 otherwise. According to [27,Lemme 2.4.1], there exists W ∈ W (π, ψ) such that W g 0 0 1 = H(g) for all g ∈ GL n−1 (F ). Notice that such a W is automatically right invariant under K r,n−1 on GL n−1 (F ) ⊂ GL n (F ). Now we can easily compute the required integral:

Close fields and Langlands parameters
This section is based on Deligne's comparison of the Galois groups of close fields. According to [18, (3.5.1)], the isomorphism (47) gives rise to an isomorphism of profinite groups which is unique up to inner automorphisms. Since both W F and W F can be described in terms of automorphisms of the residue field o F /p F ∼ = o F /p F , (63) restricts to an isomorphism We fix such an isomorphism (63), and hence (64) as well. Another choice would correspond to another separable closure of F , so that is harmless when it comes to Langlands parameters. Take r < l and recall the map W F /Gal(F s /F ) l → F × /1 + p r F from local class field theory. By [18, Proposition 3.6.1], the following diagram commutes: Now we are ready to complete the proof of Theorem 3.2, and hence of our main result Theorem 3.3. Proof of Theorem 3.2 when char(F ) = p > 0.
Because we already proved Theorem 3.2 for F , this gives That the isomorphism C[S φ /Z φ , κ φ ] ∼ = e χ G C[S φ ] is of the required form and that it is unique up to twists by characters of S φ /Z φ follows from the corresponding statements over F and (73).