On the unicity of types in special linear groups

Let $F$ be a non-archimedean local field. We show that any representation of a maximal compact subgroup of $\mathbf{SL}_N(F)$ which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell--Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible $N$-dimensional projective representations of the Weil group of $F$.


Introduction
Let G be a connected reductive group defined over a non-archimedean local field F with ring of integers O, and let G = G(F ). Given a supercuspidal representation π of G, we say that a type for π is a pair (J, λ) consisting of an irreducible representation λ of a compact open subgroup J of G such that the only irreducible representations of G which contain λ upon restriction to J are the twists of π by an unramified character of G.
In many cases, including those of G = GL N and G = SL N with which this paper will be concerned, it is known that there exists a type for every supercuspidal representation of G [BK93a, BK93b,BK94]; this construction of types is completely explicit, and results in a unique conjugacy class of maximal simple types which are contained in π. These maximal simple types are defined from strata (a very specific equivalence class of such strata; see section 3), which are essentially the data of a hereditary O-order A in Mat N (F ) and an algebraic extension E/F of degree dividing N. The order A has a lattice period e A , which coincides with the ramification degree of the extension E/F . For a supercuspidal representation π of GL N (F ) or SL N (F ), we denote by e π the lattice period of the associated hereditary order.
In this paper, we complete the classification of types for a large class of supercuspidal representations of SL N (F ) -those which are essentially tame, which is to say those supercuspidal representations π for which e π is coprime to N. We show that the only types for such a representation are the maximal simple types, together with those types obtained from simple representation theoretic renormalizations of maximal simple types.
Theorem. Let π be an essentially tame supercuspidal representation of SL N (F ). Then the number of SL N (F )-conjugacy classes of types (K, τ ) for π with K ⊂ SL N (F ) a maximal compact subgroup is precisely e π , and any two such types for π are conjugate by an element of GL N (F ). Each of these types is of the form τ = Ind K J µ for (J, µ) a maximal simple type contained in π.
This generalizes the previous result of Paskunas [Pas05] (which is due to Henniart for N = 2 [BM02]) that any supercuspidal representation π of GL N (F ) contains a unique conjugacy class of types defined on maximal compact subgroups, as well as subsuming a previous result of the author for N = 2 and F of odd residual characteristic [Lat15]. We note that while it should be expected that the result is true without the assumption that π is essentially tame, there are some serious arithmetic difficulties which arise if one drops this assumption (namely, for non-essentially tame supercuspidals it is possible for a maximal simple type to intertwine with its twist by some character of large level; our method of proof seems to be poorly suited to dealing with this problem).
We also give an application of this result, which explicitly describes an inertial form of the local Langlands correspondence for essentially tame projective Galois representations. Let I F ⊂ W F denote the inertia and Weil group of some separable algebraic closureF /F .
Theorem. There exists a canonical surjective, finite-to-one map iner from the set of SL N (F )-conjugacy classes of types (K, τ ) for essentially tame supercuspidal representations of SL N (F ) with K ⊂ SL N (F ) maximal compact, and the set of equivalence classes of N-dimensional projective representations of I F of F which extend to an essentially tame L-parameter for SL N (F ).
Given an essentially tame irreducible projective representation ϕ : W F → PGL N (C), let Π be the L-packet of supercuspidal representations of SL N (F ) associated to ϕ, and let π ∈ Π. Then the fibre of iner above ϕ| I F is of cardinality e π · |Π|.

Acknowledgements
The work contained in this paper was supported by an EPSRC studentship as well the Heilbronn Institute for Mathematical Research, and is based on a part of my UEA PhD thesis; I would like to thank Shaun Stevens for his supervision. I am also grateful to Colin Bushnell for pointing out a mistake in a previous draft of the paper, and to Maarten Solleveld for a number of helpful comments.

Notation
Let F be a non-archimedean local field with ring of integers O = O F , maximal ideal p = p E and residue field k = k E of cardinality q F and characteristic p. We write G for the group GL N (F ) andḠ = SL N (F ). Given H ⊂ G a closed subgroup, we let H = H ∩Ḡ.
All conjugacies taking place in the paper will be from the left action; for x ∈ H ⊂ G and g ∈ G, we write g x = gxg −1 , and given a representation σ of H, we let g σ be the representation of g H which acts as g σ( g x) = σ(x).
All representations under consideration will be defined over the complex numbers. For a group H, we denote by Rep(H) the category of smooth representations of H, and by Irr(H) the set of isomorphism classes of irreducible objects in Rep(H). Any representations we consider will be assumed to be smooth.
We denote by X(F ) the group of complex characters χ : F × → C × , and also fix notation for two subgroups of this. We write X nr (F ) for the subgroup of characters χ which are unramified, i.e. for which χ| O × is trivial, and X N (F ) for the subgroup of characters χ for which χ N is unramified.
Given subgroups J, J ′ of G and irreducible representations λ, λ ′ of J and J ′ , respectively, we denote the intertwining of (J, λ) and (J ′ , λ ′ ) by

The Bushnell-Kutzko theory
We begin by recalling the necessary background on the theory of types, which will underlie all of the work in this paper. We make no attempt to be comprehensive; the reader should consult [BK93a, BK93b, BK94] for a complete account.

Strata
Let V be an N-dimensional F -vector space, and let A = End F (V ). Then A × = Aut F (V ) ≃ G. We also fix, once and for all, a level 1 additive character ψ of F , i.e. a character trivial on p but not on O.
A hereditary O-order in A is an O-order A such that every left A-lattice is A-projective. Given such an order A, let P = P A denote its Jacobson radical; thus P is a two-sided invertible fractional ideal of A, and there exists a unique integer e A = e A/O F called the lattice period of A such that ̟A = P e A .
To a hereditary order A, we associate a number of subgroups. Firstly, let K A = {x ∈ G | x A = A}, which we call the normalizer of A. This is an open, compact-modulocentre subgroup of G which contains as its unique maximal compact subgroup the group U A := A × . This group U A admits a filtration by compact open subgroups, given by U k Via A, we may put a valuation on A by setting v A (x) = max{n ∈ Z | x ∈ P n }, where we take v A (0) = ∞.
A stratum in A is a quadruple [A, n, r, β] consisting of a hereditary O-order A, integers n > r ≥ 0, and β ∈ A an element such that v A (β) ≥ −n. Such a stratum defines a character ψ β of U r+1 A /U n+1 A by ψ β (x) = ψ • tr(β(x − 1)). We say that two strata [A, n, r, β] and [A, n ′ , r, β ′ ] are equivalent if the cosets β + P −r A and β ′ + P −r A are equal.
We will be specifically interested in certain classes of strata. Say that a stratum [A, n, r, β] is pure if E := F [β] is a field, E × ⊂ K A and n = −v A (β). We say that a pure stratum is simple if it satisfies a further technical condition r < −k 0 (β, A); see [BK93a,(1.4.5)].
Given a simple stratum [A, n, r, β], we may consider V as an E-vector space. This leads to an E-algebra B β = End E (V ) and a hereditary O E -order B β = A ∩ B β in B β with Jacobson radical Q β = P ∩ B β . As before, we may consider

Tame corestriction
These groups H m (β, A) admit a rather special class of characters known as simple characters. Again, the definitions are technical; see [BK93a,(3.2)]. We simple note that, for each simple stratum [A, n, r, β] and each integer m ≥ 0, one obtains a set C(A, m, β) of simple characters of H m+1 (β, A), satisfying a number of desirable properties. Key among these is the "intertwining implies conjugacy" property: if θ ∈ C(A, m, β) and θ ′ ∈ C(A, m ′ , β ′ ) are such that I G (θ, θ ′ ) = ∅, then m = m ′ and there exists a g ∈ G such that C(A ′ , m, β ′ ) = C( g A, m, g β) and θ ′ = g θ [BK93a, (3.5.11)].
Of particular interest to us is the case that m = 0. Here, we have the following: Theorem 3.1 ( [BH13]). Let π be a supercuspidal representation of G. Then there exists a simple stratum [A, n, 0, β] and a simple character θ ∈ C(A, 0, β) such that π contains θ. The simple character θ is uniquely determined up to G-conjugacy.
3.4. Essentially tame supercuspidal representations of G Definition 3.2. Let π be a supercuspidal representation of G, containing a maximal simple type (J = J(β, A), λ) corresponding to the simple stratum [A, n, 0, β]. We say that π is essentially tame if e A is coprime to N.
Note that this is well-defined, by the intertwining implies conjugacy property. The main property of essentially tame supercuspidal representations which we will require is that their conjugacy classes of simple characters are rather well-behaved: , and suppose that there exists a g ∈ G such that g θ = θ ′ . Then C(A, 0, β) = C(A, 0, β ′ ) and θ = θ ′ .

Simple types in G
We now consider those representations of J(β, A) which contain θ ∈ C(A, 0, β). We approach this problem in several stages. Fix a simple stratum [A, n, 0, β] and a simple character θ ∈ C(A, 0, β). There exists a Heisenberg extension η of θ: this is the unique irreducible representation η of J 1 (β, A) which contains θ upon restriction to H 1 (β, A); in fact, η restricts to H 1 (β, A) as a sum of copies of θ [BK93a, (5.1.1)].
Next, we say that a β-extension of η is an extension of η to J 1 (β, A) which is intertwined by B × β . By [BK93a, (5.2.2)], there always exists a β-extension of η.
. This brings us to the main definition: In practice, there is no need to distinguish between these two cases: the second is essentially a degenerate case of the first, with θ = 1, E = F and B β = A.
We will be interested in the maximal simple types; these are simple types (J, λ) [BK93a] is the following: Theorem 3.5. Let (J, λ) be a maximal simple type in G.
(i) There exists a supercuspidal representation π of G with Hom J (π| J , λ) = 0, and any irreducible representation

Conversely, any supercuspidal representation of G contains some maximal simple type, and if
Given a maximal simple type (J, λ) with λ = κ ⊗ σ, then there is a convenient way of recovering the representation σ. Given an irreducible representation ρ of a group H containing J 1 = J 1 (β, A), the space Hom J 1 (κ, ρ) carries a natural J-action given by j · f = ρ(j) • f • κ(j) −1 , for j ∈ J and f ∈ Hom J 1 (κ, ρ). Since f is J 1 -equivariant, this action is trivial on J 1 and so this defines a functor K κ : Rep(H) → Rep(J/J 1 ) given by Hom J 1 (κ, −). This is an exact functor which, in particular, maps admissible representations of G to finite-dimensional representations of J/J 1 . Given a simple type (J, λ = κ ⊗ σ), one has K κ (λ) = σ.

Simple types inḠ
We now describe the passage, via Clifford theory, from maximal simple types in G to the corresponding objects inḠ. The results in this section are established in [BK93b,BK94].
Let π be a supercuspidal representation of G, and let (J, λ) be a maximal simple type contained in π. The representation π|Ḡ is, by Clifford theory, isomorphic to a multiplicity-free direct sum of representations which are G-conjugate to some supercuspidal representationπ ofḠ. Every supercuspidal representation ofḠ arises in this way.
Definition 3.6. A maximal simple type inḠ is a pair of the form (J, µ) wherē J = J ∩Ḡ and µ is an irreducible subrepresentation of λ|J for some maximal simple type (J, λ) in G.

Types
We now interpret the constructions of the two preceding sections in a slightly more general context.
Definition 3.8. Let π be a supercuspidal representation of a p-adic group G. A [G, π] G -type is a pair (J, λ) consisting of a compact open subgroup J of G and an irreducible representation λ of J such that, for any irreducible representation π ′ of G, one has that Hom J (π ′ | J , λ) = 0 if and only if there exists an unramified character ω of G such that π ′ ≃ π ⊗ ω.
In the case that G =Ḡ, the only unramified characters of G are of the form ω = χ•det for χ ∈ X nr (F ). In the case that G =Ḡ, there are no non-trivial unramified characters, and so the condition simply becomes π ′ ≃ π. From this, it is simple to check that the maximal simple types discussed above are [G, π] G -types for the appropriate choices of G and π.
While we do not go into the details here, we note that this definition makes sense due to more theoretical reasons: a [G, π] G -type is a means of describing the block containing π in the Bernstein decomposition of Rep(G) in terms of a finite-dimensional representation of a compact group; see [BK98].
In this paper, we will completely classify [Ḡ,π]Ḡ-types whenπ is an essentially tame supercuspidal representation ofḠ. The above notion of a type turns out to be inconvenient for these purposes. Indeed, from a [G, π] G -type (J, λ), there are two simple ways of producing new types: forming the pair ( g J, g λ) for some g ∈ G; or forming the pair (K, τ ), where K ⊃ J is compact open and τ is an irreducible subrepresentation of Ind K J λ. We therefore make the following modified definition: It is these archetypes which are amenable to a clean classification. We will often abuse notation, and speak of an archetype (K, τ ) as being a conjuacy class of types, together with the fixed choice of representative (K, τ ).

The main results
Our goal is to show that, given an essentially tame supercuspidal representationπ ofḠ, any [Ḡ,π]Ḡ-type which is defined on a maximal compact subgroup ofḠ must be induced from a maximal simple type contained inπ. The key to this result is the following: Theorem 4.1. Let π be an essentially tame supercuspidal representation of G, and let π be an irreducible subrepresentation of π| K . Suppose that there exists a [Ḡ,π]Ḡ-type of the form (K,τ ). Then there exists an irreducible subrepresentation τ of This is the main technical result of the paper; we delay its proof until section 5 in order to first discuss its consequences.
Any essentially tame supercuspidal representationπ ofḠ is obtained as a subrepresentation of π|Ḡ for some π, and by [Pas05] we know that any [G, π] G -type of the form (K, τ ) for some maximal compact subgroup K of G must be of the form τ ≃ Ind K J λ for some maximal simple type (J, λ) in G. By Frobenius reciprocity, we therefore realizeτ as a subrepresentation of Any subrepresentation of this representation is of the form IndK J µ for some maximal simple type (J, µ) inḠ. We therefore conclude that: Corollary 4.2. Letπ be an essentially tame supercuspidal representation ofḠ, and let (K,τ ) be a [Ḡ,π]Ḡ-archetype. Then there exists a maximal simple type (J, µ) with J ⊂K such thatτ ≃ IndK J µ.
This brings us to our main theorem: Proof. We have already established the first claim in (i). To see thatτ is contained in π with multiplicty one, note that by Frobenius reciprocity the multiplicity with which τ appears inπ|K is equal to the multiplicity with which µ appears. By Since IḠ(µ) =J , the summands in this latter space are non-zero if and only if g ∈J; hence there exists a unique non-zero summand which is one-dimensional.
Similarly, if (K ′ ,τ ′ ) is another [Ḡ,π]Ḡ-archetype, then there exist maximal simple types (J, µ) and (J ′ , µ ′ ) which induce to giveτ andτ ′ . As before, these two types are conjugate and so, without loss of generality we may redefineK ′ so that µ = µ ′ . We fix a standard set of representatives ofḠ-conjugacy classes of maximal compact subgroups ofḠ which containJ. We already have one such group inK. The maximal simple type (J, µ) comes from a simple stratum [A, n, 0, β] withK ⊃Ū A ; let ̟ E be a uniformizer of E = F [β]. ThenJ is contained in each of the groups ̟ j EK , for 0 ≤ j ≤ e A − 1. We claim that there do not exist any otherḠ-conjugacy classes of maximal compact subgroups ofḠ into whichJ admits a containment.
Let ν denote the matrix with ν i,i+1 = 1 for 1 ≤ i ≤ N − 1, ν N,1 = ̟ F and ν i,j = 0 otherwise; then ν is a uniformizer of a degree N totally ramified extension of F . The N compact open subgroups ν jK , 0 ≤ j ≤ N − 1 form a system of representatives of the N conjugacy classes of maximal compact subgroups ofḠ. There exists a choice ̟ E of uniformizer of E such that ̟ j EK ⊂ ν Nj/e AK for each 0 ≤ j ≤ e A − 1. The groupJ/J 1 ≃ SL N/[E:F ] (k E ) contains the kernel of the norm map N k L /k E on some degree N/[E : F ] extension k L /k E . This kernel is a cyclic group of order q N/e A −1 q−1 . Suppose thatJ were contained in ν kK for some value of k other than the e A values constructed above. Then one would haveJ ⊂ e A −1 i=1 ν jN/e AK ∩ ν kK . This group is equal toŪ C for some hereditary O-order C of lattice period e A + 1 (note that no issue arises if e A = N; we have already constructed all possible archetypes).
By Zsigmondy's theorem, unless N/e A = 2 and q = 2 i − 1 or N/e A = 6 and q = 2, there exists a prime r dividing q N/e A − 1 but not dividing q s − 1 for any 1 ≤ s ≤ N/e A . If N/e A = 6 and q = 2, let r = 63, and if N/e A = 2 and q = 2 i − 1, let r = 4. While in the latter two cases r is composite, it will be coprime to q s −1 for every q ≤ s ≤ N/e A , which suffices for our purposes. Thus, via the embedding ker N k L /k F ֒→J/J 1 , one obtains in each case an order r element ofJJ 1 , which lifts to give an order r element ofJ. The inclusion where the latter map is the diagonal embedding, mapsJ/J 1 to a block-diagonal group, the blocks of which are pairwise Galois conjugate. So each of the blocks of GL N/e A (k F ) contains an order r element. However, as one also hasJ ⊂Ū C , one again obtains an order r element of U C /U 1 C ≃ e A +1 i=1 GL N i (k F ), for some partition N = N 1 + · · · + N e A +1 of N. Among these N i , there will be e A − 1 which are equal to N/e A , and the remaining two are distinct from N/e A . Hence in the image of ker N k E /k F ֒→ U C /U 1 C , one obtains an order r element in a block, which is actually contained in the standard parabolic subgroup of GL N/e A (k F ) corresponding to the Levi subgroup GL N l (k F ) × GL N k (k F ), for some l + k = N/e A . But the order of this group is −1 − 1) . So r must divide one of these factors. Clearly r cannot divide q t for any t; otherwise r could not divide q N/e A − 1. Also, as N l − i, N k − i < N/e A for all relevant i, our choice of r guarantees that r may not divide |GL N l (k F ) × GL N k (k F )|. This gives the desired contradiction, and so we conclude thatJ only admits a containment into the e A conjugacy classes of maximal compact subgroups ofḠ which were constructed above. This proves (iv).
Finally, to see (iii), note that we have already shown that given any two [Ḡ,π]Ḡarchetypes of the form (K,τ ) and ( ̟ j EK ,τ ′ ) (we have seen that it is no loss of generality to take our archetypes to be of this form), there exists a maximal simple type (J, µ) arising from the simple srtatum [A, n, 0

Proof of Theorem 4.1
It remains for us to prove Theorem 4.1. Let us begin by fixing some notation, on top of that retained from the statement of the theorem. Let [A, n, 0, β] be a simple stratum, and let θ ∈ C(A, 0, β) be such that π contains θ. Let κ be a fixed β-extension of κ, and suppose that π contains the maximal simple type λ = κ ⊗ σ defined on J = J 0 (β, A). As usual, denote by E the field extension F [β]/F , by B the algebra End E (V ), and by B the hereditary O E -order A ∩ B. Without loss of generality, we may assume that J ⊂ U A ⊂ K.

First approximation
We begin by taking the naïve approach, and attacking the problem via Clifford theory. This allows us to show that the representation Ind K Kτ contains only irreducible subrepresentations which are, in some sense, rather close to being types.
We fix, once and for all, an irreducible subrepresentation Ψ of Ind K Kτ such that Ψ is contained in π. Note that such a Ψ clearly exists: by Frobenius reciprocity we have Hom K (Ind K Kτ , Res G K π) = HomK(τ , ResḠ K Res Ḡ G π) = 0, and so some irreducible subrepresentation of Ind K Kτ is contained in π. Lemma 5.1. Suppose that π ′ is an irreducible representation of G which contains Ψ. Then there exists a χ ∈ X N (F ) such that π ′ ≃ π ⊗ (χ • det).

Decompositions of π| K
Let Λ be an extension of λ to E × J such that c-Ind G E × J Λ ≃ π. It will occasionally be convenient for us to work with slight modifications of λ and Λ. Let It follows from the fact that I G (λ) = I G (Λ) = E × J that both ρ andρ are irreducible, thatρ is an extension of ρ, and that π ≃ c-Ind G K Aρ . We therefore obtain two decompositions of the representation π| K : from the realization π ≃ c-Ind G E × J Λ we obtain the decomposition (5.2.1) while from the realization π ≃ c-Ind G K Aρ we obtain the decomposition (5.2.2) It is decomposition (5.2.1) in which we will be most interested. However, the double coset space E × J\G/K is far too complicated for us to work with directly. We therefore approach the problem via decomposition (5.2.2). Following Paskunas [Pas05,Lemma 5.3], we fix a system of coset representatives. Namely, any coset K A g ′ K in K A \G/K admits a diagonal representative g = (̟ a 1 , . . . , ̟ a N ) such that, for all 0 ≤ i ≤ e A , one has a i(N/e A )+1 ≥ · · · ≥ a (i+1)N/e A ≥ 0, and one of the following holds: (i) a j(N/e A )+1 = a (j+1)N/e A , for some 0 ≤ j < e A ; or (ii) (a) a i(N/e A )+1 = a (i+1)N/e A for all 0 ≤ i < e; (b) a 1 ≥ 2; and (c) there exists 1 ≤ j ≤ N such that a k > 0 if k < j, and a k = 0 if k ≥ j, for all 1 ≤ k ≤ N. For the remainder of the proof, we will always take our coset representative g to be of the above form.
Definition 5.2. Let g ∈ G be a coset representative of the above form, which is such that Given an irreducible subrepresentation ξ of π| K , there exists some coset representative g as above such that ξ ֒→ Ind K g U A ∩K Res g ρ. We say that ξ is a representation of type A (respectively, type B) if g is a coset representative of type A (respectively, type B).
In the case that K A gK = K A K, the representation Ind K g J∩K Res g J g J∩K g λ is equal to Ind K J λ, which is the unique [G, π] G -archetype. We are thus reduced to three possibilities: • the representation Ψ is isomorphic to Ind K J λ; or • the representation Ψ is of type A; or • the representation Ψ is of type B.
In each of the latter two cases, we will argue to obtain a contradiction. It follows that Ψ is a [G, π] G -type; whence the desired result.

Case 1: Ψ is of type A
In the case that Ψ is of type A, we may exploit the failure of the map U A ∩ g −1 K → U A /U 1 A to be surjective in order to turn the problem into one regarding the finite group J/J 1 . Denote by H the image in J/J 1 of J ∩ g −1 K. The crucial result is the following observation of Paškūnas: The use of this is as follows. If Ψ is contained in Ind K g J∩K Res g J g J∩K g (κ ⊗ σ), then there exists an irreducible subrepresentation ξ of σ| H such that Ψ is contained in Ind K g J∩K Res g J g J∩K g (κ⊗ξ). This latter representation is contained in Ind K g J∩K Res g J g J∩K g (κ⊗ σ ′ ), and hence so is Ψ. There are two cases to consider. We first examine the case that σ ′ may be taken to be non-cuspidal.
Lemma 5.5. Suppose that there exists a non-cuspidal irreducible representation σ ′ of J/J 1 such that Ψ is an irreducible subrepresentation of Ind K g J∩K Res g J g J∩K g (κ ⊗ σ ′ ). Then there exists a non-cuspidal irreducible representation π ′ of G which contains Ψ.
Proof. Let Σ be any non-cuspidal irreducible representation of J/J 1 . Restricting to H 1 , the representation κ ⊗ Σ is isomorphic to a sum of copies of θ, and so any irreducible representation π ′ of G containing κ ⊗ Σ must contain the simple character θ. If such a representation π ′ were supercuspidal, then it would contain some maximal simple type (J, λ ′ ), with λ ′ containing θ. Since a supercuspidal representation may only contain a single conjugacy class of simple characters, it must be the case that λ ′ = κ ⊗ σ ′′ for some cuspidal representation σ ′′ of J/J 1 . Performing a Mackey decomposition, we obtain representation π contains the simple characters θ and θ(χ • det); hence θ is conjugate to θ(χ • det) and, since e A is coprime to N, this implies that θ = θ(χ • det), hence χ is trivial on det H 1 = det J 1 .
This will enable us to perform a simple counting argument in order to show that Ψ may not be of type A. Before completing this argument, we first consider the type B case.

Case 2: Ψ is of type B
In the case that Ψ is of type B we require a different approach, for which we must differentiate between two cases.
Suppose first that k 0 (β, A) = −1. Then H 1 (β, A) = U 1 B β H 2 (β, A) and so we may view a non-trivial character µ of (1 + p E )/(1 + p 2 E ) as a character of H 1 /H 2 via the composition On the other hand, if k 0 (β, A) = −1 then the above approach no longer works. Instead, let [A, n, 1, γ] be a simple stratum equivalent to the pure stratum [A, n, 1, β].
To combine these two cases, we let µ be as above if k 0 (β, A) = −1, and let µ = φ −1 β−γ otherwise. As noted by Paškūnas during the proofs of [Pas05, Propositions 7.3,7.16], in each of these two cases we have θµ = θ on H 1 ∩ g −1 K. Moreover, in each case µ is trivial on H 2 .
Lemma 5.7. The representation Ψ cannot be of type B.
Proof. Since Ψ is an irreducible subrepresentation of Ind K g H 1 ∩K g θ|g H 1 ∩K and θµ = θ on H 1 ∩ g −1 K, we see that Ψ is also a subrepresentation of Ind K g H 1 ∩K g (θµ)|g H 1 ∩K ; this latter representation is in turn a subrepresentation of Res G K c-Ind G H 1 θµ. Since any irreducible subquotient of c-Ind G K Ψ is supercuspidal representation of the form π ⊗ (χ • det) for some χ ∈ X N (F ) by Lemma 5.1, there exists a supercuspidal representation of this form which contains θµ. As a supercuspidal representation contains a unique conjugacy class of simple characters, we see that θ(χ•det) is conjugate to θµ.
If χ is trivial on det H 1 then θ is conjugate to θµ, which is shown to be impossible during the proof s of [Pas05, Propositions 7.3, 7.16]. So χ is non-trivial on det H 1 . Since g is a type B coset representative, we must have e A > 1; hence χ is also non-trivial on det H 2 . But since µ is trivial on H 2 we see that θ| H 2 is conjugate to θ(χ • det)| H 2 . As e A is coprime to N these two characters must actually be equal, implying that χ is trivial on det H 2 ; this is a contradiction.

Conclusion
We have seen that Ψ may not be of type B. So suppose for contradiction that Ψ is of type A. By Lemma 5.6, there are two possibilities. If Ψ is contained in a noncuspidal irreducible representation of G, we immediately obtain a contradiction to Lemma 5.1. So suppose that the only irreducible representations σ ′ of J/J 1 for which Ψ is contained in Ind K g J∩K Res g J g J∩K g (κ ⊗ σ ′ ) are cuspidal representations of the form σ ′ ≃ σ ⊗ (χ • det) for some χ ∈ X N (F ) which is trivial on det J 1 . There are at most gcd(N, q F − 1) such characters χ.
We first take care of the simple case where the extension E/F is totally ramified. Then, by [Pas05, Corollary 6.6], the image H in J/J 1 of J ∩ g −1 K is contained in some proper parabolic P subgroup of J/J 1 . Let P op denote the parabolic subgroup opposite to P , and let U be its unipotent radical. Then the restriction to U of Ind J/J 1 H σ| H surjects onto Ind U H∩U Res J/J 1 H∩U σ; this latter representation is isomorphic to a sum of copies of the regular representation of J/J 1 since H intersects trivially with U. Hence there must exist a non-cuspidal representation of J/J 1 which identifies with σ upon restriction to H; this is a contradiction.
So we may assume that E/F is not totally ramified. Since any irreducible representation of J/J 1 which becomes isomorphic to σ upon restriction to H must be isomorphic to σ ⊗ (χ • det) for some χ ∈ X N (F ) with χ trivial on det J 1 , such a irreducible representation σ ′ also agrees with σ upon restriction to H = H · SL N/[E:F ] (k E ) ⊂ J/J 1 . Write Ξ = σ| H . Then Ind As E/F is not totally ramified, k E /k F is a non-trivial extension. Then there exists a proper subextension k of k E which contains k F and is of maximal degree among such extensions of k F such that H contains only k-rational points of J/J 1 (by combining [Pas05,Lemma 6.5] and [Pas05, Corollary 6.6]). Thus, if f = f (E/F ) is the residue class degree of E/F then k ≃ F q f −1 F , and so we may certainly take as a lower bound for [J/J 1 : H ] the number This is no less than q F .
So the representation Ψ may not be of type A. We conclude that Ψ ≃ Ind K J λ, completing the proof of Theorem 4.1.

The local Langlands correspondence
We now give a Galois theoretic interpretation of our unicity results, via the local Langlands correspondence. This allows us to completely describe the fibres of an inertial form of the local Langlands correspondence forḠ.
LetF /F be a separable algebraic closure of F with absolute Galois group Gal(F /F ), and let W F ⊂ Gal(F /F ) be the Weil group: this is the pre-image of Z under the canonical map Gal(F /F ) → Gal(k/k) ≃Ẑ. Let I F = ker(Gal(F /F ) → Gal(k/k) be the inertia group; this is the maximal compact subgroup of W F . Fix a choice Φ of geometric Frobenius element in W F , i.e. an element which maps to −1 ∈Ẑ under the above projection.
Given a p-adic group G, denote byĜ its Langlands dual group. In particular, if G = GL N (F ) thenĜ = GL N (C), and if G = SL N (F ) thenĜ = PGL N (C).
The local Langlands correspondence for G gives a unique natural bijective correspondence rec G : Irr sc (G) → L sc (G) between the set Irr sc (G) of isomorphism classes of supercuspidal representations of G and the set L sc (G) of isomorphism classes of of irreducible representations W F → GL N (C) such that the image of Φ is semisimple [HT01]. (Of course, the local Langlands correspondence for G is more general than this; however, we will only be interested in such representations).
From this, following [LL79,GK82], it is possible to deduce the local Langlands correspondence forḠ. Denote by Irr sc (Ḡ) the set of isomorphism classes of irreducible subrepresentationsπ of π|Ḡ, for π ∈ Irr sc (G), and denote by L sc (Ḡ) the set of projective representations W F → PGL N (C) which lift to an element of L sc (G). Let R : Irr sc (G) → Irr sc (Ḡ) be a map which associates to each π an irreducible subrepresentation of π|Ḡ. Then there exists a unique surjective, finite-to-one map recḠ : Irr sc (Ḡ) → L sc (Ḡ) such that the following diagram commutes for all such choices of R: Here, the map L sc (G) → L sc (Ḡ) is given by composition with the natural projection This map recḠ is the local Langlands correspondence for (the supercuspidal representations of)Ḡ. Its finite fibres are the L-packets in Irr sc (G). ; we now reinterpret this understanding in terms of the local Langlands correspondences for G andḠ. The first step is to establish a form of converse to Theorem 4.1.

Types and L-packets
Proposition 6.1. Let π be an essentially tame supercuspidal representation of G, and let (K, τ ) be the unique [G, π] G -archetype. Letπ be an irreducible subrepresentation of π|Ḡ. Then there exists a g ∈ G and an irreducible componentτ of g τ |gK such that ( gK ,τ ) is a [Ḡ,π]Ḡ-archetype.
Proof. Without loss of generality, assume thatπ ≃ c-IndḠ Kμ , whereμ = c-IndK J µ for some maximal simple type (J, µ) (if not, replaceπ with a G-conjugate for which we may do so; clearly the desired result is true forπ if and only if it is true for every G-conjugate ofπ). Let τ |K = jτ j . We first show that any π ′ ∈ Irr(Ḡ) containing someτ j must appear in the restriction toḠ of π. We have a non-zero map in j HomK(τ j , π ′ |K) = HomK(τ |K, π ′ |K) = HomḠ(c-IndḠ K Res K K τ, π ′ ), and so π ′ is a subquotient of Res Ḡ G c-Ind G K τ . Every irreducible subquotient of c-Ind G K τ is a twist of π, and hence coincides with π upon restriction toḠ, and so any irreducible representation π ′ must be of the required form. Hence the possible representations π ′ all lie in a single G-conjugacy class of irreducible representations ofḠ. Let g ∈ G be such that g π ′ ≃π, hence π ′ ≃ c-IndḠ gK gμ , and choose j so that π ′ containsτ j . We claim that ( gK , gτ j ) is the required type.
It suffices to show that any G-conjugate ofπ containing ( gK , gτ j ) is isomorphic toπ. Suppose that, for some h ∈ G we have HomgK( gπ , gτ j ) = 0. The representation hπ is of the form hπ ≃ c-IndḠ hJ h µ, and so gτ j is induced from some maximal simple type (J ′ , µ ′ ), say. So we have So h µ and µ ′ intertwine inḠ, and are thereforeḠ-conjugate. Hence π ′ is in fact G-conjugate toπ, i.e.π ≃ π ′ and the result follows.
Theorem 6.2. Let π be an essentially tame supercuspidal representation of G, and let (K, τ ) be the unique [G, π] G -archetype. Let Π be the L-packet of irreducible subrepresentations of π|Ḡ. Then the set of [Ḡ,π]Ḡ-archetypes forπ ∈ Π is equal to the set of archetypes of the form ( gK , gτ ) for g ∈ G.
Proof. We show that the union of the sets of [Ḡ,π]Ḡ-types of the form (K,τ ), asπ ranges over Π, is equal to the set of irreducible subrepresentations of τ |K; the general result then follows easily. Let (K,τ ) be such an archetype. By Theorem 4.3,τ is of the required form. Conversely, the irreducible subrepresentations of τ | K are pairwise K-conjugate by Clifford theory, and so if one of them is a type for some element of Π then they all must be. By Proposition 6.1, at least one of them must be a type for someπ ∈ Π.

The inertial correspondence
For G = G orḠ, let I et (G) denote the set of representations I F →Ĝ which are of the form ϕ| I F for some ϕ ∈ L et (G); we call such representations essentially tame inertial types. We begin by recalling the inertial Langlands correspondence for G: Note that while the statement of [Pas05, Corollary 8.2] is not stated in this language, it is trivial to show that the two statements are equivalent. It is the above form of the statement which admits a reasonable generalization toḠ.
As a notational convenience, we transfer some notation to the setting of L-parameters and inertial types. Given ϕ ∈ I et (Ḡ), letφ ∈ L et (G) be a lift of some extension of ϕ to W F . Write ℓ ϕ = length(rec −1 G (φ)|Ḡ), and e ϕ for the lattice period of the hereditary order A such that rec −1 (φ contains a simple character in A(A, 0, β) for some β.
We come to our main result: Theorem 6.4 (The essentially tame inertial Langlands correspondence for SL N (F )). There exists a unique surjective map inerḠ : A et (Ḡ) ։ I et (Ḡ) with finite fibres such that, for any map T assigning to a supercuspidal representationπ ofḠ one of the [Ḡ,π]Ḡ-archetypes, the following diagram commutes: Each of the fibres of inerḠ consists of the full orbit under G-conjugacy of an archetype, with the fibre above an inertial type ϕ being of cardinality e ϕ ℓ ϕ .
Moreover, for any map R assigning to each [G, π] G -archetype a [Ḡ,π]Ḡ-archetype, for π an irreducible subquotient of π|Ḡ, there is a commutative diagram Proof. Let S be any map which assigns to each archetype (K,τ ) in A et (Ḡ) the irreducible subrepresentationπ = c-IndḠ Kτ . Let inerḠ denote the composition Res W F I F •recḠ • S. Let ϕ ∈ I et (Ḡ), and letφ be an extension of ϕ to W F . Let Π = rec −1 (φ). Then Π = {π i } is an L-packet of supercuspidal representations of G consisting of the set of irreducible subrepresentations of some supercuspidal representation π of G. By Theorem 6.2, the finite set {(K i ,τ i } of [Ḡ,π i ]Ḡ-archetypes, asπ i ranges through Π is precisely the set of archetypes given by the irreducible subrepresentations of ( gK , g τ |gK), for g ∈ G. As eachπ i is an archetype, it follows that for all irreducible representationsπ ofḠ, we have thatπ contains someτ i upon restriction to K i if and only ifπ ∈ Π, if and only if rec(π)| I F ≃ ϕ. So the map inerḠ is well-defined, and is the unique map map making the first diagram commute.
We now consider the fibres of inerḠ. Let ϕ ∈ I et (Ḡ). Each of the archetypes in iner −1 (ϕ) is represented by a representation of the formτ = IndK J µ, for some maximal simple type (J, µ) contained in an essentially tame supercuspidal representation, and some maximal compact subgroupK ofḠ which containsJ. Moreover, any Gconjugate of (K,τ ) is also contained in the fibre above ϕ. Conversely, we have seen that any two archetypes in the same fibre of inerḠ are G-conjugate.
So it remains only to calculate the cardinality of iner −1 G (ϕ). Letφ ∈ L et (Ḡ) be an extension ofφ, and letπ be contained in the L-packet rec −1 G (φ). The cardinality of this L-packet is length(π|Ḡ), where π is any representation of G such thatπ ֒→ π|Ḡ, i.e. #rec −1 G (φ) = ℓ ϕ . So the fibre iner −1 G (ϕ) is equal to the disjoint union of the sets of archetypes contained in each of the ℓ ϕ elements of rec −1 (φ). Since any two elements of this L-packets are G-conjugate, any two elements admit the same number of archetypes, which is e ϕ by Theorem 4.3. So we conclude that #iner −1 G (ϕ) = e ϕ ℓ ϕ .
The commutativity of the second diagram is simply a translation of Theorem 6.2 into the language of the inertial correspondence.