Supercuspidal unipotent representations: L-packets and formal degrees

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $\gamma$-factors of L-parameters. This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensueing L-packets, in terms of data from the affine Dynkin diagram of G. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.


Introduction
Let K be a non-archimedean local field and let G be a connected reductive K-group. Roughly speaking, a representation of the reductive p-adic group G(K) is unipotent if it arises from a unipotent representation of a finite reductive group associated to a parahoric subgroup of G(K). Among all (irreducible) smooth G(K)-representations, this is a very convenient class, which can be studied well with classification, parabolic induction and Hecke algebra techniques. The work of Lusztig [Lus3,Lus4] goes a long way towards a local Langlands correspondence for such representations, when G is simple and adjoint.
In this paper we focus on supercuspidal unipotent G(K)-representations. For this to work well, we assume throughout that G splits over an unramified extension of K. Our main goal is a local Langlands correspondence for such representations, with as many nice properties as possible. We will derive that from the following result, which says that one can determine the L-parameters of supercuspidal unipotent representations of a simple algebraic group by comparing formal degrees and adjoint γ-factors.
Theorem 1. [Ree1,FeOp,Feng] Consider a simple K-group G which splits over an unramified extension. For each irreducible supercuspidal unipotent G(K)-representation π, there exists a discrete unramified local Langlands parameter λ ∈ Φ(G(K)) such that (0.1) fdeg(π, q) = C π γ(λ, q) for some C π ∈ Q × as rational functions in q with Q-coefficients. (Here q denotes the cardinality of the residue field of K, and one makes the terms of (0.1) into functions of q by varying the field K.) Furthermore: • λ is essentially unique, in the sense that its image in the collection Φ(G sc (K)) of L-parameters for the simply connected cover of G(K) is unique. • When G is adjoint, the map π → λ agrees with Lusztig's parametrization of unipotent representations [Lus3,Lus4].
We will make the above parametrization of supercuspidal unipotent representations more precise and generalize it to connected reductive K-groups. Let Irr(G(K)) cusp,unip denote the collection of irreducible supercuspidal unipotent representations of G(K). Denote the Weil group of K by W K . Let Φ nr (G(K)) be the set of unramified Lparameters for G(K) and let Φ(G(K)) cusp be the set of cuspidal enhanced L-parameters for G(K). (See Section 1 for the definitions of these and related objects.) Our main result can be summarized as follows: Theorem 2. Let G be a connected reductive K-group which splits over an unramified extension. There exists a bijection Irr(G(K)) cusp,unip −→ Φ nr (G(K)) cusp π → (λ π , ρ π ) with the properties: (1) When G is semisimple, the formal degree of π equals the adjoint γ-factor of λ π , up to a rational factor which depends only on ρ π .
(3) Equivariance with respect to W K -automorphisms of the root datum.
(4) Compatibility with almost direct products of reductive groups. (5) Let Z(G) s be the maximal K-split central torus of G and let H be the derived group of G/Z(G) s . When Z(G) s (K) acts trivially on π ∈ Irr(G(K)) cusp,unip , we can regard π as a representation of (G/Z(G) s )(K) and restrict to a representation π H of H(K). Then λ π has image in the Langlands L-group of G/Z(G) s and the canonical map sends λ π to λ π H . (6) The map in (5) provides a bijection between the intersection of Irr(G(K)) cusp,unip with the L-packet of λ π and the intersection of Irr(H(K)) cusp,unip with the Lpacket of λ π H . For a given π the properties (1), (2) and (4) determine λ π uniquely, modulo weakly unramified characters.
The bijection exhibited in Theorem 2 is of course a good candidate for a local Langlands correspondence (LLC) for supercuspidal unipotent representations, and we will treat it as such. The second bullet of Theorem 1 says that Lusztig's parametrization of supercuspidal unipotent representations of simple adjoint K-groups can be completely characterised by comparing formal degrees and adjoint γ-factors. Thus we can base Theorem 2 either entirely on [Lus3,Lus4] or entirely on [FeOp], that is equivalent. In particular our LLC is determined already by formal degrees of supercuspidal unipotent representations in combination with the functoriality properties (2) and (4).
When G is semisimple we obtain much finer results, summarized in Theorem 2.2. In that setting we explicitly describe the number of cuspidal enhancements of λ π and the number of supercuspidal representations in the L-packet of λ π , with combinatorial data coming from the affine Dynkin diagrams of G and G ∨ .
Strengthening and complementing Theorem 2, we will prove a conjecture by Hiraga, Ichino and Ikeda (cf. [HII,Conjecture 1.4]) for unitary supercuspidal unipotent representations G(K). It relates formal degrees and adjoint γ-factors more precisely than Theorem 1.
Theorem 3 shows in particular that all supercuspidal members of one unipotent Lpacket have the same formal degree (up to some rational factor), as expected in the local Langlands program.
Let us discuss the contents of the paper and the proofs of the main results in more detail. In Section 1 we fix the notations and we recall some facts about reductive groups, enhanced Langlands parameters and cuspidal unipotent representations. Let Ω be the fundamental group of G, interpreted as a group of automorphisms of the affine Dynkin diagram of G. We denote the action of a Frobenius element of W K on G ∨ by θ, so that the group of weakly unramified characters of G(K) can be expressed as Z(G ∨ ) W K ∼ = (Ω θ ) * . In Section 2 we make Theorem 2 more precise for semisimple K-groups, counting the involved objects in terms of subquotients of the finite abelian group (Ω θ ) * . A large part of the paper is dedicated to proving Theorem 2.2, in bottom-up fashion.
In Sections 3-11 we prove most of the claims for simple adjoint groups. The majority can be derived quickly from the tables [Lus3,§7] and [Lus4,§11], which contain a lot of information about the parametrization of Irr(G(K)) cusp,unip from Theorem 1. A simple group of type E 8 , F 4 or G 2 is both simply connected and adjoint, so Ω is trivial. Then Theorem 2.2 is contained entirely in [Lus3], and we need not spend any space on it.
The main novelty in this part is the equivariance of the LLC with respect to W Kautomorphisms of the root datum (part (3) of Theorem 2), that was not discussed in the sources on which we rely here. In some remarks we already take a look at certain non-adjoint simple groups. This concerns cases where we can only check Theorem 2.(3) by direct calculations. In Section 12 we explain which parts of Sections 3-11 are needed where, and we complete the proof of the main theorem for adjoint (unramified) groups.
In Sections 13 and 14 we generalize Theorem 2.2 form adjoint semisimple to all semisimple groups. In particular, we investigate what happens when an adjoint K-group G ad is replaced by a covering group G. It is quite easy to see how Irr(G(K)) cusp,unip behaves. Namely, several unipotent cuspidal representations of G ad (K) coalesce upon pullback to G(K), and then decompose as a direct sum of a few irreducible unipotent cuspidal representations of G(K). With some technical work, we prove that the same behaviour (both qualitatively and quantitatively) occurs for enhanced L-parameters.
The proof of the main theorem for reductive K-groups (Section 15) can roughly be divided into two parts. First we deal with the case that the connected centre of G is anisotropic. We reduce to the derived group of G which is semisimple, and use the results we establish for semisimple groups. To deal with general connected reductive groups, we note that the connected centre is an almost direct product of its maximal split and maximal anisotropic subtori. Applying Hilbert's theorem 90 to the maximal split torus, we obtain a corresponding decomposition of the group of K-rational points. This enables us to reduce to the cases of tori (well-known) and of reductive K-groups with anisotropic connected centre.
We attack the HII conjecture in Section 16. For simple adjoint groups, the second author already proved Theorem 3 in [Opd]. Starting from that and using the proof of Theorem 1, we extend Theorem 3 to all unramified reductive K-groups. Finally, in the appendix we explore the behaviour of L-parameters and adjoint γ-factors under Weil restriction. Whereas L-functions are always preserved, it turns out that adjoint γ-factors sometimes change under Weil restriction. Nevertheless, we can use these computations to prove that the HII conjecture are always stable under restriction of scalars. That is, if L/K is a finite separable extension of non-archimedean local fields and the HII conjectures hold for a reductive L-group, then they also hold for the reductive K-group obtained by restriction of scalars (and conversely).

Preliminaries
Throughout this paper we let K be a non-archimedean local field with finite residue field F of cardinality q K = |F|. We fix a separable closure K s of K and we let K nr ⊂ K s be the maximal unramified extension of K. The residue field F of K nr is an algebraic closure of F. There are isomorphisms of Galois groups Gal(K nr /K) Gal(F/F) Ẑ . The geometric Frobenius element Frob, whose inverse induces the automorphism x → x q K for any x ∈ F, is a topological generator of Gal(F/F). Let I K = Gal(K s /K nr ) be the inertia subgroup of Gal(K s /K) and let W K be the Weil group of K. We fix a lift of Frob in Gal(K s /K), so that W K = I K Frob .
Unless otherwise stated, G denotes an unramified connected reductive linear algebraic group over K. By unramified we mean that G is a quasi-split group defined over K and that G splits over K nr . The group G(K nr ) of K nr -points of G is often denoted by G = G(K nr ). Let Z(G) be the centre of G, and write G ad := G/Z(G) for the adjoint group of G.
We fix a K-Borel B and maximally split maximal K-torus S ⊂ B which splits over K nr . We denote by θ the finite order automorphism of X * (S) corresponding to the action of Frob on S = S(K nr ). Let R ∨ be the coroot system of (G, S) and define the abelian group Ω = X * (S)/ZR ∨ .
Let G ∨ be the complex dual group of G. Then Z(G ∨ ) can be identified with Irr(Ω) = Ω * , and Ω is naturally isomorphic to the group X * (Z(G ∨ )) of algebraic characters of Z(G ∨ ). In particular The isomorphism classes of inner forms of G over K are naturally parametrized by the elements of the continuous Galois cohomology group where F denotes the automorphism of G ad := G ad (K nr ) by which Frob acts on G ad . A cocycle in Z 1 c (F, G ad ) is determined by the image u ∈ G ad of F . The K-rational structure of G corresponding to such a u ∈ G ad is given by the action of the inner twist F u := Ad(u) • F ∈ Aut(G) of the K-automorphism F on G. We will denote this K-rational form of G by G u , and the corresponding group of K-points by G Fu .
The cohomology class ω ∈ H 1 c (F, G ad ) of the cocycle is represented by the F -twisted conjugacy class of u. By a theorem of Kottwitz [Kot1,Thǎ] and by (1.1) there is a natural isomorphism Let G 1 be the kernel of the Kottwitz homomorphism G → X * (Z(G ∨ )) [Kot2,PaRa]. This map is W K -equivariant and yields a short exact sequence We say that a character χ of G Fω is weakly unramified if χ is trivial on G Fω 1 , and we denote by X * wr (G Fω ) the abelian group of weakly unramified characters. Since G is unramified there are natural isomorphisms [Hai,§3.3 This can be regarded as a special case of the local Langlands correspondence. The identity components of the groups in (1.3) are isomorphic to the group of unramified characters of G Fω (which is trivial whenever G is semisimple).
satisfying certain requirements [Bor]. We say that λ is unramified if λ(w) = (1, w) for every w ∈ I K and that λ is discrete if the image of λ is not contained in the L-group of any proper Levi subgroup of G Fω . We denote the G ∨ -conjugacy class of L-parameters (resp. unramified L-parameters and discrete L-parameters) for G Fω by Φ(G Fω ) (resp. Φ nr (G Fω ) and Φ 2 (G Fω )). The group Z(G ∨ ) acts naturally on the set of L-parameters, by This descends to an action of and A λ /Z(G ∨ ) θ is finite if and only if λ is discrete. Let A λ be the component group of the full pre-image of (1.5) in the simply connected covering (G ∨ ) sc of the derived group of G ∨ . Equivalently, A λ can also be described as the component group of ) denotes the group of 1-coboundaries for group cohomology, that is, the set of maps Fix a complex character ζ of the centre Z(G ∨ sc ) of G ∨ sc whose restriction onto Z( L G ad ) = Z(G ∨ sc ) θ corresponds to ω via the Kottwitz isomorphism. If ω is given as an element of Ω ad (not just in (Ω ad ) θ ), then there is a preferred way to define a character of Z(G ∨ sc ), namely via the Kottwitz isomorphism of the K-split form of G. In particular ω = 1 corresponds to the trivial character.
Let Irr(A λ , ζ) be the set of irreducible representations of A λ whose restriction to Z(G ∨ sc ) is a multiple of ζ. The set of enhanced L-parameters for G Fω is We note that the existence of a ρ ∈ Irr(A λ , ζ) is equivalent to λ being relevant [Bor,§8.2.ii] for the inner twist G ω of the quasi-split K-group G [ABPS, Lemma 1.6].
The unipotent element u λ := λ 1, 1 1 0 1 ∈ G ∨ can also be regarded as an element of the unipotent variety of G ∨ sc , and then We say that ρ is a cuspidal representation/enhancement of A λ , or that (λ, ρ) is a cuspidal (enhanced) L-parameter for G Fω , if (u λ , ρ) is a cuspidal pair for Z 1 G ∨ sc (λ(W K )) [AMS,Definition 6.9]. Equivalently, ρ determines a Z 1 G ∨ sc (λ(W K ))-equivariant cuspidal local system on the conjugacy class of u λ . This is only possible if λ is discrete (but not every discrete L-parameter admits cuspidal enhancements). We denote the set of G ∨conjugacy classes of cuspidal enhanced L-parameters for G Fω by Φ(G Fω ) cusp .
The (Ω θ ) * -action (1.4) extends to enhanced L-parameters by The extended action preserves both discreteness and cuspidality.
Let Irr(G Fω ) be the set of irreducible smooth G Fω -representations on complex vector spaces. The group (Ω θ ) * acts on Irr(G Fω ) via (1.3) and tensoring with weakly unramified characters. It is expected that under the local Langlands correspondence (LLC) this corresponds precisely to the action (1.9) of (Ω θ ) * on Φ e (G Fω ). In other words, the conjectural LLC is (Ω θ ) * -equivariant.
Furthermore, the LLC should behave well with respect to direct products. Suppose that G ω is the almost direct product of K-subgroups G 1 and G 2 . Along the quotient map q : G 1 × G 2 → G ω one can pull back any representation π of G ω (K) to a representation π • q of G 1 (K) × G 2 (K). Since q need not be surjective on K-rational points, this operation may destroy irreducibility. Assume that π is irreducible and that π 1 ⊗π 2 is any irreducible constituent of π • q. Then the image of the L-parameter λ π of π under the map should be the L-parameter λ π 1 × λ π 2 of π 1 ⊗ π 2 . In this case A λπ is naturally a subgroup of A λπ 1 × A λπ 2 . We say that a LLC (for some class of representations) is compatible with almost direct products if, when (λ π , ρ π ) denotes the enhanced L-parameter of π and G ω = G 1 G 2 is an almost direct product of reductive K-groups, (1.10) λ π 1 × λ π 2 = q ∨ (λ π ) and ρ π 1 × ρ π 2 | A λπ contains ρ π .
We also want the LLC to be equivariant with respect to automorphisms of the root datum, in a sense which we explain now. Let be the based root datum of G, where ∆ is the basis determined by the Borel subgroup B ⊂ G. Since S and B are defined over K, the Weil group W K acts on this based root datum. When G is semisimple, any automorphism of R(G, S) is completely determined by its action on the basis ∆. Then we call it an automorphism of the Dynkin diagram of (G, S), or just a diagram automorphism of G. When G is simple and not of type D 4 , the collection of such diagram automorphisms is very small: it forms a group of order 1 (type A 1 , B n , C n , E 7 , E 8 , F 4 , G 2 or a half-spin group) or 2 (type A n , D n , E 6 with n > 1, except half-spin groups).
Suppose that τ is an automorphism of R(G, S) which commutes with the action of W K . Via the choice of a pinning of G ∨ (that is, the choice of a nontrivial element in every root subgroup for a simple root), τ acts on G ∨ . Then τ also acts on the collection of Langlands parameters for the inner forms of G ω (K). It also acts on enhancements, by τ * ρ = τ • ρ. The action of τ on G ∨ is uniquely determined up to inner automorphisms, so the action on enhanced L-parameters is canonical. Considering ω ∈ Ω ad as an element of Irr(Z(G ∨ sc )), we can define τ (ω) = ω • τ −1 . Then τ maps enhanced L-parameters relevant for G Fω to enhanced L-parameters relevant for G F τ (ω) .
Let u ∈ G ad represent ω, so that Lift τ to a K nr -automorphism of G stabilizing S and B (this can be done in a way which is unique up to inner automorphisms). Then, for g ∈ G Fu : u) . In particular, for every representation π of G Fω we obtain a representation τ · π = π • τ −1 of G F τ (ω) . Equivariance with respect to W K -automorphisms of the root datum means: if (λ π , ρ π ) is the enhanced L-parameter of π then (1.13) (τ · λ π , τ * ρ π ) is the enhanced L-parameter of τ · π, for all τ ∈ Aut(R(G, S)) which commute with W K . When G is semisimple, we also call this equivariance with respect to diagram automorphisms. We note that it suffices to check this for automorphisms of R(G, S) which fix ω ∈ Ω ad . Indeed, if we know all those cases, then we can get equivariance with respect to diagram automorphisms by defining the LLC for other groups G F ω via the LLC for G Fω and a τ with τ (ω) = ω .
We define a parahoric subgroup of G to be the stabilizer in G 1 of a facet (say f) of the Bruhat-Tits building of (G, K nr ), and we typically denote it by P. Then P fixes f pointwise. If f is F ω -stable, it determines a facet of the Bruhat-Tits building of (G ω , K), and P Fω is the associated parahoric subgroup of G Fω . All parahoric subgroups of G Fω arise in this way.
Let P u be the pro-unipotent radical of P, that is, the kernel of the reduction map from P to the associated reductive group P over F. Then P Fω u is the pro-unipotent radical of P Fω , and the quotient (1.14) P Fω /P Fω u = P Fω ∼ = P Fω is a connected reductive group over F. Unipotent representations of finite reductive groups like (1.14) were classified in [Lus1]. An irreducible representation π of G Fω is called unipotent if there exists a parahoric subgroup P Fω such that the restriction of π to P Fω contains a representation pulled back from a cuspidal unipotent representation of (1.14). We denote the set of irreducible unipotent G Fω -representations by Irr(G Fω ) unip .
In this paper we are mostly interested in supercuspidal G Fω -representations, which form a collection denoted Irr(G Fω ) cusp . Among these, the supercuspidal unipotent representations form a subset Irr(G Fω ) cusp,unip which was described quite explicitly in [Lus3]. Every such G Fω -representation arise from a cuspidal unipotent representation σ of the finite reductive quotient (1.14) of some parahoric subgroup P Fω . For a given finite reductive group there are only few cuspidal unipotent representations, and the number of them does not change when (1.14) is replaced by an isogenous F-group. It turns out that every such σ can be extended to a representation, say σ N , of the normalizer of P Fω in G Fω . By [Opd] there is a natural isomorphism where the right hand side denotes the stabilizer of P in the abelian group Ω θ . It follows that, at least when G is semisimple, is not compact, (1.16) remains true if the right hand side is replaced by a direct integral over (Ω θ,P ) * . Furthermore it is known from [Lus3] that every representation ind G Fω N G Fω (P Fω ) (χ ⊗ σ N ) is irreducible and supercuspidal. Hence (when Ω θ is finite) Every element of Irr(G Fω ) unip,cusp arises in this way, from a pair (P, σ) which is unique up to G Fω -conjugation. We denote the packet of irreducible supercuspidal unipotent G Fω -representations associated to the conjugacy class of (P, σ) via (1.16) and (1.17) by Irr(G Fω ) [P,σ] . In other words, these are precisely the irreducible quotients of ind G Fω P Fω (σ). The group (Ω θ,P ) * acts simply transitively on Irr(G Fω ) [P,σ] , by tensoring with weakly unramified characters. The choice of σ N determines an equivariant bijection . We normalize the Haar measure on G Fω as in [GrGa] and [FeOp,§2.3]. Recall the formal degree of ind G Fω P Fω (σ) equals dim(σ)/vol(P Fω ). When (Ω θ ) * is finite, (1.17) implies that for any π ∈ Irr(G Fω ) [P,σ] .
We will make ample use of Lusztig's arithmetic diagrams I/J [Lus3,§7]. This means that I is the affine Dynkin diagram of G (including the action of W K ), and that J is a W Kstable subset of I. This provides a convenient way to parametrize parahoric subgroups of G up to conjugacy. The W K -action on I boils down to that of the Frobenius element, and only the maximal Frob-stable subsets J I can correspond to parahoric subgroups of G Fω that possess a cuspidal unipotent representation. The above entails that Irr(G Fω ) cusp,unip depends only on some combinatorial data attached to G and F ω : the affine Dynkin diagram I, the Lie types of the parahoric subgroups of G associated to the subsets of I, the group Ω θ and its action on I.
In this section we count the number of enhancements of L-parameters in (2.1), and we find explicit formulas for the numbers of supercuspidal representations in the associated L-packets. To this end we define four numbers: • a is the number of λ ∈ Φ 2 nr (G Fω ) which satisfy (0.1) and have a G Fω -relevant cuspidal enhancement; • b is the number of G Fω -relevant cuspidal enhancements of λ; • a is defined as |Ω P,θ | times the number of G Fω -conjugacy classes of F ω -stable maximal parahoric subgroups P ⊂ G such that there exists a cuspidal unipotent representation σ of P Fω for which ind G Fω P Fω σ has an irreducible summand satisfying (0.1); • b is the number of cuspidal unipotent representations σ of P Fω such that deg(σ ) = deg(σ).
Lemma 2.1. When G is adjoint, simple and K-split, the above numbers a, b, a , b agree with those introduced (under the same names) in [Lus3,6.8].
Proof. Our b is defined just as that of Lusztig. Under these conditions on G, all P as above are conjugate to P, so a = |Ω P,θ |. From [Lus3,1.20] we see that Ω P,θ equalsΩ u over there, so the two versions of a agree.
With b Lusztig counts pairs (C, F) consisting of a unipotent conjugacy class in C in Z G ∨ (λ(Frob)) and a cuspidal local system F on C, such that Z(G ∨ ) acts on F according to the character defined by G ω via the Kottwitz isomorphism (1.2). The set of such (C, F) is naturally in bijection with the set of extensions of λ| W F to a G Fω -relevant cuspidal Lparameter [AMS]. The equate Lusztig's b to ours, we need to show the following. Given G ω and s = λ(Frob), there exists at most one unipotent class in Z G ∨ (s) supporting a G Fω -relevant cuspidal local system.
Recall from [Ste,§8.2] that Z G ∨ (s) is a connected reductive complex group (because G ∨ is simply connected). For the existence of cuspidal local system on unipotent classes Z G ∨ (s) has to be semisimple, so the semisimple element s = λ(Frob) must have finite order and must correspond to a single node in the affine Dynkin diagram of G ∨ [Ree2,§2.4]. As G ∨ is simple, this implies that Z G ∨ (s) has at most two simple factors. For every complex simple group which is not a (half-)spin group, there exists at most one unipotent class supporting a cuspidal local system, whereas for (half-)spin groups there are at most two such unipotent classes [Lus2]. (There are two precisely when the vector space to which the spin group is associated has as dimension a square triangular number bigger than 1.) It follows that the required uniqueness holds whenever G ∨ does not have Lie type B n or D n . The G Fω -relevance of the cuspidal local system (i.e. the Z(G ∨ )-character ω) imposes another condition, limiting the number of possibilities even further. With a tedious verification of all the cases [Lus3,] one can see that in fact the uniqueness of unipotent classes holds for all simple adjoint G. Alternatively, this can derived from Theorem 1.
This uniqueness of unipotent classes also means that our a just counts the number of possibilities for λ W F , or equivalently for s = λ(Frob). The geometric diagram in [Lus3, §7] determines a unique node v(s) of the affine Dynkin diagram I of G ∨ , and hence completely determines the image of s in G ∨ ad . Then the possibilities for s ∈ G ∨ modulo conjugacy are parametrized by the orbit of v(s) in I under the group Ω for G ∨ ad , see [Ree2,§2.2] and [Lus3,§2]. Since G ∨ is simple, this coincides with the orbit of v(s) under the group of all automorphisms of I. The cardinality of the latter orbit is used as the definition of a in [Lus3], so it agrees with our a.
Assume for the moment that G is simple (but not necessarily split or adjoint). Then sθ = λ(Frob) ∈ G ∨ θ has finite order, and s determines a vertex v(s) in the fundamental domain for the Weyl group W (G ∨ , S ∨ ) θ acting on S ∨ . The order n s of v(s) is indicated by the label in the corresponding Kac diagram [Kac,Ree2]. We can also realize v(s) as a node in Lusztig's geometric diagrams [Lus3,§7]. They are denoted as "Ĩ/J", whereĨ is a basis of the root system of the complex group (G ∨ ) θ . The complement of J inĨ is one node, the one corresponding to v(s).
In first approximation, the semisimple group G is a product of simple groups, and thus the above yields a description of the possibilities for λ(Frob) = sθ, v(s) ∈ G ∨ ad and n s = ord(v(s)).
Theorem 2.2. Let G ω be an unramified semisimple K-group.
Cuspidal unipotent representations of G Fω can exist only if J ⊂ A n−1 is empty and ω ∈ Ω has order n. Then G Fω is an anisotropic form of P GL n (K), so isomorphic to D × /K × where D is a division algebra of dimension n 2 over Z(D) = K.
The parahoric P Fω is the unique maximal compact subgroup of G Fω , so Ω P = Ω and a = |Ω P | = n. The cuspidal unipotent representations of G Fω are precisely its weakly unramified characters. There are n of them, naturally parametrized by Z(G ∨ ) via the LLC. Hence a b = n and b = 1.

Projective unitary groups
Take G = P U n , of adjoint type 2 A n−1 , with G ∨ = SL n (C). Now θ = τ is the unique nontrivial diagram automorphism of A n−1 . When n is odd, the groups Ω θ , Ω θ , (Ω * ) θ and (Ω θ ) * are all trivial. When n is even, and all these groups have order 2. When n is even, the nontrivial element of Ω θ acts on 2 A n−1 by a rotation of order 2.
When n is not divisible by four, there is a canonical way to choose the ω ∈ Ω definining the inner form, namely ω ∈ Ω θ . When n is divisible by four, the non-quasi-split inner form G Fω cannot be written with a θ-fixed ω. For that group we just pick one ω ∈ Ω \ Ω 2 . Then the diagram automorphism τ sends G Fω to G F ω −1 , a different group which counts as the same inner form. So equivariance with respect to diagram automorphisms is automatic, unless n is congruent to 2 modulo 4.
The subset J ⊂ 2 A n−1 has to consist of two (possibly empty) F ω -stable subdiagrams 2 A s and 2 A t , with s + t + 2 = n (or s + 1 = n if t = 0 and n is even). The analysis depends on whether or not s equals t, so we distinguish those two possibilities.
When n is odd, no parahoric subgroup associated to another subset of 2 A n−1 gives rise to a cuspidal unipotent representation with the same formal degree as that coming from J. When n is even, the parahoric subgroup associated to J = 2 A t 2 A s does have such a cuspidal unipotent representation, and the subsets J, J of 2 A n−1 form one orbit for Ω θ . This leads to a = |Ω θ,P | = 1. The group G Fω has only one cuspidal unipotent representation with the given formal degree, so that one is certainly fixed by τ .
The cuspidal enhancements of λ are naturally in bijection with the cuspidal local systems supported on unipotent classes in Z SLn(C) (λ(Frob)). The centralizer of the semisimple element λ(Frob) = yθ ∈ L G in SL n (C) is the classical group associated to the bilinear form given by y times the antidiagonal matrix with entries 1 on the antidiagonal. This implies an isomorphism where the Lie type depends on the index of the bilinear form and can be read off from [Lus4,]. To get A λ , we have to add Z(SL n (C)) to (4.1), and then to take the centralizer of λ(SL 2 (C)). The inclusion of Z(SL n (C)) does not make a difference, because in (1.7) we already fixed the restriction of representations of A λ to that group. Since both Sp 2q (C) and SO p (C) admit at most one cuspidal pair (u, ρ) [Lus2,§10], λ has at most one cuspidal enhancement. In other words, b = 1. When n is odd, Theorem 1 produces a unique L-parameter. When n is even, Theorem 1 gives one or two L-parameters. The action of (Ω θ ) * ∼ = Z(G ∨ )/Z(G ∨ ) 2 on L-parameters is by multiplying λ(Frob) with an element of Z(G ∨ ). An element zI n ∈ Z(G ∨ )\Z(G ∨ ) 2 can be written as (1−θ)(z 1/2 U ), where U ∈ GL n (C) θ has determinant z −n/2 = −1. When (4.1) contains a nontrivial special orthogonal group, we can choose U in Z GLn(C) (λ(Frob)), which shows that zλ(Frob) is conjugate to λ(Frob) within (4.1). By [Lus4,§11.3], this condition on (4.1) is equivalent to s = t (and n even), which we already assumed here. With Theorem 1 it follows that in that case there is only one L-parameter with the required adjoint γ-factor. Notice that here Remark. When n is even, some groups isogenous to G = P U n have trivial Ω θ , for instance H = SU n . In other words, the image of H Fω → G Fω does not contain representatives for the nontrivial element of Ω θ . For s = t, the pullback of the G Fω -representation π associated to J = 2 A s 2 A t to H Fω decomposes as a direct sum of two irreducible representations, associated to J and to J = 2 A t 2 A s . Since J and J are stable under τ , τ stabilizes both these H-representations.
The isotropy group of λ as a L-parameter λ H for H Fω is bigger than for G Fω , for zλ H = λ H and elements of SL n (C) which send λ to zλ also stabilize λ H . From the above we see that one such new element in the isotropy group is The group A λ H can be obtained from (4.2) in the same way as described after (4.1). The group (4.2) has precisely two cuspidal pairs, which should be matched with the two direct summands of the pullback of π. Note that the action of τ on (4.1) is (up to some inner automorphism) the unique nontrivial diagram automorphism of that group. In Sp 2q (C) × O p (C) that diagram automorphism becomes inner, which implies that τ fixes both cuspidal pairs for this group. In particular, the aforementioned matching of these with H Fω -representation is automatically τ -equivariant.
The case J = 2 A s 2 A s (with 2s + 2 = n) Now Ω θ,P = Ω θ is nontrivial and a = 2. There are two cuspidal unipotent representations containing σ, parametrized by the two extensions σ 1 , σ 2 of σ to N G Fω (P Fω ). Then Consider the action of τ = θ on G. We may take it to be the action of F , only without the Frobenius automorphism of K nr /K. It stabilizes G Fω , unless n is divisible by four and G Fω is not quasi-split (a case we need not consider, for there equivariance with respect to diagram automorphisms is automatic). Then (1.12) shows that the action of τ on G Fω reduces to the action of this Frobenius automorphism on the matrix coefficients.
Since N G Fω (P Fω )\P Fω contains τ -fixed elements (they are easy to find knowing the explicit form of τ ), τ fixes σ 1 and σ 2 . Thus τ fixes both cuspidal unipotent representations under consideration.
We checked that the diagram automorphism τ = θ fixes all L-parameters under consideration in this section. Every such L-parameters has only one cuspidal enhancement. Hence τ fixes everything on the Galois side, which means that our LLC is τ -equivariant for the representations in this section.

Odd orthogonal groups
Further Lusztig's geometric diagram J has two (possibly empty) components of type C n ± . The L-parameter λ is described in [FeOp,§3.2]. One observes that By [Lus3,.56] b = b = 1 and s = 0 is equivalent to n + = n − . We conclude that
Let us also look at the action of (Ω θ ) * λ on this pair of enhancements of λ. For this we need to exhibit a g ∈ G ∨ such that gλ(Frob)g −1 = −1 · λ(Frob). For that we can look The required g must lie in Z G ∨ (v(s)) \ Z G ∨ (s), so its image in Pin 2p (C) does not lie in Spin 2p (C). Therefore conjugation by g is an outer automorphism of Z G ∨ (λ(Frob)). Every outer automorphism of Spin 2q (C) acts nontrivially on the centre of that group (but fixes −1), and hence exchanges the two cuspidal local systems supported by the unipotent class from λ. Thus (Ω θ ) * λ acts transitively on the set of relevant cuspidal enhancements of λ.

Inner forms of even orthogonal groups
We consider G = P O 2n , of adjoint type D n . Then G ∨ = Spin 2n (C) and Let τ be the standard diagram automorphism of D n of order 2. Then (Ω * ) τ = {1, −1} is the kernel of the projection Spin 2n (C) → SO 2n (C). Apart from that Ω * contains elements and − . In the associated Clifford algebra, is the product of the elements of the standard basis of C 2n . We write Ω = Irr(Ω * ) = {1, η, ρ, ηρ}, where η is fixed by τ and η(−1) = 1. Furthermore we decree that ρ( ) = 1. So ρ has order 2 if n is even and order 4 if n is odd, while τ interchanges ρ and ηρ. The action of Ω {1, τ } on the affine Dynkin diagram D n can be picturized as To check τ -equivariance, the following elementary lemma is useful.
Lemma 7.1. Let n be even and let X be a set with a simply transitive Ω * -action.
Proof. First we show that τ fixes a point of X. Take any x ∈ X and consider τ (x) ∈ X.
If τ (x) = x, we are done. When τ (x) = −x, the element x is fixed by τ , for Suppose that τ (x) = x. We compute But Ω * ∼ = (Z/2Z) 2 since n is even, so we have a contradiction. For similar reasons τ (x) = − x is impossible. Thus X always contains a τ -fixed point, say x 0 . Then the map For the group P O 2n there are five different kinds of subsets J of D n which can give support to cuspidal unipotent representations.
The case J = D n Here Ω P = 1 and a = 1. We must have ω = 1, for otherwise P cannot be F ω -stable. There are four ways to embed J in D n , and they are all associate under Ω.
By [Lus3,§7.40] the geometric diagram has type D p D p , so n is even. The element s = λ(Frob) is a lift of the diagonal matrix I n ⊕ −I n ∈ SO 2n (C) in Spin 2n (C). It follows that (Ω) * λ = 1 and This group has (at most) one cuspidal pair on which Z(G ∨ ) acts as 1, so a = 1 and b = 1.
Remark. Let us rename P O 2n as G ad , and investigate what happens when it is replaced by an isogenous group G, which in particular can be the simply connected cover G sc = Spin 2n . In this remark we will endow objects associated to G ad with a subscript ad.
As Ω sc = 1, the four elements of Ω ad · J define four non-conjugate F ω -stable parahoric subgroups of G Fω sc . Hence the pullback of the unique π ∈ Irr(G Fω ad ) [P,σ] from above to G Fω sc decomposes as a direct sum of four irreducible representations, parametrized by the four elements of Ω ad · (P, σ) or, equivalently, by the four Ω ad -associates of P. We note that the diagram automorphism τ fixes two of these (P, σ) and exchanges the other two.
For G = SO 2n we find two direct summands of π, parametrized by {P, ρP} and both τstable. For G a half-spin group of rank n, π also becomes a direct sum of two irreducible representations upon pulling back to G Fω . Then they are parametrized by {P, ηP}. The diagram automorphism τ exchanges these two half-spin groups, so it does not extend to an automorphism of the (absolute) root datum of such a group.
On the Galois side, the above (λ ad , ρ ad ) determines a single L-parameter λ sc for G Fω sc . The centralizer of λ sc (Frob) is larger than that of λ ad (Frob): Since G Fω is K-split, it suffices to consider enhancements of λ sc that are trivial on Z(G ∨ ). The component group of λ sc for G Fω is identified as where w now has order two and a capital S indicates the subgroup of elements that can be realized by an element of a Spin group (not just in a Pin group). The component group for λ ad as a L-parameter λ for G = SO 2n lies inbetween the above two: It is known that ρ ad is the unique alternating character of A λ ad and of Z G ∨ ad (λ ad )/Z(G ∨ ad ). It can be extended in two ways to an enhancement ρ of λ, a representation of (7.3). Since τ fixes (7.3) pointwise, it also fixes ρ. In particular we can match these two ρ's with the set {P, ηP} (from the p-adic side for G = SO 2n ) in a τ -equivariant way.
Both these extensions ρ are symmetric with respect to the two almost direct factors, so they are stabilized by w. With Clifford theory follows that ρ ad can be extended in precisely four ways to a representation of Z G ad ∨ (λ sc )/Z(G ad ∨ ), and hence to a ρ sc ∈ Irr(A λsc ). These four extensions differ only by characters of By Lemma 7.1 the set of enhancements ρ sc of λ sc is Ω * {1, τ }-equivariantly in bijection with the Ω-orbit of P.
For G a half-spin group of rank n and λ ad considered as a L-parameter λ for G Fω , Z G ad ∨ (λ) is an index two subgroup of Z G ad ∨ (λ ad ), which contains Z G ad ∨ (λ sc ) and differs from (7.5). So ρ ad can be extended in two ways to an enhancement ρ of this λ. We note that τ maps (λ, ρ) to an enhanced L-parameter for the other half-spin group of rank n.
The case J = D s D t with s, t > 2 and s = t Here Ω P = {1, η} and the F ω -stability of P forces ω ∈ {1, η}. In particular a = 2 and b = 1. Now [N G Fω (P Fω ) : P Fω ] = 2 and there are precisely two extensions of σ from P Fω to N G Fω (P Fω ). They differ by a sign on N G Fω (P Fω ) \ P Fω . Since η stabilizes J and P Fω , The two Langlands parameters built from J and the unipotent class associated to σ differ by an element of Ω * . From [Lus3,] we see that the geometric diagram has type D p D q with p = q. The element λ(Frob) is a lift of −I 2p ⊕ I 2q to Spin 2n (C). It is conjugate to −λ(Frob) ∈ Spin 2n (C) by a lift of −1 ⊕ I 2n−2 ⊕ −1 to g ∈ G ∨ . As λ(Frob) is not conjugate to λ(Frob), we obtain The unipotent class from λ [FeOp,§3.2], in the group only supports a (unique) cuspidal local system if n = p + q is even. Then Z(G ∨ ) acts as 1 if n is divisible by 4 and as η if n ≡ 2 modulo 4. So a = 2 and b = 1. As τ fixes the above λ(Frob), it stabilizes both the L-parameters, and then also their enhancements.
In particular the LLC is τ -equivariant in this case.
Remark. Again we work out what changes if we replace G by G sc = Spin 2n . Any π ∈ Irr(G Fω ) cusp,unip as above decomposes a direct sum of the irreducibles upon pulling back to G Fω sc . These are parametrized by {P, ρP}, the set of G Fω sc -conjugacy classes of parahoric subgroups of G which are G Fω -conjugate to P. Since τ stabilizes P, it fixes all four elements of Irr(G Fω sc ) under consideration. Regarding λ as a L-parameter λ sc for G Fω sc , we get This group admits two cuspidal pairs (u λ , ρ sc ) on which Z(G ∨ ) acts as 1 or η. Notice that τ fixes some elements of Z G ∨ (λ sc ) \ Z G ∨ (λ), for example a lift of I 2p−1 ⊕ −I 2 ⊕ I 2q−1 to Spin 2n (C). Hence τ fixes all enhanced L-parameters for G Fω sc involved here.
The Ω * -orbit of λ forms a set X as in Lemma 7.1. The four extensions of σ from P Fω to N G Fω (P Fω ) also form a set X as in Lemma 7.1, and we may identify it with Irr(G Fω ) [P,σ] . Now Lemma 7.1 yields a Ω * {1, τ }-equivariant bijection X ←→ X , which fulfills all the conditions we impose on the LLC.
The case J = 2 A s The involvement of the diagram automorphism of A s implies that ω = ρ or ω = ηρ. These two are interchanged by τ . This points to an easy recipe to make the LLC τequivariant in this case: construct it in some Ω * -equivariant way for ω = ρ, and then define if for ω = ηρ by imposing τ -equivariance.
There are four ways to embed J in D n , two of them are F ω -stable and the other two are F ωη -stable. We have Ω P = {1, ω}, so a = 2 and b = 1.
According to [Lus3,§7.46], just as in the case J = D n , n is even and the geometric diagram has type D n/2 D n/2 . As over there, Ω * λ = Ω * and a = 1. The group Z G ∨ (λ(Frob)) is as in (7.1). It admits two cuspidal pairs on which Z(G ∨ ) acts as ω (so b = 2). Let the unipotent element u be as in [FeOp,§3.2] and assume that ω = ρ. In terms of Spin 2n (C) 2 , the cuspidal pairs are of the form (u × u, ρ 1 ⊗ ρ 2 ), where ρ 1 and ρ 2 differ only by the nontrivial diagram automorphism of Spin 2n (C).
The case J = D t 2 A s D t with s, t > 1 Here 2t + s = n − 1. As for J = 2 A s , ω ∈ {ρ, ηρ} and τ -equivariance of the LLC is automatic in this case. We have Ω P = Ω, so a = 4 and b = 1.
By [Lus3,] the geometric diagram has type D p D q with p > q ≥ 0 and p + q = n. The unipotent class from λ is given in [FeOp,§3.2 When q = 0, the four possibilities for λ(Frob) are non-conjugate and central in G ∨ , so a = 4. The given unipotent class in G ∨ = Spin 2n (C) supports just one cuspidal local system on which Z(G ∨ ) acts as ω, so b = 1. We also note that Ω * λ = 1 = (Ω/Ω P ) * . When q > 0, λ(Frob) and λ(Frob) ∈ Spin 2n (C) are not conjugate, but gλ(Frob)g −1 = −λ(Frob) is achieved by taking for g a lift of −1 ⊕ I 2n−2 ⊕ −1. Hence The group Z G ∨ (λ(Frob)) is given by (7.5). The unipotent class and ω impose that we only look at cuspidal pairs on which −1 ∈ Z(G ∨ ) acts nontrivially. Like in the case J = 2 A s there are four of them, two relevant for G Fρ and two relevant for G Fηρ . Let ρ 1 , ρ 2 denote cuspidal enhancements for Spin 2m (C) with different central characters, nontrivial on −1, and m ∈ {p, q}. Then the enhancements for ω = ρ are ρ 1 ⊗ ρ 2 and ρ 2 ⊗ ρ 1 , and the enhancements for ω = ηρ are ρ i ⊗ ρ i . The same analysis as in the case J = 2 A s shows that Ω * λ acts simply transitively on the G Fω -relevant enhancements of λ.
The exceptional automorphisms of D 4 All the diagram automorphisms of order 2 are conjugate to τ , so equivariance of the LLC with respect to those follows in the same way as equivariance with respect to τ .
Let τ 1 and τ 2 = τ 2 1 be the order 3 diagram automorphisms of D 4 . The subset J = D t 2 A s D t with s > 0 cannot appear here, as s + 1 needs to be of the form b(b + 1)/2 to support a cuspidal unipotent representation. Therefore we must have J = D s D t with s + t = 4. The finite reductive groups of type D 1 , D 2 and D 3 (these are actually of type A) do not admit cuspidal unipotent representations, so only the case J = D 4 remains. There a = b = a = b = 1, so it involves only one representation of G Fω and only one enhanced L-parameter, and these must be fixed by τ 1 and τ 2 .

Outer forms of even orthogonal groups
Let us look at G = P O * 2n , the quasi-split adjoint group of type 2 D n . Then G ∨ = Spin 2n (C) and in L G the Frobenius elements act nontrivially, by the standard automorphism θ = τ of D n of order 2.
Let E/K be the quadratic unramified field extension over which the quasi-split group G Fω splits, and let Frob be the associated field automorphism. From (1.11) we see that where Frob acts on the coefficients of g as a matrix. In particular the action of τ = θ on G Fω reduces to the action of the field automorphism Frob.
There are precisely two extensions of σ from P Fω to N G Fω (P Fω ). Since η stabilizes P and commutes with τ , one can find τ -fixed elements in N G Fω (P Fω ) \ P Fω (see the case G = P O 2n , J = D s D t and a = 2). This entails that τ stabilizes both extensions of σ to N G Fω (P).
From [Lus4,§11.4] we see that the geometric diagram has type B p × B q with p = q and p+q +1 = n. We can represent the image of λ(Frob)θ −1 in SO 2n (C) by the diagonal matrix −I 2p+1 ⊕ I 2q+1 . One finds One checks that λ(Frob) is not G ∨ -conjugate to λ(Frob), so (Ω θ ) * λ = (Ω θ /Ω θ,P ) * = 1 and a = 2. One can obtain A λ from (8.1) by intersecting with the centralizer of λ(SL 2 (C)) and adding Z(G ∨ ). But since G Fω is quasi-split, we may ignore the addition of the centre and just look at cuspidal pairs for (8.1) on which Z(G ∨ ) θ acts trivially. The unipotent class from λ is given in [FeOp,§3.2]. One sees quickly from the classification in [Lus2] that that unipotent class admits a unique cuspidal local system which is equivariant for (8.1), so b = 1.
The case J = 2 D t Here Ω θ,P = {1}, so ω = 1, a = 1 and b = 1. The description of λ from s > 0 remains valid, only now p = q. Let w ∈ G ∨ be a lift of 0 In −In 0 ∈ SO 2n (C). Picking suitable representatives, we can achieve that Thus (Ω θ ) * fixes the equivalence class of λ, a = 1 and (Ω θ ) * λ = (Ω θ /Ω θ,P ) * . In the same way as above one sees that b = 1. This case involves a unique object on both sides of the LLC, and the LLC matches them in an obviously τ -equivariant way.
Remark. Let G sc = Spin * 2n be the simply connected cover of G. When we pull back a G Fω -representation associated to (P, σ) as above to G Fω sc , it decomposes as a direct sum of two irreducible representations, one associated to (P, σ) and one to (ηP, η * σ). The diagram automorphism τ stabilizes P and ηP, so it fixes both these representations of G Fω sc .
On the Galois side, we can consider λ as a L-parameter λ sc for G Fω sc . Its stabilizer is larger than (8.1): The unipotent class from λ supports two cuspidal local systems which are equivariant under (8.2). The diagram automorphism τ induces an inner automorphism of (8.1) and (8.1) (namely, conjugation by λ(Frob)θ), so it stabilizes both these cuspidal enhancements of λ sc .
Consequently equivariance with respect to diagram automorphisms is automatic in this case.
We have Ω θ,P = Ω θ = {1, η} and [N G Fω (P) : P] = a = 2. The element λ(Frob) and its G ∨ -centralizer are as in (8.1), only with different conditions on p and q. In particular a = 2 as above. The unipotent class from λ is given in [FeOp,§3.2], and it differs from the above case J = D s 2 D t . For this class, only cuspidal A λ -representation of dimension > 1 have to be considered. The classification of cuspidal local systems for spin groups in [Lus2,§14] shows that (8.1) admits precisely one on which Z(G ∨ ) acts as ρ, so b = 1.
The case J = 2 A s As in the previous case we take ω = ρ. There are four ways to embed thia J in 2 D n , all conjugate under Ω θ {1, θ}. When n is even, none of these is F ω -stable, so n has to be odd. Then two of these P's are F ω -stable and Ω θ,P = {1}. Hence N G Fω (P Fω ) = P Fω and a = 1.
The element λ(Frob) and its G ∨ -centralizer are still as in (8.1), but with p = q. Just as above for J = 2 D t , one finds (Ω θ ) * λ = (Ω θ /Ω θ,P ) * and a = 1. The analysis of enhancements of λ from the case J = 2 D t 2 A s 2 D t remains valid, so b = 1.
Remark. Let us consider the pullback of one of the above G Fω -representations to G Fω sc . It decomposes as a sum of two irreducibles, parametrized by (P, σ and (ηP, η * σ). Notice that τ does not stabilize these two parahoric subgroups of G, rather, it sends them to F ρη -stable parahoric subgroups. Just as in the remark to the case J = D s 2 D t , one can show that for G sc the L-parameter λ admits two relevant enhancements. Both are fixed by τ , except for the action of Z(G ∨ ) on the enhancements, which τ changes from ρ to ηρ.
The exceptional group of type 3 D 4 Here θ is a diagram automorphism of D 4 of order 3. We have G ∨ = Spin 8 (C), Z(G ∨ ) θ = {1} and Ω θ = {1}. In particular there is a unique inner form, the quasi-split adjoint group of type 3 D 4 .
From the geometric diagrams in [Lus4,] and [Ree2,§4.4] we see that Z G ∨ (λ(Frob)) is either Spin 4 (C) or G 2 (C). Both these groups admit a unique cuspidal pair, so b = 1. Thus, given the formal degree we find exactly one cuspidal unipotent representation of G Fω and exactly one cuspidal enhanced L-parameter. In particular these are fixed by any diagram automorphism of D 4 , making the LLC for these representations equivariant with respect to diagram automorphisms.

Inner forms of E 6
Let G be the split adjoint group of type E 6 . Then G ∨ also has type E 6 and We write Ω = Irr(Ω * ) = {1, ζ, ζ 2 } and we let τ be the nontrivial diagram automorphism of E 6 . There are two possibilities for J ⊂ E 6 .
According to [Lus1,Theorem 3.23] the representation σ k can be realized as the eigenspace, for the eigenvalue e 2kπi3 q 3 , of a Frobenius element F acting on the top l-adic cohomology of a variety X w . Here w is an element of the Weyl group of E 6 which stabilizes the subsystem of type A 2 A 2 A 2 . The action of θ on the σ k comes from its action on X w , the variety of Borel subgroups B of E 6 (F q ) such that B and F (B) are in relative position w. For the particular w used here, X w is θ-stable. Since E 6 is split, F acts on it by a field automorphism applied to the coefficients. The induced action on X w commutes with the θ-action, because F and θ commute as automorphisms of the Dynkin diagram of E 6 . In particular θ stabilizes every eigenspace for F , and θ stabilizes both σ 1 and σ 2 .
Recall that the centralizer of the semisimple element s = λ(Frob) ∈ L G in the simply connected group SL n (C) is connected [Ste,§8.2]. From [Lus3,7.22] we see that s corresponds to the central node v(s) of E 6 . Its centralizer is a complex connected group of type A 2 A 2 A 2 . The root lattice of A 2 A 2 A 2 has index 3 in the root lattice of E 6 , so Z G ∨ (s) has centre of order 3|Ω * | = 9. Hence Z G ∨ (s) is the quotient of the simply connected group SL 3 (C) 3 by a central subgroup C of order 3, such that the projection of C on any of the 3 factors SL 3 (C) is nontrivial. Consequently (9.1) A λ ∼ = (Z/3Z) 3 /C.
Since ω = 1, we only have to look at enhancements of λ which are trivial on Z(G ∨ ), so we may replace Z G ∨ (s) by The centre of the latter group has order 3, and it is generated by the image v(s) of s. The group SL 3 (C) 3 has 2 3 = 8 cuspidal pairs, corresponding to the characters of Z(SL 3 (C) 3 ) ∼ = (Z/3Z) 3 which are nontrivial on each of the 3 factors. Dividing out CZ(G ∨ ) leaves only 2 of these characters. Since τ fixes v(s), it stabilizes sZ(G ∨ ) and fixes both cuspidal enhancements of λ. Thus our LLC for these objects is θ-equivariant.
Remark. Let us investigate what happens when G is replaced by its simply connected cover G sc and λ is regarded as a L-parameter λ sc for G Fω sc . The centralizer of λ sc (Frob) in G ∨ is bigger than that of s. From [Ree2, Proposition 2.1] we get a precise description, namely Z G ∨ (s) {1, w, w 2 }, where the Weyl group element w cyclically permutes the factors of In particular both the cuspidal representations ρ of A λ can be extended in 3 ways to characters of A λsc . As ρ is τ -stable the diagram automorphism group τ acts on the set of extensions of ρ to A λsc . There are 3 such extensions and τ has order 2, so it fixes (at least) one extension, say ρ sc . From the actions on the root systems we see that τ (w) = w 2 . If χ is a nontrivial character of w , then ρ sc ⊗ χ is another extension of ρ and τ (ρ sc ⊗ χ) = ρ sc ⊗ χ 2 .
Thus τ permutes the other two extensions of ρ. Notice that this 3-element set of extensions is, as a τ -space, isomorphic to the set of standard parahoric subgroups of G which are G Fω -conjugate to P.
By the same argument, τ fixes exactly one the three enhanced L-parameters associated to [P, σ i ], say (λ i , ρ i ). Decreeing that π i corresponds to (λ i , ρ i ), we obtain a Ω * τequivariant bijection between Irr(G Fω ) [P,σ i ] and the associated triple in Φ(G Fω ) cusp .
From [Lus4] we see that only J = 2 E 6 supports cuspidal unipotent representations. The group P Fω has one self-dual cuspidal unipotent representation σ 0 , for which a = a = b = b = 1. This representation and its L-parameter are determined uniquely by the geometric diagram [Lus4,11.7], so the objects are fixed by τ .
Also, P Fω has two other cuspidal unipotent representations σ 1 and σ 2 . For σ 1 and σ 2 we have a = a = 1 and b = b = 2 [Lus4,11.6]. The same reasoning as for the inner forms of E 6 with J = E 6 , relying on [Lus1], shows that τ stabilizes both σ i .
Here λ(Frob) = sθ, where s ∈ (G ∨ ) θ is associated to the central node of the affine Dynkin diagram of G ∨ . The orders of θ and of the image of s in G ∨ ad (2 and 3, respectively) are coprime, so Thus the component group of the L-parameter λ associated to σ 1 , σ 2 is obtained from (9.1) by taking θ-invariants. That removes Z(G ∨ ) from (9.1), but then the very definition of A λ says that we have to include the centre again. It follows that In (1.7) we already fixed the Z(G ∨ )-character of every relevant representation of A λ (namely, the trivial central character), so it suffices to consider the representations of the subgroup generated by s. Its irreducible cuspidal representations are precisely the two nontrivial characters. Since s is fixed entirely by θ, so are these two enhancements of λ. We conclude that also in this case the LLC is θ-equivariant.

Groups of Lie type E 7
Let G be the split adjoint group of type E 7 . Then G ∨ also has type E 6 and |Ω| = 2. From the tables [Lus3,.14] we see that two subsets of the affine Dynkin diagram E 7 are relevant for our purposes.
The case J = E 6 This J only gives rise to supercuspidal unipotent representations of G Fω if ω is nontrivial. The associated parahoric subgroup satisfies Ω P = Ω. In [Lus3,7.12 and 7.13] a = a = 2 and b = b ∈ {1, 2}. In view of Theorem 1, Ω * in each case permutes the two involved L-parameters λ. Hence (Ω/Ω P ) * = {1} is precisely the stabilizer of λ and any of its G Fω -relevant enhancements.
The case J = E 7 By [Lus3, 7.14] a = a = 1, Ω P = 1 and ω = 1, so the group G Fω is split. In particular every relevant representation of A λ is trivial on Z(G ∨ ). The group A λ /Z(G ∨ ) is isomorphic to Z/4Z [Ree1,p. 34] and is generated by the element λ(Frob), which has order four in the derived group of G ∨ [Ree2]. The nontrivial element of Ω * = Z(G ∨ ) sends λ(Frob) to a different but conjugate element of G ∨ . Suppose that g ∈ G ∨ achieves this conjugation. Then conjugation by g stabilizes λ(Frob)Z(G ∨ ), so it fixes A λ /Z(G ∨ ) pointwise. Hence the action of Ω * on the enhancements of λ is trivial, and (Ω/Ω P ) * stabilizes them all.

Adjoint unramified groups
First we wrap up our findings for unramified simple adjoint groups, then we prove Theorem 2.2 for all unramified adjoint groups.
Proposition 12.1. Theorem 2.2 holds for all unramified simple adjoint K-groups G.
All possibilities for (P, σ) up to conjugacy are tabulated in [Lus3,§7] and [Lus4,§11]. These lists show that the G Fω -conjugacy class of P is uniquely determined by λ. Hence the (Ω θ ) * -orbits on the set of solutions π of (0.1) are parametrized by the cuspidal unipotent P Fω -representations with the same formal degree as σ. In particular there are b such orbits. For inner forms of split groups the numbers a, b, a , b and the equality ab = a b are known from [Lus3,§7]. For outer forms we have exhibited these numbers in Sections 4, 8 and 10. That b = 1 for classical groups is known from [Lus1]. The equality b = φ(n s ) can be seen from [Lus3,§7] and [Lus4,§11].
In the adjoint case all parahorics admitting cuspidal unipotent representations with the same formal degree are conjugate, so a = |Ω θ,P |. By Theorem 1 (Ω θ ) * acts transitively on the set of λ with the same adjoint γ-factor. Therefore (Ω θ ) * /(Ω θ ) * λ acts simply transitively on it, and a is as claimed in Theorem 2.2.(5). In the previous sections we checked that (Ω θ ) * λ always contains (Ω θ /Ω θ,P ) * . This entails that we can find a bijection as in part (1), which is (Ω θ ) * -equivariant as far as π and λ are concerned, but maybe not on the relevant enhancements of λ. Notice that by Theorem 1 our method determines λ uniquely up to (Ω θ ) * (given π). We remark that in some of the cases a canonical λ can be found by closer inspection, for instance see Section 3. For classical groups twisted endoscopy can be used to canonically associate a L-parameter to every irreducible admissible representation, see [Art, Mok, KMSW].
In the other cases b = b and (Ω θ ) * λ = (Ω θ /Ω θ,P ) * . Usually b = b = 1, then A λ has only one relevant cuspidal representation ρ and (Ω θ /Ω θ,P ) * is the stabilizer of (λ, ρ). When b = b > 1, G must be an exceptional group. For Lie types G 2 , F 4 , E 8 and 3 D 4 the group (Ω θ ) * is trivial, so there is nothing left to prove. For Lie types E 6 and E 7 see Sections 9, 10 and 11. This shows part (3) and part (1) except the equivariance with respect to diagram automorphisms. But the latter was already verified in the previous sections (notice that in Sections 5 and 6 the Dynkin diagrams do not have nontrivial automorphisms). Now only part (6) on the Galois side remains. By the earlier parts, there are precisely b = φ(n s ) orbits under (Ω θ ) * . Since all solutions λ for (0.1) are in the same (Ω θ ) *orbit, the orbits on Φ nr (G Fω ) cusp can be parametrized by enhancements of one λ. More precisely, such orbits are parametrized by any set of representatives for the action of (Ω θ ) * λ on the G Fω -relevant enhancements of λ. Proposition 12.2. Theorem 2.2 holds for all unramified adjoint K-groups G.
Proof. Every adjoint linear algebraic group is a direct product of simple adjoint groups. It is clear that everything in Theorem 2.2 (apart from diagram automorphisms) factors naturally over direct products of groups. Here the required compatibility with (almost) direct products, as in (1.10), says that the enhanced L-parameter of a π ∈ Irr(G Fω ) cusp,unip is completely determined by what happens for the simple factors of G. In particular, Proposition 12.1 establishes Theorem 2.2, except equivariance with respect to diagram automorphisms, for all unramified adjoint groups which are inner forms of a K-split group.
Consider an unramified adjoint K-group with simple factors G i : Suppose that a diagram automorphism τ maps G 1 to G 2 . Then G 1 and G 2 are isomorphic, and τ also maps G ∨ 1 to G ∨ 2 inside G ∨ . Let Γ 1 be the stabilizer of G 1 in the group of diagram automorphisms of G.
Assume that the θ-action on the set {G 1 , . . . , G d } is trivial. Then Theorem 2.2 for G follows directly from Proposition 12.1 for the G i , except possibly for part (2). But we can make the LLC from Theorem 2.2.(1) τ -equivariant by first constructing it for G 1 in a Γ 1 -equivariant way, and then defining it for G 2 by imposing τ -equivariance.
When θ acts nontrivially on the set of direct factors of G, the above enables us to reduce to the case where the G i form one orbit under the group θ . Then clearly For a more precise formulation, let K (d) be the unramified extension of K of degree d. Then the K-group G ω is the restriction of scalars, from K (d) to K, of the K (d) -group G ω 1 . Lemma A.3 says that there is a natural bijection ) cusp . Proposition A.5 says that Theorem 2.2 for the K-group G is equivalent to Theorem 2.2 for the simple, adjoint K (d) -group G 1 , via (12.2). Now θ has been replaced by θ d , which stabilizes G 1 , so we can apply the method from the case with trivial θ-action on the set of simple factors of G.

Semisimple unramified groups
Let G be a semisimple unramified K-group, and let G ad be its adjoint quotient. We will compare the numbers a, b, a and b for G with those for G ad , which we denote by a subscript ad.
Let π ∈ Irr(G Fω ) cusp,unip be contained in the pullback of π ad to G Fω . It is known [Lus1,§3] that unipotent cuspidal representations of a finite reductive group are essentially independent of the isogeny type of the group. So every (P ad , σ ad ) lifts uniquely to (P, σ) and π ∈ Irr(G Fω ) [P,σ] . The packets of cuspidal unipotent representations of these parahoric subgroups satisfy When G ad is adjoint and simple, Lusztig's classification shows that the formal degree (of π ad ) determines a unique conjugacy class of F ω -stable parahoric subgroups of G Fω ad which gives rise to one or more supercuspidal unipotent representations with that formal degree. Via a factorization as in (12.1) this extends to all unramified adjoint G ad . This need not be true when G is not adjoint, but then still all such parahoric subgroups are associate by elements of Ω θ ad . It follows that the number of G Fω -conjugacy classes of such parahoric subgroups is precisely [Ω θ ad /Ω P,θ ad : Ω θ /Ω P,θ ] = g . By (1.18) the group (Ω θ,P ) * acts simply transitively on the set of irreducible G Fω -representations containing σ. It follows that (13.2) a = |Ω θ,P |g = [Ω θ,P ad ad : Ω θ,P ] −1 g a ad , and that a b equals the number of supercuspidal unipotent G Fω -representations with the same formal degree as π. By Theorem 2.2. (3) λ ad is naturally in bijection with the (Ω θ ad ) * -orbit of λ ad . In particular (13.4) a ad = |(Ω θ ad ) * λ ad | = |N λ ad |. Also, (Ω θ ad ) * λ ad has b ad elements and acts simply transitively on the set of G Fω ad -relevant cuspidal enhancements of λ ad .
Proof. (1) By Theorem 2.2 for G ad , (Ω θ ad ) * λ ad is precisely the collection of L-parameters for G Fω ad with a given adjoint γ-factor. Consequently every lift of a G ∨ -conjugate of λ is G ∨ ad -conjugate to an element of (Ω θ ad ) * λ ad , and (Ω θ ) * λ is the collection of L-parameters for G Fω with the same adjoint γ-factor as λ.
Hence the stabilizer of λ in (Ω θ ) * is precisely the image in (Ω θ ) * of the (Ω θ ad ) * -stabilizer of the orbit (Ω θ ad /Ω θ ) * λ ad . That works out as (3) We saw in (13.3) that (N λ ad ) * acts simply transitively on (Ω θ ad ) * λ ad , so it also acts transitively on (Ω θ ) * λ. The stabilizer of λ in this group is (N λ ad /N λ ad ∩ Ω) * , the image of (Ω ad /Ω) * in (N λ ad ) * . Then the quotient group In the setting of Lemma 13.1, A λ contains A λ ad as a normal subgroup. We want to compare these subgroups of G ∨ sc , and the cuspidal local systems which they support, more precisely.
Lemma 13.2. The group A λ /A λ ad is isomorphic to (Ω θ ad /Ω θ N λ ad ) * . Proof. First we determine which lifts of λ to a L-parameter for G Fω ad are G ∨ sc -conjugate. To be conjugate, they have to be related by elements of (Ω θ ad ) * λ ad = (Ω θ ad /N λ ad ) * . To be lifts of the one and the same λ, they may differ only by elements of ker(G ∨ sc → G ∨ ) = (Ω ad /Ω) * . Therefore two lifts of λ are conjugate if and only if they differ by element of the intersection where G ∨ sc acts by conjugation, via the natural map to the derived group of G ∨ . The above implies Next we compare the cuspidal enhancements of λ and λ ad . Since A λ contains A λad as normal subgroup, it acts on A λ ad (by conjugation) and it acts on Irr(A λ ad ). For ρ ad ∈ Irr(A λ ad ), we let (A λ ) ρ ad be its stabilizer in A λ . Lemma 13.3. Every irreducible cuspidal representation ρ ad of A λ ad extends to a representation of (A λ ) ρ ad .
Proof. Recall from Theorem 2.2.(3) for G ad that the stabilizer of (λ ad , ρ ad ) in (Ω θ ad ) * is (Ω θ ad /Ω θ,P ad ) * . Under the isomorphism from Lemma 13.2 or (13.5), the stabilizer of ρ ad in A λ /A λ ad corresponds to (13.7) (Ω θ ad /Ω θ N λ ad ) * ∩ (Ω θ ad /Ω θ,P ad ) * = (Ω θ ad /Ω θ Ω θ,P ad ) * . In the proof of Proposition 12.2 we checked that everything for G ad factors as a direct product of objects associated to simple adjoint groups. In particular ρ ad is a tensor product of cuspidal representations ρ i of groups A i associated to L-parameters λ i for adjoint simple groups G Fω i . Thus it suffices to show that every such ρ i extends to a representation of (A λ ) ρ ad . The action of A λ on ρ i factors through the almost direct factor of G Fω which corresponds to G Fω i . This enables us to reduce to the case where G is almost simple, which we assume for the remainder of this proof. Now we can proceed by classification, using [Lus3,§7]. By (13.7) and [AMS,Theorem 1.2] we have to consider projective representations of (Ω θ ad /Ω θ Ω θ,P ad ) * . In almost all cases this group is cyclic, because Ω θ ad is cyclic. Every 2-cocycle (with values in C × ) of a cyclic group is trivial, so then by [AMS, Proposition 1.1.a] ρ ad extends to a representation of (A λ ) ρ ad .
The only exceptions are the inner forms of split groups of type D 2n , for those Ω θ ad ∼ = (Z/2Z) 2 . The group (Ω θ ad /Ω θ Ω θ,P ad ) * can only be non-cyclic if G is simply connected and Ω θ,P ad = 1, which forces P to be of type D 2n . For this case, see the remark to J = D n in Section 7.
It requires more work to relate the numbers of G Fω -relevant cuspidal enhancements of λ (i.e. b) and of λ ad (i.e. b ad ), in general their ratio is less than [A λ : A λ ad ].
Proof. It follows directly from [AMS,Definition 6.9] that an irreducible A λ -representation is cuspidal if and only if its restriction to A λ ad is a direct sum of cuspidal representations. Such a situation can be analysed with a version of Clifford theory [AMS,Theorem 1.2]. Briefly, this method entails that first we exhibit the A λ -orbits of cuspidal representations in Irr(A λ ad ). In every such orbit we pick one representation ρ ad and we determine its stabilizer (A λ ) ρ ad . By a choice of intertwining operators, ρ ad can be extended to a projective representation ρ ad of (A λ ) ρ ad . Then the set of the irreducible A λ -representations that contain ρ ad is in bijection with the set of irreducible representations (say τ ) of a twisted group algebra of (A λ ) ρ ad /A λ ad . The bijection sends τ to If ρ ad can be extended to a (linear) representation of (A λ ) ρ ad , the aforementioned twisted group algebra becomes simply the group algebra of (A λ ) ρ ad /A λ ad . In that simpler case, the desired number of cuspidal irreducible A λ -representations is the sum, over the A λorbits of the appropriate A λ ad -representations, of the numbers For this equality we use that A λ /A λ ad is abelian, which is immediate from Lemma 13.2. Let us make this explicit. By (13.7) (A λ ) ρ ad is the inverse image of (Ω θ ad /Ω θ Ω θ,P ad ) * in A λ under (13.5). Notice that this does not depend on ρ ad , every other relevant enhancement gives the same stabilizer. It follows that the quotient group (13.10) acts freely on the collection of G Fω -relevant cuspidal enhancements of λ ad . Now we can compute the number of A λ -orbits of such enhancements: (13.11) b ad |(Ω θ,P ad /Ω θ,P N λ ad ) * | −1 = b ad |Ω θ,P N λ ad | |Ω θ,P ad | −1 . Lemma 13.3 ρ ad can be extended to a representation ρ ad of (A λ ) ρ ad . It follows from (13.9) that every A λ -orbit of G Fω -relevant cuspidal enhancements of λ ad accounts for the same number of G Fω -relevant cuspidal enhancements of λ, namely (13.12) By [AMS,Theorem 1.2] b is the product of (13.11) and (13.12).
Lemma 13.5. Fix a G Fω -relevant cuspidal ρ ad ∈ Irr(A λ ad ). There exists a bijection between: • the set of ρ ∈ Irr(A λ ) that contain ρ ad , • the set of G Fω -conjugacy classes of parahoric subgroups of G that are G Fω adconjugate to P, which is equivariant for Ω θ ad /Ω θ,P ad Ω θ and with respect to diagram automorphisms. Proof. By (13.8) and Lemma 13.3 every ρ ∈ Irr(A λ ) which contains ρ ad is of the form for a unique (13.14) ωΩ θ,P ad Ω θ ∈ Ω θ ad /Ω θ,P ad Ω θ = Irr (Ω θ ad /Ω θ,P ad Ω θ ) * . On the other hand, the group in (13.14) parametrizes the G Fω -conjugacy classes of G Fω adconjugates of P. Decreeing that (13.13) corresponds to ωPω −1 , we obtain the required bijection and the Ω θ ad /Ω θ,P ad Ω θ -equivariance. Notice that the set of (standard) parahoric subgroups of G is the direct product of the analogous sets for the almost direct simple factors of G. Together with the explanation at the start of the proof of Lemma 13.3 this entails that for equivariance with respect to diagram automorphisms it suffices to check the cases where G is almost simple.
We only have to consider the Lie types A n , D n and E 6 , for the others only have trivial diagram automorphisms. Among these, we only have to look at the parahoric subgroups P with Ω θ,P = Ω θ , or equivalently at the J that are not Ω θ -stable. That takes care of the inner forms of type A n and of the outer forms of type E 6 . For the outer forms of type A n (J = 2 A s 2 A t ), the inner forms of D n (J = D n and J = D s D t ), the outer forms of D n (J = 2 D t and J = 2 A s ) and the inner forms of E 6 (J = E 6 ) see the remarks in Sections 4, 7, 8 and 9.
14. Proof of main theorem for semisimple groups Proposition 12.2 proves Theorem 2.2 for unramified adjoint groups. When we replace an adjoint group by a group in the same isogeny class, several unipotent cuspidal representations of G Fω ad coalesce and then decompose as a sum of g irreducible unipotent cuspidal representations of G Fω . Similarly, several enhanced L-parameters for G Fω ad coincide, and they can be further enhanced in g ways to elements of Φ nr (G Fω ) cusp .
Recall that λ is the image of λ ad under G ∨ sc → G ∨ . When G is simple, Theorem 1 says that (Ω θ ) * λ is the unique (Ω θ ) * -orbit of L-parameters for G Fω with the same adjoint γ-factor as λ ad (up to a rational number). It follows quickly from the definitions that adjoint γ-factors are multiplicative for almost direct products of reductive groups, cf. [GrRe,§3]. From (1.19) we see that the formal degrees of supercuspidal unipotent representations are also multiplicative for almost direct products, up to some rational numbers C π (which can be made explicit, see [HII,§1] and [Opd,§4.6]). Hence the uniqueness of (Ω θ ) * λ in the above sense also holds for semisimple G, provided we impose the compatibility with almost direct products from (1.10). Together with (14.1) this proves Theorem 2.2.(2).
By Lemma 13.1 the λ ad which coalesce to λ form precisely one orbit under (Ω θ ad /Ω θ ∩N λ ad ) * . From the proof of Lemma 13.2 we see that the restriction of a relevant cuspidal representation ρ of A λ to A λ ad contains precisely enhancements of λ ad in one (Ω θ ad /Ω θ N λ ad ) * -orbit. From Lemma 13.1.
Part (4) can be observed from the adjoint case and (13.1). For Part (5), we note by (13.1) and (13.2) On the other hand, by Lemmas 13.1 and 13.4 Thus Theorem 2.2.(5) for G ad implies that a b = ab. Now we can construct a LLC for Irr(G Fω ) unip,cusp . Every π in there corresponds to a unique (Ω θ ad /Ω θ ) * -orbit in Irr(G Fω ad ) unip,cusp . Then Proposition 12.2 gives an orbit (14.3) (Ω θ ad /Ω θ ) * (λ ad , ρ ad ) ⊂ Φ nr (G Fω ad ) cusp . By Lemma 13.1.(2) that determines a single λ ∈ Φ nr (G Fω ) and from Lemma 13.2 we get one A λ -orbit (14.4) (Ω θ ad /Ω θ N λ ad ) * ρ ad ⊂ Irr(A λ ad ). But (14.3) does not yet determine a unique representation of A λ , in general several extensions of ρ ad to ρ ∈ Irr(A λ ) are possible. By Lemma 13.5 we can match these ρ's with the G Fω -conjugacy classes of parahoric subgroups of G that are G Fω ad -conjugate to P, in a way which is equivariant for Ω θ ad and for diagram automorphisms. For π ∈ Irr(G Fω ) [P,σ] we now choose the ρ which corresponds to the G Fω -conjugacy class of the P. Above we saw that π and (λ, ρ) have the same isotropy group in (Ω θ ) * , so we get a well-defined map from (Ω θ ) * π to (Ω θ ) * (λ, ρ). This map is equivariant for (Ω θ ) * and for all diagram automorphisms that stabilize the domain.
For another π ∈ Irr(G Fω ) unip,cusp we construct (λ , ρ ) ∈ Φ nr (G Fω ) cusp in the same way. We only must take care that, if λ = λ, we select a ρ that we did not use already. Since a b = ab, this procedure yields a bijection Irr(G Fω ) unip,cusp → Φ nr (G Fω ) cusp .
As explained above, at the same time this determines bijections for all diagram automorphisms τ of G. The union of all these bijections is the LLC for all the involved representations, and then it is equivariant with respect to diagram automorphisms. From parts (1) and (3) we get the number of (Ω θ ) * -orbits on the Galois side of the LLC, namely g φ(n s ), just as on the p-adic side. Since the L-parameters with the same adjoint γ-factor form just one (Ω θ ) * -orbit, the orbits of enhanced L-parameters can be parametrized by enhancements of λ, just as in the adjoint case.

Proof of main theorem for reductive groups
First we check that Theorem 2 is valid for any K-torus, ramified or unramified. Of course, the local Langlands correspondence for K-tori is well-known, due to Langlands.
Proposition 15.1. Let T be a K-torus and write T = T(K nr ).
(1) The unipotent representations of T(K) are precisely its weakly unramified characters.
(2) The LLC for Irr(T(K)) unip is injective, and has as image the collection of Lparameters λ : (2) is equivariant for (Ω I K * ) Frob and with respect to W K -automorphisms of the root datum.
The target in part (2) is the analogue of Φ nr (G Fω ) for tori. As A λ = 1, we can ignore enhancements here. Proof.
(1) The kernel T 1 of the Kottwitz homomorphism [Kot2,§7] T → X * ((T ∨ ) I K ) has finite index in the maximal bounded subgroup of T . By [PaRa,Appendix,Lemma 5] T 1 equals the unique parahoric subgroup of T . Then T(K) 1 = T Frob 1 is the unique parahoric subgroup of T(K) = T Frob . The finite reductive quotient T Frob is again a torus, so its only cuspidal unipotent representation is the trivial representation. Hence the unipotent T(K)-representations are precisely the characters of T(K) that are trivial on T(K) 1 , that is, the weakly unramified characters.
(3) From Z(T ∨ ) ∼ = Ω * we see that Now it is clear that the LLC for Irr(T(K)) unip is equivariant under (15.1). Since the LLC for tori is natural, it is also equivariant with respect to all automorphisms of X * (T) that define automorphisms of T(K). These are precisely the automorphisms of the root datum (X * (T), ∅, X * (T), ∅, ∅) that commute with W K .
Next we consider the case that G is an unramified reductive K-group such that Z(G) • (K) is anisotropic. By [Pra, Theorem BTR], that happens if and only if Z(G Fω ) is compact. Equivalently, every unramified character of G Fω is trivial. We will divide the proof of Theorem 2 for such groups over a sequence of lemmas.
Lemma 15.2. Suppose that Z(G) • is K-anisotropic, and let Ω der be the Ω-group for G der . Then Ω θ der = Ω θ . Proof. By (1.1) This implies Lemma 15.3. Suppose that Z(G) • is K-anisotropic. The inclusion G Fω der → G Fω induces a bijection Irr(G Fω ) cusp,unip −→ Irr(G Fω der ) cusp,unip . Proof. These two groups have the same affine Dynkin diagram I. For any proper subset of I, the two associated parahoric subgroups, of G and of G der , have the same Lie type. By Lemma 15.2 G Fω and G Fω der also have the same Ω θ -group. Now the classification of supercuspidal unipotent representations, as in (1.17) and further, is the same for G Fω and for G Fω der . The explicit form (1.18) shows that the ensueing bijection is induced by . Then there exists a g ∈ G ∨ such that gλ g −1 = λ as maps . Replace λ by the equivalent parameter λ = z 2 gλ g −1 z −1 2 . These parameters are unramified, so λ | W K = λ| W K . But λ| SL 2 (C) = λ | SL 2 (C) still only holds as maps SL 2 (C) → G ∨ /Z(G ∨ ) • . In any case, λ 1, ( 1 1 0 1 ) and λ 1, ( 1 1 0 1 ) determine the same unipotent class (in G ∨ and in G ∨ /Z(G ∨ ) • ). Consequenty λ and λ are G ∨ -conjugate.
Lemma 15.5. Suppose that Z(G) • is K-anisotropic. Let λ ∈ Φ nr (G Fω ) and let λ der be its image in Φ nr (G Fω der ). Then A λ = A λ der . Proof. Recall the construction of A λ from (1.5) and (1.6). It says that A λ der is the component group of Since λ is unramified, the difference with λ der resides only in the image of the Frobenius element (see the second half of the proof of Lemma 15.4). To centralize λ der means to centralize λ, up to adjusting λ(Frob) by an element of Z(G ∨ ) • . Together with (15.2) this implies that In particular A λ der = π 0 Z 1 (G der ) ∨ sc )(λ der ) = π 0 Z 1 G ∨ sc (λ) = A λ . Proposition 15.6. Theorem 2 holds whenever Z(G) • is K-anisotropic.
Assume that G is the almost direct product of K-groups G 1 and G 2 . Then (15.3) G der = G 1,der G 2,der and Z(G) are also almost direct products, and there are epimorphisms of K-groups Notice that the connected centres of G 1 and G 2 are K-anisotropic. The composition of the maps in (15.4) factors via G 1 × G 2 → G, so our LLC is also compatible with that almost direct product. Conversely, Lemma 15.3 and Theorem 2.2.(4) leave no choice for the LLC in this case, it just has to be the same as for Irr(G ω der (K)) cusp,unip . We already know from Theorem 2.2 that for the latter the L-parameters are uniquely determined modulo (Ω θ ) * by properties (1), (2) and (4) in Theorem 2. Hence the same goes for the LCC for Irr(G ω (K)) cusp,unip . Now G may be any unramified reductive K-group. Let Z(G ω ) s be the maximal Ksplit central torus of G ω . Recall that the K-torus Z(G ω ) • is the almost direct product of Z(G ω ) s and a K-anisotropic torus Z(G ω ) a [Spr,Proposition 13.2.4]. These central subgroups do not depend on ω, so may denote them simply by Z(G) s and Z(G) a .
Proof. Consider the short exact sequence of K-groups [Spr,Proposition 13.2.2] T is a K-split torus. In the short exact sequences of complex groups the Lie algebra of T ∨ maps isomorphically to the Lie algebra of Z(G) s ∨ . Hence As (1 − θ)T ∨ = 1, we find that This implies that every element of Proof of Theorem 2. Recall that by Hilbert 90 the continuous Galois cohomology group H 1 c (K, Z(G ω ) s ) is trivial. The long exact sequence in Galois cohomology yields a short exact sequence By [Spr,Proposition 13.2.2] the connected centre of G ω /Z(G ω ) s is K-anisotropic, so from Proposition 15.6 we already know Theorem 2 for G ω /Z(G ω ) s . Every (cuspidal) unipotent representation of G ω (K) restricts to a unipotent character of Z(G ω ) s (K). From Proposition 15.1.a we know that those are precisely the weakly unramified characters of Z(G ω ) s (K). This torus is K-split, so all its weakly unramified characters are in fact unramified. Since C × is divisible, every χ ∈ X nr (Z(G ω ) s (K)) can be extended to an unramified character of G ω (K). Thus every π ∈ Irr(G ω (K)) cusp,unip can be made trivial on Z(G ω ) s (K) by an unramified twist.
We just saw that (15.5) induces a short exact sequence Thus we can reformulate the above as a bijection (15.6) On the Galois side of the LLC there is a short exact sequence This induces maps between L-parameters for these groups. It also induces a short exact sequence whose terms can be interpreted as the sets of weakly unramified characters of the associated K-groups (or of their inner forms). As Φ 2 nr (Z(G ω ) s ) ∼ = Z(G ω ) s ∨ , (15.8) and (15.7) show that the map Φ 2 nr (G ω (K)) → Φ 2 nr (Z(G ω ) s ) is surjective with fibres Φ 2 nr (G ω /Z(G ω ) s ). With (15.8) we obtain a bijection (15.9) Both for G ω (K) and for G ω /Z(G ω ) s (K) the component groups of L-parameters are computed in the simply connected cover of G ∨ der = G/Z(G) s ∨ der , see (1.5). By Lemma 15.7 and (1.6) the group A λ for λ ∈ Φ G ω /Z(G ω ) s (K) is the same as the component group for λ as L-parameter for G ω (K). Any z ∈ Z(G ∨ ) θ is made from central elements of G ∨ , so A zλ for G ω (K) equals A λ for G ω (K), and then also for G ω /Z(G ω ) s (K). This says that (15.9) extends to a bijection between the spaces of enhanced L-parameters. Recall from (1.8) that cuspidality of the enhancements is defined via the group Z 1 (G ω ) ∨ sc (λ(W K )), which is the same for G ω (K) as for G ω /Z(G ω ) s (K). Hence (15.9) extends to a bijection (15.10) Comparing (15.10) with (15.6) and invoking Theorem 2 for G ω /Z(G ω ) s (K), we obtain a bijection (15.11) Irr G ω (K) cusp,unip ←→ Φ nr (G ω (K)) cusp .
We note that the LLC for unipotent characters of tori is compatible with almost direct products, that follows readily from Proposition 15.1. Consider G as the almost direct product of Z(G ω ) s and G ω der Z(G ω ) a , where Z(G ω ) a denotes the maximal Kanisotropic subtorus of Z(G ω ) • . Let π ∈ Irr(G ω (K)) cusp,unip with enhancement Lparameter (λ π , ρ π ). In terms of (15.6) we write π = π der ⊗ χ and in terms of (15.9) we write λ π = λ π der λ χ and ρ π = ρ π der . Then . The naturality of the LLC for weakly unramified characters entails that the L-parameter Lemmas 15.3,15.4 and 15.5 show that, to analyse the enhancement L-parameter of π der | (G ω der Z(G ω )a)(K) , it suffices to consider the restriction to G ω der (K). Then we are back in the case of semisimple groups, and the constructions in the proof of Theorem 2.2, see especially (14.3), were designed such that the L-parameter of π der | G ω der (K) is the image of λ π der in Φ nr (G ω der (K)). Similarly, the constructions in Section 13 and their wrap-up after (14.4) show that the enhancement for π der | G ω der (K) contains ρ λπ der . The above says that (15.11) is compatible with the almost direct product G = G ω der Z(G ω ) a Z(G ω ) s . Now the same argument as in and directly after (15.3) and (15.4) shows that Theorem 2.(4) holds.
We will prove (16.1) with a series of lemmas, of increasing generality. By Proposition A.5 we do not have to worry about restriction of scalars.
Proof. By [GrGa,Proposition 6.1.4] the normalization of the Haar measure on G ω (K) is respected by direct products of reductive K-groups. It follows that all the terms in (16.1) behave multiplicatively with respect to direct products. Using that and Proposition A.5, we can follow the strategy from the proof of Proposition 12.2 to reduce to the case of simple adjoint groups. For such groups (16.1) was proven in [Opd,Theorem 4.11].
Proof. As in Sections 13 and 14, we consider the covering map G → G ad . We will show that it adjusts both sides of (16.1) by the same factor. From (1.16) and (1.17) we see that By [DeRe,§5.1] the normalization of the Haar measure is such that By [MaGe,Proposition 1.4.12.c] the cardinality of the group of f-points of a connected reductive group does not change when we replace it by an isogenous f-group. In particular |P ad Fω | = |P Fω |. From [Lus1,§3] we know that σ can also be regarded as a cuspidal unipotent representation σ ad of P ad Fω , and then of course dim(σ ad ) = dim(σ).
Proof. Recall that in Section 15 we established Theorem 2 for G via restriction to G der . By [Lus1,§3], the cuspidal unipotent representations of P Fω can be identified with those of P Fω der . We denote σ as P Fω der -representation by σ der . In Lemma 15.2 we checked that Ω θ,P = Ω θ,P der . Let P a be the image of Z(G) a (K nr ) = Z(G) • (K nr ) in P, a f-anisotropic torus. Then P a × P der is isogenous to P, and [MaGe,Proposition 1.4.12.c] tells us that |P a Fω | |P der Fω | = |P Fω |.
The action of W K on Z(g ∨ ) can be considered the composition of the adjoint representation of L Z(G) a and id W K (as L-parameter for Z(G) a (K)). From the definition A.13 we see that (16.13) γ(s, Ad • λ π , ψ) = γ(s, Ad • λ π der , ψ)γ(s, Ad • id W K , ψ).
Recall from Lemma 15.5 and the proof of Proposition 15.6 that ρ π can be identified with ρ π der . Since Z(G) a (K) is a torus, L-parameters for that group do not need enhancements. Formally, we can say that the enhancement of id W K is the trivial one-dimensional representation of A id W K = π 0 (Z(G) ∨ a ) = 1. As in the proof of Lemma 15.5 we see that By (15.2) this equals Now we compare the right hand sides of (16.1) for π and π der : It was shown in [HII,Lemma 3.5] that Then (16.14) becomes which equals (16.12). In combination with Lemma 16.2 for G der that gives (16.1) for π ∈ Irr(G ω (K)) cusp,unip .
Finally, suppose that ψ a : K → C × is another nontrivial additive character. We may assume that ψ a (x) = ψ(ax) for some fixed a ∈ K × . When we replace ψ by ψ a , we must also replace the Haar measure µ G ω ,ψ on G ω (K) by µ G ω ,ψa . In [HII,(1.1)] the latter is defined as By [HII, Lemmas 1.1 and 1.3] Thus (16.1) for ψ is equivalent with (16.1) for ψ a , and hence with Theorem 3.
Appendix A. Restriction of scalars and adjoint γ-factors Let L/K be a finite separable extension of non-archimedean local fields. In this appendix we will first investigate to what extent local factors are inductive for W L ⊂ W K . This issue is well-known for Weil group representations, but more subtle for representations of Weil-Deligne groups. We could not find in the literature, although it probably is known to some experts. Then we will check that this applies to the Langlands parameters obtained from restriction of scalars of reductive groups. Then we will show that the HII conjectures are stable under Weil restriction.
We follow the conventions of [Tate] for local factors. Let ψ : K → C × be a nontrivial additive character . We endow K with the Haar measure that gives the ring of integers o K volume 1, and similarly for L. For s ∈ C let ω s : W K → C × be the character w → w s . For any (finite dimensional) W K -representation V , by definition L(s, V ) = L(ω s ⊗ V ) and (s, V, ψ) = (ω s ⊗ V, ψ).
We endow objects associated to L with a subscript L, to distinguish them from objects for K (without subscript). The restriction of ω s from W K to W L equals ω s (as defined purely in terms of L), so for any W L -representation V L : As concerns representations of the Weil-Deligne group W K × SL 2 (C), we only consider those which are admissible, that is, finite dimensional and the image of W K consists of semisimple automorphisms. In view of [Tate,§4.1.6] that is hardly a restriction for local factors. It has the advantage that the category of such representations is semisimple, so all the local factors are additive and make sense for virtual admissible representations of W K × SL 2 (C). (The definitions of these local factors will be recalled in the proof of the next proposition.) Recall that the order of ψ is the largest n ∈ Z such that ψ(k) = 1 for all k ∈ K of valuation ≥ −n. Let ψ L : L → C × be the composition of ψ with the trace map for L/K. We recall from [Ser,Proposition III.3.7] that the order of ψ L is determined by the order of ψ and the (co)different of L/K. Proposition A.1. Let (τ L , V L ) be an admissible virtual representation of W L ×SL 2 (C).
(1) L-functions of Weil-Deligne representations are inductive: (2) Denote the cardinality of the residue field of L by q L . Then (s, ind for all s ∈ C.
In particular, -factors of Weil-Deligne representations are inductive in the following cases: • virtual representations of dimension zero; • unramified extensions L/K (up to a sign, which is 1 if ord(ψ) is even).
Proof. Since these local factors are additive, we may assume that (ρ L , V L ) is an actual representation. We write N L = d(τ L | SL 2 (C) ) ( 0 1 0 0 ) ∈ End C (V L ) and The function L, from Weil group representations to C × , is additive and inductive [Tate,§3.3.2]. The latter means that . Let E be the maximal unramified subextension of L/K, and define N E , τ E etcetera in the same way as for K. Since I E = I K , From (A.2), (A.3), (A.4) and (A.5) we deduce that L(s, τ ) = L(ω s ⊗ ker N I K ) = L ω s ⊗ ind W K W E (ker N I E E ) = L(s, τ E ). The extension L/E is totally ramified, so q L = q E . We can take Frob E in W L , then it is also a Frobenius element of W L . From and (A.3) we obtain L(s, τ E ) = L(s, τ L ).
(2) We write coker N = V / ker N and coker N L = V L / ker N L . These are representations of W K and W L , respectively. From (A.1) and (A.2) we see that As L/E is totally ramified, q L = q E and Frob L = Frob E . From (A.1) we see that Hence the rightmost term in (A.7) is the same for τ E and for τ L .
As in (A.5), we find that With elementary linear algebra one checks that Since Frob [E:K] is a Frobenius element of W E , we see that here the rightmost term in (A.7) is the same for τ and for τ E , which we already know is the same as for τ L . By [Tate,§3.4], (V, ψ) is additive and inductive in degree 0 (i.e. for virtual W Krepresentations V of dimension 0). Consider the virtual W L -representation By [Tate,(3.2 . In view of (A.7) and the above analysis of the rightmost term in that formula, (A.10) equals (s, τ, ψ) (s, τ L , ψ L ) −1 , as asserted. In particular we see that (s, τ, ψ) = (s, τ L , ψ L ) if dim(V L ) = 0, proving the inductivity in degree zero. When L/K is unramified, we can simplify (A.10). In that case [Ser,Proposition III.3.7 and Theorem III.5.1] show that ord(ψ L ) = ord(ψ). Furthermore the W K -representation C[W K /W L ] is a direct sum of unramified characters, which makes it easy to calculate its -factor. Pick a ∈ K × with valuation −ord(ψ), so that the additive character ψ a : k → ψ(ak) has order zero. From [Tate,(3.4.4) and §3.2.6] we obtain This holds whenever L/K is unramified. When in addition ord(ψ) is even, the sign on the right hand side of (A.12) disappears, and we obtain an equality of -factors.
With this definition it is obvious that our claims about γ-factors follow from parts (1) and (2).
In the remainder of the paper G can be any connected reductive K-group. Let Ad denote the adjoint representation of L G on For any λ ∈ Φ(G(K)), Ad • λ is an admissible representation of W K × SL 2 (C) on g ∨ /Z(g ∨ ) W K . We refer to the local factors of Ad • λ as the adjoint local factors of λ (all of them tacitly with respect to the chosen Haar measure on K).
Supppose that L/K is a finite separable field extension, that H is a reductive L-group and that G is the restriction of scalars, from L to K, of H. Then G(K) = H(L) and, according to [Bor,Proposition 8.4], Shapiro's lemma yields a natural bijection (A.14) Φ(G(K)) −→ Φ(H(L)).
It is desirable that (A.14) preserves L-functions, -factors and γ-factors -basically that is an aspect of the well-definedness of these local factors. Recall from [Spr,§12.4.5] that G = ind W L W K H ∼ = H [L:K] as L-groups. This yields isomorphisms of W K -groups (A.15) G ∨ ∼ = ind W K W L (H ∨ ) ∼ = (H ∨ ) [L:K] . We regard H ∨ as a subgroup of G ∨ , embedded as the factor associated to the identity element in W K /W L . From the proof of Shapiro's lemma one gets an explicit description of (A.14): it sends λ ∈ Φ(G(K)) to λ W L ×SL 2 (C) ∈ Φ H(L) .
Lemma A.2. Let L be a finite separable extension of K and denote the bijection Φ(G(K)) → Φ H(L) from (A.14) by λ → λ H . Then Ad • λ can be regarded as ind W K ×SL 2 (C) W L ×SL 2 (C) (Ad • λ H ). In particular the adjoint local factors of λ and λ H are related as in Proposition A.1 (with V L = h ∨ /Z(h ∨ ) W L ).
Proof. The proof of Shapiro's lemma and (A.14) entail that every λ ∈ Φ(G(K)) is of the form λ = ind W K ×SL 2 (C) W L ×SL 2 (C) λ H for a λ H ∈ Φ H(L) . To make sense of this induction, we regard λ (resp. λ H ) as a 1-cocycle with values in G ∨ (resp. H ∨ ) and we apply (A.15). Then Ad • λ : W K × SL 2 (C) → Aut g ∨ /Z(g ∨ ) W K equals ind W K ×SL 2 (C) W L ×SL 2 (C) (Ad • λ H ). Knowing that, Proposition A.1 applies. Lemma A.2 shows that adjoint L-functions are always preserved under restriction of scalars (most likely that was known already). Surprisingly, it also shows that adjoint -factors and adjoint γ-factors are usually not preserved under Weil restriction, only if L/K is unramified (and a minor condition on the order of ψ).
We will deduce from Proposition A.1 that the HII conjectures [HII] are stable under Weil restriction: they hold for G(K) if and only if they hold for H(L). For that statement to make sense, we need a way to transfer enhancements of L-parameters from G(K) to H(L): Lemma A.3. The map (A.14) extends naturally to a bijection Φ e (G(K)) → Φ e H(L) , which preserves cuspidality.
Proof. For Φ(H(L)) the equivalence relation on L-parameters and the component groups come from the conjugation action of H ∨ and for Φ(G(K)) they come from the conjugation action of G ∨ ∼ = (H ∨ ) [L:K] . But (A.14) means that a L-parameter for G(K) depends (up to equivalence) only on its coordinates in one factor H ∨ of G ∨ , so the conjugation action of the remaining factors of G ∨ can be ignored. Consequently (A.14) induces a bijection between the component groups A λ on both sides, and (A.14) extends to a bijection (A.16) Φ e (G(K)) → Φ e H(L) .
For G(K) cuspidality of enhancements of λ is formulated via (A.17) Z G ∨ sc (λ(W K )) = Z ind W K W L (H ∨ )sc (λ(W K )) ∼ = Z H ∨ sc (λ(W L )). The right hand side is just the group in which we detect cuspidality of enhancements of λ W L ×SL 2 (C) . Therefore (A.16) respects cuspidality.
We endow our reductive p-adic groups with the Haar measures as in [GrGa] and [HII, Correction]. By [GrGa,Proposition 6.1.4] the normalization of the Haar measures on H(L)/Z(H) s (L) is respected by restriction of scalars of reductive groups, so it agrees with the chosen Haar measure on G(K)/Z(G) s (K). This implies: Lemma A.4. Let π ∈ Rep(G(K)) be admissible and square-integrable modulo centre. The formal degree of π as G(K)-representation is the same as when we consider π as H(L)-representation.
Lemma A.4 agrees with the entire setup in [HII] if the additive characters on K and L have order zero. But that is not always tenable. Namely, if ψ : K → C × has order zero, then ψ L (the composition of ψ with the trace map for L/K) need not have order zero. More precisely, when ord(ψ) = 0, [Ser,Theorem III.5.1] says that ψ L has order zero if and only if L/K is unramified.
When working with an additive character ψ of arbitrary order, we need to use the Haar measure µ G/Z(G)s,ψ on G(K)/Z(G) s (K) defined in [HII,(1.1) and correction].
Proposition A.5. Let π ∈ Irr(G(K)) be square-integrable modulo centre. Let (λ, ρ) ∈ Φ e (G(K)) be an enhanced L-parameter associated to π, and let (λ π H , ρ π H ) ∈ Φ e (H(L)) be its image under the map from Lemma A.3. The following are equivalent: • The HII conjecture (16.1) holds for π as G(K)-representation, with respect to any nontrivial additive character of K. • The HII conjecture (16.1) holds for π as H(L)-representation, with respect to any nontrivial additive character of L.
Proof. If the HII conjecture holds for π ∈ Irr(G(K)) with respect to one nontrivial additive character of K, then by [HII, Lemma 1.1 and 1.3] it holds with respect to all nontrivial additive characters of K. The same applies to π as representation of H(L). Therefore we may assume that ord(ψ) = 0, and it suffices to consider the additive character ψ L of L. Let e L/K and f L/K denote the ramification index and the residue degree of L/K, respectively. We endow L with the discrete valuation whose image is Z ∪ {∞}. The restriction of this valuation to K equals e L/K times the valuation of K.
(1) and Lemma A.2 the L-functions in (A.20) do not change if we replace λ π by λ π H . So all terms on the right hand side of (A.20), except possibly the -factor, are inert under (λ π , ρ π ) → (λ π H , ρ π H ). In [Tate,(3.4 In the body of the paper we only use Proposition A.5 for unramified extensions L/K. That case can be proven more elementarily, without Artin conductors and discrimants.