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The local Langlands correspondence for inner forms of SL\(_{n}\)
 AnneMarie Aubert^{1},
 Paul Baum^{2},
 Roger Plymen^{3, 4} and
 Maarten Solleveld^{5}Email authorView ORCID ID profile
https://doi.org/10.1186/s4068701600794
© The Author(s) 2016
 Received: 3 April 2015
 Accepted: 3 August 2016
 Published: 5 December 2016
Abstract
Let F be a nonarchimedean local field. We establish the local Langlands correspondence for all inner forms of the group \(\mathrm{SL}_n (F)\). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for \(\mathrm{SL}_n (F)\) enhanced with an irreducible representation of an Sgroup and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of \(\mathrm{SL}_n (F)\) up to equivalence. An analogous result is shown in the archimedean case. For padic fields, this is based on the work of Hiraga and Saito. 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 \(\mathrm{GL}_n (F)\), when the fields are close enough compared to the depth of the representations.
Keywords
 Representation theory
 Local Langlands conjecture
 Division algebra
 Close fields
Mathematics Subject Classification
 20G05
 22E50
1 Background
Let F be a local field and let D be a division algebra with centre F, of dimension \(d^2 \ge 1\) over F. Then \(G = \mathrm{GL}_m (D)\) is the group of Frational points of an inner form of \(\mathrm{GL}_{md}\). We will say simply that G is an inner form of \(\mathrm{GL}_n (F)\), where \(n = md\). It is endowed with a reduced norm map Nrd\(: \mathrm{GL}_m (D) \rightarrow F^\times \). The group \(G^{\sharp } := \ker (\text {Nrd} : G \rightarrow F^\times )\) is an inner form of \(\mathrm{SL}_n (F)\). (The split case \(D=F\) is allowed here.) In this paper, we will complete the local Langlands correspondence for \(G^{\sharp }\).
We sketch how it goes and which part of it is new. For any reductive group over a local field, say H, let \(\mathrm{Irr}(H)\) denote the set of (isomorphism classes of) irreducible admissible Hrepresentations. Let \(\Phi (H)\) be the collection of (equivalence classes of) Langlands parameters for H, as defined in [13].
In general, more subtle component groups \(\mathcal S_{\phi ^{\sharp }}\) are needed, see [3, 32, 48]. Our enhanced Lparameters will be pairs \((\phi ^{\sharp }, \rho )\) consisting of a Langlands parameter \(\phi ^{\sharp }\) for \(G^{\sharp }\), enhanced with a \(\rho \in \mathrm{Irr}(\mathcal S_{\phi ^{\sharp }})\). We consider such enhanced Lparameters for \(G^{\sharp }\) modulo the equivalence relation coming from the natural \(\mathrm{PGL}_n (\mathbb {C})\)action. When two Lparameters \(\phi _1^{\sharp }\) and \(\phi _2^{\sharp }\) are conjugate, there is a canonical bijection \(\mathrm{Irr}(\mathcal S_{\phi _1^{\sharp }}) \rightarrow \mathrm{Irr}(\mathcal S_{\phi _2^{\sharp }})\), coming from conjugation by any \(g \in \mathrm{PGL}_n (\mathbb {C})\) with \(g^{1} \phi _1^{\sharp } g = \phi _2^{\sharp }\). With this in mind, it makes sense to speak about \(\mathrm{Irr}(\mathcal S_{\phi ^{\sharp }})\) for \(\phi ^{\sharp } \in \Phi (G^{\sharp })\).
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 righthand 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 \(\mathrm{Irr}(H)\) on the lefthand side. Such a statement was proven for unipotent representations of simple padic groups in [41].
Let us make this explicit for inner forms of \(\mathrm{GL}_n (F)\), respectively, \(\mathrm{SL}_n (F)\). We define the equivalence classes of such inner forms to be in bijection with the isomorphism classes of central simple Falgebras of dimension \(n^2\) via \(\mathrm{M}_m (D) \mapsto \mathrm{GL}_m (D)\), respectively, \(\mathrm{M}_m (D) \mapsto \mathrm{GL}_m (D)_{\mathrm{der}}\). This equivalence relation can also be motivated with Galois cohomology, see Sect. 2.
As Langlands dual group, we take \(\mathrm{GL}_n (\mathbb {C})\), respectively, \(\mathrm{PGL}_n (\mathbb {C})\). Langlands parameters for \(\mathrm{GL}_n (F)\) (respectively, \(\mathrm{SL}_n (F)\)) 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 \(\mathrm{PGL}_n (\mathbb {C})\), 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 \(\mathrm{SL}_n (\mathbb {C})\) on the dual groups and on the collections of Langlands parameters for \(\mathrm{GL}_n (F)\) or \(\mathrm{SL}_n (F)\).
Hence, \(\mathcal S_\phi \) (resp. \(\mathcal S_{\phi ^{\sharp }}\)) has more irreducible representations than \(S_\phi \) (resp. \(S_{\phi ^{\sharp }}\)). Via the Langlands correspondence, the additional ones are associated with irreducible representations of nonsplit inner forms of \(\mathrm{GL}_n (F)\) (resp. \(\mathrm{SL}_n (F)\)). For example, consider a Langlands parameter \(\phi \) for \(\mathrm{GL}_2 (F)\) which is elliptic, that is, whose image is not contained in any torus of \(\mathrm{GL}_2 (\mathbb {C})\). Then \(\mathcal S_\phi = Z(\mathrm{SL}_2 (\mathbb {C})) \cong \{\pm 1\}\). The pair \((\phi ,\mathrm {triv}_{\mathcal S_\phi })\) parametrizes an essentially squareintegrable representation of \(\mathrm{GL}_2 (F)\) and \((\phi ,\mathrm {sgn}_{\mathcal S_\phi })\) parametrizes an irreducible representation of the nonsplit inner form \(D^\times \), 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:
Theorem 1.1

pairs \((G,\pi )\) with \(\pi \in \mathrm{Irr}(G)\) and G an inner form of \(\mathrm{GL}_n (F)\), considered up to equivalence;

pairs \((\phi ,\rho )\) with \(\phi \in \Phi (\mathrm{GL}_n (F))\) and \(\rho \in \mathrm{Irr}(\mathcal S_\phi )\).
For these Langlands parameters, \(\mathcal S_\phi = \mathcal Z_\phi \) and a character of \(\mathcal Z_\phi \) determines an inner form of \(\mathrm{GL}_n (F)\) 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 \(\Pi _\phi \) contains the unique representation \(\mathrm {rec}_{D,m}^{1}(\phi )\) of \(G = \mathrm{GL}_m (D)\) if \(\phi \) is relevant for G, and no Grepresentations 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 Lpacket \(\Pi _{\phi ^{\sharp }} (G^{\sharp })\) if there are more than one. Already for tempered representations of \(\mathrm{SL}_2 (F)\), the enhanced LLC is not canonical, see [6, Example 11.3]. It is possible to determine a unique parametrization of \(\mathrm{Irr}(\mathrm{SL}_n (F))\) by fixing a Whittaker datum and of \(\mathrm{Irr}(G^{\sharp })\) by adding more data [32], but this involves noncanonical choices.
Secondly, the quaternion algebra \({\mathbb {H}}\) 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:
Theorem 1.2

pairs \((G^{\sharp },\pi )\) with \(\pi \in \mathrm{Irr}(G^{\sharp })\) and \(G^{\sharp }\) an inner form of \(\mathrm{SL}_n (F)\), considered up to equivalence;

pairs \((\phi ^{\sharp },\rho )\) with \(\phi ^{\sharp } \in \Phi (\mathrm{SL}_n (F))\) and \(\rho \in \mathrm{Irr}(\mathcal S_{\phi ^{\sharp }})\),
 (a)
The group \(G^{\sharp }\) determines \(\rho \big _{\mathcal Z_{\phi ^{\sharp }}}\) and conversely.
 (b)
The correspondence satisfies the desired properties from [13, § 10.3], with respect to restriction from inner forms of \(\mathrm{GL}_n (F)\), temperedness and essential square integrability of representations.
This theorem supports more general conjectures on Lpackets and the LLC for nonsplit groups, cf. [3, § 3] and [32, § 5.4].
In the archimedean case, the classification of \(\mathrm{Irr}(\mathrm{SL}_m (D))\) is well known, at least for \(D \ne {\mathbb {H}}\). The main value of our result lies in the strong analogy with the nonarchimedean 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 \({\mathbb {H}}\) satisfies Nrd\(({\mathbb {H}}^\times ) = \mathbb {R}_{>0}\), whereas for all other local division algebras D with centre F the reduced norm map is surjective, that is, Nrd\((D^\times ) = F^\times \). Of course, there are various ad hoc ways to restore the bijectivity in Theorem 1.2, for example by decreeing that \(\mathrm{SL}_m ({\mathbb {H}})\) appears twice among the equivalence classes of inner forms of \(\mathrm{SL}_{2m}(\mathbb {R})\). This can be achieved in a natural way with strong inner forms, as in [1]. But one may also argue that for \(\mathrm{SL}_m ({\mathbb {H}})\) one would actually be better off without any component groups.
For padic 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 padic case. Let F be a local field of characteristic \(p ,\; \mathfrak o_F\) its ring of integers and \(\mathfrak p_F\) the maximal ideal of \(\mathfrak o_F\). There exist finite extensions \({\widetilde{F}}\) of \({\mathbb {Q}}_p\) which are lclose to F, which means that \(\mathfrak o_F / \mathfrak p_F^l\) is isomorphic to the corresponding ring for \({\widetilde{F}}\). Let \({\widetilde{D}}\) be a division algebra with centre \({\widetilde{F}}\), such that D and \({\widetilde{D}}\) have the same Hasse invariant. Let \(K_r\) be the standard congruence subgroup of level \(r \in \mathbb {N}\) in \(\mathrm{GL}_m (\mathfrak o_D)\), and let \(\mathrm{Irr}(G,K_r)\) be the set of irreducible representations of \(G = \mathrm{GL}_m (D)\) with nonzero \(K_r\)invariant vectors. Define \({\widetilde{K_r}} \subset \mathrm{GL}_m ({\widetilde{D}})\) and \(\mathrm{Irr}(\mathrm{GL}_m ({\widetilde{D}}),{\widetilde{K_r}})\) in the same way.
Theorem 1.3
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 \(\epsilon \) and \(\gamma \)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 \(\phi \), we let \(d(\phi ) \in \mathbb {R}_{\ge 0}\) be the smallest number such that \(\phi \notin \Phi _{d(\phi )}(\mathrm{GL}_n (F))\). That is, the smallest number such that \(\phi \) is nontrivial on the \(d(\phi )\)th ramification group of the Weil group of F with respect to the upper numbering. For a supercuspidal representation \(\pi \) of \(\mathrm{GL}_n (F)\), let \(d (\pi )\) be its normalized level, as in [14].
Proposition 1.4
2 Inner forms of \(\mathrm{GL}_n (F)\)
Let F be a local field and let D be a division algebra with centre F, of dimension \(\dim _F (D) = d^2\). The Fgroup \(\mathrm{GL}_m (D)\) is an inner form of \(\mathrm{GL}_{md}(F)\), and conversely every inner form of \(\mathrm{GL}_n (F)\) is isomorphic to such a group.
As G is an inner form of \(\mathrm{GL}_n (F) ,\; \check{G} = \mathrm{GL}_n (\mathbb {C})\) and the action of Gal\((\overline{F} / F)\) on \(\mathrm{GL}_n (\mathbb {C})\) determined by G is by inner automorphisms. Therefore, we may take as Langlands dual group \({}^{L}G = \check{G} = \mathrm{GL}_n (\mathbb {C})\).

\(\check{M} = \prod _{i=1}^l \mathrm{GL}_{n_i}(\mathbb {C})^{e_i}\) and \(M = \prod _{i=1}^l \mathrm{GL}_{m_i}(D)^{e_i}\) are standard Levi subgroups of \(\mathrm{GL}_n (C)\) and \(\mathrm{GL}_m (D)\), respectively;

\(\phi = \prod _{i=1}^l \phi _i^{\otimes e_i}\) with \(\phi _i \in \Phi (\mathrm{GL}_{m_i}(D))\) and im\((\phi _i)\) not contained in any proper Levi subgroup of \(\mathrm{GL}_{n_i}(\mathbb {C})\);

\(\phi _i\) and \(\phi _j\) are not equivalent if \(i \ne j\).
Now we analyse the nontempered part of Irr(G). Let \(M_1\) be a Levi subgroup of G containing M such that \(\omega \) is squareintegrable 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 \(\omega _1 = I_{P_1}^{M_1} (\omega )\) is irreducible and independent of \(P_1\), while by the aforementioned results the restriction of \(\omega _1\) to the derived group of \(M_1\) is tempered. Furthermore, the absolute value of the character of \(\omega _1\) 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 \((P_2,\omega _1)\) satisfies the hypothesis of the Langlands classification [34, 37]. That result says that \(I_{P_2}^G (\omega _1)\) has a unique irreducible quotient \(L(P_2,\omega _1)\) and that every irreducible Grepresentation can be obtained in this way, from data that are unique up to Gconjugation. This provides a canonical bijection between \(\mathrm{Irr}(G)\) and the righthand side of (12).
By construction, \(L(P,\omega )\) is essentially squareintegrable if and only if \(M = P = G\), which happens precisely when the image of \(\phi \) is not contained in any proper Levi subgroup of \(\mathrm{GL}_n (\mathbb {C})\). By the uniqueness part of the Langlands classification [34, Theorem 3.5.ii] \(L(P,\omega )\) is tempered if and only if \(\omega \) is squareintegrable modulo centre, which by the above is equivalent to boundedness of \(\phi \in \Phi (G)\).
In the archimedean case, Langlands [37] himself established the correspondence between the irreducible admissible representations of \(\mathrm{GL}_m (D)\) 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 \(\mathrm{GL}_n (\mathbb {C})\) by the Borel–Weil theorem and for \(\mathrm{GL}_n (\mathbb {R})\) and \(\mathrm{GL}_m ({\mathbb {H}})\) by [11, § 12].
The above method to go from essentially squareintegrable 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 \(\mathrm{GL}_2 (\mathbb {R})\) and \({\mathbb {H}}\). Therefore, it is justified to say that (11)–(14) hold in the archimedean case.
Lemma 2.1
A Langlands parameter \(\phi \in \Phi (\mathrm{GL}_n (F))\) is relevant for \(G = \mathrm{GL}_m (D)\) if and only if \(\ker \chi _G \supset Z(\mathrm{SL}_n (\mathbb {C})) \cap C(\phi )^{\circ }\).
Proof
This can be derived with [2, Corollary 2.2] and [29, Lemma 9.1]. However, we prefer a more elementary proof.
Theorem 2.2
Proof
In the archimedean case, the above argument does not suffice, because some characters of \(Z(\mathrm{SL}_n (\mathbb {C}))\) do not parametrize an inner form of \(\mathrm{GL}_n (F)\). We proceed by direct calculation, inspired by [37, § 3].
Suppose that \(F = \mathbb {C}\). Then \(\mathbf {W}_F = \mathbb {C}^\times \) and im\((\phi )\) is just a real torus in \(\mathrm{GL}_n (\mathbb {C})\). Hence, \(Z_{\mathrm{GL}_n (\mathbb {C})}(\phi )\) is a Levi subgroup of \(\mathrm{GL}_n (\mathbb {C})\) and \(C(\phi ) = Z_{\mathrm{SL}_n (\mathbb {C})}(\phi )\) is the corresponding Levi subgroup of \(\mathrm{SL}_n (\mathbb {C})\). All Levi subgroups of \(\mathrm{SL}_n (\mathbb {C})\) are connected, so \(\mathcal S_\phi = C(\phi ) / C(\phi )^{\circ } = 1\). Consequently, \(\Phi ^e (\text {inn } \mathrm{GL}_n (\mathbb {C})) = \Phi (\mathrm{GL}_n (\mathbb {C}))\), and the theorem for \(F = \mathbb {C}\) reduces to the Langlands correspondence for \(\mathrm{GL}_n (\mathbb {C})\).
Thus, we checked that for every \(\phi \in \Phi (\mathrm{GL}_n (\mathbb {R})) ,\; \mathrm{Irr}(\mathcal S_\phi )\) parametrizes the equivalence classes of inner forms G of \(\mathrm{GL}_n (\mathbb {R})\) for which \(\phi \) is relevant. To conclude, we apply the LLC for G. \(\square \)
3 Inner forms of \(\mathrm{SL}_n (F)\)
Lemma 3.1
[29, Lemma 12.1], see also [22, Theorem 4.1] when \(G^{\sharp } = \mathrm{SL}_n (F)\).
Two Lpackets \(\Pi _{\phi _1^{\sharp }}(G^{\sharp })\) and \(\Pi _{\phi _2^{\sharp }}(G^{\sharp })\) are either disjoint or equal, and the latter happens if and only if \(\phi _1^{\sharp }\) and \(\phi _2^{\sharp }\) are \(\mathrm{PGL}_n (\mathbb {C})\)conjugate (i.e. equal in \(\Phi (G^{\sharp }))\).
Theorem 3.2
Proof
Now we suppose that char\((F) = 0\) and that the representation \(\Pi _\phi (G)\) is tempered. In the archimedean case, the cocycle \(\kappa _{\phi ^{\sharp }}\) 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 \(\phi ^{\sharp } \in \Phi (G^{\sharp })\), with a lift \(\phi \in \Phi (G)\). Let Y be a connected set of unramified twists \(\phi _\chi \) of \(\phi \), such that \(C(\phi _\chi ) = C(\phi )\) and \(C(\phi _\chi ^{\sharp }) = C(\phi ^{\sharp })\) for all \(\phi _\chi \in Y\). 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 \(\mathbf {W}_F\) under \(\phi \)) of a torus in a complex reductive group, there is an element of the maximal compact subtorus which has the same centralizer.
The proof of the case char\((F) > 0\) requires more techniques; we dedicate Sects. 4–6 to it.
We note that the actions of \(\mathrm{PGL}_n (\mathbb {C})\) on the various \(\Phi ^e (G^{\sharp })\) combine to an action on \(\Phi ^e (\text {inn } \mathrm{SL}_n (F))\). With the collection of equivalence classes \(\Phi ^e (\text {inn } \mathrm{SL}_n (F))\), we can formulate the local Langlands correspondence for all such inner forms simultaneously.
First we consider the nonarchimedean case. As for \(\mathrm{GL}_n (F)\), we fix one group in every equivalence class of inner forms. We choose the groups \(\mathrm{GL}_m ( [L/F,\chi ,\varpi _F] )^{\sharp }\) with \([L/F,\chi ,\varpi _F]\) as in (8) and call these the standard inner forms of \(\mathrm{SL}_n (F)\).
Theorem 3.3
 (a)
Suppose that \(\rho \) sends \(\exp (2 \pi i/n) \in Z(\mathrm{SL}_n (\mathbb {C}))\) to a primitive dth root of unity z. Then \(G_\rho ^{\sharp } = \mathrm{GL}_m ( [L/F,\chi ,\varpi _F] )^{\sharp }\), where \(md = n\) and \(\chi : \mathrm {Gal}(L/F) \rightarrow \mathbb {C}^\times \) sends the Frobenius automorphism to z.
 (b)
Suppose that \(\phi ^{\sharp }\) is relevant for \(G^{\sharp }\) and lifts to \(\phi \in \Phi (G)\). Then the restriction of \(\Pi _\phi (G)\) to \(G^{\sharp }\) is \(\bigoplus _{\rho \in \mathrm{Irr}(\mathcal S_{\phi ^{\sharp }},\chi _{G^{\sharp }})} \pi (\phi ^{\sharp },\rho ) \otimes \rho \).
 (c)
\(\pi (\phi ^{\sharp },\rho )\) is essentially squareintegrable if and only if \(\phi ^{\sharp } (\mathbf {W}_F \times \mathrm{SL}_2 (\mathbb {C}))\) is not contained in any proper parabolic subgroup of \(\mathrm{PGL}_n (\mathbb {C}))\).
 (d)
\(\pi (\phi ^{\sharp },\rho )\) is tempered if and only if \(\phi ^{\sharp }\) is bounded.
Proof
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 \(\mathrm{GL}_n (F)\) (which were discussed after (13)) in combination with [46, Proposition 2.7]. \(\square \)
Let us formulate an archimedean analogue of Theorem 3.3, that is, for the groups \(\mathrm{SL}_n (\mathbb {C}), \mathrm{SL}_n (\mathbb {R})\) and \(\mathrm{SL}_m ({\mathbb {H}})\). In view (27), we cannot expect a bijection, and part (b) has to be adjusted.
Theorem 3.4
 (a)
The preimage of \(\mathrm{Irr}(\mathrm{SL}_n (F))\) consists of the \((\phi ^{\sharp },\rho )\) with \(\mathcal Z_{\phi ^{\sharp }} \subset \ker \rho \), and the map is injective on this domain. The preimage of \(\mathrm{Irr}(\mathrm{SL}_{n/2}({\mathbb {H}}))\) consists of the \((\phi ^{\sharp },\rho )\) such that \(\rho \) is not trivial on \(\mathcal Z_{\phi ^{\sharp }}\), and the map is twotoone on this domain.
 (b)
Suppose that \(\phi ^{\sharp }\) is relevant for \(G^{\sharp } = \mathrm{SL}_m (D)\) and lifts to \(\phi \in \Phi (G)\). Then the restriction of \(\Pi _\phi (G)\) to \(G^{\sharp }\) is irreducible if \(D = \mathbb {C}\) or \(D = {\mathbb {H}}\) and is isomorphic to \(\bigoplus _{\rho \in \mathrm{Irr}(\mathcal S_{\phi ^{\sharp }} / \mathcal Z_{\phi ^{\sharp }})} \pi (\phi ^{\sharp },\rho ) \otimes \rho \) in case \(D = \mathbb {R}\).
 (c)
\(\pi (\phi ^{\sharp },\rho )\) is essentially squareintegrable if and only if \(\phi ^{\sharp } (\mathbf {W}_F)\) is not contained in any proper parabolic subgroup of \(\mathrm{PGL}_n (\mathbb {C}))\).
 (d)
\(\pi (\phi ^{\sharp },\rho )\) is tempered if and only if \(\phi ^{\sharp }\) 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 \(\mathrm{GL}_m (D)\) is so. For \(\mathrm{SL}_n (F)\), the intertwining operators admit a canonical normalization in terms of Whittaker functionals [29, pp. 17 and 69], so the definition (24) of \(\pi (\phi ^{\sharp },\rho )\) can be made canonical. For \(\mathrm{SL}_m ({\mathbb {H}})\), the definition (27) clearly leaves no room for arbitrary choices.
Part (a) and part (b) for \(D = \mathbb {R}\) follow as in the nonarchimedean case, except that for \(D = {\mathbb {H}}\) the preimage of \(\pi (\phi ^{\sharp },\rho )\) is in bijection with \(\mathrm{Irr}(\mathcal S_\phi , e_{{\mathbb {H}}^\times })\). To prove part (b) for \(D = \mathbb {C}\) and \(D = {\mathbb {H}}\), it suffices to remark that \(\mathrm {Res}_{G^{\sharp }}^G\) preserves irreducibility, as \(G = G^{\sharp } Z(G)\). The proof of parts (c) and (d) carries over from Theorem 3.3.
\(\square \)
4 Characterization of the LLC for some representations of \(\mathrm{GL}_n (F)\)
In this section, F is any local nonarchimedean field. It is known from [27] that generic representations of \(\mathrm{GL}_n (F)\) can be characterized in terms of \(\gamma \)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.
Lemma 4.1
Proof
We recall from [16, (1.5)] that a stratum is a quadruple \([\mathfrak A,m,m',\beta ]\) consisting of a hereditary \(\mathfrak o_F\)order \(\mathfrak A\) in \(\mathrm{M}_n(F)\), integers \(m>m'\ge 0\), and an element \(\beta \in \mathrm{M}_n(F)\) with \(\mathfrak A\)valuation \(\nu _{\mathfrak A}(\beta )\ge m\). A stratum of the form \([\mathfrak A,m,m1,\beta ]\) is called fundamental [16, (2.3)] if the coset \(\beta +\mathfrak p^{1}\mathfrak P^{1m}\) does not contain a nilpotent element of \(\mathrm{M}_n(F)\). We remark that the formulation in [14] is slightly different because the notion of a fundamental stratum there allows m to be 0.
 (a)
\(\pi \) contains the trivial character of \(U^1(\mathfrak A)\), or
 (b)
there is a fundamental stratum \([\mathfrak A,m,m1,\beta ]\) in \(\mathrm{M}_n(F)\) such that \(\pi \) contains the character \(\psi _\beta \) of \(U^m(\mathfrak A)\).
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.
Proposition 4.2
Proof
We will check that (38) is actually an equality. The case where \(d(\pi )=0\) is easy, so we only consider \(d(\pi )>0\).
Let \(\mathfrak A'\) be any hereditary \(\mathfrak o_F\)order \(\mathfrak A'\) in \(\mathrm{M}_n(F)\), and define \(m_{\mathfrak A'}(\pi )\) to be the least nonnegative integer m such that the restriction of \(\pi \) to \(U^{m+1}(\mathfrak A')\) contains the trivial character. Then choose \(\mathfrak A'\) so that \(m_{\mathfrak A'}(\pi )/e(\mathfrak A')\) is minimal, and let \([\mathfrak A', m_{\mathfrak A'}(\pi ),m_{\mathfrak A'}(\pi )1,\beta ]\) be a stratum occurring in \(\pi \). By [14, Theorem \(2'\)], this is a fundamental stratum. By [14, (3.4)], we may assume that the integers \(e(\mathfrak A')\) and \(m_{\mathfrak A'}(\pi )\) are relatively prime. Hence, we may apply [14, (3.13)]. We find that \(\mathfrak A'\) is principal and that every irreducible representation \(\varrho \) of \(\mathfrak K(\mathfrak A')\) which occurs in the restriction of \(\pi \) to \(\mathfrak K(\mathfrak A')\), and such that the restriction of \(\varrho \) to \(U^{m_{\mathfrak A'}(\pi )}(\mathfrak A')\) contains \(\psi _\beta \), is nondegenerate. In particular, we have \(d(\varrho ')=m_{\mathfrak A'}(\pi )\).
We conclude that (38) is indeed an equality, which together with (37) shows that \(d(\varrho ')=d(\pi )\). \(\square \)
Lemma 4.3

\(\pi \in \mathrm{Irr}(\mathrm{GL}_n (F), K_r)\),

\(d(\pi ) \le r1\).
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 \(P_{o,(r1)+}\), where o denotes the origin in the standard apartment of the Bruhat–Tits building of \(\mathrm{GL}_n (F)\). From the definition in [42, § 3.4], we read off that any \(\pi \in \mathrm{Irr}(\mathrm{GL}_n (F), K_r)\) has depth \(\le r1\).
Theorem 4.4
 (a)
for \(n' = n1\) and every generic \(\pi ' \in \mathrm{Irr}(\mathrm{GL}_{n'}(F),K_{2r1,n'})\);
 (b)
for every \(n'\) such that \(1 \le n' < n\), and for every supercuspidal representation \(\pi '\) in \(\mathrm{Irr}(\mathrm{GL}_{n'}(F),K_{2r1,n'})\).
Proof
5 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 padic fields to groups over local fields of positive characteristic. It was worked out for inner forms of \(\mathrm{GL}_n (F)\) by Badulescu [7].
Let F and \({\widetilde{F}}\) be two local nonarchimedean fields, which we think of as being similar in a way that will be made precise below. Let \(G = \mathrm{GL}_m (D)\) be a standard inner form of \(\mathrm{GL}_n (F)\) and let \({\widetilde{G}} = \mathrm{GL}_m ({\widetilde{D}})\) be the standard inner form of \(\mathrm{GL}_n ({\widetilde{F}})\) with the same Hasse invariant as G.
Theorem 5.1
 (a)
\(\overline{\zeta _r^G}\) respects twists by unramified characters and its effect on central characters is that of (48).
 (b)
For irreducible representations, \(\overline{\zeta _r^G}\) preserves temperedness, essential square integrability and cuspidality.
 (c)Let be P a parabolic subgroup of G with a Levi factor M which is standard, and let \({\widetilde{P}}\) and \({\widetilde{M}}\) be the corresponding subgroups of \({\widetilde{G}}\). Thencommutes.$$\begin{aligned} \begin{array}{ccc} \mathrm{Mod}(G,K_r) &{} \xrightarrow {\; \overline{\zeta _r^G} \;} &{} \mathrm{Mod}({\widetilde{G}},{\widetilde{K}}_r) \\ \uparrow I_P^G &{} &{} \uparrow I_{{\widetilde{P}}}^{{\widetilde{G}}} \\ \mathrm{Mod}(M,K_r \cap M) &{} \xrightarrow {\; \overline{\zeta _r^M} \;} &{} \mathrm{Mod}({\widetilde{M}},{\widetilde{K}}_r \cap {\widetilde{M}}) \end{array} \end{aligned}$$
 (d)
\(\overline{\zeta _r^G}\) commutes with the formation of contragredient representations.
 (e)
\(\overline{\zeta _r^G}\) preserves the Lfunctions, \(\epsilon \)factors and \(\gamma \)factors.
Proof
The existence of the isomorphism \(\zeta _r^G\) is [7, Théorème 2.13]. The equivalence of categories follows from that and (46).
(b) By [7, Théorème 2.17], \(\overline{\zeta _r^G}\) 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 \(\mathrm{Mod}(G,K_r)\) is obtained with parabolic induction from a squareintegrable modulo centre representation in \(\mathrm{Mod}(M,M \cap K_r)\), and then apply part (c).
(e) For the \(\gamma \)factors, see [7, Théorème 2.19].
Corollary 5.2
Theorem 5.1 (except part e) also holds for the corresponding subgroups of elements with reduced norm 1.
Proof
Using the isomorphisms \(\zeta _r^{M^{\sharp }}\), this can be proven in the same way as Theorem 5.1 itself. For part (b) one can use that an irreducible Grepresentation is tempered (resp. essentially squareintegrable or cuspidal) if and only if all its \(G^{\sharp }\)constituents are so [46, Proposition 2.7].
As preparation for the next section, we will show that in certain special cases the functors \(\overline{\zeta _r^G}\) preserve the Lfunctions, \(\epsilon \)factors and \(\gamma \)factors of pairs of representations, as defined in [31].
Theorem 5.3
Remark
It will follow from Theorem 6.1 that the above remains valid with any natural number instead of \(n1\) (except that the Lfunctions 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 \(l \ge n^2r + 4\).
Proof
\(U_{n1} \backslash \mathrm{GL}_{n1}(F) / K_{r,n1} \rightarrow \mathbb {C}\).
Now the idea is to transfer these functions to objects over \({\widetilde{F}}\) by means of the Iwasawa decomposition as in [39, § 3], and to show that neither side of (41) changes.
6 Close fields and Langlands parameters
We remark that the obtained bound \(l > 2^{n1} r\) appears nevertheless to be much larger than necessary. We expect that the result is valid whenever \(l > r\), but we did not manage to prove that.
Theorem 6.1
Proof
Now we fix \(n>1\) and we assume the theorem for all \(n' < n\). Consider a supercuspidal \(\pi \in \mathrm{Irr}(\mathrm{GL}_n (F),K_r)\) with Langlands parameter \(\phi = \mathrm {rec}_{F,n}(\pi ) \in \Phi _l (\mathrm{GL}_n (F))\). By the construction of the local Langlands correspondence for general linear groups, \(\mathrm{SL}_2 (\mathbb {C}) \subset \ker \phi \) and \(\phi \) is elliptic. By Theorem 5.1 \(\overline{\zeta _r^{\mathrm{GL}_n (F)}}(\pi ) \in \mathrm{Irr}\big ( \mathrm{GL}_n ({\widetilde{F}}), {\widetilde{K}}_r \big )\) is also supercuspidal and its central character is related to that of \(\pi \) via (48).

the parameters of supercuspidal representations;

the parameters of generalized Steinberg representations;

compatibility with unramified twists;

compatibility with parabolic induction followed by forming Langlands quotients.
To determine the Langlands parameters of elements of \(\mathrm{Irr}(\mathrm{GL}_n (F),K_r)\) via the above method, one needs only representations (possibly of groups of lower rank) that have nonzero \(K_r\)invariant vectors. We checked that in every step of this method the effect of \(\overline{\zeta _r^{\mathrm{GL}_n (F)}}\) on the Langlands parameters is given by \(\Phi ^\zeta _l\). Hence, the diagram of the theorem commutes for all representations in \(\mathrm{Irr}(\mathrm{GL}_n (F),K_r)\). \(\square \)
Because the LLC for inner forms of \(\mathrm{GL}_n (F)\) is closely related to that for \(\mathrm{GL}_n (F)\) itself, we can generalize Theorem 6.1 to inner forms.
Theorem 6.2
Proof
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 \(\mathrm{char}(F) = p > 0\).
Acknowledgements
The authors wish to thank Ioan Badulescu for interesting emails about the method of close fields, Wee Teck Gan for explaining some subtleties of inner forms and Guy Henniart for pointing out a weak spot in an earlier version of Theorem 4.4. We also thank the referee for his comments, which helped to clarify and improve the paper. Paul Baum was partially supported by NSF Grant DMS1200475. Maarten Solleveld was partially supported by a NWO Vidi Grant No. 639.032.528.
Declarations
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Authors’ Affiliations
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