Unicity of types for supercuspidal representations of p-adic SL(2)

We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $\pi$, we give an additional explicit description of the number of archetypes admitted by $\pi$ in terms of its ramification. We also describe a relationship between archetypes for $\mathbf{GL}_2$ and $\mathbf{SL}_2$ in terms of $L$-packets, and deduce an inertial Langlands correspondence for $\mathbf{SL}_2$.


Introduction
The local Langlands conjectures provide natural correspondences between certain representations of the Weil-Deligne group of a nonarchimedean local field F and (packets of) smooth, irreducible representations of certain reductive groups defined over F . While this is theoretically very important, it is rather difficult to obtain explicit information from this correspondence. The theory of types arose as a means of obtaining an explicit understanding of the representation theory of p-adic groups, partly in the hope that this would allow one to obtain explicit information on the Weil group side of the Langlands correspondence. Following Bernstein, one may factorize the category of smooth representations of a reductive p-adic group G into a direct product of full subcategories consisting of representations whose irreducible subquotients, viewed as representations of the Weil-Deligne group via local Langlands, identify upon restriction to the inertia group. This suggests an approach to obtaining explicit information about the representation theory of G by restriction of these natural families of representations to the compact open subgroups of G. Formally, given a Bernstein component R of the category of smooth representations of G, one says that a type for R is a smooth irreducible representation λ of a compact open subgroup J of G, such that an irreducible representation π of G contains λ upon restriction to J if and only if π ∈ R. Similarly, we say that (J, λ) is typical for R if any irreducible representation π containing λ must be contained in R. Thus, types provide a means of transferring problems regarding the (infinite-dimensional) representation theory of G into problems regarding the (finite-dimensional) representation theory of its compact open subgroups, allowing for more explicit results. One would then hope to be able to transfer this information to the Weil group side of the Langlands correspondence in order to obtain results about Weil-Deligne representations in terms of their restriction to inertia. Results in this direction have been highly influential, allowing for the proof of an inertial local Langlands correspondence for GL N (F ) in [Pas05], and for progress towards the Breuil-Mézard conjecture, as in [BM02].
As with the representation theory of G, the natural approach to the construction of types is to proceed by parabolic induction -that is, to construct types for the supercuspidal representations of all Levi subgroups of G, and then to find some operation compatible with parabolic induction which will allow for the construction of types for all of the Bernstein components of G (which we now know to be given by the Bushnell-Kutzko theory of covers, as in [BK98]). Considerable progress has been made in this direction: there are now constructions of types for all of the Bernstein components of G when G is a general or special linear group ( [BK93a], [BK93b], [BK94], [BK99] and [GR02]), when G is a classical group in odd residual characteristic ( [Ste08] and [MS14]), and when G is an inner form of a general linear group ([Séc05] [SS08] and [SS12]), and constructions of types for all tamely ramified supercuspidal representations of arbitrary groups G over fields of characteristic zero ( [Yu01] and [Kim07]).
In each of these cases, the approach has been to construct, by a series of successively stronger approximations, an explicit type for each Bernstein component of G. Moreover, all constructions of types for supercuspidals to date have led to resolutions of the long-standing folklore conjecture that any supercuspidal representation of G should be compactly induced from an irreducible representation of an open, compact-modulo-centre subgroup of G. However, it is a priori unclear from the abstract definition of a type that there should not exist other examples of types, which are perhaps less suited to applications. This naturally raises the question of the "unicity of types", which asks whether or not this is the case.
Call any type arising from the constructions discussed above a "Bushnell-Kutzko type". It is simple to see that, given a Bushnell-Kutzko type (J, λ) and a maximal compact subgroup K of G which contains J, the irreducible components of the representation obtained by inducing λ to K are typical representations. One then wishes to show that repeating this process for all Bushnell-Kutzko types provides a complete list of typical representations of maximal compact subgroups of G. As well as being of interest in itself as a resolution of the theory of types for G, such a result also turns out to be precisely what is required in order to allow applications to results such as inertial Langlands correspondences and Breuil-Mézard-like conjectures.
The question of unicity was first considered by Henniart in the appendix to [BM02], precisely in order to allow an application to the Breuil-Mézard conjecture, where he obtains a positive answer for all representations of GL 2 (F ) (at least when the cardinality of the residue field is at least 3). Since then, the result has been extended to cover all supercuspidal representations of GL N (F ) by Paskunas in [Pas05] and to cover all representations of GL 3 (F ) ([Nad14]), and most representations of GL 4 (F ), as well as all representations of GL N (F ) of depth zero by Nadimpalli in work to appear.
In this paper, we take the first steps towards unicity for special linear groups, providing a positive answer for supercuspidal representations of SL 2 (F ) when F is of odd residual characteristic. The main difference between the cases of GL 2 (F ) and SL 2 (F ) is that there are now two conjugacy classes of maximal compact subgroups. Given an irreducible representation π of SL 2 (F ), there will clearly always be a typical representation of at least one of these maximal compact subgroups, obtained by inducing up a Bushnell-Kutzko type for π, but not necessarily on both. In order to deal with these complications, we introduce the notion of an archetype, which is an SL 2 (F )-conjugacy class of typical representations (K , τ ) for some maximal compact subgroup K . With this in place, we are able to give a natural extension of the unicity of types to this setting, providing a positive answer for supercuspidal representations in section 2.
In section 3, we go on to provide a more explicit description of the archetypes for the supercuspidal representations of SL 2 (F ), allowing the following refinement of our unicity result: Theorem 1.1. Let π be a supercuspidal representation of SL 2 (F ), for F a nonarchimedean local field of odd residual characteristic. If π is of integral depth, then there exists a unique archetype for π, while if π is of half-integral depth then there exist precisely two archetypes for π, which are GL 2 (F )-conjugate but not SL 2 (F )-

conjugate.
We also provide, in Proposition 3.3, a description of the relationship between supercuspidal archetypes in GL 2 (F ) and SL 2 (F ) in terms of the local Langlands correspondence, which in some sense says that archetypes are functorial with respect to restriction from GL 2 (F ) to SL 2 (F ). This allows us to deduce in Corollary 3.4 an extension of Paskunas' inertial Langlands correspondence to our setting.
Our method is to transfer Henniart's results on GL 2 (F ) over to SL 2 (F ), with the key step being to show that any archetype for an irreducible representationπ of SL 2 (F ) must be isomorphic to an irreducible component of the restriction of the unique archetype for some irreducible representation π of GL 2 (F ) containingπ upon restriction. This is achieved in Lemma 2.2. From this, it is mostly a case of performing simple calculations to deduce in Theorem 2.4 that our unicity result holds. The explicit counting result on the number of archetypes contained in a supercuspidal representation follows easily from Theorem 2.4, while we are able to prove in Lemma 3.2 a form of converse to Lemma 2.2 for the supercuspidal representations, which allows us to easily deduce the remaining results.
While we have avoided doing so in this paper, one could have proved the same results by essentially copying the methods used by Henniart for GL 2 (F ). One may show unicity with respect to a fixed choice of maximal compact subgroup by following Henniart's approach, making only the necessary changes, with the only additional complication being the proof that the integral depth supercuspidal representations admit only a single archetype. For the positive depth representations, this is achievable using a minor variation of Henniart's arguments, but the depth zero representations require more work. The author knows of two approaches in this case: to use the branching rules found in [Nev13], or to argue using covers (in the sense of [BK98]). The problem with this approach is that there is necessarily a large amount of duplication of effort. While one would expect that such an approach could be made to work for arbitrary N, this would require reproving most of the results found in [Pas05] with only minor modifications.
On the other hand, the approach taken in this paper is largely general, and already gives partial progress towards a general proof of the unicity of types for SL N (F ). In particular, the proof of Lemma 2.1 goes through in the general setting without any additional difficulties, suggesting the possibility of applying the results of [Pas05] in a similar manner to our use of Henniart's arguments in order to prove an analogue of Lemma 2.2, which the author is hopeful of managing in the near future. In particular, this would lead easily to a positive answer to the question of unicity. The remaining results should then follow without too much difficulty in a similar manner to that here. In particular, this should allow for the following extension of our explicit results on supercuspidals. Given a supercuspidal representationπ of SL N (F ) and a supercuspidal representation π of GL N (F ) which containsπ upon restriction, we define the ramification degree ofπ to be the number eπ such that there are N/eπ characters χ of F × such that π ≃ π ⊗ (χ • det). This is independent of the choice of π. Then we make the following conjecture: Conjecture 1.2. Letπ be a supercuspidal representation of SL N (F ). Then there are precisely eπ archetypes forπ, which are GL N (F )-conjugate.

Acknowledgements
This paper has resulted from my first year of PhD work at the University of East Anglia. It has benefited hugely from discussions with my advisor, Shaun Stevens, who is due thanks for his constant encouragement and generosity with ideas. I am also grateful to the EPSRC for providing my research studentship.

Notation
Throughout, F will denote a nonarchimedean local field of odd residual characteristic p. We will denote by O = O F the ring of integers of F , and write p = p F for its maximal ideal. The residue field will be denoted by k = k F = O/p, and we will write q for the cardinality of k. We fix once and for all a choice ̟ of uniformizer of F , i.e. an element such that ̟O = p.
When working in generality, we will use G to denote an arbitrary p-adic group defined over F , by which we will mean the group G = G(F ) of F -rational points of some connected reductive algebraic group G defined over F . We will always denote by G the general linear group GL 2 (F ). We fix notation for a number of important subgroups of G. We will write K = GL 2 (O) for the standard maximal compact subgroup, T for the split maximal torus of diagonal matrices, and B for the standard Borel subgroup of upper triangular matrices. We also writeḠ for the special linear group SL 2 (F ) and, given a closed subgroup H of G, we letH denote the subgroup H ∩Ḡ ofḠ. We also denote by T 0 the compact part of the torus, i.e. the group of diagonal matrices with entries in O × , and by B 0 = B ∩ K the group of upper triangular matrices with entries in O. We will denote by η the matrix 0 1 ̟ 0 , so that we may takeK and ηKη −1 as representatives of the twoḠ-conjugacy classes of maximal compact subgroups inḠ.
We use the notation g x = gxg −1 for conjugation, similarly denoting by g X = { g x | x ∈ X} the action of conjugation on a set. Given a representation σ of a closed subgroup H of G, we denote by g σ the representation of g H given by We write Rep(G) for the category of smooth representations of G, and Irr(G) for the set of isomorphism classes of irreducible representations in Rep(G). Given a closed subgroup H of G, we write Ind G H σ for the smooth induction of σ to G, and c-Ind G H σ for the compact induction. We write Res G H π for the restriction of π to H, or simply π ⇂ H for brevity when it is unnecessary to make clear the functor. Given subgroups H, H ′ of G and representations λ, λ ′ of H, H ′ , respectively, we write Given a parabolic subgroup P of G with Levi decomposition P = MN , we denote the normalized parabolic induction of an irreducible representation ζ of M to G by Ind G M,P ζ. By this, we mean Ind G M,P ζ = Ind G Pζ ⊗ δ −1/2 P , whereζ is the inflation of ζ to P and δ P is the modular character of P.
Finally, we denote by X(F ) the group of complex characters χ : F × → C × . We will be interested in two subgroups of this: the group X nr (F ) of unramified characters in X(F ) (i.e. those which are trivial on O × ), and the group X N (F ) of order N characters in X(F ) (i.e. those χ ∈ X(F ) such that χ N = 1).

The Bernstein decomposition and types
The Bernstein decomposition, which was first introduced in [Ber84], allows us to give a factorization of the category Rep(G), which suggests a natural approach to its study. Given an irreducible representation π of G, there exists a unique G-conjugacy class of smooth irreducible representations σ of Levi subgroups M of G such that π is isomorphic to an irreducible subrepresentation of Ind G M,P σ, for some parabolic subgroup P of G with Levi factor M. We call this equivalence class the supercuspidal support of π, and denote it by scusp(π). We put a further equivalence relation on the set of possible supercuspidal supports, by saying that The inertial support of π is then the inertial equivalence class of scusp(π). If scusp(π) = (M, σ), then we write [M, σ] G for the inertial support of π.
With this in place, let B(G) denote the set of inertial equivalence classes of supercuspidal supports, and, for s ∈ B(G), let Rep s (G) denote the full subcategory of Rep(G) consisting of representations such that all irreducible subquotients have inertial support s, and write Irr s (G) for the set of isomorphism classes of irreducible representations in Rep s (G). Bernstein then shows that More generally, given a subset S of B(G), let Rep S (G) = s∈S Rep s (G) and Irr S (G) = s∈S Irr s (G). This allows us to define the notion of a type in generality: (i) We say that (J, λ) is S-typical if, for any smooth irreducible representation π of G, we have that Hom J (π ⇂ J , λ) = 0 ⇒ π ∈ Irr S (G).
In the case that S = {s} is a singleton, we will simply speak of s-types rather than {s}-types.
In the cases of interest to us, the Bernstein components of Rep(G) admit particularly simple descriptions: if Rep s (G) contains a supercuspidal representation π, then Irr s (G) = {π ⊗ (χ • det) | χ ∈ X nr (F )}. The situation forḠ is even simpler: asḠ has no unramified characters, Irr s (Ḡ) is a singleton whenever it contains a supercuspidal representation.
We now introduce the slightly modified notion of an archetype, which is more suited to studying the unicity of types in groups other than GL N (F ).
Definition 1.4. Let S ⊂ B(G). An S-archetype is a G-conjugacy class of Stypical representations (K , τ ) for K a maximal compact subgroup of G. Given a representative (K , τ ) of an archetype, we write G (K , τ ) for the full conjugacy class.
Remark 1.5. It may seem odd to define an archetype as a conjugacy class of typical representations rather than as a conjugacy class of types. However, for us, the difference turns out to be unimportant: the unicity of types will allow us to see that typical representations of maximal compact subgroups are types in almost all cases (indeed, for all representations not contained in the restriction of the Steinberg representation of G). The reason for working with typical representations rather than types is that it allows us to include these "Steinberg" representations in the general picture, despite them admitting no type of the form (K , τ ).
There is one obvious way of constructing archetypes: Lemma 1.6. Let π be an irreducible representation of a p-adic group G of inertial support s. Let (J, λ) be an s-type, and let K be a maximal compact subgroup of G containing J. Then the irreducible components of τ := c-Ind K J λ are representatives of s-archetypes. Moreover, if τ is irreducible then it is an s-type.
Proof. Using Frobenius reciprocity, it is clear that if an irreducible representation π ′ of G contains τ , then it must contain λ, hence the first claim. The second claim simply follows by the transitivity of induction.
The question of the unicity of types is then whether there are any archetypes other than those induced from Bushnell-Kutzko types. For G, Henniart answers this in the appendix to [BM02]: Theorem 1.7. Suppose q = 2. Let π be an irreducible representation of G of inertial support s. Let G (K, τ ) be an s-archetype. Then there exists a Bushnell-Kutzko type (J, λ) with J ⊂ K such that τ ֒→ c-Ind K J λ. Moreover, unless π is a twist of the Steinberg representation St G , the representation c-Ind K J λ is irreducible and hence (K, τ ) is an s-type.

Bushnell-Kutzko types
We now describe the explicit construction, due to Bushnell and Kutzko, of types for the irreducible representations of G andḠ. In this section, we discuss the types for supercuspidal representations, which are the simple types constructed in [BK93a], [BK93b] and [BK94]. The construction of these types is by a series of successively stronger approximations of a type, and is rather technical in nature. We omit as many details as possible; the full details for our case of N = 2 may be found in the appendix of [BM02], or in [BH06]. The starting points for the construction are the hereditary O-orders. For our purposes, we may simply say that the G- We also require the Jacobson radicals of these hereditary orders. The radical of M is P M = Mat 2 (p), and the radical of I is the ideal P I of matrices which are strictly upper-triangular modulo p. Given a hereditary order A, we may then define a filtration of U A by compact open subgroups, by setting U n A = 1 + P n A , for n ≥ 1. There is an integer e A called the O-lattice period associated to each hereditary order; it is the positive integer e A such that P e A A = ̟A. The construction of the simple types (J, λ) is then by simple strata. Roughly speaking, any type (J, λ) for a supercuspidal representation π of G is constructed via a triple [A, n, β] consisting of a hereditary O-order A, the integer n such that n/e A is the depth of π, and an element β of P −n A such that E := F [β] is a field. For our purposes, it suffices to know that . We will also briefly make use of certain filtration subgroups of J: These constructions lead, for each supercuspidal representation π of G, to an irreducible representation λ of a compact open subgroup J of G, such that (J, λ) is a [G, π] G -type and there exists a unique extension Λ of λ to the G-normalizerJ of λ such that π ≃ c-Ind G J Λ. Any s-type arising from these constructions is a (maximal) G-simple type. The other main fact that we will require is the "intertwining implies conjugacy" property ([BK93a], Theorem 5.7.1), which says that, if we have two maximal simple types (J, λ) and (J ′ , λ ′ ) such that I G (λ, λ ′ ) = ∅, then (J, λ) and (J ′ , λ ′ ) must actually be G-conjugate.
In our case, the simple types inḠ are easily obtained from those in G. Let π be a supercuspidal representation of G, so that π ⇂Ḡ splits into a finite sum of supercuspidal representations ofḠ. Choose a simple type (J, λ) extending to (J, Λ) such that π ≃ c-Ind G J Λ, so that we may perform a Mackey decomposition to obtain π ⇂Ḡ≃ Ḡ \G/J c-IndḠ gJ gλ , whereλ = λ ⇂J. This is a finite length sum, and the summands will generally be reducible of finite length. However, in our case all ramification is tame and this is actually a decomposition into irreducibles, with one family of exceptions: for the unramified twists of the "exceptional depth zero" supercuspidal representation of G, which under local Langlands corresponds to the triple imprimitive representation of the Weil group, each of the above summands is reducible of length 2. We then define the (maximal)Ḡ-simple types to be the irreducible components of the representations gλ , for (J, λ) running over the G-simple types. Given such aḠ-simple type (J, µ), we have that IḠ(µ) =J; thus they induce up to a supercuspidal representation ofḠ, and it is clear that this gives a construction of all of the supercuspidals ofḠ. Just as in the case of G, we have an intertwining implies conjugacy property: if two maximalḠ-simple types (J, µ) and (J ′ , µ ′ ) are such that IḠ(µ, µ ′ ) = ∅, then there exists a g ∈Ḡ such that (J ′ , µ ′ ) ≃ ( gJ , g µ) ([BK93a], Theorem 5.3 and Corollary 5.4).

The local Langlands correspondence for supercuspidals
Some of our results on supercuspidals will require a basic understanding of the relevant local Langlands correspondences, which we quickly recall here. Fix once and for all a choiceF /F of separable algebraic closure. We have a natural projection Gal(F /F ) ։ Gal(k/k), constructed by viewing Gal(F /F ) as an inverse limit over finite Galois extensions. The kernel of this map map is the inertia group of F , which we denote by I F . Let W F denote the Weil group, which as an abstract group is given by the subgroup of Gal(F /F ) generated by I F and the Frobenius elements, and is then topologized so that I F is an open subgroup of W F , on which the subspace topology coincides with its topology inherited from Gal(F /F ). While in general one requires the full Weil-Deligne group, we will only consider the Langlands correspondence for supercuspidal representations; thus we may simply work with the Weil group.
For a p-adic group G, let Irr scusp (G) denote the set of equivalence classes of supercuspidal representations of G. Let L 0 (G) denote the set of irreducible L-parameters for G, which is the same as the set of irreducible Frobenius-semisimple representations W F → GL 2 (C), i.e. those irreducible representations under which some fixed Frobenius element of W F acts semisimply. Then the local Langlands correspondence for G provides a unique natural bijection rec : Irr scusp (G) ↔ L 0 (G), which preserves L-and ε-factors, as well as mapping supercuspidal representations to irreducible L-parameters, among a list of other properties.
From this, as shown in [LL79] and [GK82], one may deduce a Langlands correspondence for the supercuspidal representations ofḠ, which suffices for our purposes. Let L 0 (Ḡ) be the image of L 0 (G) under the natural map Hom(W F , GL 2 (C)) → Hom(W F , PGL 2 (C)). Then we define the local Langlands correspondence rec on Irr scusp (Ḡ) by requiring that, for any map R which sends a supercuspidal representation π of G to one of the irreducible components of Res Ḡ G π, the diagram commutes. The (supercuspidal) L-packets are then simply the finite fibres of the map rec, which are precisely the sets of irreducible components of the restrictions toḠ of supercuspidal representations of G.
One may use the local Langlands correspondence to give an alternative description of G-inertial equivalence classes of supercuspidal representations: two supercuspidal representations π and π ′ of G are inertially equivalent if and only if rec(π) ⇂ I F ≃ rec(π ′ ) ⇂ I ′ F .

The main unicity result
We now begin working towards the main results, beginning with a description of the relationship between archetypes in G and those inḠ.
Lemma 2.1. Let π be a supercuspidal representation of G, letπ be an irreducible component of π ⇂Ḡ, and suppose thatπ admits an archetypeḠ(K,τ ). Let Ψ be an irreducible subquotient of Ind K Kτ which is contained in π ⇂ K , and let S = Proof. We first note that such a Ψ clearly exists: let ω π denote the central character of π, and write ω 0 π for its restriction to O × . Letτ be the extension to O ×K ofτ by ω 0 π . Then, by Frobenius reciprocity, some irreducible quotient of c-Ind K O ×Kτ must be contained in π upon restriction to K. From now on, Ψ will always denote this representation.
Proof. By Lemma 2.1, it remains only to rule out the possibility that Ψ is contained in a representation of inertial support [G, π ⊗ (χ • det)] G , for some non-trivial χ ∈ X 2 (F ). Indeed, this would show that Ψ is [G, π] G -typical, and the unicity of types for G would immediately imply that Ψ represents a [G, π] G -archetype. We now argue by cases.
As π is a supercuspidal representation, we may write π ≃ c-Ind G J Λ, with (J, Λ) extending a maximal simple type (J, λ) contained in π. Suppose for contradiction that Ψ is not a type. As noted by Henniart in the appendix to [BM02], paragraphs A. 2.4 -A.2.7, A.3.6 -A.3.7 and A.3.9 -A.3.11, every irreducible component of π ⇂ K other than τ appears in the restriction to K of either a parabolically induced representation, or some other supercuspidal π µ , in a different inertial equivalence class to that of π, which we may now describe explicitly. There are three further subcases which we treat separately.
Suppose first that Ψ is contained in some parabolically induced representation. We may therefore find a character ζ of T such that Ψ is isomorphic to some irreducible component of Res G K Ind G T,B ζ. As Ind K Kτ projects onto Ψ, we therefore have Hom G (π, IndḠ T ,B Res T T ζ).
Here, n is the integer such that c-IndḠ Kτ ≃π ⊕n , which exists by Proposition 5.2 of [BK98], and the final equality follows from a Mackey decomposition with the summation involved being trivial as BḠ = G. Henceπ is contained in some parabolically induced representation, which provides a contradiction by Lemma 2.1. Now suppose Ψ does not appear as an irreducible component of the restriction to K of any parabolically induced representation, and suppose furthermore that π is of integral depth n. In this case, we may construct a new supercuspidal representation containing every irreducible component of π ⇂ K other than the archetype τ . Let E/F be the unique unramified quadratic extension of F , and choose an embedding O × E ⊂ K. Let µ be any level 1 character of E × trivial on F × , and let (J, λ) be a simple type for π. Then the pair (J, λ ⊗µ) is again a maximal simple type contained in some supercuspidal representation π µ lying in a different inertial equivalence class to that of π, and any irreducible component of π ⇂ K other than τ must be contained in π µ upon restriction; in particular, we must have σ ֒→ π µ ⇂ K . But then π µ must be isomorphic to an unramified twist of π ⊗ (χ • det), for some (non-trivial by assumption) χ ∈ X 2 (F ), which is to say that their archetypes must coincide. The archetype for π µ is c-Ind K J λ ⊗ µ, and the archetype for π ⊗ (χ • det) is (c-Ind K J λ) ⊗ (χ • det). If these two representations are isomorphic, then we must have λ ⊗µ ≃ λ ⊗(χ• det), as I K (λ ⊗ µ, λ ⊗ (χ • det)) = ∅, and if g intertwines λ ⊗ µ with λ ⊗ (χ • det), then g intertwines λ ⇂ J 1 with itself, and so g ∈ J. As λ ⊗ µ ≃ λ ⊗ (χ • det), we may use Schur's lemma to obtain 0 = Hom J (λ ⊗ µ, λ ⊗ (χ • det)) ⊆ End J 1 (λ ⇂ J 1 ) = C, and hence Hom J (λ ⊗ µ, λ ⊗ (χ • det)) contains the identity map, so that we must have µ = χ • det on O × E . However, there are only two quadratic characters χ of F × while there are q + 1 ≥ 4 such characters µ. Choosing µ non-quadratic, we obtain a contradiction.
Finally, consider the case where π is of half-integral depth and Ψ is not contained in any parabolically induced representation, we argue essentially as before. We assume further that π is of depth at least 3 2 ; for π of depth 1 2 Henniart shows that this case never arises. Let E/F be the ramified quadratic extension associated to the simple type for π, and choose an embedding O × E ⊂ U I ⊂ K. For µ, we take a level 2 character of E × trivial on O × F , and construct π µ as before. Letting (J, λ) be a simple type for π, so that λ is one-dimensional, the pair (J, λ ⊗ µ) is a simple type for some supercuspidal π µ in a different inertial equivalence class to that of π, and Ψ must appear in the restriction to K of π µ . Then, up to an unramified twist, π µ ≃ π ⊗(χ•det) for χ a nontrivial quadratic character of F × , and so the archetypes c-Ind K J λ ⊗ µ and (c-Ind K J λ) ⊗ (χ • det) coincide. Then I K (λ ⊗ µ, λ ⊗ (χ • det)) = ∅, and if g ∈ I K (λ ⊗ µ, λ ⊗ (χ • det)), then g ∈ I K (λ ⇂ J 2 , λ ⇂ J 2 ) = J, as π is of depth at least 3 2 , and λ is one-dimensional so that λ ⇂ J 2 is a simple character. It follows that we must have λ ⊗ µ ≃ λ ⊗ (χ • det), and so µ ≃ χ • det. But then µ is of level 2 while χ • det is tame and hence of level at most 1, giving a contradiction and completing the proof.
To see (ii), it remains to check that, given two distinct simple types (J , µ) and (J ′ , µ ′ ) contained inπ which are, moreover, contained in the same conjugacy class of maximal compact subgroups, these simple types provide the same archetypes through induction. Thus, we may as well assume thatJ,J ′ ⊂K. As (J, µ) and (J ′ , µ ′ ) are s-types, π will appear as a subquotient of the induced representations c-IndḠ J µ and c-IndḠ J ′ µ ′ ; hence we will have and so IḠ(µ, µ ′ ) = ∅. As π is supercuspidal then (J , µ) and (J ′ , µ ′ ) will be simple types, and so by the intertwining implies conjugacy property there will exist a g ∈Ḡ such that g (c-IndK J µ) ≃ c-Ind gK J ′ µ ′ . AsJ ′ is contained in at most one maximal compact subgroup in eachḠ-conjugacy class, we must actually have gK ′ =K, and so (J , µ) and (J ′ , µ ′ ) induce to the same archetype.

An explicit description of supercuspidal archetypes and a result on L-packets
Having completed the proof of Theorem 2.4, we now provide a more explicit description of the theory of archetypes for supercuspidal representations. Given a supercuspidal representationπ ofḠ, we define the ramification degree eπ ofπ to be 1 ifπ is of integral depth, or 2 ifπ is of half-integral depth. Then we obtain the following corollary to Theorem 2.4: Corollary 3.1. Letπ be a supercuspidal representation ofḠ. Then the number of [Ḡ,π]Ḡ-archetypes is precisely eπ.
Proof. It remains only for us to count the number of archetypes obtained by inducing a maximal simple type contained inπ up to maximal compact subgroups. Ifπ is ramified then, up to conjugacy, any simple type forπ is defined on a group contained in the Iwahori subgroupŪ I ofḠ, which is itself contained in bothK and ηK ; hence ramified supercuspidals admit two archetypes. Ifπ is unramified, it suffices to show that the subgroupJ on which any simple type µ forπ is defined embeds into precisely oneḠ-conjugacy class of maximal compact subgroups. Without loss of generality, we may as well assume thatJ ⊆Ū M . We have ker N E/F ⊆J ⊆Ū M , where N E/F is the norm map on the (unramified) quadratic extension E/F associated toπ. Suppose for contradiction that we also haveJ ⊆ ηŪ M . As the group ker N E/F contains the group µ q+1 of (q + 1)-th roots of unity, we would therefore also have µ q+1 ⊂Ū M ∩ ηŪ M =Ū I . However, the Iwahori subgroup contains no order q + 1 elements, giving the desired contradiction.
Thus, the only way in which one might obtain two archetypes when eπ = 1 is if π contains simple types which are G-conjugate but notḠ-conjugate; this clearly cannot be the case by the intertwining implies conjugacy property.
This completely describes the number of archetypes contained in any supercuspidal representation ofḠ. We now prove a complementary result, which allows us to describe the relationship between the theories of archetypes forḠ and G. We first require a converse result to Lemma 2.2.
Lemma 3.2. Let π be a supercuspidal representation of G, let s = [G, π] G , and let G (K, τ ) be the unique s-archetype. Letπ be an irreducible component of π ⇂Ḡ. Then there exists a g ∈ G and an irreducible componentτ of g τ ⇂gK such thatḠ( gK ,τ ) is an archetype forπ.
Proof. We may assume without loss of generality, by conjugating by η if necessary, thatπ = c-IndḠ K ρ, where ρ = c-IndK J µ is the induction toK of aḠ-simple type. Let {τ j } be the (finite) set of irreducible components of τ ⇂K. We first show that any π ′ ∈ Irr(Ḡ) containing one of theτ j upon restriction must appear in the restriction toḠ of π. We have 0 = j HomK(τ j , π ′ ) = HomK(Res K K τ, ResḠ K π ′ ) = HomḠ(c-IndḠ K Res K K τ, π ′ ), and so we obtain π ′ և c-IndḠ K Res K K τ ֒→ Res Ḡ G c-Ind G K τ . Every irreducible subquotient of the representation c-Ind G K τ is a twist of π, and hence coincides with π upon restriction toḠ, so that any such representation π ′ must be a subrepresentation of the restriction toḠ of π. Hence the possible representations π ′ all lie in a single G-conjugacy class of representations ofḠ. Let g ∈ G be such that g π ′ ≃π, so that π ′ ≃ c-IndḠ gK g ρ, 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( hπ , gτ j ) = 0. The representation hπ is of the form hπ = c-IndḠ hJ h µ and, using Lemma 2.3, we see that the representation Then h µ and µ ′ must intertwine inḠ, and the intertwining implies conjugacy property shows that the types h µ and µ ′ must actually beḠ-conjugate, and hence hπ is G-conjugate to g π ′ ≃π. Therefore hπ ≃π, and the result follows.
We are then able to give a description of the relationship between the archetypes in the two groups G andḠ in terms of L-packets.
Proposition 3.3. Let π be a supercuspidal representation of G, let s = [G, π] G , and let G (K, τ ) be the unique s-archetype. Let Π be the L-packet of irreducible components of π ⇂Ḡ. Then the set of archetypes for the representations in Π is precisely the set of theḠ(K ,τ ), for (K ,τ ) an irreducible component of either τ ⇂K or η τ ⇂ηK.
Proof. We show that the set of typical representations ofK for someπ ∈ Π is equal to the set of irreducible components of τ ⇂K; the general result then follows immediately. Let (K,τ ) be an archetype for someπ ∈ Π. Applying Lemma 2.2, τ is of the required form. Conversely, the irreducible components of τ ⇂K are all K-conjugate by Clifford theory, and so if one of them is a type for some element of Π then they all must be. Applying Lemma 3.2, at least one of these irreducible components must be a type for someπ ∈ Π.
Corollary 3.4 (Inertial local Langlands correspondence for SL 2 (F )). Let ϕ : I F → PGL 2 (C) be a representation extending to an irreducible L-parameterφ : W F → PGL 2 (C). Then there exists a finite set {(K i , τ i )} of smooth irreducible representations τ i of maximal compact subgroups K i ofḠ such that, for all smooth, irreducible, infinite-dimensional representations π ofḠ, we have that π contains some τ i upon restriction to K i if and only if rec(π) ⇂ I F ≃ ϕ. Furthermore, this set is unique up tō G-conjugacy.
Proof. Let Π = rec −1 (φ) be the L-packet corresponding toφ, so that Π is the set of irreducible components upon restriction toḠ of some supercuspidal representation σ of G. Let ψ = rec(σ), so that, by Corollary 8.2 of [Pas05], there exists a unique smooth irreducible representation τ of K such that, for all smooth, irreducible, infinite-dimensional representations ρ of G, we have that ρ contains τ upon restriction to K if and only if rec(ρ) ⇂ I ′ F ≃ ψ ⇂ I ′ F . Then G (K, τ ) is the unique archetype for σ, and the set {Ḡ(K i , τ i )} of archetypes for Π is precisely that represented by the finite set of irreducible components of τ ⇂K and η τ ⇂ηK. Let S be the set of G-inertial equivalence classes of representations in Π. Then, as each of the (K i , τ i ) is an archetype, it follows that, for all smooth, irreducible, infinite-dimensional representations π ofḠ, we have that π contains one of the τ i upon restriction to K i if and only if [Ḡ, π]Ḡ ∈ S, if and only if π ∈ Π, if and only if rec(π) ⇂ I F ≃ ϕ, as required.