The noncommutative geometry of inner forms of p-adic special linear groups

SMOOTH DUALS OF INNER FORMS OF GLn AND SLn arXiv:1505.04361v2 [math.RT] 5 Apr 2019 ANNE-MARIE AUBERT, PAUL BAUM, ROGER PLYMEN, AND MAARTEN SOLLEVELD Abstract. Let F be a local non-archimedean field. We prove that every Bernstein component in the smooth dual of each inner form of the general linear group GLn (F ) is canonically in bijection with the extended quotient for the action, given by Bernstein, of a finite group on a complex torus. For inner forms of SLn (F ) we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections satisfy naturality properties with respect to the tempered dual, parabolic induction, central character, and the local Langlands correspondence. Contents Introduction 1. Preliminaries 2. Bernstein tori 3. Hecke algebras 4. Spectrum preserving morphisms and stratified equivalences 5. Extended quotients for inner forms of GLn 6. Twisted extended quotients for inner forms of SLn 7. Relation with the local Langlands correspondence Appendix A. Twisted extended quotients References 1 4 8 10 13 18 21 27 35 38 Introduction Let X be a complex affine variety. Denote the coordinate algebra of X by O(X). The Hilbert Nullstellensatz asserts that X 7→ O(X) is an equivalence of categories   unital commutative  finitely generated ∼ nilpotent-free C-algebras affine complex algebraic varieties !op O(X) 7→ X Date: April 8, 2019. 2010 Mathematics Subject Classification. 20G25, 22E50. The fourth author is supported by a NWO Vidi grant ”A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528). We thank the referee for the exceptionally detailed and accurate report. 1 2 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD where op denotes the opposite category. A finite type algebra is a C-algebra A with a given structure as an O(X)-module such that A is finitely generated as an O(X)-module. A compatibility is required between the algebra structure of A and the given action of O(X) on A. However, A is not required to be unital. Due to the above equivalence of categories, any finite type algebra can be viewed as a slightly non-commutative affine algebraic variety. This will be the point of view of the paper. Following this point of view, each Bernstein component in the smooth dual of any reductive p-adic group G is a non-commutative affine algebraic variety. In a series of papers we have examined the question “What is the geometric structure of any given Bernstein component?” Our proposed answer to this (which has been verified for all the classical split reductive p-adic groups) is based on the notion of extended quotient. In the present paper we show that for inner forms of SLn it is necessary to use a twisted extended quotient (see the Appendix). The twisting is given by a family of 2-cocycles. Let F be a local non-archimedean field. Let D be a central simple F -algebra with dimF (D) = d2 . Then GLm (D) is an inner form of GLmd (F ) and the derived group GLm (D)der is an inner form of SLmd (F ). The main result of this paper is Theorem 1. Let G be an inner form of GLn (F ) or SLn (F ). Let Irrs (G) be any Bernstein component in the smooth dual of G. Let Ts //Ws and (Ts //Ws )♮ be the appropriate extended quotient and twisted extended quotient. Then: (1) If G is an inner form of GLn (F ) there exists a bijection Irrs (G) ←→ Ts //Ws . (2) If G is an inner form of SLn (F ) there exists a family of 2-cocycles ♮ and a bijection Irrs (G) ←→ (Ts //Ws )♮ . (3) In either case, the bijection satisfies naturality properties with respect to the tempered dual, parabolic induction, and central character. We remark that • Ts //Ws is the non-commutative affine variety whose coordinate algebra is the crossed product algebra O(Ts ) ⋊ Ws where Ts , Ws are respectively the complex torus and finite group acting on the torus which Bernstein assigns to Irrs (G). The crossed product algebra O(Ts ) ⋊ Ws is a finite O(Ts /Ws )-algebra. • (Ts //Ws )♮ is the non-commutative affine variety whose coordinate algebra is the twisted crossed product algebra O(Ts ) ⋊♮ Ws . The twisted crossed product algebra O(Ts ) ⋊♮ Ws is a finite O(Ts /Ws )-algebra. Example 7.7 shows that for inner forms of SL5 there are Bernstein components where the twisting is non-trivial. We observe that it is somewhat remarkable that many of the subtleties of the representation theory of GLm (D), SLm (D) are captured by the algebras O(Ts ) ⋊ Ws O(Ts ) ⋊♮ Ws and their geometric realizations Ts //Ws , (Ts //Ws )♮ . The proof of Theorem 1 uses a weakening of Morita equivalence called stratified equivalence, see [ABPS7]. The proof is achieved by combining the theory of types with an analysis of the structure of Hecke algebras. Outline of the proof. INNER FORMS 3 The proof of Theorem 1 for GLm (D) consists of two steps. • Step 1: For each point s in the Bernstein spectrum B(GLm (D)), the ideal H(G)s in the Hecke algebra H(GLm (D)) is Morita equivalent to an affine Hecke algebra, see [ABPS4]. • Step 2: Using the Lusztig asymptotic algebra, a stratified equivalence is constructed between this affine Hecke algebra and the associated crossed product algebra. The irreducible representations of the crossed product algebra (in a canonical way) are the required extended quotient. The proof of Theorem 1 for SLm (D) is achieved by introducing an intermediate group SLm (D) · Z(GLm (D)): SLm (D) ⊂ SLm (D) · Z(GLm (D)) ⊂ GLm (D), where Z(GLm (D)) is the center of GLm (D). It consists in two analogous steps. • Step 1: For each point s in the Bernstein spectrum B(SLm (D)), the ideal H(SLm (D))s in the Hecke algebra H(SLm (D)) is Morita equivalent to a twisted affine Hecke algebra. • Step 2: Using the Lusztig asymptotic algebra, a stratified equivalence is constructed between this twisted affine Hecke algebra and the associated twisted crossed product algebra. The irreducible representations of the twisted crossed product algebra (in a canonical way) are the required twisted extended quotient. In Section 7, we prove that the bijections of Theorem 1 are compatible with the local Langlands correspondence, in the sense below. Let G be an inner form of GLn (F ) or SLn (F ), and let Ǧ denote GLn (C) or PGLn (C), respectively. The set of Ǧ-conjugacy classes of Langlands parameters (resp. enhanced Langlands parameters) for G is denoted by Φ(G) (resp. Φe (G)). We may identify Φe (G) with Φ(G) when G is an inner form of GLn (F ). Let L be a set of representatives for the conjugacy classes of Levi subgroups of G. For L in L, we denote by Irrcusp (L) the set of isomorphism classes of supercuspidal irreducible representations of L and we let Φ(L)cusp be its image in Φe (L). The group W (G, L) = NG (L)/L, quotient by L of the normalizer of L in G, acts naturally on both sets. Thus we may consider the extended quotients Irrcusp (L)//W (G, L) and Φ(L)cusp //W (G, L), as well as their twisted versions. It follows directly from the definitions that G (Irrcusp (L)//W (G, L)) ♮ = (Ts //Ws )♮ , s where the disjoint union runs over all s ∈ B(G) coming from supercuspidal Lrepresentations. Theorem 2. The following statements hold: • If G = GLm (D) there exists a canonical, bijective, commutative diagram / Φ(G) O Irr(G) o O F  Irr (L)//W (G, L) o cusp L∈L / F  Φ(L) //W (G, L). cusp L∈L 4 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD • If G = SLm (D) there exists a family of 2-cocycles ♮ and a (canonical up to permutations within L-packets) bijective commutative diagram / Φe (G) O Irr(G) o O F  (Irr (L)//W (G, L)) ♮ o cusp L∈L /F  (Φ(L) cusp //W (G, L))♮ . L∈L 1. Preliminaries We start with some generalities, to fix the notations. Let G be a connected reductive group over a local non-archimedean field F of residual characteristic p. All our representations are assumed to be smooth and over the complex numbers. We write Rep(G) for the category of such G-representations and Irr(G) for the collection of isomorphism classes of irreducible representations therein. Let P be a parabolic subgroup of G with Levi factor L. The Weyl group of L is W (G, L) = NG (L)/L. It acts on equivalence classes of L-representations π by (w · π)(g) = π(w̄gw̄−1 ), where w̄ ∈ NG (L) is a chosen representative for w ∈ W (G, L). We write Wπ = {w ∈ W (G, L) | w · π ∼ = π}. Let ω be an irreducible supercuspidal L-representation. The inertial equivalence class s = [L, ω]G gives rise to a category of smooth G-representations Reps (G) and a subset Irrs (G) ⊂ Irr(G). Write Xnr (L) for the group of unramified characters L → C× . Then Irrs (G) consists of all irreducible constituents of the parabolically induced representations IPG (ω ⊗ χ) with χ ∈ Xnr (L). We note that IPG always means normalized, smooth parabolic induction from L via P to G. The set IrrsL (L) with sL = [L, ω]L can be described explicitly, namely by Xnr (L, ω) = {χ ∈ Xnr (L) : ω ⊗ χ ∼ (1) = ω}, (2) IrrsL (L) = {ω ⊗ χ : χ ∈ Xnr (L)/Xnr (L, ω)}. Several objects are attached to the Bernstein component Irrs (G) of Irr(G) [BeDe]. Firstly, there is the torus Ts := Xnr (L)/Xnr (L, ω), which is homeomorphic to IrrsL (L). Secondly, we have the groups NG (sL ) ={g ∈ NG (L) | g · ω ∈ IrrsL (L)} ={g ∈ NG (L) | g · [L, ω]L = [L, ω]L }, Ws :={w ∈ W (G, L) | w · ω ∈ IrrsL (L)} = NG (sL )/L. Of course Ts and Ws are only determined up to isomorphism by s, actually they depend on sL . To cope with this, we tacitly assume that sL is known when considering s. The choice of ω ∈ IrrsL (L) fixes a bijection Ts → IrrsL (L), and via this bijection the action of Ws on IrrsL (L) is transferred to Ts . The finite group Ws can be thought of as the “Weyl group” of s, although in general it is not generated by reflections. Let Cc∞ (G) be the vector space of compactly supported locally constant functions G → C. The choice of a Haar measure on G determines a convolution product * INNER FORMS 5 on Cc∞ (G). The algebra (Cc∞ (G), ∗) is known as the Hecke algebra H(G). There is an equivalence between Rep(G) and the category Mod(H(G)) of H(G)-modules V such that H(G) · V = V . We denote the collection of inertial equivalence classes for G by B(G), the Bernstein spectrum of G. The Bernstein decomposition Y Rep(G) = Reps (G) s∈B(G) induces a factorization in two-sided ideals M H(G) = s∈B(G) H(G)s . From now on we discuss things that are specific for G = GLm (D), where D is a central simple FQ -algebra. We write dimF (D) = d2 . P Every Levi subgroup L of G is conjugate to j GLm̃j (D) for some m̃j ∈ N with j m̃j = m. Hence every irreducible L-representation ω can be written as ⊗j ω̃j with ω̃j ∈ Irr(GLm̃j (D)). Then ω is supercuspidal if and only if every ω̃j is so. As above, we assume that this is the case. Replacing (L, ω) by an inertially equivalent pair allows us to make the following simplifying assumptions: Condition 1.1. • if m̃i = m̃j and [GLm̃j (D), ω̃i ]GLm̃j (D) = [GLm̃j (D), ω̃j ]GLm̃j (D) , then ω̃i = ω̃j ; N • ω = Qi ωi⊗ei , such Q that ωi and ωj are not inertially equivalent if i 6= j; • L = i Lei i = i GLmi (D)ei , embedded diagonally in GLm (D) such that factors Li with the same (mi , ei ) are in subsequent positions; • as representatives for the elements of W (G, L) we take permutation matrices; • P is the parabolic subgroup of G generated by L and the upper triangular matrices; • if mi = mj , ei = ej and ωi is isomorphic to ωj ⊗ γ for some character γ of GLmi (D), then ωi = ωj ⊗ γχ for some χ ∈ Xnr (GLmi (D)). Most of the time we will not need the conditions for stating the results, but they are useful in many proofs. Under Conditions 1.1 we consider Y e
Lj j . (3) Mi = GLmi ei (D) naturally embedded in ZG j6=i Q Then i Mi is a Levi subgroup of G containing L. For s = [L, ω]G we have Y Y (4) Ws = NQi Mi (L)/L = NMi (Lei i )/Lei i ∼ Sei , = i i a direct product of symmetric groups. Writing si = [Li , ωi ]Li , the torus associated to s becomes Y Y Ts = (Tsi )ei = (5) Ti , i (6) i Tsi = Xnr (Li )/Xnr (Li , ωi ). By our choice of representatives for W (G, L), ωi⊗ei is stable under NMi (Lei i )/Lei i ∼ = Q e e i i Sei . If Ri ⊂ X∗ ( i Z(Li ) ) denotes the coroot system of (Mi , Z(Li ) ), we can identify Sei with W (Ri ). The action of Ws on Ts is just permuting coordinates in the standard way and (7) Ws = Wω . 6 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD The reduced norm map D → F gives rise to a group homomorphism Nrd : G → F × . We denote its kernel by G♯ , so G♯ is also the derived group of G. For subgroups H ⊂ G we write H ♯ = H ∩ G♯ . In [ABPS4] we determined the structure of the Hecke algebras associated to types for G♯ , starting with those for G. As an intermediate step, we did this for the group G♯ Z(G), where Z(G) ∼ = F × denotes the centre of G. The advantage is that the comparison between G♯ and G♯ Z(G) is easy, while G♯ Z(G) ⊂ G can be treated as an extension of finite index. In fact it is a subgroup of finite index if p does not divide md. In case p does divide md, the quotient G/G♯ Z(G) is compact and similar techniques can be applied. For an inertial equivalence class s = [L, ω]G we define Irrs (G♯ ) as the set of irres ducible G♯ -representations that are subquotients of ResG G♯ (π) for some π ∈ Irr (G), and Reps (G♯ ) as the collection of G♯ -representations all of whose irreducible subquotients lie in Irrs (G♯ ). We want to investigate the category Reps (G♯ ). It is a product of finitely many Bernstein blocks for G♯ , see [ABPS4, Lemma 2.2]: Y ♯ (8) Reps (G♯ ) = Rept (G♯ ). ♯ t ≺s ♯ We note that the Bernstein components Irrt (G♯ ) which are subordinate to one s (i.e., such that t♯ ≺ s) form precisely one class of L-indistinguishable components: every L-packet for G♯ which intersects one of them intersects them all. Analogously we define Reps (G♯ Z(G)), and we obtain Y Reps (G♯ Z(G)) = Rept (G♯ Z(G)), t≺s where the t are inertial equivalence classes for G♯ Z(G). The restriction of t to G♯ is a single inertial equivalence class t♯ , and by [ABPS4, (43)]: Tt♯ = Tt /Xnr (Nrd(Z(G))). (9) For π ∈ Irr(G) we put X G (π) := {γ ∈ Irr(G/G♯ ) : γ ⊗ π ∼ = π}. The same notation will be used for representations of parabolic subgroups of G which admit a central character. For every γ ∈ X G (π) there exists a nonzero intertwining operator (10) I(γ, π) ∈ HomG (π ⊗ γ, π) = HomG (π, π ⊗ γ −1 ), which is unique up to a scalar. As G♯ ⊂ ker(γ), I(γ, π) can also be considered as an element of EndG♯ (π). As such, these operators determine a 2-cocycle κπ by (11) I(γ, π) ◦ I(γ ′ , π) = κπ (γ, γ ′ )I(γγ ′ , π). By [HiSa, Lemma 2.4] they span the G♯ -intertwining algebra of π: (12) End ♯ (ResG♯ π) ∼ = C[X G (π), κπ ], G G where the right hand side denotes the twisted group algebra of X G (π). Furthermore by [HiSa, Corollary 2.10] M ∼ HomC[X G (π),κπ ] (ρ, π) ⊗ ρ (13) ResG G♯ π = ρ∈Irr(C[X G (π),κπ ]) INNER FORMS 7 as representations of G♯ × X G (π). The analogous groups for s = [L, ω]G and sL = [L, ω]L are X L (s) := {γ ∈ Irr(L/L♯ Z(G)) : γ ⊗ ω ∈ [L, ω]L }, X G (s) := {γ ∈ Irr(G/G♯ Z(G)) : γ ⊗ IPG (ω) ∈ Reps (G)}. The role of the group Ws for Reps (G♯ ) is played by Ws♯ := {w ∈ W (G, L) | ∃γ ∈ Irr(L/L♯ Z(G)) such that w(γ ⊗ ω) ∈ [L, ω]L } By [ABPS4, Lemma 2.3] Ws♯ = Ws ⋊ R♯s , where R♯s = Ws♯ ∩ NG (P ∩ (14) while [ABPS4, Lemma 2.4.d] says that Y i Mi )/L. X G (s)/X L (s) ∼ = R♯s . (15) For another way to view X G (s), we start with Stab(s) := {(w, γ) ∈ NG (L)/L × Irr(L/L♯ Z(G)) | w(γ ⊗ ω) ∈ [L, ω]L }. The normal subgroup Ws has a complement: Y Stab(s) = Stab(s, P ∩ Mi ) ⋉ Ws := Stab(s)+ ⋉ Ws i Y Stab(s)+ := {(w, γ) ∈ NG (P ∩ Mi )/L × Irr(L/L♯ Z(G)) | w(γ ⊗ ω) ∈ [L, ω]L } i By [ABPS4, Lemma 2.4.a] projection of Stab(s) on the second coordinate gives an isomorphism X G (s) ∼ = Stab(s)/Ws ∼ = Stab(s)+ (16) In particular Stab(s)+ /X L (s) ∼ = R♯s . (17) As in [ABPS4, (159)–(161)] we choose χγ ∈ Xnr (L)Ws for (w, γ) ∈ Stab(s)+ , such that w(ω) ⊗ γ ∼ = ω ⊗ χγ . (18) Notice that χγ is unique up to Xnr (L, ω). Furthermore we choose an invertible −1 −1 −1 J(γ, ω ⊗ χ−1 γ ) ∈ HomL (ω ⊗ χγ , w (ω) ⊗ γ ). (19) This generalizes (10) in the sense that J(γ, ω ⊗ χ−1 γ ) = I(γ, ω) if γ ∈ X L (ω) and χγ = 1. Let Vω denote the vector space underlying ω. We may assume that (20) χγ = γ and J(γ, ω ⊗ χ−1 γ ) = idVω if γ ∈ Xnr (L/L♯ Z(G)). 8 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD 2. Bernstein tori We will determine the Bernstein tori for G♯ Z(G) and G♯ , in terms of those for G. The group X L (s) acts on Ts = IrrsL (L) by π 7→ π ⊗ γ. By [ABPS4, Proposition 2.1] L ResL L♯ (ω) and ResL♯ (ω ⊗ χ) with χ ∈ Xnr (L) have a common irreducible subquotient if and only if there is a γ ∈ X L (s) such that ω ⊗ χ ∼ = ω ⊗ χγ . As in (19) we choose a nonzero J(γ, ω) ∈ HomL (ω, ω ⊗ χγ γ −1 ) = HomL (ω ⊗ γ, ω ⊗ χγ ). Then J(γ, ω) ∈ HomL♯ (ω, ω ⊗ χγ ) and for every irreducible subquotient σ ♯ of ResL L♯ (ω) (21) γ ∗ (σ ♯ ⊗ χ) : m 7→ J(γ, ω) ◦ (σ ♯ ⊗ χ)(m) ◦ J(γ, ω)−1 is an irreducible subquotient representation of −1 L ). ResL L♯ (ω ⊗ χχγ ) = ResL♯ (ω ⊗ χχγ γ This prompts us to consider (22) X L (s, σ ♯ ) := {γ ∈ X L (s) | γ ∗ σ ♯ ∼ = σ ♯ ⊗ χγ }. By [ABPS4, Lemma 4.14] (23) σ♯ ⊗ χ ∼ = σ ♯ for all χ ∈ Xnr (L, ω). Hence the group (22) is well-defined, that is, independent of the choice of the χγ . For γ ∈ X L (ω) (21) reduces to σ ♯ ⊗ χ, so γ ∈ X L (s, σ ♯ ). By (20) the same goes for γ ∈ Xnr (L/L♯ Z(G)), so there is always an inclusion (24) X L (ω)Xnr (L/L♯ Z(G)) ⊂ X L (s, σ ♯ ). We gathered enough tools to describe the Bernstein tori for G♯ and G♯ Z(G). Recall ♯ that sL = [L, ω]L , Ts ∼ = Xnr (L)/Xnr (L, ω). Let Ts be the restriction of Ts to L♯ , that is, (25) Ts♯ := Ts /Xnr (G) = Ts /Xnr (L/L♯ ) ∼ = Xnr (L♯ )/Xnr (L, ω), where Xnr (L/L♯ ) denotes the group of unramified characters of L which are trivial on L♯ . Proposition 2.1. Let σ ♯ be an irreducible subquotient of ResL L♯ (ω) and write ♯ ♯ ♯ ♯ ♯ t = [L Z(G), σ ]G♯ Z(G) and t = [L , σ ]G♯ . (a) X L (s, σ ♯ ) depends only on sL , not on the particular σ ♯ . (b) Xnr (L, ω){χγ | γ ∈ X L (s, σ ♯ )} is a subgroup of Xnr (L) which contains Xnr (L/L♯ Z(G)). (c) Tt ∼ = Ts /{χγ | γ ∈ X L (s, σ ♯ )} ∼ = Xnr (L♯ Z(G))/Xnr (L, ω){χγ | γ ∈ X L (s, σ ♯ )}. ♯ L ♯ (d) Tt♯ ∼ = Ts /{χγ | γ ∈ X (s, σ )} ∼ = Xnr (L♯ )/Xnr (L, ω){χγ | γ ∈ X L (s, σ ♯ )}. Proof. (a) By [ABPS4, Proposition 2.1] every two irreducible subquotients of ResL L♯ (ω) are direct summands and are conjugate by an element of L. Given γ ∈ L X (s), pick mγ ∈ L such that γ ∗ σ♯ ∼ = (ω(mγ )−1 ◦ σ ♯ ◦ ω(mγ )) ⊗ χγ = (mγ · σ ♯ ) ⊗ χγ . INNER FORMS 9 For any other irreducible summand τ = mτ · σ ♯ of ResL L♯ (ω) we compute γ ∗ τ = γ ∗ (mτ · σ ♯ ) = J(γ, ω) ◦ ω(mτ )−1 ◦ σ ♯ ◦ ω(mτ ) ◦ J(γ, ω)−1 ♯ −1 = (χγ γ −1 ⊗ ω)(m−1 ◦ (χγ γ −1 ⊗ ω)(mτ ) τ ) ◦ J(γ, ω) ◦ σ ◦ J(γ, ω) ∼ = ω(m−1 ) ◦ (mγ · σ ♯ ) ⊗ χγ ◦ ω(mτ ) τ ∼ = (mτ mγ · σ ♯ ) ⊗ χγ . As L/L♯ is abelian, we find that mτ mγ · σ ♯ ∼ = mγ mτ · σ ♯ and that γ ∗τ ∼ = (mγ mτ · σ ♯ ) ⊗ χγ = mγ · τ ⊗ χγ . Writing Lτ = {m ∈ L | m · τ ∼ = τ }, we deduce the following equivalences: ∼ γ ∗ σ♯ ∼ = σ ♯ ⊗ χγ ⇔ mγ ∈ Lσ♯ ⇔ mγ ∈ mτ Lσ♯ m−1 τ = Lτ ⇔ γ ∗ τ = τ ⊗ χγ . This means that X L (s, σ ♯ ) = X L (s, τ ). (b) By (20) and (24) Xnr (L/L♯ Z(G)) ⊂ {χγ | γ ∈ X L (s, σ ♯ )}. In view of the uniqueness property of χγ the map X L (s) → Xnr (L)/Xnr (L, ω) : γ 7→ χγ is a group homomorphism with kernel X L (ω). Hence the χγ form a subgroup of Xnr (L)/Xnr (L, ω), isomorphic to X L (s)/X L (ω). (c) Consider the family of L♯ Z(G)-representations {σ ♯ ⊗ χ | χ ∈ Xnr (L)}. ∼ σ ♯ ∈ Irr(L♯ Z(G)). From [ABPS4, We have to determine the χ for which σ ♯ ⊗ χ = Lemma 4.14] we see that this includes all the elements of Xnr (L, ω)Xnr (L/L♯ Z(G)). By [ABPS4, Proposition 2.1.b] and part (a), all the remaining χ come from {χγ | γ ∈ X L (s, σ ♯ )}. This gives the first isomorphism, and the second follows with part (b). (d) This is a consequence of part (c) and (9).  Proposition 2.1 entails that for every inertial equivalence class t = [L♯ Z(G), σ ♯ ]G♯ Z(G) ≺ s = [L, ω]G the action (21) of X L (s, σ ♯ ) leads to Tt ∼ = Ts /X L (s, σ ♯ ). However, some of the tori Tt = TtL = Irr [L♯ Z(G),σ♯ ]L♯ Z(G) (L♯ Z(G)) ♯ associated to inequivalent σ ♯ ⊂ ResL L♯ (ω) can coincide as subsets of Irr(L Z(G)). This is caused by elements of X L (s) \ X L (s, σ ♯ ) via the action (21). With (23), (22) and (13) we can write [
(26) IrrsL (L♯ Z(G)) = TtL = Ts × Irr(C[X L (ω), κω ]) /X L (s), tL ≺sL where (ω ⊗ χ, ρ) ∈ Ts × Irr(C[X L (ω), κω ]) corresponds to HomC[X L (ω),κω ] (ρ, ω ⊗ χ) ∈ Irr(L♯ Z(G)). 10 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD With (9) we can deduce a similar expression for L♯ : [
Tt♯ = Ts♯ × Irr(C[X L (ω), κω ]) /X L (s) IrrsL (L♯ ) = ♯ tL ≺sL L (27)
= Ts × Irr(C[X L (ω), κω ]) /X L (s)Xnr (L♯ Z(G)/L♯ ). In the notation of (26) and (27) the action of γ ∈ X L (s) becomes (28) γ · (ω ⊗ χ, ρ) = (ω ⊗ χχγ , φω,γ ρ), where φω,γ is yet to be determined. Any γ ∈ X L (ω) can be adjusted by an element of Xnr (L, ω) to achieve χγ = 1. Then (23) shows that φω,γ ρ ∼ = ρ for all γ ∈ X L (ω). Lemma 2.2. For γ ∈ X L (s), φω,γ ρ is ρ tensored with a character of X L (ω), which we also call φω,γ . Then X L (s) → Irr(X L (ω)) : γ 7→ φω,γ is a group homomorphism. Proof. Let Nγ ′ be a standard basis element of C[X L (ω), κω ]. In view of (21) φω,γ ρ is given by (29) Nγ ′ 7→ J(γ, ω)I(γ ′ , ω)J(γ, ω)−1 ∈ HomL (ω ⊗ γ ′ χγ , ω ⊗ χγ ). Since these are irreducible L-representations, there is a unique λ ∈ C× such that J(γ, ω)I(γ ′ , ω)J(γ, ω)−1 = λ−1 I(γ ′ , ω ⊗ χγ ), (φω,γ ρ)(Nγ ′ ) = ρ(λI(γ ′ , ω)) = λρ(Nγ ′ ). Moreover the relation (30) I(γ1′ , ω ⊗ χγ )I(γ2′ , ω ⊗ χγ ) = κω⊗χγ (γ1′ , γ2′ )I(γ1′ γ2′ , ω ⊗ χγ ) also holds with J(γ, ω)I(γi′ , ω)J(γ, ω)−1 instead of I(γi′ , ω)– a basic property of conjugation. It follows that γ ′ 7→ λ defines a character of X L (ω) which implements the action ρ 7→ φω,γ ρ. As φω,γ comes from conjugation by J(γ, ω ⊗ χ) and by (30), γ 7→ φω,γ is a group homomorphism.  A straightforward check, using the above proof, shows that (31) HomC[X L (ω),κω ] (ρ, ω ⊗ χ) → HomC[X L (ω),κω ] (φω,γ ρ, ω ⊗ χχγ ) f 7→ J(γ, ω ⊗ χ) ◦ f is an isomorphism of L♯ Z(G)-representations. 3. Hecke algebras We will show that the algebras H(G♯ Z(G))s and H(G♯ )s are stratified equivalent [ABPS7] with much simpler algebras. In this section we recall the final results of [ABPS4], which show that up to Morita equivalence these algebras are closely related to affine Hecke algebras. In section 4 we analyse the latter algebras in the framework of [ABPS7]. Our basic affine Hecke algebra is called H(Ts , Ws , qs ). By definition [ABPS4, (119)] it has a C-basis {θx [w] : x ∈ X ∗ (Ts ), w ∈ Ws } such that • the span of the θx is identified with the algebra O(Ts ) of regular functions on Ts ; • the span of the [w] is the finite dimensional Iwahori–Hecke algebra H(Ws , qs ); INNER FORMS 11 • the multiplication between these two subalgebras is given by (32) f [s] − [s](s · f ) = (qs (s) − 1)(f − (s · f ))(1 − θ−α )−1 f ∈ O(Ts ), for a simple reflection s = sα ; • the algebra is well-defined for any array of parameters qs = (qs,i )i in C× . The parameters qs,i that we will use are given explicitly in [Séc, Théorème 4.6]. Thus H(Ts , Ws , qs ) is a tensor product of affine Hecke algebras of type GLe , but written in such a way that the torus Ts appears canonically in it (i.e. independent of the choice of a base point of Ts ). Sécherre and Stevens [Séc, SéSt2] showed that (33) H(G)s is Morita equivalent with H(Ts , Ws , qs ). From [SéSt1] we know that there exists a simple type (K, λ) for [L, ω]M , and in [SéSt2] it was shown to admit a G-cover (KG , λG ). We denote the associated central idempotent of H(K) by eλ , and similarly for other irreducible representations. Then Vλ = eλ Vω . For the restriction process we need an idempotent that is invariant under X G (s). To that end we replace λG by the sum of the representations γ ⊗ λG with γ ∈ X G (s), which we call µG . In [ABPS4, (91)] we constructed an idempotent eµG ∈ H(G) which is supported on the compact open subgroup KG ⊂ G. It follows from the work of Sécherre and Stevens [SéSt2] that eµG H(G)eµG is Morita equivalent with H(G)s . In [ABPS4, (128)] we defined a finite dimensional subspace X Vµ := eγ⊗λ Vω G γ∈X (s) of Vω which is stable under the operators I(γ, ω) with γ ∈ X L (s). By [Séc] and [ABPS4, Theorem 4.5.d] (34) eµ H(G)eµ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ⊗C CR♯ ). G G s X G (s) The groups and Xnr (G) act on eµG H(G)eµG by pointwise multiplication of functions G → C with characters of G. However, for technical reasons we use the action (35) αγ (f )(g) = γ −1 (g)f (g) f ∈ H(G), γ ∈ Irr(G/G♯ ), g ∈ G. The action on the right hand side of (34) preserves the tensor factors, and on ♯ EndC (CR♯s ) it is the natural action of X G (s)/X L (s) ∼ = Rs . Although eµG looks like the idempotent of a type, it is not clear whether it is one, because the associated KG -representation is reducible and no more suitable compact subgroup of G is in sight. Let eµG♯ (respectively eµG♯ Z(G) ) be the restriction of eµG : G → C to G♯ (resp. G♯ Z(G)). We normalize the Haar measure on G♯ (resp. G♯ Z(G)) such that it becomes an idempotent in H(G♯ ) (resp. H(G♯ Z(G))). In [ABPS4, Lemma 3.3] we constructed a certain finite set [L/Hλ ], consisting of representatives for a normal subgroup Hλ ⊂ L. Consider the elements X e♯λG := aeµG a−1 ∈ H(G), a∈[L/Hλ ] X e♯λ ♯ := aeµG♯ Z(G) a−1 ∈ H(G♯ Z(G)), (36) a∈[L/Hλ ] G Z(G) X aeµG♯ a−1 ∈ H(G♯ ). e♯λ ♯ := G a∈[L/Hλ ] 12 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD It follows from [ABPS4, Lemma 3.12] that they are again idempotent. Notice that e♯λG detects the same category of G-representations as eµG , namely Reps (G). In the proof of [ABPS4, Proposition 3.15] we established that (34) extends to an isomorphism (37) e♯ H(G)e♯ ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ⊗C CR♯ ) ⊗ M[L:H ] (C). λG s λG λ Theorem 3.1. [ABPS4, Theorem 4.13] Let G = GLm (D) be an inner form of GLn (F ). Then for any s ∈ B(G): (a) H(G♯ Z(G))s is Morita equivalent with its subalgebra M e♯λ ♯ H(G♯ Z(G))e♯λ ♯ = aeµG♯ Z(G) a−1 H(G♯ Z(G))aeµG♯ Z(G) a−1 G Z(G) G Z(G) (b) Each of the algebras a∈[L/Hλ ] aeµG♯ Z(G) a−1 H(G♯ Z(G))aeµG♯ Z(G) a−1 is isomorphic to
X L (s) H(Ts , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s . (38) (c) Under these isomorphisms the action of Xnr (G) on H(G♯ Z(G))s becomes the action of Xnr (L/L♯ ) ∼ = Xnr (G) on (38) via translations on Ts . Recall from (25) that Ts♯ is the restriction of Ts to L♯ . With this torus we build an affine Hecke algebra H(Ts♯ , Ws , qs ) for G♯ . Theorem 3.2. [ABPS4, Theorem 4.15] (a) H(G♯ )s is Morita equivalent with M e♯λ ♯ H(G♯ )e♯λ ♯ = G G a∈[L/Hλ ] aeµG♯ a−1 H(G♯ )aeµG♯ a−1 (b) Each of the algebras aeµG♯ a−1 H(G♯ )aeµG♯ a−1 is isomorphic to
X L (s) H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s . Let us describe the above actions of the group X G (s) explicitly. The action on a−1 H(G♯ Z(G))aeµ (39) aeµ a−1 ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ). G♯ Z(G) G♯ Z(G) does not depend on a ∈ [L/Hλ ] because αγ (af a−1 ) = a(αγ (f ))a−1 f ∈ H(G). The isomorphism (17) yields an action α of Stab(s)+ on (39). Theorem 3.3. [ABPS4, Lemmas 3.5 and 4.11] (a) The action of Stab(s)+ on H(Ts , Ws , qs ) ⊗ EndC (Vµ ) in Theorem 3.1 preserves both tensor factors. On H(Ts , Ws , qs ) it is given by −1 α(w,γ) (θx [v]) = χ−1 γ (x)θw(x) [wvw ] x ∈ X ∗ (Ts ), v ∈ Ws , and on EndC (Vµ ) by −1 −1 α(w,γ) (h) = J(γ, ω ⊗ χ−1 γ ) ◦ h ◦ J(γ, ω ⊗ χγ ) . (b) The subgroup of elements that act trivially is X L (ω, Vµ ) = {γ ∈ X L (ω) | I(γ, ω)|Vµ ∈ C× idVµ }. Its cardinality equals [L : Hλ ]. INNER FORMS 13 (c) Part (a) and Theorem 3.1.c also describe the action of Stab(s)+ Xnr (G) on H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) in Theorem 3.2. The subgroup of elements that act trivially on this algebra is X L (ω, Vµ )Xnr (G) = X L (ω, Vµ )Xnr (L/L♯ ). 4. Spectrum preserving morphisms and stratified equivalences We will show that the Hecke algebras obtained in Theorems 3.1 and 3.2 fit in the framework of spectrum preserving morphisms and stratified equivalence of finite type algebras, see [ABPS7]. First we exhibit an algebra that interpolates between (H(Ts , Ws , qs ) ⊗ EndC (Vµ ))X L (s) ⋊ R♯s L and (O(Ts ) ⋊ Ws ⊗ EndC (Vµ ))X (s) ⋊ R♯s . Recall that Conditions 1.1 are in force and write Y G Y Y Ts = Ti , Rs = Ri , W s = W (Ri ) = Sei . i i i i Let qi be the restriction of qs : X ∗ (Ts ) ⋊ Ws → R>0 to X ∗ (Ti ) ⋊ W (Ri ). Recall Lusztig’s asymptotic Hecke algebra J(X ∗ (Ti ) ⋊ W (Ri )) from [Lus2, Lus3]. We remark that, although in [Lus2] it is supposed that the underlying root datum is semisimple, this assumption is shown to be unneccesary in [Lus3]. This algebra is unital and of finite type over O(Ti )W (Ri ) . It has a distinguished C-basis {txv | x ∈ X ∗ (Ti ), v ∈ W (Ri )} and the tx with x ∈ X ∗ (Ti )W (Ri ) are central. We define O J(X ∗ (Ts ) ⋊ Ws ) = J(X ∗ (Ti ) ⋊ W (Ri )). i This is a unital finite type algebra over O(Ts )Ws , in fact for several different O(Ts )Ws module structures. Lusztig [Lus3, §1.4] defined injective algebra homomorphisms (40) φi,q φi,1 i H(Ti , W (Ri ), qi ) −−−→ J(X ∗ (Ti ) ⋊ W (Ri )) ←−− O(Ti ) ⋊ W (Ri ) with many nice properties. Among these, we record that (41) φi,qi and φi,1 are the identity on C[X ∗ (Ti )W (Ri ) ] ∼ = O(Xnr (Z(Mi ))). There exist O(Ti )W (Ri ) -module structures on J(X ∗ (Ti )⋊W (Ri )) for which the maps (40) are O(Ti )W (Ri ) -linear, namely by letting O(Ti )W (Ri ) act via the map φi,qi or via φi,1 . Taking tensor products over i in (40) and with the identity on EndC (Vµ ) gives injective algebra homomorphisms (42) φqs : H(Ts , Ws , qs ) ⊗ EndC (Vµ ) → J(X(Ts ) ⋊ Ws ) ⊗ EndC (Vµ ), φ1 : O(Ts ) ⋊ Ws ⊗ EndC (Vµ ) → J(X(Ts ) ⋊ Ws ) ⊗ EndC (Vµ ). The maps φqs and φ1 are O(Ts )Ws -linear with respect to the appropriate module structure on J(X(Ts ) ⋊ Ws ). Lemma 4.1. Via (42) the action of Stab(s)+ on H(Ts , Ws , qs ) ⊗ EndC (Vµ ) from Theorem 3.3 extends canonically to an action on J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ), which stabilizes the subalgebra O(Ts ) ⋊ Ws ⊗ EndC (Vµ ). 14 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Proof. In X ∗ (Ts ) ⋊ W (G, L) every w ∈ R♯s normalizes the subgroup X ∗ (Ts ) ⋊ Ws . The group automorphism xv 7→ wxvw−1 (43) of X ∗ (Ts ) ⋊ Ws only permutes the subgroups X ∗ (Ti ) ⋊ W (Ri ). In particular it stabilizes the set of simple reflections. With the canonical representatives for Ws♯ in G from (14), conjugation by w stabilizes the type for sL (it is a product of simple types in the sense of [Séc, §4]). Hence (43) preserves qs . Thus (43) can be factorized as Y Y
X ∗ (Ti ) ⋊ W (Ri ) (44) ωj with ωj ∈ Aut j i:Ri =Rj ,qi =qj The function qs takes the same value on all simple (affine) roots associated to the group for one j in (44), so the algebra O (45) J(X ∗ (Ti ) ⋊ W (Ri )) i|Ri =Rj ,qi =qj is of the kind considered in [Lus3, §1]. Then ωj is an automorphism which fits in a group called Ω in [Lus3, §1.1], so it gives rise to an automorphism of the algebra (45). In this way the group R♯s ∼ = Stab(s)+ /X L (s) acts naturally on J(X ∗ (Ts )⋊Ws ). W s Since Ts is central in Ts ⋊ Ws , every χ ∈ TsWs gives rise to an algebra automorphism of J(X ∗ (Ts ) ⋊ Ws ): (46) txv 7→ χ(x)txv x ∈ X ∗ (Ts ), v ∈ Ws . Thus we can make Stab(s)+ act on J(X ∗ (Ts ) ⋊ Ws ) by (w, γ) · txv = χ−1 γ (x)twxvw −1 x ∈ X ∗ (Ts ), v ∈ Ws . The action of Stab(s)+ on EndC (Vµ ) may be copied to this setting, so we can define the following action on J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ): −1 −1 −1 α(w,γ) (txv ⊗ h) = χ−1 γ (x)twxvw −1 ⊗ J(γ, ω ⊗ χγ ) ◦ h ◦ J(γ, ω ⊗ χγ ) . Of course the above also works with the label function 1 instead of qs . That yields a similar action of Stab(s)+ on O(Ts ) ⋊ Ws ⊗ EndC (Vµ ), namely (47) −1 −1 −1 ⊗ J(γ, ω ⊗ χ−1 α(w,γ) (xv ⊗ h) = χ−1 γ ) ◦ h ◦ J(γ, ω ⊗ χγ ) , γ (x) wxvw where xv ∈ X ∗ (Ts ) ⋊ Ws . It follows from [Lus3, §1.4] that φqs and φ1 are now Stab(s)+ -equivariant.  Lemma 4.2. The O(Ts )Ws -algebra homomorphisms φqs and φ1 from (42) are spectrum preserving with respect to filtrations, in the sense of [ABPS7]. Proof. It suffices to consider the map φqs , for the same reasoning will apply to φ1 . Our argument is a generalization of [BaNi, Theorem 10], which proves the analogous statements for J(X ∗ (Ti ) ⋊ W (Ri )). Recall the function (48) a : X ∗ (Ts ) ⋊ Ws → Z≥0 from [Lus3, §1.3]. For fixed n ∈ Z≥0 , the subspace of J(X ∗ (Ti ) ⋊ W (Ri )) spanned by the txv with a(xv) = n is a two-sided ideal, let us call it J i,n . Then M J(X ∗ (Ti ) ⋊ W (Ri )) = J i,n n≥0 INNER FORMS and the sum is finite by [Lus1, §7]. Moreover M Hi,n := φ−1 i,qi 15 k≥n J i,k
is a two-sided ideal of H(Ti , W (Ri ), qi ). According to [Lus2, Corollary 3.6] the morphism of O(Ti )W (Ri ) -algebras Hi,n /Hi,n+1 → J i,n induced by φi,qi is spectrum preserving. For any irreducible J i,n -module MJi the Hi,n -module φ∗qi ,i (MJi ) i , which is an irreducible Hi,n /Hi,n+1 -module. has a distinguished quotient MH P Let n be a vector with coordinates ni ∈ Z≥0 and put |n| = i ni . We write n ≤ n′ if ni ≤ n′i for all i. We define the two-sided ideals N Jn = Ni J i,ni ⊗ EndC (Vµ ) ⊂ J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ), i,ni ⊗ End (V ) ⊂ H(T , W , q ) ⊗ End (V ), Hn = C µ s s s C µ iH P n′ . Hn+ = H ′ ′ n ≥n,|n |=|n|+1 It follows from the above that the morphism of O(Ts )Ws -algebras O (49) (Hi,ni /Hi,ni +1 ) ⊗ EndC (Vµ ) ∼ = Hn /Hn+ → J n i induced by φqs is spectrum preserving, and that every irreducible J n -module MJ has a distinguished quotient MH which is an irreducible Hn /Hn+ -module. Next we define, for n ∈ Z≥0 : M M (50) J n := J n , Hn := Hn . |n|=n |n|=n The aforementioned properties of the map (49) are also valid for Hn /Hn+1 → J n , (51) which shows that φqs is spectrum preserving with respect to the filtrations (Hn )n≥0 and (⊕m≥n J m )n≥0 .  By Lemma 4.1 and (17) (52)
X L (s) H(Ts , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s ,
X L (s) J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s ,
X L (s) O(Ts ) ⋊ Ws ⊗ EndC (Vµ ) ⋊ R♯s are unital finite type O(Ts )Stab(s) -algebras, while φqs and φ1 provide morphisms between them. Theorem 4.3. (a) The above morphisms between the O(Ts )Stab(s) -algebras (52) are spectrum preserving with respect to filtrations. (b) The same holds for the three algebras of (52) with Ts♯ instead of Ts . Proof. (a) We use the notations from the proof of Lemma 4.2. Since Lusztig’s afunction is constant on two-sided cells [Lus3, §1.3] and conjugation by elements of R♯s preserves the set of simple (affine) reflections in X ∗ (Ts ) ⋊ Ws : a(wxvw−1 ) = a(xv) for all x ∈ X ∗ (Ts ), v ∈ Ws , w ∈ R♯s . 16 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Hence J n and Hn are stable under the respective actions α and (51) is Stab(s)+ L equivariant. Let MJ be an irreducible J n -module and regard it as a (J n )X (s) module via the map J n → J n from (50). By Clifford theory (see [RaRa, Appendix]) its decomposition is governed by a twisted group algebra of the stabilizer of MJ in X L (s). Since (51) is X L (s)-equivariant and MH is a quotient of MJ , the decomL position of MH as module over (Hn /Hn+1 )X (s) is governed by the same twisted group algebra in the same way. Therefore (51) restricts to a spectrum preserving L morphism of O(Ts )Ws ×X (s) -algebras (Hn /Hn+1 )X L (s) → (J n )X L (s) . Now a similar argument with Clifford theory for crossed product algebras shows that (Hn /Hn+1 )X L (s) ⋊ R♯s → (J n )X L (s) ⋊ R♯s is a spectrum preserving morphism of O(Ts )Stab(s) -algebras. By definition [BaNi, §5], this means that the map
X L (s) ⋊ R♯s → (53) φ′qs : H(Ts , Ws , qs ) ⊗ EndC (Vµ )
X L (s) J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s induced by φqs is spectrum preserving with respect to filtrations. The same reasoning is valid with O(Ts )⋊ Ws instead of H(Ts , Ws , qs ) – it is simply the case qs = 1 of the above. (b) Recall that Ts ∼ = Xnr (L)/Xnr (L, ω). The torus Ts /Xnr (L/L♯ Z(G)) can be identified with (54) Xnr (L♯ Z(G))/Xnr (L, ω). Since the elements of Xnr (L, ω) are trivial on Z(L) ⊃ Z(G) and L♯ ∩ Z(G) ∼ = o× F is compact, (54) factors as Xnr (L♯ )/Xnr (L, ω) × Xnr (Z(G)) = Ts♯ × Xnr (Z(G)). By Theorem 3.1 the action of Xnr (L/L♯ Z(G)) ⊂ X L (s) on the algebras (52) comes only from its action on the torus Ts . Hence these three algebras do not change if we replace Ts by (54). Equivalently, we may replace Ts by Ts♯ × Xnr (Z(G)). It follows that
X L (s) H(Ts , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s ∼ =
X L (s) O(Xnr (Z(G))) ⊗ H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s . The action of Stab(s)+ fixes O(Xnr (Z(G))) pointwise, so this equals
X L (s) ⋊ R♯s ⊗ O(Xnr (Z(G))). H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) The other two algebras in (52) can be rewritten similarly. By (41) the morphisms φqs and φ1 fix the respective subalgebras O(Xnr (Z(G))) pointwise. It follows that (53) decomposes as
X L (s) φ♯qs ⊗ id : H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s ⊗ O(Xnr (Z(G))) →
X L (s) J(X ∗ (Ts♯ ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s ⊗ O(Xnr (Z(G))), INNER FORMS 17 and similarly for φ′1 . From part (a) we know that φ′qs = φ♯qs ⊗ id and φ′1 = φ♯1 ⊗ id are spectrum preserving with respect to filtrations. So φ♯qs and
X L (s) φ♯1 : O(Ts♯ ) ⋊ Ws ⊗ EndC (Vµ ) ⋊ R♯s → have that property as well.
X L (s) J(X ∗ (Ts♯ ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s  With Theorem 4.3 we can show that the Hecke algebras for G♯ and for G♯ Z(G) are stratified equivalent (see [ABPS7]) to much simpler algebras. Recall the subgroup Hλ ⊂ L from [ABPS4, Lemma 3.3]. Theorem 4.4. (a) The algebra H(G♯ Z(G))s is stratified equivalent with M[L:Hλ ]
X L (s) O(Ts ) ⊗ EndC (Vµ ) ⋊ Ws♯ . 1 Here the action of w ∈ Ws♯ is α(w,γ) (as in Theorem ??.a) for any γ ∈ Irr(L/L♯ Z(G)) such that (w, γ) ∈ Stab(s). (b) The algebra H(G♯ )s is stratified equivalent with M[L:Hλ ]
X L (s) O(Ts♯ ) ⊗ EndC (Vµ ) ⋊ Ws♯ , 1 with respect to the same action of Ws♯ . Remark. In principle one could factorize the above algebras according to single Bernstein components for G♯ Z(G) and G♯ . However, this would result in less clear formulas. Proof. (a) Recall from Theorem 3.1 that H(G♯ Z(G))s is Morita equivalent with M[L:Hλ ]
X L (s) (55) ⋊ R♯s . H(Ts , Ws , qs ) ⊗ EndC (Vµ ) 1 Consider the sequence of algebras (56)
X L (s) ⋊ R♯s H(Ts , Ws , qs ) ⊗ EndC (Vµ )
X L (s) → J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s
X L (s) = J(X ∗ (Ts ) ⋊ Ws ) ⊗ EndC (Vµ ) ⋊ R♯s
X L (s) ← O(Ts ) ⋊ Ws ⊗ EndC (Vµ ) ⋊ R♯s . In Theorem 4.3.a we proved that the map between the first two lines is spectrum preserving with respect to filtrations. The equality sign does nothing on the level of C-algebras, but we use it to change the O(Ts )Stab(s) -module structure, such that the map from
X L (s) ⋊ R♯s (57) O(Ts ) ⋊ Ws ⊗ EndC (Vµ ) becomes O(Ts )Stab(s) -linear. By Theorem 4.3.a that map is also spectrum preserving with respect to filtrations. Every single step in the above sequence is an instance of stratified equivalence from [ABPS7] (the second step by definition), so H(G♯ Z(G))s is stratified equivalent with a direct sum of [L : Hλ ] copies of (57). Since χγ ∈ Ts in (47) is Ws -invariant, the 18 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD actions of X L (s) and Ws on O(Ts ) ⊗ EndC (Vµ ) commute. This observation and (14) allow us to identify (57) with
X L (s)

X L (s) (58) O(Ts ) ⊗ EndC (Vµ ) ⋊ Ws ⋊ R♯s = O(Ts ) ⊗ EndC (Vµ ) ⋊ Ws♯ . The description of the action of Ws♯ can be derived from Theorem 3.3. (b) This follows from Theorem 3.2 and the same proof as for part (a).  5. Extended quotients for inner forms of GLn It turns out that via (33) any Bernstein component for G can be described in a canonical way with an extended quotient. Before we prove that, we recall the parametrization of irreducible representations of H(Ts , Ws , qs ). Let Ǧs be the complex reductive group with root datum (X ∗ (Ts ), Rs , X∗ (Ts ), Rs∨ ), Q it is isomorphic to i GLei (C), embedded in Ǧ = GLmd (C) as  Y GLmi d (C)ei . Ǧs = ZǦ (Ľ) = ZGLmd (C) i Recall that a Kazhdan–Lusztig triple for Ǧs consists of: Q • a unipotent element u = i ui ∈ Ǧs ; Q qi qs • a semisimple element tq ∈ Ǧs with tq ut−1 q = u := i ui ; • a representation ρq ∈ Irr(π0 (ZǦs (tq , u))) which appears in the homology of variety of Borel subgroups of Ǧs containing {tq , u}. Typically such a triple is considered up to Ǧs -conjugation, we denote its equivalence class by [tq , u, ρq ]Ǧs . These equivalence classes parametrize Irr(H(Ts , Ws , qs )) in a natural way, see [KaLu]. We denote that by (59) [tq , u, ρq ]Ǧs 7→ π(tq , u, ρq ). Recall from [ABPS6, §7] that an affine Springer parameter for Ǧs consists of: Q • a unipotent element u = i ui ∈ Ǧs ; • a semisimple element t ∈ ZǦs (u); • a representation ρ ∈ Irr(π0 (ZǦs (t, u))) which appears in the homology of variety of Borel subgroups of Ǧs containing {t, u}. Again such a triple is considered up to Ǧs -conjugacy, and then denoted [t, u, ρ]Ǧs . Kato [Kat] established a natural bijection between such equivalence classes and Irr(O(Ts ) ⋊ Ws ), say (60) [t, u, ρ]Ǧs 7→ τ (t, u, ρ). For a more explicit description, we note that ZǦs (t) is a connected reducitive group with Weyl group Ws,t , and that (u, ρ) represents a Springer parameter for Ws,t . Via the classical Springer correspondence (61) (u, ρ) determines an irreducible Ws,t -representation π(u, ρ). Then [Kat] and (60) work out to (62) O(T )⋊W s τ (t, u, ρ) = indO(Tss )⋊Ws,t (Ct ⊗ π(u, ρ)). From [KaLu, §2.4] we get a canonical bijection between Kazhdan–Lusztig triples and affine Springer parameters: (63) [tq , u, ρq ]Ǧs ←→ [t, u, ρ]Ǧs . INNER FORMS 19 Basically it adjusts tq in a minimal way so that it commutes with u, and then there is only one consistent way to modify ρq to ρ. Via Lemma 4.2 the algebra homomorphisms (42) give rise to a bijection (64) Irr(H(Ts , Ws , qs )) ←→ Irr(O(Ts ) ⋊ Ws ). We showed in [ABPS6, (90)] that (64) is none other than the composition of (63) with (60) and the inverse of (59): (65) π(tq , u, ρq ) ←→ τ (t, u, ρ). Theorem 5.1. The Morita equivalence H(G)s ∼M H(Ts , Ws , qs ) and (64) give rise to a bijection (66) Irrs (G) ←→ Ts //Ws with the following properties: (1) Let Ts,un be the maximal compact subtorus of Ts and let Irrtemp (G) ⊂ Irr(G) be the subset of tempered representations. Then (66) restricts to a bijection Irrstemp (G) ←→ Ts,un //Ws . (2) (66) can be obtained from its restriction to tempered representations by anasL′ lytic continuation, as in [ABPS1]. For instance, suppose that σ ∈ Irrtemp (L′ ) for some standard parabolic P ′ = L′ U ′ ⊃ P = LU , and that IPG′ (σ ⊗ χ) is mapped to τ (tχ, u, ρ) for almost all unitary χ ∈ Xnr (L′ ). Then, whenever χnr (L′ ) and IPG′ (σ ⊗ χ) is irreducible, it is mapped to τ (tχ, u, ρ). (3) If π ∈ Irrstemp (G) is mapped to [t, ρ′ ] ∈ Ts,un //Ws and has cuspidal support Ws σ ∈ Ts /Ws , then Ws t is the unitary part of Ws σ, with respect to the polar decomposition Ts = Ts,un × HomZ (X ∗ (Ts ), R>0 ). (4) In the notation of (3), suppose that the parameter of ρ′ ∈ Irr(Ws,t ) in the classical Springer correspondence (61) involves a unipotent class [u] which is distinguished in a Levi subgroup M̌ ⊂ ZǦs (t). Then π = IPGM (δ), where M ⊃ L is the unique standard Levi subgroup of G corresponding to M̌ and [L,ω] δ ∈ IrrtempM (M ) is square-integrable modulo centre. Moreover (66) is the unique bijection with the properties (1)–(4). Proof. The Morita equivalence (33) gives a bijection (67) Irrs (G) ←→ Irr(H(Ts , Ws , qs )). Via Lemmas 4.2 and A.1 the right hand side is in bijection with (68) Irr(O(Ts ) ⋊ Ws ) ∼ = Ts //Ws . In this way we define the map (66). (1) It is easy to check from [Séc, Théorème 4.6] and [SéSt2, Theorem C] that the Morita equivalence H(G)s ∼M H(Ts , Ws , qs ) preserves the canonical involution and trace (maybe up to a positive scalar). Accepting that, [DeOp, Theorem 10.1] says that the ensueing bijection between Irrs (G) and Irr(H(Ts , Ws , qs )) respects the subsets of tempered representations. By [ABPS6, Proposition 9.3] the latter subset corresponds to the set of Kazhdan–Lusztig triples such that the t in (63) lies in Ts,un . (2) Consider the bijection (64) and its formulation (65). Here the representations are tempered if and only if t ∈ Ts is unitary. Thus (65) for tempered representations 20 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD determines the bijection (64), by analytic continuation (in the parameters t and tq ) of the formula. The relation between Irrs (G) and Irrstemp (G) is similar, see [ABPS2, Proposition 2.1]. Hence (67) is can also be deduced from its restriction to tempered representations, with the method from [ABPS2, §4]. (3) In [Séc, Théorème 4.6] a sL -type (KL , λL ) is constructed, with eλL H(L)eλL ∼ = O(Ts ) ⊗ EndC (Vλ ). It [SéSt2] it is shown that it admits a cover (KG , λG ) with eλG H(G)eλG ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vλ ), see also [ABPS4, §4.1]. It follows from [ABPS4, Proposition 3.15] that (67) arises from this cover of a sL -type. With [BuKu, §7] this implies that (66) translates the cuspidal support of a (π, Vπ ) ∈ Irrs (G) to the unique Ws tq ∈ Ts /Ws such that H(T ,W ,q ) eλG Vπ is a subquotient of indO(Tss ) s s (Ctq ) ⊗ Vλ . It follows from [ABPS6, (33) and Lemma 7.1] that the bijection (65) sends any tempered irreducible subquotient of H(T ,W ,q ) indO(Tss) s s (Ctq ) to an irreducible O(Ts ) ⋊ Ws -representation with O(Ts )-weights Ws (tq |tq |−1 ). The associated element of Ts //Ws is then [t = tq |tq |−1 , ρ] with ρ ∈ Irr(Ws,t ). (4) Since Ws,t is a direct product of symmetric groups, the representations ρ′ of the component groups in the Springer parameters are all trivial. By (61), (62) and (65) the H(Ts , Ws , qs )-representation associated to [t, ρ′ ] is π(tq , u, triv). Then (tq , u, triv) is also a Kazhdan–Lusztig triple for H(Ts , Ws,M , qs ) and by [KaLu, §7.8] π(tq , u, triv) = indH HM πM (tq , u, triv). By [ABPS6, Proposition 9.3] (see also [KaLu, Theorem 8.3]) πM (tq , u, triv) is essentially square-integrable and tempered, that is, square-integrable modulo centre. Since (KG , λG ) is a a cover of a sL -type (KL , λL ), there is a a [L, ω]M -type (KM , λM ) which covers (KL , λL ) and is covered by (KG , λG ). By [ABPS6, Proposition 16.6] πM (tq , u, triv) corresponds to a M -representation δ which is squareintegrable modulo centre. By [BuKu, Corollary 8.4] the bijection (67) respects parabolic induction, so π(tq , u, triv) corresponds to IPGM (δ). Now we check that (66) is canonical in the specified sense. By (1) and (2) it suffices to do so for tempered representations. For π ∈ Irrstemp (G), property (3) determines the Ws -orbit Ws t. Fix a t in this orbit. By a result of Harish-Chandra [Wal, Proposition III.4.1] there are a Levi subgroup M ⊂ G containing L and a square-integrable (modulo centre) representation δ ∈ Irr(M ) such that π is a subquotient of IPGM (δ). Moreover (M, δ) is unique up to conjugation. For t ∈ Ts,un , Ws,t is a product of symmetric groups Se and ZǦs (t) is a product of groups of the form GLe (C). Hence the Springer correspondence for Ws,t is a bijection between Irr(Ws,t ) and unipotent classes in ZǦs (t). Every Levi subgroup of GLe (C) has a unique distinguished unipotent class, and these exhaust the unipotent classes in GLe (C). Hence Irr(Ws,t ) is also in canonical bijection with the set of conjugacy classes of Levi subgroups M̌ ⊂ ZǦs (t). Viewed in this light, properties (3) and (4) entail that for every pair (M̌ , t) as above there is precisely one square-integrable modulo centre δ ∈ Irr(M ) such that INNER FORMS 21 Ws t is the unitary part of the cuspidal support of IPGM (δ). Thus (3) and (4) determine the (tempered) G-representation associated to [t, ρ] ∈ Ts,un //Ws .  6. Twisted extended quotients for inner forms of SLn Twisted extended quotients appear naturally in the description of the Bernstein components for L♯ Z(G) and L♯ . Lemma 6.1. Let sL = [L, ω]L and define a two-cocycle κω by (11). (a) Equation (13) for L determines bijections (Ts //X L (s))κω → IrrsL (L♯ Z(G)), (Ts //X L (s)Xnr (L/L♯ ))κω = (Ts //X L (s))κω /Xnr (L♯ Z(G)/L♯ ) → IrrsL (L♯ ). (b) The induced maps IrrsL (L♯ Z(G)) → Ts /X L (s) and IrrsL (L♯ ) → Ts /X L (s)Xnr (L/L♯ ) are independent of the choice of κω . (c) Let Ts,un be the real subtorus of unitary representations in Ts . The subspace L of tempered representations Irrstemp (L♯ Z(G)) corresponds to (Ts,un //X L (s))κω . sL ♯ Similarly Irrtemp (L ) is obtained by restricting the second line of part (a) to Ts,un . Proof. (a) Apart from the equality, this is a reformulation of the last page of Section 2. For the equality, we note that by (27) the action of Xnr (L♯ Z(G)/L♯ ) ∼ = Xnr (L/L♯ )/(X L (s) ∩ Xnr (L/L♯ )) on (Ts //X L (s))κω is free. Hence the isotropy groups for the action of X L (s)Xnr (L/L♯ ) are the same as for X L (s), and we can use the same 2-cocycle κω to construct a twisted extended quotient. (b) By (13) a different choice of κω in part (a) would only lead to the choice of ♯ another irreducible summand of ResG G♯ (π) for π ∈ Ts , and similarly for G Z(G). (c) Since ω is supercuspidal, the set of tempered representations in Ts = IrrsL (L) is Ts,un . In the decomposition (13), an irreducible representation of L♯ or L♯ Z(G) is tempered if and only if it is contained in a tempered L-representation ω ⊗ χ. This proves the statement for L. The claim for L♯ follows upon dividing out the free action of Xnr (L♯ Z(G)/L♯ ).  The subgroup X L (ω, Vµ ) acts trivially on O(Ts ) ⊗ EndC (Vµ ), and for that reason it can be pulled out of the extended quotient from Lemma 6.1. Lemma 6.2. There are bijections (Ts //X L (s))κω ←→ (Ts //X L (s)/X L (ω, Vµ ))κω × Irr(X L (ω, Vµ )), (Ts //X L (s)Xnr (L/L♯ ))κω ←→ (Ts //X)κω × Irr(X L (ω, Vµ )), where X = X L (s)Xnr (L/L♯ )/X L (ω, Vµ ). They fix the coordinates in Ts . Proof. In Lemma 6.1.a we saw that (Ts //X L (s))κω is in bijection with Irr(H(L♯ Z(G))sL ). By [ABPS4, (169)] H(L♯ Z(G))sL is Morita equivalent with 22 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD (H(L)sL )X L (s) and with the subalgebra (69) esL H(L)X L (s) X L (s) esL ∼ = (O(Ts ) ⊗ EndC (esL Vω )) M L L ∼ (O(Ts ) ⊗ EndC (Vµ ))X (s)/X (ω,Vµ ) . = a∈[L/Hλ ] Here the X L (s)-action on the middle term comes from an isomorphism EndC (esL Vω ) ∼ = EndC (Vµ ) ⊗ C[L/Hλ ] ⊗ C[L/Hλ ]∗ . We recall that by [ABPS4, Lemma 3.5] there is a group isomorphism (70) L/Hλ ∼ = Irr(X L (ω, Vµ )). By the above Morita equivalences, Lemma 6.1.a and Clifford theory, the set (71) {ρ ∈ Irr(C[X L (ω), κω ]) : ρ|X L (ω,Vµ ) = triv} parametrizes the irreducible representations of (69) associated to a fixed χ ∈ Ts and the trivial character of X L (ω, Vµ ). For any χ ∈ Ts , the X L (s)/X L (ω, Vµ )-stabilizer of the irreducible representation Cχ ⊗ Vµ of O(Ts ) ⊗ End(Vµ ) is X L (ω)/X L (ω, Vµ ). It follows that the irreducible representations of (69) with fixed a ∈ [L/Hλ ] and fixed O(Ts )-character χ are in L L bijection with Irr(EndC (Vµ )X (ω)/X (ω,Vµ ) ). Comparing with (71), we see that every irreducible representation of C[X L (ω)/X L (ω, Vµ ), κω ] appears in Vµ . This is equivalent to each irreducible representation of EndC (Vµ ) ⋊ X L (ω)/X L (ω, Vµ ) having nonzero vectors fixed by X L (ω)/X L (ω, Vµ ). Thus Lemma A.2 can be applied to X L (ω)/X L (ω, Vµ ) acting on O(Ts ) ⊗ EndC (Vµ ), and it shows that the irreducible representations on the right hand side of (69) are in bijection with (Ts //X L (s)/X L (ω, Vµ ))κω × Irr(X L (ω, Vµ )). The second bijection follows by dividing out the free action of Xnr (L♯ Z(G)/L♯ ), as in the proof of Lemma 6.1.a.  As a result of the work in Section 4, twisted extended quotients can also be used to describe the spaces of irreducible representations of G♯ Z(G) and G♯ . Let us extend κω to a two-cocycle of Stab(s), trivial on the normal subgroup Ws × X L (ω, Vµ ), by (72) J(γ, ω)J(γ ′ , ω) = κω (γ, γ ′ )J(γγ ′ , ω) γ, γ ′ ∈ X G (s). Theorem 6.3. (a) Lemmas A.1 and A.2 gives rise to bijections (Ts //Stab(s)/X L (ω, Vµ ))κω → Irr (O(Ts ) ⊗ EndC (Vµ ))X L (s)
⋊ Ws♯ , (Ts //Stab(s)Xnr (L/L♯ )/X L (ω, Vµ ))κω → Irr (O(Ts♯ ) ⊗ EndC (Vµ ))X L (s) (b) The stratified equivalences from 4.4 provide bijections
⋊ Ws♯ . (Ts //Stab(s))κω → (Ts //Stab(s)/X L (ω, Vµ ))κω × Irr(X L (ω, Vµ )) → Irrs (G♯ Z(G)), (Ts //Stab(s)Xnr (L/L♯ ))κω → (Ts //S)κω × Irr(X L (ω, Vµ )) → Irrs (G♯ ), where S = Stab(s)Xnr (L/L♯ )/X L (ω, Vµ ). (c) In part (b) Irrstemp (G♯ Z(G)) (respectively Irrstemp (G♯ )) corresponds to the same extended quotient, only with Ts,un instead of Ts . INNER FORMS 23 Proof. In each of the three parts the second claim follows from the first upon dividing out the action of Xnr (L♯ Z(G)/L♯ ), like in Lemma 6.1.a (a) In the proof of Lemma 6.2 we exhibited a bijection
L (Ts //Stab(s)/X L (ω, Vµ ))κω ←→ Irr (O(Ts ) ⊗ EndC (Vµ ))X (s) . With Lemma A.2 we deduce a Morita equivalence (73) (O(Ts ) ⊗ EndC (Vµ ))X L (s) ∼M (O(Ts ) ⊗ EndC (Vµ )) ⋊ (X L (s)/X L (ω, Vµ )). In the notation of (98) this means that p := pX L (s)/X L (ω,Vµ ) is a full idempotent in the right hand side of (73), that is, the two-sided ideal it generates is the entire algebra. Then p is also full in (74) (O(Ts ) ⊗ EndC (Vµ )) ⋊ (Stab(s)/X L (ω, Vµ )), which implies that (74) is Morita equivalent with
p (O(Ts ) ⊗ EndC (Vµ )) ⋊ (Stab(s)/X L (ω, Vµ )) p ∼ = (O(Ts ) ⊗ EndC (Vµ ))X L (s)/X L (ω,V ) µ ⋊ (Stab(s)/X L (s)). As a direct consequence of (14), (16) and (17), Stab(s)/X L (s) ∼ = Ws♯ . In this way we reach the algebra featuring in part (a). By the above Morita equivalence, its irreducible representations are in bijection with those of (74). Apply Lemma A.1.a to the latter algebra. (b) All the morphisms in (56) are spectrum preserving with respect to filtrations. In combination with the other remarks in the proof of Theorem 4.4.a this gives a bijection
L (75) Irrs (G♯ Z(G)) → Irr (O(Ts ) ⊗ EndC (Vµ ))X (s) ⋊ Ws♯ × [L/Hλ ]. By part (a) and (70) the right hand side of (75) is in bijection with (76) (Ts //Stab(s)/X L (ω, Vµ ))κω × Irr(X L (ω, Vµ )). The group X L (s) acts on C[L] by pointwise multiplication of functions on L. That gives rise to actions on C[L/Hλ ] and on EndC (C[L/Hλ ]) ∼ = C[L/Hλ ] ⊗ C[L/Hλ ]∗ . L Regarding C[L/Hλ ] as the algebra a∈[H/Hλ ] C, (70) leads to an isomorphism (77) (O(Ts ) ⊗ EndC (Vµ ))X L (s) ⋊ Ws♯ ⊗ C[L/Hλ ] ∼ = (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]))X L (s) ⋊ Ws♯ . We note that (76) is also the space of irreducible representations of (77). In the proof of Lemma 6.2 we encountered a bijection
L (Ts //X L (s))κω ←→ Irr (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]))X (s) . It implies a Morita equivalence (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]))X L (s) ∼M (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ])) ⋊ X L (s). 24 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Just as in the proof of part (a), this extends to (78) (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]))X L (s) ⋊ Ws♯ ∼M (O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ])) ⋊ Stab(s). Finally we apply Lemma A.1.a to the right hand side and we combine it with (78), (77) and (75). (c) The first bijection in part (b) obviously preserves the subspaces associated to Ts,un . We need to show that the second bijection sends them to Irrstemp (G♯ Z(G)). This is a property of the geometric equivalences in Theorem 4.4, as we will now check. We may and will assume that ω is unitary, or equivalently that it is tempered. The Morita equivalence between H(G♯ Z(G))s and (55) is induced by an idempotent e♯λ ♯ ∈ H(G♯ Z(G)), see Theorem 3.1. Its construction (which starts around G Z(G) (34)) shows that eventually it comes from a central idempotent in the algebra of a profinite group, so it is a self-adjoint element. Hence, by [BHK, Theorem A] this Morita equivalence preserves temperedness. The notion of temperedness in [BHK] agrees with temperedness for representations of affine Hecke algebras (see [Opd]) because both are based on the Hilbert algebra structure and the canonical tracial states on these algebras. The sequence of algebras (56) is derived from its counterpart for H(Ts , Ws , qs ) ⊗ EndC (Vµ ). By Theorem 5.1 that one matches tempered representations with Ts,un //Ws . By Clifford theory any irreducible representation π of (79) (H(Ts , Ws , qs ) ⊗ EndC (Vµ ))X L (s) ⋊ R♯s is contained in a sum of irreducible representations π̃ of H(Ts , Ws , qs ) ⊗ EndC (Vµ ), which are all in the same Stab(s)-orbit. Temperedness of π depends only on the action of the subalgebra O(Ts ) ∼ = C[X ∗ (Ts )], and in fact can already be detected on C[X] for any finite index sublattice X ⊂ X ∗ (Ts ). The analogous statement for (79) holds as well, with X = X ∗ (Ts /X L (s)), and it is stable under the action of Stab(s). Consequently π is tempered if and only if π̃ is tempered. These observations imply that the sequence of algebra homomorphisms (56) preserves temperedness of irreducible representations, and that it maps such representations of (79) to irreducible representations of (58) with O(Ts /X L (s))-weights in Ts,un /X L (s). Now we invoke this property for every a ∈ L/Hλ ∼ = Irr(X L (ω, Vµ )) and we deduce that the second map in part (b) has the required property with respect to temperedness.  We will work out what Theorem 6.3 says for a single Bernstein component of G♯ . To this end, we first analyse what parabolic induction from L♯ to G♯ looks like in the setting of Theorem 3.2. Theorem 6.4. (a) There exist idempotents esL ∈ H(L), esL♯ ∈ H(L♯ ) such that H(L♯ )s is Morita equivalent with
X L (s) L ♯ esL♯ H(L♯ )esL♯ ∼ = esL H(L)X (s)Xnr (L/L ) esL ∼ = O(Ts♯ ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]) M
X L (s)/X L (ω,Vµ ) ∼ O(Ts♯ ) ⊗ EndC (Vµ ) . = a∈[L/Hλ ] INNER FORMS 25 (b) Under the equivalences from part (a) and Theorem 3.2, the normalized parabolic induction functor ♯ IPG♯ : RepsL (L♯ ) → Reps (G♯ ) corresponds to induction from the last algebra in part (a) to M
X L (s)/X L (ω,Vµ ) H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s . a∈[L/Hλ ] Proof. Part (a) is a consequence of [ABPS4, (169)] and [ABPS4, Lemma 4.8], which shows that EndC (C[L/Hλ ]) ∼ = C[L/Hλ ] ⊗ C[X L (ω, Vµ )]. The analogue of part (b) for L and G says that IPG : RepsL (L) → Reps (G) corresponds to induction from esL H(L)esL ∼ = O(Ts ) ⊗ EndC (Vµ ⊗ C[L/Hλ ]) to e♯λG H(G)e♯λG ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vµ ⊗ C[L/Hλ ] ⊗ CR♯s ). To see that it is true, we reduce with [ABPS4, Theorem 4.5] to the algebras eλ H(L)eλ ∼ = O(Ts ) ⊗ EndC (Vλ ), L L eλG H(G)eλG ∼ = H(Ts , Ws , qs ) ⊗ EndC (Vλ ). Then we are in the situation where (KG , λG ) is a cover of a type (KL , λL ), and the statement about the induction functors follows from [ABPS4, (126)] and [BuKu, Corollary 8.4]. We note that here, for a given algebra homomorphism φ : A → B, we must use induction in the version IndB A (M ) = HomA (B, M ). However, in all the cases we encounter B is free of finite rank as a module over A and it is endowed with a canonical anti-involution f 7→ [f ∨ : g 7→ f (g −1 )]. Hence we may identify HomA (B, M ) ∼ = B ∗ ⊗A M ∼ = B ⊗A M . This shows the desired G claim for IP . ♯ Since G♯ /P ♯ ∼ = G/P, IPG = IPG♯ on RepsL (L). In particular (80) ♯ L G G ResG G♯ ◦ IP = IP ♯ ◦ ResL♯ as functors RepsL (L) → Reps (G♯ ). e♯ H(L)e♯L ♯ s , ♯ H(L )e ♯ L The functor ResL L♯ corresponds to Reses L L and ResG G♯ to restriction from e♯λG H(G)e♯λG to e♯λ ♯ H(G♯ )e♯λ ♯ , which is the algebra appearing in the statement G G of part (b). The set H(Ws , qs )⊗C[R♯s ] forms a basis for e♯λG H(G)e♯λG as a module over e♯L H(L)e♯L and for e♯λ ♯ H(G♯ )e♯λ G♯ G (81) Res e♯λ H(G)e♯λ G as a module over esL♯ H(L♯ )esL♯ . It follows that G e♯λ H(G♯ )e♯λ G♯ G♯ e♯λ H(G)e♯λ ◦ ind G e♯L H(L)e♯L G e♯λ = indes G♯ H(G♯ )e♯λ H(L L♯ ♯ )es L♯ G♯ e♯ H(L)e♯L ♯ s . ♯ H(L )e ♯ ◦ ResesL L L Comparing (80) and (81), we find that (82) ♯ IPG♯ ◦ ResL L♯ corresponds to e♯λ indes G♯ H(G♯ )e♯λ H(L♯ )es ♯ L L♯ G♯ e♯ H(L)e♯L ♯ s ♯ H(L )e ♯ ◦ ResesL L L 26 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD under the Morita equivalences from Theorems 3.2 and 6.4.a. Here both inductions can be constructed entirely in H(G♯ ). The two sides of (82) are then related by applications of the idempotents esL♯ and e♯λ ♯ (which are supported on G♯ ). Hence G the correspondence (82) preserves L♯ -subrepresentations. Since every irreducible L♯ -representation appears as a summand of an L-representation, this implies the analogue of (82) on the whole of RepsL (L♯ ).  Let t♯ = [L♯ , σ ♯ ]G♯ be an inertial equivalence class for G♯ , with t♯ ≺ s = [L, ω]G . We abbreviate φω,X L (s) = {φω,γ : γ ∈ X L (s)}, where φω,γ is as in Lemma 2.2. By (27) there is a unique X L (s)-orbit φω,X L (s) ρ ⊂ Irr(C[X L (ω), κω ]) (83) such that Tt♯ = (Ts♯ × φω,X L (s) ρ)/X L (s). Then φω,X L (s) ρ determines a unique L summand Ca of C[L/Hλ ] ∼ = [L/Hλ ] C, namely the irreducible representation of L X (ω, Vµ ) obtained by restricting ρ. Let Vσ♯ be the intersection of Vµ with the subspace of Vω on which σ ♯ is defined, and let Rt♯ be its stabilizer in R♯s . Then Rt♯ is also the stabilizer of t♯ in R♯s and Wt♯ = Ws ⋊ Rt♯ , (84) by [ABPS4, Lemma 2.3]. Via the formula (72) the operators J(γ, ω)|V ♯ determine σ a 2-cocycle κ′ω of the group W ′ = {(w, γ) ∈ Stab(s) : w ∈ Wt♯ }. (85) Since (72) is 1 on Ws , so is κ′ω . By (15) W ′ /X L (s) ∼ = Wt♯ . As Vσ♯ is associated ′ ′ L ′ to the single X (s)-orbit (83), κω ((w, γ), (w , γ )) depends only on (w, w′ ). Thus it determines a 2-cocycle κσ♯ of Wt♯ , which factors through Rt♯ ∼ = Wt♯ /Ws . Lemma 6.5. (a) The bijections in Theorem 6.3 restrict to ♯ Irrt (G♯ ) ←→ (Tt♯ //Wt♯ )κσ♯ , ♯ Irrttemp (G♯ ) ←→ (Tt♯ ,un //Wt♯ )κσ♯ , where Tt♯ ,un denotes the space of unitary representations in Tt♯ . ♯ (b) Suppose π ∈ Irrttemp (G♯ ) corresponds to [t, ρ] and has cuspidal support Wt♯ (χ ⊗ σ ♯ ) ∈ Tt♯ /Wt♯ . Then Wt♯ t is the unitary part of χ ⊗ σ ♯ , with respect to the polar decomposition Tt♯ = Tt♯ ,un × HomZ (X ∗ (Tt♯ ), R>0 ). ♯ Proof. (a) Recall that Irrt (G♯ ) consists of those irreducible representations that are ♯ ♯ contained in IPG♯ (χ ⊗ σ ♯ ) for some χ ⊗ σ ♯ ∈ Tt♯ . In Theorem 6.4.b we translated IPG♯ to induction between two algebras. The first one, Morita equivalent with H(L♯ )sL , was
X L (s) . C[L/Hλ ] ⊗ O(Ts♯ ) ⊗ EndC (Vµ ) The second algebra, Morita equivalent with H(G♯ )s , was
X L (s) C[L/Hλ ] ⊗ H(Ts♯ , Ws , qs ) ⊗ EndC (Vµ ) ⋊ R♯s . INNER FORMS 27 As above, the L♯ -representation σ ♯ determines a summand Ca of C[L/Hλ ] and a ♯ ♯ X L (s)-stable subspace Vσ♯ ⊂ Vµ . Consequently H(L♯ )[L ,σ ] is Morita equivalent with the two algebras
X L (s)
X L (s) . and O(Ts♯ ) ⊗ EndC (CR♯s · Vσ♯ ) (86) O(Ts♯ ) ⊗ EndC (Vσ♯ ) ♯ In these terms Theorem 6.4 shows that H(G♯ )t is is Morita equivalent with
X L (s) ⋊ R♯s . (87) H(Ts♯ , Ws , qs ) ⊗ EndC (CR♯s · Vσ♯ ) Here the subspaces wVσ♯ with w ∈ R♯s are permuted transitively by R♯s , so upon taking R♯s -invariants only the Rt♯ on Vσ♯ survives. Recall that, for any finite group Γ and any Γ-algebra A: (88) Γ A⋊Γ∼ = (A ⊗ EndC (CΓ)) . Applying (88) to (87) first with Γ = R♯s and subsequently with Γ = Rt♯ (in the opposite direction, taking the above transitivity into account), we find that (87) is Morita equivalent with
X L (s) (89) H(Ts♯ , Ws , qs ) ⊗ EndC (Vσ♯ ) ⋊ Rt♯ . The constructions in Section 4 restrict to stratified equivalences between (87) and (90) (O(Ts♯ ) ⊗ EndC (CR♯s · Vσ♯ ))X (O(Ts♯ ) ⊗ EndC (Vσ♯ ))X L (s) L (s) ⋊ Ws♯ , ⋊ Wt♯ . By (86) (91) Irr (O(Ts♯ ) ⊗ EndC (Vσ♯ ))X L (s)
∼ = Tt♯ . As explained above with (85), the 2-cocycle κω of Stab(s) reduces to the 2-cocycle κσ♯ for the action of Wt♯ in (90). Now we apply Lemma A.1.a to (90) and we find the first bijection. To obtain the second bijection, we use Theorem 6.3.c. (b) For the stratified equivalence between H(Ts♯ , Ws , qs ) ⊗ EndC (Vσ♯ ) and O(Ts♯ ) ⊗ EndC (Vσ♯ ) ⋊ Ws the analogous claim about the cuspidal support is property (3) of Theorem 5.1. Clifford theory relates the irreducible representations of these algebras to those of (87) and (90), in a way already discussed after (79). This implies that the desired property of the cuspidal support persists to the stratified equivalence between (87) and (90), which underlies part (a).  7. Relation with the local Langlands correspondence We show how the local Langlands correspondence (LLC) for G and G♯ can be reconstructed in terms of twisted extended quotients. Let WF be the Weil group of the local non-archimedean field F . Recall that the Langlands dual group of G = GLm (D) is Ǧ = GLmd (C). A Langlands parameter for G is continuous group homomorphism φ : WF × SL2 (C) → Ǧ such that: • φ|SL2 (C) is a homomorphism of algebraic groups. • φ(WF ) consists of semisimple elements. 28 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD • φ is relevant for G: if Ľ is a Levi subgroup of Ǧ which contains im(φ) and is minimal for that property, then (the conjugacy class of) Ľ corresponds to (the conjugacy class of) a Levi subgroup of G. We denote the collection of Langlands parameters for G, modulo conjugation by Ǧ, by Φ(G). Every smooth character of G is of the form ν ◦ Nrd, with ν a smooth character of F × . Via Artin reciprocity it determines a Langlands parameter (trivial on SL2 (C)) (92) ν̂ : WF → C× ∼ = Z(GLmd (C)). For any φ ∈ Φ(G), φν̂ is a well-defined element of Φ(G) because the image of ν̂ is central in Ǧ. Theorem 7.1. The local Langlands correspondence for G is a canonical bijection recD,m : Irr(G) → Φ(G) with the following properties: (a) π ∈ Irr(G) is tempered if and only if recD,m (π) is bounded, that is, if recD,m (π)(WF ) is a bounded subset of Ǧ. (b) The L-packet Πφ (G) is the single representation rec−1 D,m (φ). (c) recD,m is equivariant for the two actions of Irr(G/G♯ ): on Irr(G) by twisting with smooth characters and on Φ(G) by multiplication with central Langlands parameters as in (92). Proof. For the bijection and part (a) see [HiSa, §11] and [ABPS3, §2]. Ultimately it relies on the Jacquet–Langlands correspondence from [DKV, Bad]. (b) This is a direct consequence of the bijectivity. (c) Since recD,m is determined completely by its behaviour on essentially square integrable representations of Levi subgroups of G [ABPS3, (13)], it suffices to prove (c) for such representations. Via the Jacquet–Langlands correspondence the issue can be transferred to Irr(GLn (F )) with n ≤ md. For general linear groups (c) is a well-known property of the LLC, and in fact constitutes a starting point of the construction, confer [Hen, 1.2].  For s = [L, ω]G we define Φ(G)s as the image of Irrs (G) under the bijection recD,m . Similarly we define Φ(L)sL ⊂ Φ(L). Lemma 7.2. The LLC for G fits in a commutative diagram of canonical bijections Irrs (G) O  Ts //Ws o recD,m / Φ(G)s O  / Φ(L)sL //Ws Here the bottom map comes from the LLC for IrrsL (L) and the left hand side comes from Theorem 5.1. (a) Suppose that [φL ] ∈ Φ(L)sL and that ρ ∈ Irr(Ws,φL ) has as Springer parameter a unipotent class [u] ∈ ZǦs (φL ). Then there is a representative u such that the right hand side sends [φL , ρ] to a Langlands parameter φ with φ|WF = φL |WF and φ(1, ( 10 11 )) = φL (1, ( 10 11 ))u. INNER FORMS 29 (b) Two elements [t, ρ] and [t′ , ρ′ ] of Ts //Ws map to the same G-representation if and only if there exists a w ∈ Ws such that wt = t′ and the Ws,t -representations ρ, wρ′ have Springer parameters involving the same unipotent class. Proof. The top horizontal and left vertical maps have already been established as bijective and canonical. The LLC for L is the Cartesian product of the LLCs for the factors of L. Hence it is W (G, L)-equivariant, and its restriction to Ts ←→ Φ(L)sL is Ws -equivariant. The canonicity and bijectivity of the LLC for L are inherited by the bottom horizontal map in the diagram. This leaves a unique, canonical way to complete the commutative diagram. (a) To work out the map on the right hand side, it suffices to consider Y e Y e L= Li i and ω = ωi i i i such that (Li , ωi ) is not isomorphic to (Lj , ωj ) for i 6= j. Let φi : WF × SL2 (C) → GLmi d (C) be a Langlands parameter for ωi . Then Y e Y φL = φi i : WF × SL2 (C) → GLmi d (C)ei i i Q is a Langlands parameter for ω. We have Ws,φL = i Sei , where Sei is embedded in NGLe d m (C) (GLmi d (C)ei ) as permutation matrices. The unipotent class i i Y Y [u] = [ ui ] ∈ GLei mi d (C) ⊂ ZǦs (φL ) i i is determined by the standard Levi subgroup in which it is distinguished, say Y X M̌ = GLbij mi d (C)cij with cij bij = ei . i,j j Assume for the moment that ω is tempered. By Theorem 5.1 [ω, ρ] ∈ Ts,un //Ws corresponds to IPGM (δ), where Y c [L,ω] δ= δijij ∈ IrrtempM (M ) i,j is the unique square-integrable modulo centre representation such that Ws,M ω is the unitary part of the cuspidal support of δ. By construction [ABPS4,Q§2] the c Langlands parameter φ of IPGM (δ) is the same as that of δ, namely φ = i,j φijij b with φij |WF = φi ij |WF and φij (1, ( 10 11 )) = φi (1, ( 10 11 ))bij uij where uij is a distinguished unipotent element in ZGLb ij mi d φ(1, ( 10 11 )) bij (C) (GLmi d (C) ). Thus is distinguished in M̌ and φ has the asserted shape. The general case, where ω is not necessarily tempered, follows from the tempered case. The reason is that all the maps in the commutative diagram (a priori except the right hand side) can be obtained from their tempered parts by some kind of analytic continuation, as in [ABPS1] and Theorem 5.1. (b) By Theorem 7.1.b the elements of Ts //Ws are in bijection with the L-packets in Irrs (G). Two elements [t, ρ] and [t′ , ρ′ ] are equal if and only if there is a w ∈ Ws such that wt′ = t and w · ρ′ = ρ. We note also that for every t ∈ Ts the group Ws,t = W (Rs,t ) is product of symmetric groups. Hence all irreducible representations of Ws,t are parametrized by different unipotent classes in a connected complex reductive group with maximal torus Ts and root system Rs,t . So the condition becomes that ρ and w · ρ′ have the same unipotent class as Springer parameter.  30 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Let Irrcusp (L) be the space of supercuspidal L-representations and let Φ(L)cusp be its image in Φ(L). The Weyl group W (G, L) = NG (L)/L ∼ = N (Ľ)/Ľ Ǧ acts naturally on both sets. Theorem 7.3. Let L be a set of representatives for the conjugacy classes of Levi subgroups of G. The maps from Lemma 7.2 combine to a commutative diagram of canonical bijections Irr(G) O F  Irr (L)//W (G, L) o cusp L∈L recD,m / Φ(G) O /F  Φ(L) cusp //W (G, L) L∈L Here the tempered representations correspond to the bounded Langlands parameters. Proof. The action of W (G, L) on L is simply by permuting some direct factors of L, and the same for Ľ. Hence the canonical bijection Irr(L) ↔ Φ(L) is W (G, L)equivariant. The group Ws is defined as the stabilizer in W (G, L) of Ts = IrrsL (L), and by the above equivariance it is also the stabilizer of ΦsL (L). Consequently G Irrcusp (L)//W (G, L) ∼ Ts //Ws , = s=[L,ω]G G Φ(L)cusp //W (G, L) ∼ ΦsL (L)//Ws . = s=[L,ω]G Now we simply take the union of the commutative diagrams of Lemma 7.2. The characterization of temperedness and boundedness comes from Theorems 7.1.a and 6.3.c.  To formulate the LLC for G♯ , we need enhanced Langlands parameters. In fact these are already present in the LLC for G, but there the enhancement can be neglected without any problems. Recall that a Langlands parameter for G♯ = GLm (D)der is a homomorphism φ : WF ×SL2 (C) → PGLmd (C) subject to the same requirements as a Langlands parameter for G. The set of such parameters modulo conjugation by Ǧ♯ = PGLmd (C) is denoted Φ(G♯ ). We note that the simply connected cover SLmd (C) of PGLmd (C) also acts by conjugation on Langlands parameters for G♯ . An enhancement of φ is an irreducible representation ρ of π0 (ZSLmd (C) (φ)). In order that (φ, ρ) is relevant for G♯ , an extra condition is needed. For this we have to regard D as part of the data of G♯ , in other words, we must consider not just the inner form G♯ of SLmd (F ), but even the inner twist determined by (G♯ , D). The Hasse invariant of D gives a character χD of Z(SLmd (C)) ∼ = Z/mdZ with kernel mZ/mdZ. Notice that, by Schur’s lemma, every enhancement ρ of φ determines a character of Z(SLmd (C)). We define an enhanced Langlands parameter for G♯ = GLm (D)der as a pair (φ, ρ) such that ρ|Z(SLmd (C)) = χD . The collection of these, modulo conjugation by SLmd (C), is denoted Φe (G♯ ). The LLC for G♯ [ABPS3] is a bijection (93) such that Φe (G♯ ) ←→ Irr(G♯ ) : (φ, ρ) 7→ π(φ, ρ). INNER FORMS 31 (i) if φ lifts to a Langlands parameter φ̃ for G, then π(φ, ρ) is a direct summand −1 of ResG G♯ (recD,m (φ̃)), (ii) π(φ, ρ) is tempered if and only if φ is bounded, (iii) the L-packet Πφ (G♯ ) = {π(φ, ρ) : ρ ∈ Irr(π0 (ZSLmd (C) (φ))), ρ|Z(SLmd (C)) = χD } is canonically determined. As Irrs (G♯ ) is defined in terms of restriction from Irrs (G), it is a union of Lpackets for G♯ . With (i) it canonically determines a set Φe (G♯ )s of enhanced Langlands parameters for G♯ . ♯ ♯ In the same Qway as for G, the LLC for a Levi subgroup L = L ∩ G follows from that for L = i GLmi (D). It involves enhancements from the action of Y (Ľ♯ )sc = SLmd (C) ∩ GLmi d (C). i sL (L♯ ) Given sL = [L, ω]L , Irr is a union of L-packets for L♯ . Hence the corresponding set Φe (L♯ )sL of enhanced Langlands parameters is well-defined. Lemma 7.4. The LLC for G♯ and the maps from Lemma 6.1 and Theorem 6.3.b fit in the following commutative bijective diagram: / Φe (G♯ )s O Irrs (G♯ ) o O  sL (Irr (L♯ )//Ws♯ )κω  / (Φe (L♯ )sL //Ws♯ )κω O o O 
(Ts //X L (s)Xnr (L/L♯ ))κω //Ws♯ κ o ω O 
/ (Φ(L)sL //X L (s)Xnr (L/L♯ )) //Ws♯ κω κω O 
Ts //Stab(s)Xnr (L/L♯ ) κ o ω O 
/ Φ(L)sL //Stab(s)Xnr (L/L♯ ) κω O 
(Ts //Ws )//Stab(s)+ Xnr (L/L♯ ) κ o 
/ (Φ(L)sL //Ws )//Stab(s)+ Xnr (L/L♯ ) κ ω ω All these maps are canonical up to permutations within L-packets. In the last row the collection of L-packets is in bijection with (Ts //Ws )/Stab(s)+ Xnr (L/L♯ ) and with (Φ(L)sL //Ws )/Stab(s)+ Xnr (L/L♯ ). Proof. The bijection between the first and the fourth set on the left hand side is given by Theorem 6.3.b. Then Corollary A.4 and (78) give bijections to the third and fifth sets on the left, as the 2-cocycle κω is by construction (72) trivial on Ws . The bijection between the second and third sets on the left comes from Lemma 6.1.a. By Lemma 6.1.b it is canonical up to permutations within L-packets. The LLC for L is equivariant for permutations of the direct factors of L and for twisting with characters of L (because the LLC for GLm (D) is so). This gives the three lower horizontal bijections. Applying Corollary A.4 to the three lower terms 32 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD on the right hand side gives bijections between them, and shows that the two lower squares in the diagram are canonical and commutative. Similarly the LLC for L♯ is equivariant for the action of Ws♯ , which leads to the second horizontal bijection. We define the upper two maps on the right hand side as the unique bijections that make the diagram commute. Since all the other maps in the upper two squares are canonical up to permutations within L-packets, so are the last two. An L-packet for G♯ consists of the irreducible G♯ -constituents of an irreducible G-representation. In view of Lemma 7.2, the collection of L-packets in Irrs (G♯ ) is canonically in bijection with Ts //Ws . From (56) we can see how
(Ts //Ws )//Stab(s)+ Xnr (L/L♯ ) κ ω is constructed on the level of representations. We take an element π ∈ Irrs (G) and transform it to an irreducible representation of O(Ts ) ⋊ Ws by a geometric equivalence. Then we form the twisted extended quotient by Stab(s)+ , using Lemmas A.1 and A.2, which corresponds to identifying π with π ′ if they have the same restriction to G♯ Z(G), and decomposing π in irreducible G♯ Z(G)-subrepresentations. Finally we divide out the action of Xnr (L♯ Z(G)/L♯ ), thus identifying the G♯ Z(G)representations with the same restriction to G♯ . The implies the description of the L-packets in the lower left term of the commutative diagram, and hence also in the lower right term.  The bijection between the upper and the lower term on the right hand side of Lemma 7.4 can also be obtained as follows. First apply the recipe from Lemma 7.2 to Φ(L)sL //Ws , then take the twisted extended quotient with respect to Stab(s)+ , and finally divide out the free action of Xnr (L♯ Z(G)/L♯ ) to reach Φe (G♯ )sL . Let Rt♯ ,t be the root system associated to (G♯ , t) be Harish-Chandra, by means of zeros of the µ-function [Wal, §V.2]. Recall that the classical Springer correspondence was extended to Weyl groups of disconnected complex reductive groups in [ABPS5, Theorem 4.4]. Lemma 7.5. Let t♯ = [L♯ , σ ♯ ]G♯ be an inertial equivalence class subordinate to s = [L, ω]G . Lemma 6.5.a and the LLC for G♯ and for L♯ provide a commutative, bijective diagram ♯ Irrt (G♯ ) o O  (Tt♯ //Wt♯ )κσ♯ o / Φe (G♯ )t♯ O  / (Φe (L♯ )[L♯ ,σ♯ ]L♯ //Wt♯ )κ σ♯ Two elements [t, ρ], [t′ , ρ′ ] ∈ (Tt♯ //Wt♯ )κσ♯ are mapped to G♯ -representations in the same L-packet if and only if • wt′ = t for some w ∈ Wt♯ ; • the Wt♯ ,t -representations ρ and w · ρ′ have parameters (in the Springer correspondence for possibly disconnected complex reductive groups) with the same unipotent class, in the complex reductive group with maximal torus Tt♯ , root system Rt♯ ,t and Weyl group Wt♯ ,t . INNER FORMS 33 Proof. The commutative diagram is obtained from Lemma 7.4, taking (27) into account. To see whether [t, ρ] and [t′ , ρ′ ] belong to the same L-packet, Lemma 7.4 says that it suffices to look at their images in (Ts //Ws )/Stab(s)+ Xnr (L/L♯ ). Let t̃ ∈ Ts be a lift of t. Then Wt♯ ,t is the isotropy group of X L (s)Xnr (L/L♯ )(σ̃ ♯ ) ∈ Tt♯ in Ws♯ . Here σ ♯ is a projective representation of (X L (s)Xnr (L/L♯ ))t̃ = X L (ω). With Lemma A.1 we get σ ♯ ⋊ ρ ∈ Irr(C[(Stab(s)Xnr (L/L♯ )t̃ , κω ]). The intersection of (Stab(s)Xnr (L/L♯ )t̃ with Ws is Ws,t̃ = W (Rs,t̃ ). Since Ws commutes with X L (s)Xnr (L/L♯ ), the restriction of σ ♯ ⋊ ρ to Ws,t̃ is dim(σ ♯ ) times ρ|W . s,t̃ We want to show that (94) Rt♯ ,t = Rs,t̃ , although in general Wt♯ ,t is strictly larger than Ws,t̃ . Both root systems can be defined in terms of zeros of Harish-Chandra µ-functions associated to roots α ∈ Rs . The function µα (for G) is defined via intertwining operators betweeen Grepresentations, see [Wal, §IV.3 and §V.2]. These remain well-defined as intertwining operators between G♯ -representations, which implies that µα factors through Ts → Tt♯ and in this way gives the function µα for G♯ . By [Sil2, Theorem 1.6] all zeros of µα are fixed points of the reflection sα ∈ Ws . Hence µα (t) 6= 0 if sα (t̃) 6= t̃, proving (94). It follows that [t, ρ] maps to [t̃, ρ|W (R ♯ ) in (Ts //Ws )/Stab(s)+ Xnr (L/L♯ ), and t ,t similarly for [t′ , ρ′ ]. The Stab(s)+ Xnr (L/L♯ )-orbits of [t̃, ρ|W (R ♯ ) and [t̃′ , ρ|W (R ♯ ′ ) t ,t t ,t are equal if and only if there is a w ∈ Wt♯ such that wt′ = t and (wρ′ )|W (R ♯ t ,t ) = ρ|W (R ♯ ) . t ,t wρ′ By Lemma 7.2.b the last condition is equivalent to and ρ having the same unipotent class as Springer parameter. Because w is only determined up to Wt♯ ,t , these unipotent classes must be considered in the complex reductive group with maximal torus Tt♯ , root system Rt♯ ,t and Weyl group Wt♯ ,t .  As before, let L be a set of representatives for the conjugacy classes of Levi subgroups of G. Then {L♯ : L ∈ L} is a set of representatives for the conjugacy classes of Levi subgroups of G♯ . Theorem 7.6. The maps from Lemma 7.4 combine to a commutative diagram of bijections / Φe (G♯ ) O Irr(G♯ ) o O F F L∈L L∈L 
Irrcusp (L♯ )//W (G♯ , L♯ ) ♮ o O 
Irrcusp (L)//Irr(L/L♯ )W (G, L) ♮ o / / F F L∈L L∈L 
Φ(L♯ )cusp //W (G♯ , L♯ ) ♮ O 
Φ(L)cusp //Irr(L/L♯ )W (G, L) ♮ 34 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Here the family of 2-cocycles ♮ restricts to κω on Irr[L,ω]L (L). The tempered representations correspond to the bounded enhanced Langlands parameters and the entire diagram is canonical up to permutations within L-packets. Proof. The upper square follows quickly from Lemma 7.4, in the same way as Theorem 7.3 followed from Lemma 7.2. Recall from Lemma 6.1 that IrrsL (L♯ ) is in bijection with (Ts //X L (s)Xnr (L/L♯ ))κω . Here X L (s) is the stabilizer of sL = [L, ω]L in Irr(L/L♯ Z(G)). A character of L/L♯ which is ramified on Z(G) cannot stabilize sL , so X L (s)Xnr (L/L♯ ) is the stabilizer of sL in Irr(L/L♯ ). By Theorem 7.1 the LLC for L is bijective and Irr(L/L♯ )equivariant, so X L (s)Xnr (L/L♯ ) is also the stabilizer of Φ(L)sL in Irr(L/L♯ ). This implies G (Irrcusp (L)//Irr(L/L♯ ))♮ ∼ (IrrsL (L)//X L (s)Xnr (L/L♯ ))κω = sL =[L,ω]L G ∼ IrrsL (L♯ ) = Irrcusp (L♯ ), = sL =[L,ω]L and similarly for Langlands parameters. These bijections are equivariant for permutations of the direct factors of L, so applying (−//W (G, L))κω to all of them produces a commutative square as in the theorem, but with lower row G
(Irrcusp (L)//Irr(L/L♯ ))♮ //W (G, L) ♮ ←→ L∈L G
(Φ(L)cusp //Irr(L/L♯ ))♮ //W (G, L) ♮ . L∈L We apply Corollary A.4 to get the row in the theorem. The canonicity of the thus obtained commutative diagram is a consequence of the analogous property in Lemma 7.4. The temperedness/boundedness correspondence follows from the properties of the local Langlands correspondences for G, G♯ , L and L♯ .  Example 7.7. Let G = SL5 (D). Let V4 denote the non-cyclic group of order 4. Let WF denote the Weil group of F . There exists a classical Langlands parameter φ which factors through V4 : (95) φ : WF → V4 → PGL2 (C) Let τ be the cuspidal representation of D × which has, as its Langlands parameter, a lift of φ to GL2 (C). Consider the group of characters χ for which χτ ∼ = τ . This group is isomorphic to V4 and comprises the four characters {1, γ, η, γη}, where γ, η are quadratic characters. Let L = (D × )5 ∩ SL5 (D) σ = τ ⊗ 1 ⊗ γ ⊗ η ⊗ γη ∈ Irr(L) s = [L, σ]G Twisting by η corresponds to the permutation (13)(24), twisting by γη corresponds to the permutation (14)(23). The Bernstein finite group W s is isomorphic to V4 . The corresponding Bernstein variety is the quotient T s /V4 , where T s has the structure of a complex torus of dimension 4. INNER FORMS 35 The summand Hs (G) of the Hecke algebra H(G) is Morita equivalent to the twisted crossed product O(T s ) ⋊♮ V4 , where ♮ is the 2-cocycle associated to the above projective representation of V4 . Following [ABPS3], Irr(O(T s ) ⋊♮ V4 ) is the twisted extended quotient (T s //V4 )♮ . Now consider the standard projection π s : (T s //V4 )♮ → T s /V4 . Let (V4 )t denote the isotropy group of t ∈ T s . Let X = {t ∈ T s : |(V4 )t | = 2}, Y = {t ∈ T s : (V4 )t = V4 }. We note that • on the complement of (X ∪ Y )/V4 the fibre of π s has cardinality 1: the corresponding (parabolically) induced representation is irreducible • on X/V4 the fibre of π s has cardinality 2: the corresponding induced representation has two inequivalent irreducible constituents • on Y /V4 the fibre of π s has cardinality 1, because V4 admits a unique irrreducible projective representation with cocycle ♮: the corresponding induced representation has two equivalent constituents. The geometric structure of Irrs (SL5 (D)) is the variety T s /V4 with doubling on X/V4 . Appendix A. Twisted extended quotients Let Γ be a group acting on a topological space X. In [ABPS6, §2] we studied various extended quotients of X by Γ. In this paper we need the most general version, the twisted extended quotients. Let ♮ be a given function which assigns to each x ∈ X a 2-cocycle ♮x : Γx × Γx → C× , where Γx = {γ ∈ Γ : γx = x}. It is assumed that ♮γx and γ∗ ♮x define the same class in H 2 (Γx , C× ), where γ∗ : Γx → Γγx sends α to γαγ −1 . Define e♮ := {(x, ρ) : x ∈ X, ρ ∈ Irr C[Γx , ♮x ]}. X We require, for every (γ, x) ∈ Γ × X, a definite algebra isomorphism φγ,x : C[Γx , ♮x ] → C[Γγx , ♮γx ] such that: • φγ,x is inner if γx = x; • φγ ′ ,γx ◦ φγ,x = φγ ′ γ,x for all γ ′ , γ ∈ Γ, x ∈ X. We call these maps connecting homomorphisms, because they are reminiscent of a e♮ by connection on a vector bundle. Then we can define a Γ-action on X γ · (x, ρ) = (γx, ρ ◦ φ−1 γ,x ). We form the twisted extended quotient e♮ /Γ. (X//Γ)♮ := X We note that this reduces to the extended quotient of the second kind X//Γ from [ABPS6, §2] if ♮x is trivial for all x ∈ X and φγ,x is conjugation by γ. 36 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD The map e♮ → X, X induces a map (x, ρ) 7→ x π1 : (X//Γ)♮ → X/Γ which we will call the standard projection. Such twisted extended quotients typically arise in the following situation. Let A be a C-algebra such that all irreducible A-modules have countable dimension over C. Let Γ be a group acting on A by automorphisms and form the crossed product A ⋊ Γ. Let X = Irr(A). Now Γ acts on Irr(A) and we get ♮ as follows. Given x ∈ Irr(A) choose an irreducible representation (πx , Vx ) whose isomorphism class is x. For each γ ∈ Γ consider πx twisted by γ: γ · πx : a 7→ πx (γ −1 aγ). Then γ · x is defined as the isomorphism class of γ · πx . Since γ · πx is equivalent to πγx , there exists a nonzero intertwining operator (96) Tγ,x ∈ HomA (γ · πx , πγx ). By Schur’s lemma (which is applicable because dim Vx is countable) Tγ,x is unique up to scalars, but in general there is no preferred choice. For γ, γ ′ ∈ Γx there exists a unique c ∈ C× such that Tγ,x ◦ Tγ ′ ,x = cTγγ ′ ,x . We define the 2-cocycle by ♮x (γ, γ ′ ) = c. Let Nγ,x with γ ∈ Γx be the standard basis of C[Γx , ♮x ]. The algebra homomorphism φγ,x is essentially conjugation by Tγ,x , but we must be careful if some of the Tγ coincide. The precise definition is (97) φγ,x (Nγ ′ ,x ) = λNγγ ′ γ −1 ,γx if −1 Tγ,x Tγ ′ ,x Tγ,x = λTγγ ′ γ −1 ,γx , λ ∈ C× . Notice that (97) does not depend on the choice of Tγ,x . Suppose that Γx is finite and (τ, Vτ ) ∈ Irr(C[Γx , ♮x ]). Then Vx ⊗ Vτ∗ is an irreducible A ⋊ Γx -module, in a way which depends on the choice of intertwining operators Tγ,x . Lemma A.1. [ABPS6, Lemma 2.3] Let A and Γ be as above and assume that the action of Γ on Irr(A) has finite isotropy groups. (a) There is a bijection (Irr(A)//Γ)♮ ←→ Irr(A ⋊ Γ) ∗ (πx , τ ) 7→ πx ⋊ τ := IndA⋊Γ A⋊Γx (Vx ⊗ Vτ ). (b) If all irreducible A-modules are one-dimensional, then part (a) becomes a natural bijection Irr(A)//Γ ←→ Irr(A ⋊ Γ). Via the following result twisted extended quotients also arise from algebras of invariants. INNER FORMS 37 Lemma A.2. Let Γ be a finite group acting on a C-algebra A. There is a bijection {V ∈ Irr(A ⋊ Γ) : V Γ 6= 0} ←→ Irr(AΓ ) V 7→ V Γ. If all elements of Irr(A) have countable dimension, it becomes Irr(AΓ ) {(πx , τ ) ∈ (Irr(A)//Γ)♮ : HomΓx (Vτ , Vx ) 6= 0} ←→ (πx , τ ) 7→ HomΓx (Vτ , Vx ). Proof. Consider the idempotent pΓ = |Γ|−1 (98) X γ∈Γ γ ∈ C[Γ]. It is well-known and easily shown that AΓ ∼ = pΓ (A ⋊ Γ)pΓ and that the right hand side is Morita equivalent with the two-sided ideal I = (A ⋊ Γ)pΓ (A ⋊ Γ) ⊂ A ⋊ Γ. The Morita equivalence sends a module V over the latter algebra to pΓ (A ⋊ Γ) ⊗(A⋊Γ)pΓ (A⋊Γ) V = V Γ . As I is a two-sided ideal, Irr(I) = {V ∈ Irr(A ⋊ Γ) : I · V 6= 0} = {V ∈ Irr(A ⋊ Γ) : pΓ V = V Γ 6= 0} This gives the first bijection. From Lemma A.1.a we know that every such V is of the form πx ⋊ τ . With Frobenius reciprocity we calculate
∗ Γ ∼ (πx ⋊ τ )Γ = IndA⋊Γ = (Vx ⊗ Vτ∗ )Γx = HomΓx (Vτ , Vx ). A⋊Γx (Vx ⊗ Vτ ) Now Lemma A.1.a and the first bijection give the second.  Let A be a commutative C-algebra all whose irreducible representations are of countable dimension over C. Then Irr(A) consists of characters of A and is a T1 -space. Typical examples are A = C0 (X) (with X locally compact Hausdorff), A = C ∞ (X) (with X a smooth manifold) and A = O(X) (with X an algebraic variety). As a kind of converse to Lemmas A.1 and A.2, we show that every twisted extended quotient of Irr(A) appears as the space of irreducible representations of some algebras. With small modifications, the argument also works for smooth manifolds and algebraic varieties. Let Γ be a group acting on A by algebra automorphisms, such that Γx is finite for every x ∈ Irr(A). Recall that every 2-cocycle ♮ of Γ arises from a projective Γ-representation (µ, Vµ ) by µ(γ)µ(γ ′ ) = ♮(γ, γ ′ )µ(γ, γ ′ ). Let Γ act on A ⊗ EndC (Vµ ) by γ · (a ⊗ h) = γ(a) ⊗ µ(γ)hµ(γ)−1 . Lemma A.3. There are bijections Irr((A ⊗ EndC (Vµ )) ⋊ Γ) ←→ (Irr(A)//Γ)♮ , Irr((A ⊗ EndC (Vµ ))Γ ) ←→ {[x, ρ] ∈ (X//Γ)♮ : ρ appears in Vµ }. 38 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Proof. We can identify Irr(A ⊗ EndC (Vµ )) with {Cx ⊗ Vµ : x ∈ Irr(A)}. It follows directly from (96) that we can take Tγ,x = µ(γ) for all γ ∈ Γ and x ∈ Irr(A). Thus the first bijection is an instance of Lemma A.1.a. Let x ∈ Irr(A) and (τ, Vτ ) ∈ Irr(C[Γx , ♮]). Then HomΓx (τ, Cx ⊗ Vµ ) = HomΓx (τ, Vµ ), and this is nonzero if and only if τ appears in Vµ . Now an application of Lemma A.2 proves the second bijection.  Corollary A.4. In the above setting, suppose that Γ = Γ1 ⋊ Γ2 is a semidirect product. Then there is a canonical bijection (Irr(A)//Γ)♮ ←→ ((Irr(A)//Γ1 )♮ //Γ2 )♮ . Proof. The bijection is obtained from Lemma A.3 and (A ⊗ EndC (Vµ )) ⋊ Γ = ((A ⊗ EndC (Vµ )) ⋊ Γ1 ) ⋊ Γ2 It is canonical because the same 2-cocycle is used on both sides.  References [ABPS1] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “On the local Langlands correspondence for non-tempered representations”, Münster Journal of Mathematics 7 (2014), 27–50. [ABPS2] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “Geometric structure in smooth dual and local Langlands conjecture”, Japanese Journal of Mathematics 9 (2014), 99–136. [ABPS3] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “The local Langlands correspondence for inner forms of SLn ”, Res. Math. Sci. (2016) 33.2. [ABPS4] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Hecke algebras for inner forms of p-adic special linear groups, J. Inst. Math. Jussieu 16 (2017) 351–419. [ABPS5] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. 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CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu – Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France E-mail address: anne-marie.aubert@imj-prg.fr Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA E-mail address: pxb6@psu.edu School of Mathematics, Manchester University, Manchester M13 9PL, England E-mail address: roger.j.plymen@manchester.ac.uk IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, the Netherlands E-mail address: m.solleveld@science.ru.nl