aa r X i v : . [ m a t h . R T ] J un ON AN ALGORITHM TO COMPUTE DERIVATIVES
HIRAKU ATOBE
Abstract.
In this paper, we complete Jantzen’s algorithm to compute the highest deriva-tives of irreducible representations of p -adic odd special orthogonal groups or symplecticgroups. As an application, we give some examples of the Langlands data of the Aubert dualsof irreducible representations, which are in the integral reducibility case. Contents
1. Introduction 12. Representations of classical groups 33. Arthur packets 144. Derivatives of certain representations 245. Proof of Theorem 4.1 29References 341.
Introduction
The notion of derivatives of admissible representations is an influential ingredient in rep-resentation theory of p -adic classical groups. Let G be a split odd special orthogonal groupSO n +1 ( F ), or a symplectic group Sp n ( F ). Denote by P d = M d N d the standard parabolicsubgroup of G with Levi M d = GL d ( F ) × G for some classical group of the same type as G .Fix an irreducible unitary supercuspidal representation ρ of GL d ( F ). Definition 1.1.
Let π be a smooth admissible representation of G of finite length. (1) If the semisimplification of the Jacquet module
Jac P d ( π ) along P d is of the form s . s . Jac P d ( π ) = M i τ i ⊠ π i , we define the partial Jacquet module Jac ρ |·| x ( π ) with respect to ρ | · | x for x ∈ R by Jac ρ |·| x ( π ) = M iτ i ∼ = ρ |·| x π i . Mathematics Subject Classification.
Primary 22E50; Secondary 11S37.
Key words and phrases.
Jacquet module; Derivatives; Arthur packets. (2)
For a positive integer k , the k -th derivative D ( k ) ρ |·| x ( π ) with respect to ρ | · | x is definedby D ( k ) ρ |·| x ( π ) = 1 k ! Jac ρ |·| x ◦ · · · ◦ Jac ρ |·| x | {z } k ( π ) . When D ( k ) ρ |·| x ( π ) = 0 but D ( k +1) ρ |·| x ( π ) = 0 , we say that D ( k ) ρ |·| x ( π ) is the highest derivative (with respect to ρ | · | x ). It is important that when ρ | · | x is not self-dual, the highest derivative D ( k ) ρ |·| x ( π ) of anirreducible representation is also irreducible, and π is a unique irreducible subrepresentationof the parabolically induced representation ( ρ | · | x ) k ⋊ D ( k ) ρ |·| x ( π ) (see Proposition 2.7). By theseproperties, the highest derivatives have many applications. For example: • the proofs of the How duality conjecture by M´ınguez [14] and Gan–Takeda [4]; • another proof of the classification the unitary dual of general linear groups by Lapid–M´ınguez [12]; • several results on the irreducibility of parabolically induced representations by Jantzen[7] and Lapid–Tadi´c [13].Jantzen [5] and M´ınguez [15] obtained a complete description of the highest derivativesof irreducible representations of general linear groups GL n ( F ) independently. It gives an al-gorithm to compute the Zelevinsky involutions. Similarly, if one were able to compute thehighest derivatives of all irreducible representations, one can compute the Aubert dual of anyirreducible representation (see Theorem 2.13 below). Jantzen [8] suggested an algorithm tocompute the highest derivatives of irreducible representations. We will recall this algorithmin § L (( ρ |·| − x ) a , ∆ ρ [ x − , − x ] b ; T ) which is a unique irreducible (Lang-lands) subrepresentation of the standard module ( ρ | · | − x ) a × ∆ ρ [ x − , − x ] b ⋊ T , where ρ isan irreducible unitary supercuspidal representation of GL d ( F ), x is a positive half-integer,∆ ρ [ x − , − x ] is a Steinberg representation of GL dx ( F ), and T is an irreducible temperedrepresentation of a small classical group. For these notations, see § L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; T ),Jantzen gave an explicit formula for x = 1 / x > § x , i.e., the computationfor x is reduced to the one for x −
1. Hence it would be possible to solve this problem when x ∈ (1 / Z \ Z . Using this strategy, he computed some examples of the Aubert duals of certainirreducible representations in the half-integral reducibility case ([8, § L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; T ). One can also write down an explicit formula (Corollary 4.2).This corollary together with Corollary 4.4 completes Jantzen’s algorithm. When a = 0, onemight prove Corollary 4.2 by a similar argument to [8, Theorem 3.3], but we will give another N AN ALGORITHM TO COMPUTE DERIVATIVES 3 argument. A new ingredient for the proof is two results of Xu on A -packets [23, 24]. Thefirst is an estimation when derivatives of unitary representations of Arthur type are nonzero(Lemma 3.4). Namely, for a specific tuple ( a, b, T ), we will find a “good” A -parameter ψ suchthat L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; T ) belongs to the A -packet Π ψ (e.g., see Example 3.5 andProposition 3.13). To compute the highest derivative, we will use Mœglin’s construction of A -packets together with Xu’s combinational result [24, Theorem 6.1].This paper is organized as follows. In §
2, we review several results in representation theoryfor classical groups. In particular, we explain how the highest derivatives of an irreduciblerepresentation π determine its Langlands data almost completely (Theorem 2.13). Also werecall Jantzen’s algorithm to compute the highest derivatives in § §
3, we review Arthur’stheory including Xu’s lemma (Lemma 3.4) and Mœglin’s construction ( § § §
4, we state the main results (Theorem 4.1 and Corollaries 4.2, 4.4). Also in § § Acknowledgement.
The author is grateful to Professor Alberto M´ınguez for telling thenotion of derivatives and several results. He wishes to thank Professor Chris Jantzen to allowhim to work on this topic.
Notation.
Let F be a non-archimedean local field of characteristic zero. We denote by W F the Weil group of F . The norm map | · | : W F → R × is normalized so that | Frob | = q − , whereFrob ∈ W F is a fixed (geometric) Frobenius element, and q is the cardinality of the residualfield of F .Each irreducible unitary supercuspidal representation ρ of GL d ( F ) is identified with theirreducible bounded representation of W F of dimension d via the local Langlands correspon-dence for GL d ( F ). Through this paper, we fix such a ρ . For each positive integer a , theunique irreducible algebraic representation of SL ( C ) of dimension a is denoted by S a .For a p -adic group G , we denote by Rep( G ) (resp. Irr( G )) the set of equivalence classes ofsmooth admissible (resp. irreducible) representations of G . For Π ∈ Rep( G ), we write s . s . (Π)for the semisimplification of Π.2. Representations of classical groups
In this section, we recall some results on parabolically induced representations and Jacquetmodules.2.1.
Representations of GL n ( F ) . Let P = M N be a standard parabolic subgroup ofGL n ( F ), i.e., P contains the Borel subgroup consisting of upper half triangular matrices.Then the Levi subgroup M is isomorphic to GL n ( F ) × · · · × GL n r ( F ) with n + · · · + n r = n .For smooth representations τ , . . . , τ r of GL n ( F ) , . . . , GL n r ( F ), respectively, we denote thenormalized parabolically induced representation by τ × · · · × τ r = Ind GL n ( F ) P ( τ ⊠ · · · ⊠ τ r ) . When τ = · · · = τ r = τ , we write τ r = τ × · · · × τ | {z } r . HIRAKU ATOBE A segment is a symbol [ x, y ], where x, y ∈ R with x − y ∈ Z and x ≥ y . We identify[ x, y ] with the set { x, x − , . . . , y } so that x, y ] = x − y + 1. Let ρ be an irreducible(unitary) supercuspidal representation of GL d ( F ). Then the normalized parabolically inducedrepresentation ρ | · | x × · · · × ρ | · | y of GL d ( x − y +1) ( F ) has a unique irreducible subrepresentation, which is denoted by∆ ρ [ x, y ]and is called a Steinberg representation . This is an essentially discrete series representationof GL d ( x − y +1) ( F ).When τ i = ∆ ρ i [ x i , y i ] with x + y ≤ · · · ≤ x r + y r , the parabolically induced representation τ × · · · × τ r is called a standard module . The Langlands classification says that it has aunique irreducible subrepresentation, which is denoted by L ( τ , . . . , τ r ). This notation is usedalso for τ × · · · × τ r which is isomorphic to a standard module.When segments [ x , y ] , . . . , [ x t , y t ] satisfies that x i ≥ y i , x < · · · < x t , y < · · · < y t and x ≡ · · · ≡ x t mod Z , we call L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ]) a ladder representation .This notion was introduced by Lapid–M´ınguez [11]. In particular, when x i = x + i − y i = y + i − ≤ i ≤ t , the ladder representation L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ]) is alsocalled a Speh representation , which is a unitary representation.An irreducibility criterion for parabolically induced representations of Steinberg represen-tations is given by Zelevinsky.
Theorem 2.1 (Zelevinsky [25, Theorem 9.7, Proposition 4.6]) . Let [ x, y ] and [ x ′ , y ′ ] be seg-ments, and let ρ and ρ ′ be irreducible unitary supercuspidal representations of GL d ( F ) and GL d ′ ( F ) , respectively. Then the parabolically induced representation ∆ ρ [ x, y ] × ∆ ρ ′ [ x ′ , y ′ ] is irreducible unless the following conditions hold: • ρ ∼ = ρ ′ ; • [ x, y ] [ x ′ , y ′ ] and [ x ′ , y ′ ] [ x, y ] as sets; • [ x, y ] ∪ [ x ′ , y ′ ] is also a segment.In this case, if x + y < x ′ + y ′ , then there exists an exact sequence −−−−→ L (∆ ρ [ x, y ] , ∆ ρ ′ [ x ′ , y ′ ]) −−−−→ ∆ ρ [ x, y ] × ∆ ρ ′ [ x ′ , y ′ ] −−−−→ ∆ ρ [ x ′ , y ] × ∆ ρ [ x, y ′ ] −−−−→ . Here, when x = y ′ − , we omit ∆ ρ [ x, y ′ ] . For an irreducibility criterion for parabolically induced representations of Speh (resp. lad-der) representations, see [20] (resp. [12]).For a partition ( n , . . . , n r ) of n , we denote by Jac ( n ,...,n r ) the normalized Jacquet functoron Rep(GL n ( F )) with respect to the standard parabolic subgroup P = M N with M ∼ =GL n ( F ) × · · · × GL n r ( F ). The Jacquet modules of ladder representations with respect tomaximal parabolic subgroups are computed by Kret–Lapid [10, Theorem 2.1] in general. Thefollowing is a special case of this computation. Proposition 2.2.
Let L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ]) be a ladder representation of GL n ( F ) sothat x i ≥ y i , x < · · · < x t , y < · · · < y t and x ≡ · · · ≡ x t mod Z . Then unless n ≡ d , we have Jac ( n ,n − n ) ( L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ])) = 0 . Moreover: N AN ALGORITHM TO COMPUTE DERIVATIVES 5 (1)
The semisimplification of
Jac ( d,n − d ) ( L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ])) is equal to X ≤ i ≤ tx i − Jac ( n − d,d ) ( L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x t , y t ])) is equal to X ≤ i ≤ ty i +1 Definition 2.3. Let π be an irreducible representation of GL n ( F ) . (1) Suppose that s . s . Jac ( d,n − d ) ( π ) = M i ∈ I τ i ⊠ π i , s . s . Jac ( n − d,d ) ( π ) = M j ∈ J π ′ j ⊠ τ ′ j with τ i , τ ′ j and π i , π ′ j being irreducible representations of GL d ( F ) and GL n − d ( F ) , re-spectively. Then for x ∈ R , we define the left derivative L ρ |·| x ( π ) and the rightderivative R ρ |·| x ( π ) with respect to ρ | · | x by L ρ |·| x ( π ) = M i ∈ Iτ i ∼ = ρ |·| x π i , R ρ |·| x ( π ) = M j ∈ Jτ ′ j ∼ = ρ |·| x π ′ j . (2) For a positive integer k , we define the k -th left and right derivatives L ( k ) ρ |·| x ( π ) and R ( k ) ρ |·| x ( π ) with respect to ρ | · | x by L ( k ) ρ |·| x ( π ) = 1 k ! L ρ |·| x ◦ · · · ◦ L ρ |·| x | {z } k ( π ) , R ( k ) ρ |·| x ( π ) = 1 k ! R ρ |·| x ◦ · · · ◦ R ρ |·| x | {z } k ( π ) . HIRAKU ATOBE We also set L (0) ρ |·| x ( π ) = R (0) ρ |·| x ( π ) = π . (3) When L ( k ) ρ |·| x ( π ) = 0 but L ( k +1) ρ |·| x ( π ) = 0 , we call L ( k ) ρ |·| x ( π ) the highest left derivative .We also define the highest right derivative similarly. By [5, Lemma 2.1.2], for any irreducible representation π , its highest derivatives L ( k ) ρ |·| x ( π )and R ( k ′ ) ρ |·| x ( π ) are irreducible. The following result was obtained by Jantzen [5] and M´ınguez[15, Th´eor`eme 7.5] independently. We adopt the statements of [5, Propositions 2.1.4, 2.4.3,Theorems 2.2.1, 2.4.5]. For another reformulation, see [12, Theorem 5.11]. Theorem 2.4 (Jantzen [5], M´ınguez [15]) . Let π = L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x r , y r ]) be an irre-ducible representation. (1) We may assume that y ≤ · · · ≤ y r , and that if y j = y j +1 , then x j ≥ x j +1 . For ≤ j ≤ r , define n x ( j ) = { i ≤ j | x i = x } , and set n x (0) = 0 . Then with k = max j ≥ { n x ( j ) − n x − ( j ) } , the left derivative L ( k ) ρ |·| x ( π ) is highest. For ≤ m ≤ k ,if we set j m = min { j | n x ( j ) − n x − ( j ) = m } , then L ( k ) ρ |·| ( π ) is given from π = L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x r , y r ]) by replacing x j = x with x − for all j ∈ { j , . . . , j k } . (2) We may assume that x ≤ · · · ≤ x r , and that if x j = x j +1 , then y j ≥ y j +1 . For ≤ j ≤ r , define n ′ y ( j ) = { i ≥ r − j + 1 | y i = y } , and set n ′ y (0) = 0 . Thenwith k = max j ≥ { n ′ y ( j ) − n ′ y +1 ( j ) } , the right derivative R ( k ) ρ |·| y ( π ) is highest. For ≤ m ≤ k , if we set j m = min { j | n ′ y ( j ) − n ′ y +1 ( j ) = m } , then R ( k ) ρ |·| y ( π ) is given from π = L (∆ ρ [ x , y ] , . . . , ∆ ρ [ x r , y r ]) by replacing y j = y with y + 1 for all j ∈ { j , . . . , j k } . Representations of SO n +1 ( F ) and Sp n ( F ) . We set G n to be split SO n +1 ( F ) orSp n ( F ), i.e., G n is the group of F -points of the split algebraic group of type B n or C n . Fixa Borel subgroup of G n , and let P = M N be a standard parabolic subgroup of G n . Then theLevi part M is of the form GL k ( F ) × · · · × GL k r ( F ) × G n such that k + · · · + k r + n = n .For a smooth representation τ ⊠ · · · ⊠ τ r ⊠ π of M , we denote the normalized parabolicallyinduced representation by τ × · · · × τ r ⋊ π = Ind G n P ( τ ⊠ · · · ⊠ τ r ⊠ π ) . The functor Ind G n P : Rep( M ) → Rep( G n ) is exact.On the other hand, for a smooth representation π of G n , we denote the normalized Jacquetmodule with respect to P by Jac P ( π ) , and its semisimplification by s . s . Jac P ( π ). The functor Jac P : Rep( G n ) → Rep( M ) is exact.The Frobenius reciprocity asserts thatHom G n ( π, Ind G n P ( σ )) ∼ = Hom M (Jac P ( π ) , σ )for π ∈ Rep( G n ) and σ ∈ Rep( M ).The maximal standard parabolic subgroup with Levi GL k ( F ) × G n − k is denoted by P k = M k N k for 0 ≤ k ≤ n . Let R ( G n ) be the Grothendieck group of the category of smoothrepresentations of G n of finite length. Set R ( G ) = ⊕ n ≥ R ( G n ). The parabolic inductiondefines a module structure ⋊ : R ⊗ R ( G ) → R ( G ) , τ ⊗ π s . s . ( τ ⋊ π ) , N AN ALGORITHM TO COMPUTE DERIVATIVES 7 and the Jacquet functor defines a comodule structure µ ∗ : R ( G ) → R ⊗ R ( G )by Irr( G n ) ∋ π n X k =0 s . s . Jac P k ( π ) . Tadi´c established a formula to compute µ ∗ for parabolically induced representations. Thecontragredient functor τ τ ∨ defines an automorphism ∨ : R → R in a natural way. Let s : R ⊗ R → R ⊗ R be the homomorphism defined by P i τ i ⊗ τ ′ i P i τ ′ i ⊗ τ i . Theorem 2.5 (Tadi´c [21]) . Consider the composition M ∗ = ( m ⊗ id) ◦ ( ∨ ⊗ m ∗ ) ◦ s ◦ m ∗ : R → R ⊗ R . Then for the maximal parabolic subgroup P k = M k N k of G n and for an admissible represen-tation τ ⊠ π of M k , we have µ ∗ ( τ ⋊ π ) = M ∗ ( τ ) ⋊ µ ∗ ( π ) . For any irreducible representation π of G n , there exists an irreducible representation τ ⊠ · · · ⊠ τ r ⊠ σ of a Levi subgroup M = GL k ( F ) ×· · ·× GL k r ( F ) × G n of some standard parabolicsubgroup P satisfying that • τ i = ∆ ρ i [ x i , y i ] for some irreducible unitary supercuspidal representation ρ i of GL d i ( F ),and some segment [ x i , y i ] with x i ≥ y i ; • σ is an irreducible tempered representation of G n ; • x + y ≤ · · · ≤ x r + y r < π is a unique irreducible subrepresentation of the parabolically induced representa-tion τ × · · · × τ r ⋊ σ . In this case, we write π = L ( τ , . . . , τ r ; σ ) , and call it the Langlands subrepresentation of τ ×· · ·× τ r ⋊ σ . Note that L ( τ , . . . , τ r ; σ ) ∼ = L ( τ ′ , . . . , τ ′ r ′ ; σ ′ ) if and only if τ × · · · × τ r ⋊ σ ∼ = τ ′ × · · · × τ ′ r ′ ⋊ σ ′ . We refer ( τ , . . . , τ r ; σ )as the Langlands data of π . For a detail, see [9].2.3. Derivatives.Definition 2.6. Let π be a smooth representation of G n . (1) Consider s . s . Jac P d ( π ) (and a fixed irreducible supercuspidal unitary representation ρ of GL d ( F ) ). If s . s . Jac P d ( π ) = M i ∈ I τ i ⊠ π i with τ i (resp. π i ) being an irreducible representation of GL d ( F ) (resp. G n − d ), for x ∈ R , we define a partial Jacquet module Jac ρ |·| x ( π ) by Jac ρ |·| x ( π ) = M i ∈ Iτ i ∼ = ρ |·| x π i . This is a representation of G n − d . For pairs ( ρ , x ) , . . . , ( ρ t , x t ) , we also set Jac ρ |·| x ,...,ρ t |·| xt =Jac ρ t |·| xt ◦ · · · ◦ Jac ρ |·| x . HIRAKU ATOBE (2) For a non-negative integer k , the k -th derivative of π with respect to ρ | · | x is the k -th composition D ( k ) ρ |·| x ( π ) = 1 k ! Jac ρ |·| x ◦ · · · ◦ Jac ρ |·| x | {z } k ( π ) . If D ( k ) ρ |·| x ( π ) = 0 but D ( k +1) ρ |·| x ( π ) = 0 , we call D ( k ) ρ |·| x ( π ) the highest derivative of π with respect to ρ | · | x . The derivative D ( k ) ρ |·| x ( π ) is a representation of some group G n ′ of the same type as G n . Infact, it is characterized so thats . s . Jac P dk ( π ) = ( ρ | · | x ) k ⊗ D ( k ) ρ |·| x ( π ) + X i ∈ I τ i ⊗ π i where τ i is an irreducible representation of GL dk ( F ) such that τ i = ( ρ | · | x ) k .The following is essentially the same as [6, Lemma 3.1.3]. For the convenience of thereaders, we give a proof. Proposition 2.7. Let π be an irreducible representation of G n , and D ( k ) ρ |·| x ( π ) be the highestderivative. (1) There exists an irreducible representation π ′ of some group G n ′ of the same type as G n such that D ( k ) ρ |·| x ( π ) = m · π ′ with a positive integer m . Moreover, π is an irreduciblesubrepresentation of the parabolically induced representation ρ | · | x × · · · × ρ | · | x | {z } k ⋊ π ′ . (2) If ρ ∨ | · | − x = ρ | · | x (in particular, if x = 0 ), then D ( k ) ρ |·| x ( π ) is irreducible, i.e., m =1 . Moreover, in this case, π is a unique irreducible subrepresentation of the aboveparabolically induced representation.Proof. By the same argument as the proof of [22, Lemma 5.3], there exists an irreduciblerepresentation π ′ of some group G n ′ such that π ֒ → ρ | · | x × · · · × ρ | · | x | {z } k ⋊ π ′ = ( ρ | · | x ) k ⋊ π ′ . Note that Jac ρ |·| x ( π ′ ) = 0 since D ( k ) ρ |·| x ( π ) is the highest derivative. By Theorem 2.5, we seethat Jac ρ |·| x (cid:16) ( ρ | · | x ) k ⋊ π ′ (cid:17) = m k · ( ρ | · | x ) k − ⋊ π ′ , where m k = ( k if ρ ∨ | · | − x = ρ | · | x , k if ρ ∨ | · | − x ∼ = ρ | · | x . This implies that D ( k ) ρ |·| x (cid:16) ( ρ | · | x ) k ⋊ π ′ (cid:17) = c k · π ′ N AN ALGORITHM TO COMPUTE DERIVATIVES 9 with c k = ( ρ ∨ | · | − x = ρ | · | x , k if ρ ∨ | · | − x ∼ = ρ | · | x . Therefore D ( k ) ρ |·| x ( π ) = m · π ′ with a positive integer m ≤ c k . This shows (1) and the first asser-tion of (2). For the last assertion of (2), we note that if π is an irreducible subrepresentationof ( ρ | · | x ) k ⋊ π ′ , then D ( k ) ρ |·| x ( π ) = 0. However, when ρ ∨ | · | − x = ρ | · | x , we have D ( k ) ρ |·| x (cid:16) ( ρ | · | x ) k ⋊ π ′ − π (cid:17) = 0 . This means that ( ρ | · | x ) k ⋊ π ′ contains π as a unique irreducible subrepresentation (withmultiplicity one). (cid:3) ρ -data. In this subsection, we introduce the ρ -data of irreducible representations. Definition 2.8. Let π be an irreducible representation of G n . For ǫ ∈ {±} , the ρ -data of π is of the form M ǫρ ( π ) = [( x , k ) , . . . , ( x t , k t ); π ] , where x i is a real number and k i is a positive integer, defined inductively as follows. (1) If Jac ρ |·| x ( π ) = 0 for any x ∈ R , we set M ǫρ ( π ) = [ π ] (so that t = 0 and π = π ). (2) If Jac ρ |·| x ( π ) = 0 for some x ∈ R , we set x = ( max { x ∈ R | Jac ρ |·| x ( π ) = 0 } if ǫ = + , min { x ∈ R | Jac ρ |·| x ( π ) = 0 } if ǫ = − and k ≥ to be such that D ( k ) ρ |·| x ( π ) is the highest derivative of π with respect to ρ | · | x . By Proposition 2.7, we can write D ( k ) ρ |·| x ( π ) = m · π ′ for some irreduciblerepresentation π ′ . Then we define M ǫρ ( π ) = (cid:2) ( x , k ); M ǫρ ( π ′ ) (cid:3) . Let π be an irreducible representation of G n . Then one can define another irreduciblerepresentation ˆ π , which is called the Aubert dual of π (see [3]). It is known that • ˆˆ π = π ; • if π is supercuspidal, then ˆ π = π ; • D ( k ) ρ |·| x (ˆ π ) is the Aubert dual of D ( k ) ρ ∨ |·| − x ( π ).In particular, for ǫ ∈ {±} , if M ǫρ ( π ) = [( x , k ) , . . . , ( x t , k t ); π ] , then M − ǫρ ∨ (ˆ π ) = [( − x , k ) , . . . , ( − x t , k t ); ˆ π ] . We will use M − ρ ( π ) mainly. By taking the Aubert dual, several properties of M − ρ can betranslated into ones of M + ρ ∨ . The following is the most important property of M − ρ . Theorem 2.9. Let π be an irreducible representation of G n . Then the ρ -data M − ρ ( π ) can berewritten as M − ρ ( π ) = h ( x (1)1 , k (1)1 ) , . . . , ( x (1) t , k (1) t ) , . . . , ( x ( r )1 , k ( r )1 ) , . . . , ( x ( r ) t r , k ( r ) t r ); π i such that • x ( i ) j +1 = x ( i ) j − and k ( i ) j +1 ≤ k ( i ) j for ≤ i ≤ r and ≤ j ≤ t i − ; • x (1)1 < x (2)1 < · · · < x ( r )1 .Moreover, if we set τ ( i ) j = ∆ ρ [ x ( i )1 , x ( i ) j ] , then τ ( i ) j × τ ( i ) j ′ ∼ = τ ( i ) j ′ × τ ( i ) j for any ≤ j, j ′ ≤ t r , and π is an irreducible subrepresentation of t × j =1 (cid:16) τ (1) j (cid:17) k (1) j − k (1) j +1 ! × · · · × t r × j =1 (cid:16) τ ( r ) j (cid:17) k ( r ) j − k ( r ) j +1 ! ⋊ π , where we set k ( i ) t i +1 = 0 for ≤ i ≤ r .Proof. Write M − ρ ( π ) = [( x , k ) , . . . , ( x t , k t ); π ]. Note that x i +1 = x i by definition. Sup-pose that x i +1 < x i . Replacing π with the (unique) irreducible representation appearing in D ( k i − ) ρ |·| xi − ◦ · · · ◦ D ( k ) ρ |·| x ( π ), we may assume that i = 1. If we write D ( k ) ρ |·| x ◦ D ( k ) ρ |·| x ( π ) = m · π ′ ,then π ֒ → ρ | · | x × · · · × ρ | · | x | {z } k × ρ | · | x × · · · × ρ | · | x | {z } k ⋊ π ′ . If x < x but x = x − 1, then ρ | · | x × ρ | · | x ∼ = ρ | · | x × ρ | · | x so that Jac ρ |·| x ( π ) = 0. Thiscontradicts the definition of x . Hence x = x − 1. In this case, we have an exact sequence0 −−−−→ ∆ ρ [ x , x ] −−−−→ ρ | · | x × ρ | · | x −−−−→ L ( ρ | · | x , ρ | · | x ) −−−−→ . Note that Jac ρ |·| x ( L ( ρ |·| x , ρ |·| x )) = 0. Since ρ |·| x × L ( ρ |·| x , ρ |·| x ) ∼ = L ( ρ |·| x , ρ |·| x ) × ρ |·| x by [25, Theorem 4.2], we must have π ֒ → ( ρ | · | x ) k − k × ∆ ρ [ x , x ] k × ( ρ | · | x ) k − k ⋊ π ′ , where we set k = min { k , k } . If k > k so that k = k , by Theorem 2.1, we would have π ֒ → ( ρ | · | x ) k − k × ∆ ρ [ x , x ] k ⋊ π ′ , which implies that Jac ρ |·| x ( π ) = 0. This contradicts the definition of x . Hence we have k ≤ k .We conclude that if x > · · · > x a , then x j +1 = x j − k j +1 ≤ k j for 1 ≤ j ≤ a − π is a subrepresentation of∆ ρ [ x , x ] k − k × · · · × ∆ ρ [ x , x a − ] k a − − k a × ∆ ρ [ x , x a ] k a ⋊ π ′ for some irreducible representation π ′ . Now suppose that x a < x a +1 . • If x a < x a +1 < x and x a +1 = x b for any 1 ≤ b < a , by Theorem 2.1, we haveJac ρ |·| xa +1 ( π ) = 0. This contradicts the definition of x . • If x a +1 = x b for some 1 ≤ b < a , by Theorem 2.1, we have D ( k b +1) ρ |·| xb ◦ D ( k b − ) ρ |·| xb − ◦ · · · ◦ D ( k ) ρ |·| x ( π ) = 0 . This contradicts the definition of k b . N AN ALGORITHM TO COMPUTE DERIVATIVES 11 Hence we must have x a +1 > x . Therefore M − ρ ( π ) can be written as in the statement. Theabove argument also shows the other statements. (cid:3) To obtain some consequences, let us prepare a useful lemma. Lemma 2.10. Let π be an irreducible representation of G n . Write M − ρ ( π ) = [( x , k ) , . . . , ( x t , k t ); π ] .Suppose that there exists an inclusion π ֒ → ∆ ρ [ x, y ] ⋊ π ′ with some irreducible representation π ′ such that x + y < . Then there exists i ≤ t − ( x − y ) such that x i = x, x i +1 = x − , . . . , x i + x − y = y . Moreover, if i > , then x i − < x .Proof. Write M − ρ ( π ′ ) = (cid:2) ( x ′ , k ′ ) , . . . , ( x ′ t ′ , k ′ t ′ ); π ′ (cid:3) . We prove the lemma by induction on t ′ .Note that x ≤ x < − y . If t ′ = 0 or x ′ ≥ x , then i = 1 and the assertion is trivial. Now wewrite π ֒ → ( ρ | · | x ) k ⋊ π and π ′ ֒ → ( ρ | · | x ′ ) k ′ ⋊ π ′ for some irreducible representations π and π ′ . If x ′ < x and x ′ = y − 1, by Theorem 2.1, we have π ֒ → ( ρ | · | x ′ ) k ′ × ∆ ρ [ x, y ] ⋊ π ′ so that x = x ′ , k = k ′ and π ֒ → ∆ ρ [ x, y ] ⋊ π ′ . By applying the induction hypothesis, weobtain the assertion.Finally, we assume that x ′ = y − 1. Since we have an exact sequence0 −−−−→ ∆ ρ [ x, y − −−−−→ ∆ ρ [ x, y ] × ρ | · | y − −−−−→ ρ | · | y − × ∆ ρ [ x, y ] , by Theorem 2.1, we see that π can be embedded into( ρ | · | y − ) k ′ × ∆ ρ [ x, y ] ⋊ π ′ , or( ρ | · | y − ) k ′ − × ∆ ρ [ x, y − ⋊ π ′ . In the former case, we have x = y − k = k ′ and π ֒ → ∆ ρ [ x, y ] ⋊ π ′ . In the latter case,we have ( x , k ) = ( y − , k ′ − 1) or k ′ = 1 and x = y − 1, and π ֒ → ∆ ρ [ x, y − ⋊ π ′ . Inboth cases, the induction hypothesis gives the assertion. (cid:3) Corollary 2.11. Let π be an irreducible representation of G n . Write M − ρ ( π ) = h ( x (1)1 , k (1)1 ) , . . . , ( x (1) t , k (1) t ) , . . . , ( x ( r )1 , k ( r )1 ) , . . . , ( x ( r ) t r , k ( r ) t r ); π i as in Theorem 2.9. Suppose that ( x, y ) = ( x ( i )1 , x ( i ) t i ) satisfies that x + y < and x + y ≤ x ( j )1 + x ( j ) t j for any j < i . Then there exists an irreducible representation π ′ such that π ֒ → ∆ ρ [ x, y ] ⋊ π ′ . For any such π ′ , the ρ -data M − ρ ( π ′ ) is obtained from M − ρ ( π ) by replacing k ( i )1 , . . . , k ( i ) t i with k ( i )1 − , . . . , k ( i ) t i − , respectively.Proof. By the assumption, we notice that x < − y and x > x ( j )1 ≥ x ( j ) t j > y for any j < i . ByTheorems 2.9 and 2.1, one can find an irreducible representation π ′ such that π ֒ → ∆ ρ [ x, y ] ⋊ π ′ . We compute the ρ -data M − ρ ( π ′ ) = [( x ′ , k ′ ) , . . . , ( x ′ t , k ′ t ); π ′ ] by induction on P ri =1 t i as in theproof of Lemma 2.10. If x ′ ≥ x , then x (1)1 = x so that i = 1. In this case, the assertion is trivial. If x ′ < x and x ′ = y − 1, then i > x ′ , k ′ ) = ( x (1)1 , k (1)1 ). Moreover, by Theorem 2.1, we have D ( k ′ ) ρ |·| x ′ ( π ) ֒ → ∆ ρ [ x, y ] ⋊ D ( k ′ ) ρ |·| x ′ ( π ′ ) . By the induction hypothesis, we obtain the assertion.To complete the proof, it suffices to show that x ′ never equals to y − 1. Suppose that x ′ = y − 1. Then there exists an irreducible representation π ′′ such that π ′ ֒ → ρ | · | y − ⋊ π ′′ so that π ֒ → ∆ ρ [ x, y ] × ρ | · | y − ⋊ π ′′ . Since we have an exact sequence0 −−−−→ ∆ ρ [ x, y − −−−−→ ∆ ρ [ x, y ] × ρ | · | y − −−−−→ ρ | · | y − × ∆ ρ [ x, y ] , we have Jac ρ |·| y − ( π ) = 0 or π ֒ → ∆ ρ [ x, y − ⋊ π ′′ . Lemma 2.10 eliminates the latter case.In the former case, we must have x (1)1 ≤ y − < x . This implies that i > x (1)1 + x (1) t ≤ x (1)1 ≤ y − < y ≤ x + y, which contradicts our assumption. This completes the proof. (cid:3) Also we can reformulate Casselman’s tempered-ness criterion as follows. Corollary 2.12. Let π be an irreducible representation of G n . Then π is tempered if andonly if for any ρ , we can write M − ρ ( π ) = h ( x (1)1 , k (1)1 ) , . . . , ( x (1) t , k (1) t ) , . . . , ( x ( r )1 , k ( r )1 ) , . . . , ( x ( r ) t r , k ( r ) t r ); π i as in Theorem 2.9 such that x ( i )1 + x ( i ) t i ≥ for any ≤ i ≤ r .Proof. Suppose first that some ρ -data M − ρ ( π ) of the above form has an index i such that x ( i )1 + x ( i ) t i < 0. We take i so that x ( i )1 + x ( i ) t i achieves the minimum value. Then by Theorem2.9, we find an irreducible representation π ′ such that π ֒ → ∆ ρ [ x ( i )1 , x ( i ) t i ] ⋊ π ′ , or equivalently, Jac P dti ( π ) ։ ∆ ρ [ x ( i )1 , x ( i ) t i ] ⊗ π ′ . By the Casselman criterion, we see that π is not tempered.Suppose conversely that π is not tempered. By the Casselman criterion, there exist ρ , [ x, y ]and π ′ such that π ֒ → ∆ ρ [ x, y ] ⋊ π ′ with x + y < 0. By Lemma 2.10, we conclude that thereexists i such that x ( i )1 = x and x ( i ) t i ≤ y so that x ( i )1 + x ( i ) t i ≤ x + y < (cid:3) Now we compare the ρ -data with the Langlands data. Theorem 2.13. Let π be an irreducible representation of G n . Write M − ρ ( π ) = h ( x (1)1 , k (1)1 ) , . . . , ( x (1) t , k (1) t ) , . . . , ( x ( r )1 , k ( r )1 ) , . . . , ( x ( r ) t r , k ( r ) t r ); π i N AN ALGORITHM TO COMPUTE DERIVATIVES 13 and set τ ( i ) j = ∆ ρ [ x ( i )1 , x ( i ) j ] as in Theorem 2.9. Suppose that π = L ( τ , . . . , τ l ; σ ) with τ i =∆ ρ i [ x i , y i ] . Then × ≤ i ≤ lρ i ∼ = ρ τ i = × i,jx ( i )1 + x ( i ) j < (cid:16) τ ( i ) j (cid:17) k ( i ) j − k ( i ) j +1 . Moreover, { M − ρ ( σ ) } ρ can be computed from { M − ρ ( π ) } ρ by Corollary 2.11.Proof. This follows from Theorem 2.9 and Corollaries 2.11, 2.12. (cid:3) In fact, by Proposition 3.8 below, the tempered representation σ is determined almostcompletely by { M − ρ ( σ ) } ρ .Unfortunately, the map π M ǫρ ( π ) is not injective. For example, when π is supercuspidaland ρ ⋊ π is reducible, this induced representation is semisimple of length two, i.e., ρ ⋊ π = π ⊕ π and π = π . However M ǫρ ( π ) = M ǫρ ( π ) = [(0 , π ] for ǫ ∈ {±} .2.5. Jantzen’s algorithm. Let π = L ( τ , . . . , τ r ; σ ) be an irreducible representation of G n .Suppose that ρ is self-dual and x ∈ (1 / Z . We recall Jantzen’s algorithm ([8, § D ( k ) ρ |·| x ( π ) with x > τ × · · · × τ r ∼ = τ (1)1 × · · · × τ (1) r × ∆ ρ [ x − , − x ] b with b maximal. Then π ֒ → L ( τ (1)1 , . . . , τ (1) r ) × ∆ ρ [ x − , − x ] b ⋊ σ. (2) Compute the right highest derivative R ( a ) ρ |·| − x ( L ( τ (1)1 , . . . , τ (1) r )) = L ( τ (2)1 , . . . , τ (2) r ). ThenJantzen’s Claim 1 says that π ֒ → L ( τ (2)1 , . . . , τ (2) r ) ⋊ L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b , σ ) . (3) Assume for a moment that we were able to compute the highest derivative D ( k ) ρ |·| x ( L (( ρ |·| − x ) a , ∆ ρ [ x − , − x ] b , σ )) = π . Then Jantzen’s Claim 2 says that π ֒ → L ( τ (2)1 , . . . , τ (2) r , ( ρ | · | x ) k ) ⋊ π . (4) Compute the left highest derivative L ( k ) ρ |·| x ( L ( τ (2)1 , . . . , τ (2) r , ( ρ |·| x ) k )) = L ( τ (3)1 , . . . , τ (3) r ).Then D ( k ) ρ |·| x ( π ) is the highest derivative, and D ( k ) ρ |·| x ( π ) ֒ → L ( τ (3)1 , . . . , τ (3) r ) ⋊ π . (5) By Theorem 2.4, one can write τ (3)1 × · · · × τ (3) r ∼ = τ (4)1 × · · · × τ (4) r × ( ρ | · | x ) k suchthat τ (4) i has a negative central exponent, and k ≤ k .(6) There exists a unique irreducible representation π of the form L (( ρ | · | − x ) a ′ , ∆ ρ [ x − , − x ] b ′ ; σ ′ ) such that D ( k ) ρ |·| x ( π ) = π is the highest derivative. Jantzen’s Claim 3 saysthat D ( k ) ρ |·| x ( π ) ֒ → L ( τ (4)1 , . . . , τ (4) r ) ⋊ π . Assume for a moment that we could specify π . (7) There exists a unique irreducible representation L ( τ (5)1 , . . . , τ (5) r ) such that R ( a ′ ) ρ |·| − x ( L ( τ (5)1 , . . . , τ (5) r )) = L ( τ (4)1 , . . . , τ (4) r )is the highest right derivative. Moreover, τ (5) i has a negative central exponent. Then D ( k ) ρ |·| x ( π ) = L ( τ (5)1 , . . . , τ (5) r , ∆ ρ [ x − , − x ] b ′ ; σ ′ ) . In conclusion, the computation of the highest derivative D ( k ) ρ |·| x ( π ) is reduced to the one of D ( k ) ρ |·| x ( L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b , σ )). Jantzen’s strategy of this computation is an inductionon x . Namely, this computation for the case x ≥ / x − x = 1 / π of certain irreduciblerepresentations π in the half-integral reducibility case ([8, § Arthurpackets . 3. Arthur packets In his book [1], for each A -parameter ψ , Arthur defined a finite (multi-)set Π ψ consisting ofunitary representations of split SO n +1 ( F ) or Sp n ( F ). In this section, we review his theory.3.1. A -parameters. A homomorphism ψ : W F × SL ( C ) × SL ( C ) → GL n ( C )is called an A -parameter for GL n ( F ) if • ψ (Frob) ∈ GL n ( C ) is semisimple and all its eigenvalues have absolute value 1; • ψ | W F is smooth, i.e., has an open kernel; • ψ | SL ( C ) × SL ( C ) is algebraic.The local Langlands correspondence for GL n ( F ) asserts that there is a canonical bijectionbetween the set of irreducible unitary supercuspidal representations of GL n ( F ) and the setof irreducible representations of W F of bounded images. We identify these two sets, and usethe symbol ρ for their elements.Any irreducible representation of W F × SL ( C ) × SL ( C ) is of the form ρ ⊠ S a ⊠ S b , where S a is the unique irreducible representation of SL ( C ) of dimension a . We shortly write ρ ⊠ S a = ρ ⊠ S a ⊠ S and ρ = ρ ⊠ S ⊠ S . For an A -parameter ψ , the multiplicity of ρ ⊠ S a ⊠ S b in ψ is denoted by m ψ ( ρ ⊠ S a ⊠ S b ). When an A -parameter ψ is decomposed into a direct sum ψ = M i ρ i ⊠ S a i ⊠ S b i , we define τ ψ by the product of Speh representations τ ψ = × i L (cid:18) ∆ ρ i (cid:20) a i − b i , − a i + b i (cid:21) , . . . , ∆ ρ i (cid:20) a i + b i − , − a i − b i (cid:21)(cid:19) . We say that an A -parameter ψ : W F × SL ( C ) × SL ( C ) → GL k ( C ) is symplectic or ofsymplectic type (resp. orthogonal or of orthogonal type ) if the image of ψ is in Sp k ( C ) N AN ALGORITHM TO COMPUTE DERIVATIVES 15 (so that k is even) (resp. in O k ( C )). We call ψ an A -parameter for SO n +1 ( F ) if it is an A -parameter for GL n ( F ) of symplectic type, i.e., ψ : W F × SL ( C ) × SL ( C ) → Sp n ( C ) . Similarly, ψ is called an A -parameter for Sp n ( F ) if it is an A -parameter for GL n +1 ( F ) oforthogonal type with the trivial determinant, i.e., ψ : W F × SL ( C ) × SL ( C ) → SO n +1 ( C ) . For G n = SO n +1 ( F ) (resp. G n = Sp n ( F )), we let Ψ( G n ) be the set of c G n -conjugacy classesof A -parameters for G n , where c G n = Sp n ( C ) (resp. c G n = SO n +1 ( C )). We say that • ψ ∈ Ψ( G n ) is tempered if the restriction of ψ to the second SL ( C ) is trivial; • ψ ∈ Ψ( G n ) is of good parity if ψ is a sum of irreducible self-dual representations ofthe same type as ψ ;We denote by Ψ temp ( G n ) = Φ temp ( G n ) (resp. Ψ gp ( G n )) the subset of Ψ( G ) consisting of tem-pered A -parameters (resp. A -parameters of good parity). Also, we put Φ gp ( G n ) = Φ temp ( G n ) ∩ Ψ gp ( G n ). Set Ψ ∗ ( G ) = ∪ n ≥ Ψ ∗ ( G n ) and Φ ∗ ( G ) = ∪ n ≥ Φ ∗ ( G n ) for ∗ ∈ {∅ , temp . gp } .For ψ ∈ Ψ( G n ), the component group is defined by S ψ = π (Cent c G n (Im( ψ )) /Z ( c G n )).This is an elementary two abelian group. It can be described as follows. Let ψ ∈ Ψ( G ). Forsimplicity, we assume that ψ is of good parity. Hence we can decompose ψ = ⊕ ti =1 ψ i , where ψ i is an irreducible representation (which is self-dual of the same type as ψ ). We define an enhanced component group A ψ as A ψ = t M i =1 ( Z / Z ) α ψ i . Namely, A ψ is a free Z / Z -module of rank t with a basis { α ψ i } associated with the irreduciblecomponents { ψ i } . Then there exists a canonical surjection A ψ ։ S ψ whose kernel is generated by the elements • z ψ = P ti =1 α ψ i ; and • α ψ i + α ψ i ′ such that ψ i ∼ = ψ i ′ .Let c S ψ and c A ψ be the Pontryagin duals of S ψ and A ψ , respectively. Via the surjection A ψ ։ S ψ , we may regard c S ψ as a subgroup of c A ψ . For η ∈ c A ψ , we write η ( α ψ i ) = η ( ψ i ). Byconvention, we understand that m ψ ( ρ ⊠ S ) = 1 and η ( ρ ⊠ S ) = 1.Let ψ, ψ ′ ∈ Ψ gp ( G ). When there is a canonical inclusion A ψ ′ ֒ → A ψ , for η ∈ c A ψ , we denoteits restriction by η ′ ∈ d A ψ ′ . For example, when ψ ′ = ψ − ψ + ψ ′ with ψ , ψ ′ being irreducible,then we set η ′ ( ψ ′ ) = η ( ψ ).Let Irr unit ( G n ) (resp. Irr temp ( G n )) be the set of equivalence classes of irreducible unitary(resp. tempered) representations of G n . The local main theorem of Arthur’s book is as follows. Theorem 3.1 ([1, Theorem 2.2.1, Proposition 7.4.1]) . Let G n be a split SO n +1 ( F ) or Sp n ( F ) . (1) For each ψ ∈ Ψ( G n ) , there is a finite multi-set Π ψ over Irr unit ( G n ) with a map Π ψ → c S ψ , π 7→ h· , π i ψ satisfying certain (twisted and standard) endoscopic character identities. We call Π ψ the A -packet for G n associated with ψ . (2) When ψ = φ ∈ Φ temp ( G n ) , the A -packet Π φ is in fact a subset of Irr temp ( G n ) . More-over, the map Π φ ∋ π 7→ h· , π i φ ∈ c S φ is bijective, Π φ ∩ Π φ ′ = ∅ for φ = φ ′ , and Irr temp ( G n ) = G φ ∈ Φ temp ( G n ) Π φ . When π ∈ Π φ with η = h· , π i φ ∈ c S φ , we write π = π ( φ, η ) . (3) If ψ = ⊕ i ρ i ⊠ S a i ⊠ S b i , set φ ψ, = M ib i ≡ ρ i ⊠ S a i . Then for any σ ∈ Π φ ψ, , the unique irreducible subrepresentation of × ib i ≡ ,b i =1 L (cid:18) ∆ ρ i (cid:20) a i − b i , − a i + b i (cid:21) , . . . , ∆ ρ i (cid:20) a i − , − a i − (cid:21)(cid:19) × × ib i ≡ L (cid:18) ∆ ρ i (cid:20) a i − b i , − a i + b i (cid:21) , . . . , ∆ ρ i h a i − , − a i i(cid:19) ⋊ σ belongs to Π ψ . Remark 3.2. (1) The map Π ψ ∋ π 7→ h· , π i ψ ∈ c S ψ is not canonical when G = Sp n ( F ) .To specify this, we implicitly fix an F × -orbit of non-trivial additive characters of F through this paper. (2) In general, the map Π ψ ∋ π 7→ h· , π i ψ ∈ c S ψ is neither injective nor surjective. (3) In general, Π ψ can intersect with Π ψ ′ even when ψ = ψ ′ . However, [18, 4.2 Corollaire] says that if Π ψ ∩ Π ψ ′ = ∅ , then ψ d ∼ = ψ ′ d , where ψ d = ψ ◦ ∆ is the diagonal restrictionof ψ , i.e., ∆ : W F × SL ( C ) → W F × SL ( C ) × SL ( C ) is defined by ∆( w, g ) = ( w, g, g ) . The following is a deep result of Mœglin. Theorem 3.3 (Mœglin [19]) . The A -packet Π ψ is multiplicity-free, i.e., it is a subset of Irr unit ( G ) . Xu proved the following key lemma, whose proof uses the theory of endoscopy. Lemma 3.4 (Xu [23, Proposition 8.3 (ii)]) . Let ψ = ⊕ i ∈ I ρ i ⊠ S a i ⊠ S b i ∈ Ψ gp ( G ) . For fixed x ∈ R , if D ( k ) ρ |·| x ( π ) = 0 for some π ∈ Π ψ , then k ≤ { i ∈ I | ρ i ∼ = ρ, x = ( a i − b i ) / } . The following is the first observation. Example 3.5. Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix x ∈ (1 / Z with x > . Suppose that ρ ⊠ S x +1 is self-dual of the same type as φ . N AN ALGORITHM TO COMPUTE DERIVATIVES 17 (1) Consider π = L (∆ ρ [ x − , − x ] b ; π ( φ, η )) . By Theorem 3.1 (3), we have π ∈ Π ψ with ψ = φ + ( ρ ⊠ S x ⊠ S ) b . In particular, by Lemma 3.4, if D ( k ) ρ |·| x ( π ) = 0 , then k ≤ m φ ( ρ ⊠ S x +1 ) . (2) Assume that x = 1 , and consider π = L (( ρ | · | − ) a , ∆ ρ [0 , − b ; π ( φ, η )) . By Theorem 3.1 (3), if a ≤ m φ ( ρ ) , then π ∈ Π ψ with ψ = φ − ρ a + ( ρ ⊠ S ⊠ S ) a + ( ρ ⊠ S ⊠ S ) b . In particular, by Lemma 3.4, if D ( k ) ρ |·| ( π ) = 0 , then k ≤ m φ ( ρ ⊠ S ) . Highest derivatives of tempered representations. In [6, 7, 8], Jantzen studied thederivatives of irreducible representations of G n . To do this, he used the extended Mœglin–Tadi´c classification, which characterizes irreducible tempered representations by their cuspidalsupports and the behavior of Jacquet modules. See [6]. Since the behavior of Jacquet modulesof irreducible tempered representations are known by the previous paper [2], one can easilytranslate Jantzen’s results in terms of the local Langlands correspondence (Theorem 3.1 (2)).The highest derivatives of tempered representations are given as follows. Proposition 3.6 ([7, Theorem 3.1]) . Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix a positive half-integer x ∈ (1 / Z . Suppose that ρ ⊠ S x +1 is self-dual of the same type as φ . Denote m = m φ ( ρ ⊠ S x +1 ) ≥ by the multiplicity of ρ ⊠ S x +1 in φ . (1) When x > , we have D ( m ) ρ |·| x ( π ( φ, η )) = π ( φ − ( ρ ⊠ S x +1 ) m + ( ρ ⊠ S x − ) m , η ) . It is the highest derivative if it is nonzero. Moreover, D ( m ) ρ |·| x ( π ( φ, η )) = 0 if and only if • m > ; • ρ ⊠ S x +1 is self-dual of the same type as φ ; • φ ⊃ ρ ⊠ S x − and η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ) ,where we understand that φ ⊃ ρ ⊠ S and η ( ρ ⊠ S ) = 1 when ρ is self-dual of theopposite type to φ . (2) Suppose that m > , x > and D ( m ) ρ |·| x ( π ( φ, η )) = 0 . If m is odd, then D ( m − ρ |·| x ( π ( φ, η )) = π (cid:0) φ − ( ρ ⊠ S x +1 ) m − + ( ρ ⊠ S x − ) m − , η (cid:1) . If m is even, D ( m − ρ |·| x ( π ( φ, η )) is equal to L (cid:0) ∆ ρ [ x − , x ]; π (cid:0) φ − ( ρ ⊠ S x +1 ) m + ( ρ ⊠ S x − ) m − , η (cid:1)(cid:1) . In particular, in both cases, D ( m − ρ |·| x ( π ( φ, η )) is irreducible and the highest derivative.Moreover, D ( m − ρ |·| x ( π ( φ, η )) is tempered if and only if m is odd. (3) When x = 0 , set k = [ m/ to be the largest integer which is not greater than m/ .Then D ( k ) ρ ( π ( φ, η )) = c k · π (cid:16) φ − ρ k , η (cid:17) with c k = ( k − if ρ is of the same type as φ and m is even , k otherwise . This is the highest derivative. Remark 3.7. (1) In [2] , the highest derivatives of irreducible tempered representationswas also considered, but Proposition 6.3 and Remark 6.4 in that paper are mistakes. (2) One can also prove Proposition 3.6 by using results in [2] . (3) When ρ is not self-dual, x ≥ and m = m φ ( ρ ⊠ S x +1 ) > , we have m φ ( ρ ∨ ⊠ S x +1 ) = m and π ( φ, η ) = ∆ ρ [ x, − x ] m ⋊ π ( φ , η ) , where φ = φ − ( ρ ⊕ ρ ∨ ) m ⊠ S x +1 , and η = η |A φ . In this case, the highest derivativeis D ( m ) ρ |·| x ( π ( φ, η )) = ∆ ρ [ x − , x ] m ⋊ π ( φ , η ) . Here, when x = 0 , we omit ∆ ρ [ x − , x ] . As a consequence, we have the following. Proposition 3.8. Let π = π ( φ, η ) be an irreducible tempered representation of G n . Fix x ∈ (1 / Z and assume that ρ ⊠ S | x | +1 is self-dual of the same type as φ . (1) Suppose that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ) , ( x, k ); M − ρ ( π ) (cid:3) where x > and π = π ( φ , η ) tempered ( t can be zero). Then x = 1 / , or φ contains ρ ⊠ S x − withmultiplicity greater than or equal to k . Moreover, if φ contains ρ ⊠ S x +1 and η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ) , then k is even. Set φ = φ − ( ρ ⊠ S x − ) k + ( ρ ⊠ S x +1 ) k , and define η ∈ d A φ so that η ( ρ ′ ⊠ S a ) = η ( ρ ′ ⊠ S a ) for any ( ρ ′ , a ) = ( ρ, x + 1) and η ( ρ ⊠ S x +1 ) = ( η ( ρ ⊠ S x +1 ) if φ ⊃ ρ ⊠ S x +1 ,η ( ρ ⊠ S x − ) otherwise . Here, we understand that φ ⊃ ρ ⊠ S and η ( ρ ⊠ S ) = 1 when x = 1 / . Then M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ); M − ρ ( π ( φ , η )) (cid:3) . (2) Suppose that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ) , ( x, k ); M − ρ ( π ) (cid:3) where x = 0 and π = π ( φ , η ) tempered ( t can be zero). Set φ = φ + ρ k . Then there exists η ∈ d A φ with η |A φ = η such that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ); M − ρ ( π ( φ , η )) (cid:3) . (3) Suppose that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ) , ( − x, k ); M − ρ ( π ) (cid:3) where x > and π = π ( φ , η ) tempered. Then N AN ALGORITHM TO COMPUTE DERIVATIVES 19 • k = 1 ; • there exists ≤ t ′ ≤ t such that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t ′ , k t ′ ) , ( x, k ′ ) , ( x − , , . . . , ( − x, M − ρ ( π ) (cid:3) for some odd k ′ ; • φ contains ρ ⊠ S x − with multiplicity greater than or equal to k ′ ; • φ does not contain ρ ⊠ S x +1 .Set φ = φ + ( ρ ⊠ S x +1 ) k ′ +1 − ( ρ ⊠ S x − ) k ′ − , and define η ∈ d A φ so that η ( ρ ′ ⊠ S a ) = η ( ρ ′ ⊠ S a ) for any ( ρ ′ , a ) = ( ρ, x + 1) and η ( ρ ⊠ S x +1 ) = − η ( ρ ⊠ S x − ) . Then M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t ′ , k t ′ ); M − ρ ( π ( φ , η )) (cid:3) . Proof. We show (1) and (2) so that x ≥ 0. If an irreducible representation π satisfies that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ); M − ρ ( π ) (cid:3) , then M − ρ ( π ) = (cid:2) ( x, k ); M − ρ ( π ) (cid:3) . By applyingCorollary 2.12 to π and π , we see that π is also tempered. If π = π ( φ , η ), the relationbetween ( φ , η ) and ( φ , η ) is given in Proposition 3.6. Hence we obtain (1) and (2).We show (3) so that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t , k t ) , ( − x, k ); M − ρ ( π ) (cid:3) with x > 0. Byapplying Corollary 2.12 to π , we see that there exists 1 ≤ t ′ ≤ t such that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t ′ , k t ′ ) , ( x, k ′ ) , ( x − , k ′ ) , . . . , ( − x, k ′ x +1 ); M − ρ ( π ) (cid:3) with k ′ ≥ · · · ≥ k ′ x +1 = k . Take an irreducible representation π such that M − ρ ( π ) = (cid:2) ( x , k ) , . . . , ( x t ′ , k t ′ ); M − ρ ( π ) (cid:3) .Then M − ρ ( π ) = (cid:2) ( x, k ′ ) , ( x − , k ′ ) , . . . , ( − x, k ′ x +1 ); M − ρ ( π ) (cid:3) . Since π is tempered, byCorollary 2.12, we see that π is also tempered. Write π = π ( φ , η ). Then by Proposi-tion 3.6, we have • φ contains ρ ⊠ S x +1 with even multiplicity 2 m > • φ contains ρ ⊠ S x − and η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ); • k ′ = 2 m − k ′ = · · · = k ′ x +1 = 1; • φ = φ − ( ρ ⊠ S x +1 ) m +( ρ ⊠ S x − ) m − so that φ contains ρ ⊠ S x − with multiplicitygreater than 2 m − k ′ − (cid:3) When ρ ⊠ S | x | +1 is not self-dual of the same type as φ , a similar (and easier) statementholds. We leave the detail for readers. We note that: • When ρ ⊠ S | x | +1 is self-dual of the opposite type to φ , the case (3) cannot occur. • When ρ is not self-dual, the case (1) cannot occur.3.3. Irreducibility of certain induced representations. Using the highest derivatives,Jantzen proved some irreducibility of parabolically induced representations. For φ ∈ Φ gp ( G ),we denote the multiplicity of ρ ⊠ S a in φ by m φ ( ρ ⊠ S a ). For consistency, we define m φ ( ρ ⊠ S ) =1 and η ( ρ ⊠ S ) = +1 if ρ is self-dual of the opposite type to φ . Theorem 3.9 ([7, Theorem 4.7]) . Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix a ∈ Z with a ≥ suchthat ρ ⊠ S a is self-dual of the same type as φ . Consider ρ | · | a − ⋊ π ( φ, η ) . (1) If m φ ( ρ ⊠ S a − ) = 0 , then ρ | · | a − ⋊ π ( φ, η ) is irreducible. (2) If m φ ( ρ ⊠ S a − ) = 1 , then (a) ρ | · | a − ⋊ π ( φ, η ) is reducible if m φ ( ρ ⊠ S a ) = 0 or η ( ρ ⊠ S a ) = η ( ρ ⊠ S a − ) ; (b) ρ | · | a − ⋊ π ( φ, η ) is irreducible if m φ ( ρ ⊠ S a ) > and η ( ρ ⊠ S a ) = η ( ρ ⊠ S a − ) . (3) If m φ ( ρ ⊠ S a − ) ≥ , then ρ | · | a − ⋊ π ( φ, η ) is reducible. We also need the irreducibility of other parabolically induced representations. Proposition 3.10. Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix x ∈ (1 / Z with x ≥ such that ρ ⊠ S x +1 is self-dual of the same type as φ . (1) ∆ ρ [ x − , − x ] ⋊ π ( φ, η ) is irreducible if and only if φ contains both ρ ⊠ S x +1 and ρ ⊠ S x − , and η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ) . (2) If φ contains both ρ ⊠ S x +1 and ρ ⊠ S x − , but if η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ) , then ∆ ρ [ x − , − x ] ⋊ L (∆ ρ [0 , − π ( φ, η )) is irreducible.Proof. The only if part of (1) is proven in [2, Proposition 5.2]. The if part is similar to thatproposition. Assume that φ contains both ρ ⊠ S x +1 and ρ ⊠ S x − and η ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x − ). Suppose that ∆ ρ [ x − , − x ] ⋊ π ( φ, η ) is reducible. Take an irreducible quotient π ′ of ∆ ρ [ x − , − x ] ⋊ π ( φ, η ). Since the Langlands subrepresentation appears in the standardmodule as subquotients with multiplicity one, we see that π ′ is tempered. Moreover, π ′ ∈ Π φ ′ with φ ′ = φ + ρ ⊠ ( S x +1 + S x − ). We write π ′ = π ( φ ′ , η ′ ). When D ( k ) ρ |·| x ( π ( φ, η )) is the highestderivative, by Proposition 3.6, we see that D ( k +1) ρ |·| x ( π ′ ) is equal to the irreducible induction∆ ρ [ x − , − ( x − ⋊ D ( k ) ρ |·| x ( π ( φ, η )). If η ′ ( ρ ⊠ S x +1 ) = η ′ ( ρ ⊠ S x − ), then D ( k +1) ρ |·| x ( π ′ ) istempered such that its L -parameter φ ′′ does not contain ρ ⊠ S x +1 , whereas, if ∆ ρ [ x − , − ( x − ⋊ D ( k ) ρ |·| x ( π ( φ, η )) is tempered, then its L -parameter must contain ρ ⊠ S x +1 . This is acontradiction, so that we have η ′ ( ρ ⊠ S x +1 ) = η ′ ( ρ ⊠ S x − ). In addition, we have equationsof the highest derivatives D ( l +2) ρ |·| x − ◦ D ( k +1) ρ |·| x ( π ′ ) = ∆ ρ [ x − , − ( x − ⋊ D ( l ) ρ |·| x − ◦ D ( k ) ρ |·| x ( π ( φ, η )) if x ≥ ,D ( l +2) ρ |·| ◦ D ( k +1) ρ |·| ( π ′ ) = D ( l ) ρ |·| ◦ D ( k ) ρ |·| ( π ( φ, η )) if x = 32 ,D ( l +1) ρ |·| ◦ D ( k +1) ρ |·| ( π ′ ) = 2 · D ( l ) ρ |·| ◦ D ( k ) ρ |·| ( π ( φ, η )) if x = 1 . In each case, by Proposition 3.6, ∆ ρ [ x − , − x ] appears in exactly one of the left or right handside. This is a contradiction.(2) is proven similarly to [8, Lemma 6.3]. We give a sketch of the proof. • For φ ′ = φ − ρ ⊠ ( S x +1 + S x − ), we have π ( φ, η ) ֒ → ∆ ρ [ x, − ( x − ⋊ π ( φ ′ , η ′ ) . • If we set π = L (∆ ρ [ x − , − x ]; π ( φ, η )), then π ֒ → ∆ ρ [ x − , − x ] × ∆ ρ [ x, − ( x − ⋊ π ( φ ′ , η ′ ). This implies that π ֒ → L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − ⋊ π ( φ ′ , η ′ ) or π ֒ → ∆ ρ [ x, − x ] × ∆ ρ [ x − , − ( x − ⋊ π ( φ ′ , η ′ ). Since π is non-tempered, the former casemust hold. • Put m = m φ ( ρ ⊠ S x +1 ). Since L ρ |·| x ( L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − R ρ |·| − x ( L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − N AN ALGORITHM TO COMPUTE DERIVATIVES 21 with ρ | · | x , we see that D ( m − ρ |·| x ( π ) is the highest derivative, and D ( m − ρ |·| x ( π ) ֒ → L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − ⋊ D ( m − ρ |·| x ( π ( φ ′ , η ′ )) . • Set λ = L (∆ ρ [ x − , − x ] ; π ( φ, η )). Since up to semisimplification∆ ρ [ x − , − x ] ⋊ π ( φ, η ) =∆ ρ [ x − , − x ] × ∆ ρ [ x, − ( x − ⋊ π ( φ, η )= L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − ⋊ π ( φ, η ) ⊕ ∆ ρ [ x, − x ] × ∆ ρ [ x − , − ( x − ⋊ π ( φ, η ) , we see that λ is a subrepresentation of L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − ⋊ π ( φ, η ). Inparticular, D ( m ) ρ |·| x ( λ ) ֒ → L (∆ ρ [ x − , − x ] , ∆ ρ [ x, − ( x − ⋊ D ( m ) ρ |·| x ( π ( φ, η ))is the highest derivative. • Since λ ֒ → ∆ ρ [ x − , − x ] ⋊ π , we must have D ( m ) ρ |·| x ( λ ) ⊂ ∆ ρ [ x − , − ( x − ⋊ D ( m − ρ |·| x ( π ) . In particular, since λ ֒ → ( ρ | · | x ) m × ∆ ρ [ x − , − ( x − ⋊ D ( m − ρ |·| x ( π ), we have λ ֒ → ∆ ρ [ x, − ( x − × ( ρ | · | x ) m − ⋊ D ( m − ρ |·| x ( π ) or λ ֒ → L (∆ ρ [ x − , − ( x − , ρ | · | x ) × ( ρ | · | x ) m − ⋊ D ( m − ρ |·| x ( π ). Since x ≥ D ( m ) ρ |·| x ( λ ) = 0, the former case must hold.Moreover, we see that λ ֒ → ∆ ρ [ x, − ( x − ⋊ π. • Since ∆ ρ [ x, − ( x − × π ( φ, η ) ։ π , we see that λ is the unique irreducible (Langlands)quotient of ∆ ρ [ x, − ( x − ⋊ π . Since this quotient appear in subquotients withmultiplicity one, ∆ ρ [ x, − ( x − ⋊ π must be irreducible.This completes the proof. (cid:3) Mœglin’s construction. To obtain Theorem 3.3, Mœglin constructed A -packets Π ψ concretely. In this subsection, we review her construction in a special case. For more precision,see [16, 17, 19] and [23].Let ψ ∈ Ψ gp ( G n ). We assume that ψ is of the form ψ = φ + r M i =1 ρ ⊠ S a i ⊠ S b i ! such that(a) φ ∈ Φ gp ( G ) such that φ ρ ⊠ S d for any d ≥ a i ≥ b i for any 1 ≤ i ≤ r ;(c) if a i − b i > a j − b j and a i + b i > a j + b j , then i > j .Note that the last condition may not determine an order on { ( a , b ) , . . . , ( a r , b r ) } uniquely.Once we fix such an order, we write ρ ⊠ S a i ⊠ S b i < ψ ρ ⊠ S a j ⊠ S b j if i < j . Fix η ∈ d A φ such that if φ = ⊕ i φ i is a decomposition into irreducible representations,then η ( ψ i ) = η ( ψ j ) whenever φ i ∼ = φ j . Let l = ( l , . . . , l r ) ∈ Z r and η = ( η , . . . , η r ) ∈ {± } r such that 0 ≤ l i ≤ b i / η ( z φ ) r Y i =1 ( − [ b i / l i η b i i = 1 . For these data, Mœglin constructed a representation π < ψ ( ψ, l, η, η ) of G n . If the order < ψ isfixed, we may write π ( ψ, l, η, η ) = π < ψ ( ψ, l, η, η ). Theorem 3.11 (Mœglin) . Notation is as above. (1) The representation π ( ψ, l, η, η ) is irreducible or zero. Moreover, Π ψ = { π ( ψ, l, η, η ) | l, η, η as above } \ { } . (2) If π ( ψ, l, η, η ) ∼ = π ( ψ, l ′ , η ′ , η ′ ) = 0 , then l = l ′ , η = η ′ , and η i = η ′ i unless l i = b i / . (3) Assume further that a i − b i ≥ a i − + b i − for any i > . Then π ( ψ, l, η, η ) is nonzeroand is a unique irreducible subrepresentation of r × i =1 L (cid:18) ∆ ρ (cid:20) a i − b i , − a i + b i (cid:21) , . . . , ∆ ρ (cid:20) a i − b i l i − , − a i + b i l i (cid:21)(cid:19) ⋊ π ( φ, η ) , where φ = φ + r M i =1 ρ ⊠ ( S a i − b i +2 l i +1 + · · · + S a i + b i − l i − ) ! , and η ∈ c S φ ⊂ c A φ is given by η |A φ = η and η ( ρ ⊠ S a i − b i +2 l i +2 c − ) = ( − c − η i for ≤ c ≤ b i − l i . (4) Suppose that ψ ′ = φ + r M i =1 ρ ⊠ S a ′ i ⊠ S b i ! ∈ Ψ gp ( G ) also satisfies the above assumptions (a)–(c). Assume further that a ′ i ≥ a i for any i ≥ and that a ′ i − b i ≥ a ′ i − + b i − for any i > . Then π ( ψ, l, η, η ) = Jac ρ |·| a ′ r − ,...,ρ |·| ar +12 ◦ · · · ◦ Jac ρ |·| a ′ − ,...,ρ |·| a ( π ( ψ ′ , l, η, η )) . For the proof, see [23, § Example 3.12. Let us consider the situation of Example 3.5 with x ≥ . Set ψ = φ + ( ρ ⊠ S x ⊠ S ) b . Write ψ = φ + r M i =1 ρ ⊠ S a i ⊠ S b i ! as above such that a ≤ · · · ≤ a r . Define • l ∈ Z r so that l i = 1 if b i = 2 , and l i = 0 if b i = 1 ; • η ∈ {± } r so that η i = η ( ρ ⊠ S a i ) if b i = 1 ; • η ∈ d A φ by η = η |A φ .Then it is easy to see that π ( ψ, l, η, η ) = L (∆ ρ [ x − , − x ] b ; π ( φ, η )) . Similarly, we have the following proposition, which is a key observation. N AN ALGORITHM TO COMPUTE DERIVATIVES 23 Proposition 3.13. Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix x ∈ (1 / Z with x ≥ . Suppose that ρ ⊠ S x +1 is self-dual of the same type as φ . Assume that m φ ( ρ ⊠ S x +1 ) = 0 , m φ ( ρ ⊠ S x − ) = 0 and η ( ρ ⊠ S x +1 ) η ( ρ ⊠ S x − ) = ( − b +1 . Then L (∆ ρ [ x − , − x ] b ; π ( φ, η )) ∈ Π ψ with ψ = φ − ρ ⊠ ( S x +1 + S x − ) + ( ρ ⊠ S x ⊠ S ) b +1 . More precisely, when we write ψ = φ + r M i =1 ρ ⊠ S a i ⊠ S b i ! as above such that a ≤ · · · ≤ a r , define • l ∈ Z r so that l i = 0 for any ≤ i ≤ r ; • η ∈ {± } r so that η i = ( η ( ρ ⊠ S a i ) if b i = 1 , ( − { j
2. Take ψ ′ = ψ − ρ ⊠ S x ⊠ S + ρ ⊠ S x +2 ⊠ S − M ≤ i ≤ ra i ≥ x +1 ρ ⊠ S a i + M ≤ i ≤ ra i ≥ x +1 ρ ⊠ S a ′ i such that a ′ r > a ′ r − > · · · are sufficiently large so that m ψ ′ ( ρ ⊠ S x +1 ) = m ψ ′ ( ρ ⊠ S x +3 ) = 0.By Theorem 3.11 (4), we have π ( ψ, l, η, η ) = J ◦ J ( π ( ψ ′ , l, η, η )) with J = Jac ρ |·| x ,ρ |·| x +1 ,J = (cid:18) Jac ρ |·| a ′ r − ,...,ρ |·| ar +12 (cid:19) ◦ Jac ρ |·| a ′ r − − ,...,ρ |·| ar − ! ◦ · · · . Since l i = 0 for any i , we may apply the induction hypothesis to π ( ψ ′ , l, η, η ). Hence π ( ψ ′ , l, η, η ) = L (∆ ρ [ x − , − x ] b − ; π ( φ ′ , η ′ )), where φ ′ = ψ ′ − ( ρ ⊠ S x ⊠ S ) b − ρ ⊠ S x +2 ⊠ S + ρ ⊠ ( S x − + S x +1 + S x +3 )= φ + ρ ⊠ ( S x +1 + S x +3 ) − M ≤ i ≤ ra i ≥ x +1 ρ ⊠ S a i + M ≤ i ≤ ra i ≥ x +1 ρ ⊠ S a ′ i . Note that m φ ′ ( ρ ⊠ S x +1 ) = 2, m φ ′ ( ρ ⊠ S x +3 ) = 1 and η ′ ( ρ ⊠ S x − ) η ′ ( ρ ⊠ S x +1 ) = ( − b . Take κ ∈ { , } such that b ≡ κ mod 2. By Proposition 3.10, ∆ ρ [ x − , − x ] ⋊ L (∆ ρ [ x − , − x ] − κ ; π ( φ ′ , η ′ )) is irreducible. Note that any irreducible subquotient of ∆ ρ [ x − , − x ] c isof the form L (∆ ρ [ x − , − x ]; ∆ ρ [ x, − ( x − α × (∆ ρ [ x, − x ] × ∆ ρ [ x − , − ( x − β with α + β = c . Considering the Langlands data for π ( ψ ′ , l, η, η ), we see that it is a subrep-resentation of L (∆ ρ [ x − , − x ]; ∆ ρ [ x, − ( x − b − κ ⋊ L (∆ ρ [ x − , − x ] − κ ; π ( φ ′ , η ′ )) . Note that this induced representation is irreducible. Indeed, it is unitary so that it is semisim-ple, but since it is subrepresentation of a standard module, it has a unique irreducible sub-representation. Hence π ( ψ, l, η, η ) is equal to L (∆ ρ [ x − , − x ]; ∆ ρ [ x, − ( x − b − κ ⋊ J ◦ J ( L (∆ ρ [ x, − ( x − − κ ; π ( φ ′ , η ′ ))) . Since Jac ρ |·| x +1 ( π ( φ ′ , η ′ )) = 0, we have J ◦ J ( L (∆ ρ [ x, − ( x − − κ ; π ( φ ′ , η ′ ))) = L (∆ ρ [ x, − ( x − − κ ; J ◦ J ( π ( φ ′ , η ′ ))) . By [2, Theorem 4.2], we have J ◦ J ( π ( φ ′ , η ′ )) = L (∆ ρ [ x, − ( x − π ( φ, η )) . Therefore, π ( ψ, l, η, η ) = L (∆ ρ [ x − , − x ]; ∆ ρ [ x, − ( x − b − κ ⋊ L (∆ ρ [ x, − ( x − − κ ; π ( φ, η )) ∼ = L (∆ ρ [ x, − ( x − b ; π ( φ, η )) , as desired. (cid:3) When x = 1, we have the following more general version. Proposition 3.14. Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Suppose that ρ is self-dual of the same typeas φ . Assume that m φ ( ρ ) = 0 , m φ ( ρ ⊠ S ) = 0 and η ( ρ ) η ( ρ ⊠ S ) = ( − b +1 . If a ≤ m φ ( ρ ) − , then L (( ρ | · | − ) a , ∆ ρ [0 , − b ; π ( φ, η )) ∈ Π ψ with ψ = φ − ρ a + ( ρ ⊠ S ⊠ S ) a − ( ρ + ρ ⊠ S ) + ( ρ ⊠ S ⊠ S ) b +1 . In particular, if D ( k ) ρ |·| ( π ) = 0 , then k ≤ m φ ( ρ ⊠ S ) − . The proof is the same as the one of Proposition 3.13, but it requires Mœglin’s constructionmore generally since ψ contains ρ ⊠ S ⊠ S . We omit the detail.4. Derivatives of certain representations Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Fix x ∈ (1 / Z with x > ρ ⊠ S x +1 is self-dual of thesame type as φ . Let π = L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; π ( φ, η )). By Jantzen’s algorithm recalledin § D ( k ) ρ |·| x ( π ). In this section, we give an algorithm to compute this for x ≥ N AN ALGORITHM TO COMPUTE DERIVATIVES 25 Statements. The following is an algorithm. Theorem 4.1. Suppose that x ≥ . (1) If m φ ( ρ ⊠ S x +1 ) = 0 , m φ ( ρ ⊠ S x − ) = 0 and η ( ρ ⊠ S x +1 ) η ( ρ ⊠ S x − ) = ( − b +1 ,set ψ = φ − ρ ⊠ ( S x +1 + S x − ) + ( ρ ⊠ S x ⊠ S ) b +1 . Otherwise, set ψ = φ + ( ρ ⊠ S x ⊠ S ) b . Then L (∆ ρ [ x − , − x ] b ; π ( φ, η )) ∈ Π ψ . (2) Put m = m ψ ( ρ ⊠ S x +1 ) , m ′ = m ψ ( ρ ⊠ S x − ) , and l = min { a − m ′ , } . Then D ( l + m ) ρ |·| x ( L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; π ( φ, η ))) is the highest derivative. Moreover,it is a unique irreducible subrepresentation of ( ρ | · | − x ) a − l ⋊ D ( m ) ρ |·| x ( L (∆ ρ [ x − , − x ] b ; π ( φ, η ))) . (3) If we set ψ ′ = ψ − ( ρ ⊠ S x +1 ) m +( ρ ⊠ S x − ) m , then D ( m ) ρ |·| x ( L (∆ ρ [ x − , − x ] b ; π ( φ, η ))) ∈ Π ψ ′ . In particular, if we write ψ = φ + r M i =1 ρ ⊠ S a i ⊠ S b i ! , ψ ′ = φ + r M i =1 ρ ⊠ S a ′ i ⊠ S b ′ i ! such that a ≤ · · · ≤ a r , a ′ ≤ · · · ≤ a ′ r , and such that φ ρ ⊠ S d for any d > , then π ( ψ, l, η, η ) = L (∆ ρ [ x − , − x ] b ; π ( φ, η )) ,π ( ψ ′ , l ′ , η ′ , η ′ ) = D ( m ) ρ |·| x ( L (∆ ρ [ x − , − x ] b ; π ( φ, η ))) for some data. These are related as follows: • η = η ′ = η |A φ ; • if b ′ i = 1 , then l ′ i = 0 ; • if b ′ i = b j = 2 , then l ′ i = l j if and only if m is even; • if b ′ i = 1 and a ′ i = 2 x − , then η ′ i = η ( ρ ⊠ S a i ) ; • if b ′ i = 1 and a ′ i = 2 x − , or if b ′ i = 2 and l ′ i = 0 , then η ′ i = η ( ρ ⊠ S x − ) or η ′ i = ( − m ψ ( ρ ⊠ S x ⊠ S ) η ( ρ ⊠ S x +1 ) . These values are equal to each other if φ ⊃ ρ ⊠ ( S x − + S x +1 ) . (4) By Example 3.12 and Proposition 3.13, we can obtain the Langlands data for D ( m ) ρ |·| x ( L (∆ ρ [ x − , − x ] b ; π ( φ, η ))) . We can also write down this theorem more explicitly. Let κ ∈ { , } such that κ ≡ b mod 2.Set m d = m φ ( ρ ⊠ S d ) for d ≥ 1. For d ≥ 3, define δ d = (cid:26) m d m d − = 0 and η ( ρ ⊠ S d ) = η ( ρ ⊠ S d − ) , . Put φ ′ = φ − ( ρ ⊠ S x +1 ) m x +1 + ( ρ ⊠ S x − ) m x +1 . Let η ′ ∈ d A φ ′ be the pullback of η via the canonical map A φ ′ ֒ → A φ ։ S φ . The followingis a reformulation of Theorem 4.1, which follows from Example 3.12 and Proposition 3.13immediately. Corollary 4.2. The notation is as above. Let π = L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; π ( φ, η )) . Then D ( k ) ρ |·| x ( π ) is the highest derivative with k = ( max { a − m x − + m x +1 , m x +1 − } if m x − m x +1 = 0 , κ = δ x +1 , max { a − m x − + m x +1 , m x +1 } otherwise . Moreover: (1) Suppose that m x +1 = 0 . Set l = max { a − m x − , } . Then D ( l ) ρ |·| x ( π ) = L (( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ, η )) . (2) Suppose that m x − m x +1 = 0 . Set l = ( max { a − m x − , } if κ = δ x +1 , max { a − m x − + 1 , } if κ = δ x +1 . (a) If κ = δ x +1 , and if m x +1 is odd, then D ( l + m x +1 − ρ |·| x ( π ) = L (cid:16) ( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ ′ + ρ ⊠ ( S x +1 − S x − ) , η ′ ) (cid:17) . (b) If κ = δ x +1 , and if m x +1 is even, then D ( l + m x +1 − ρ |·| x ( π ) = L (cid:16) ( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b +1 ; π ( φ ′ − ( ρ ⊠ S x − ) , η ′ ) (cid:17) . (c) If κ = δ x +1 , and if m x +1 is odd and b > , then D ( l + m x +1 ) ρ |·| x ( π ) = L (cid:16) ( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b − ; π ( φ ′ + ρ ⊠ ( S x +1 + S x − ) , η ′ ) (cid:17) . (d) If κ = δ x +1 , and if m x +1 is even or b = 0 , then D ( l + m x +1 ) ρ |·| x ( π ) = L (cid:16) ( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ ′ , η ′ ) (cid:17) . (3) Suppose that m x − = 0 . (a) If m x +1 is even or b = 0 , then D ( a + m x +1 ) ρ |·| x ( π ) = L (∆ ρ [ x − , − x ] b ; π ( φ ′ , η ′ b )) , where η ′ b ( ρ ⊠ S x − ) = ( − b η ( ρ ⊠ S x +1 ) . (b) If m x +1 is odd and b > , then D ( a + m x +1 ) ρ |·| x ( π ) = L (∆ ρ [ x − , − x ] b − ; π ( φ ′ + ρ ⊠ ( S x − + S x +1 ) , η ′ b )) , where η ′ b ( ρ ⊠ S x − ) = ( − b η ( ρ ⊠ S x +1 ) and η ′ b ( ρ ⊠ S x +1 ) = η ( ρ ⊠ S x +1 ) . Remark 4.3. When x = 1 / , we formally understand that m = 1 , η ( ρ ⊠ S ) = +1 and a = 0 .Then Corollary 4.2 still holds for x = 1 / ( [8, Theorem 3.3] ). Note that when x = 1 / , thecase (3) does not appear. Recall that for k ′ ≥ 0, the parabolically induced representation ( ρ | · | x ) k ′ ⋊ D ( k ) ρ |·| x ( π ) has aunique irreducible subrepresentation π ′ . When k ′ = 0 (resp. k ′ = k ), we have π ′ = D ( k ) ρ |·| x ( π )(resp. π ′ = π ). When 0 < k ′ < k , the following formula follows from Corollary 4.2 immediately. N AN ALGORITHM TO COMPUTE DERIVATIVES 27 Corollary 4.4. Let π = L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; π ( φ, η )) be as in Corollary 4.2, and D ( k ) ρ |·| x ( π ) be its highest derivative. Set a = ( min { a, m x − } if κ = δ x +1 , min { a, m x − − } if κ = δ x +1 . Suppose that < k ′ < k . Then the unique irreducible subrepresentation π ′ of ( ρ | · | x ) k ′ ⋊ D ( k ) ρ |·| x ( π ) is of the form L (( ρ | · | − x ) a k ′ , ∆ ρ [ x − , − x ] b k ′ ; π ( φ k ′ , η k ′ )) , where ( a k ′ , b k ′ , φ k ′ , η k ′ ) isgiven as follows. (1) Suppose that m x +1 = 0 . Then k = a − m x − > and ( a k ′ , b k ′ , φ k ′ , η k ′ ) = ( k ′ + m x − , b, φ, η ) . (2) Suppose that m x − m x +1 = 0 . (a) If κ = δ x +1 , then ( a k ′ , b k ′ , φ k ′ , η k ′ ) is equal to ( a , b, φ ′ + ( ρ ⊠ S x +1 ) k ′ +1 − ( ρ ⊠ S x − ) k ′ +1 , η ′ ) if k ′ < m x +1 − , k ′ m x +1 mod 2 , ( a , b + 1 , φ ′ + ( ρ ⊠ S x +1 ) k ′ − ( ρ ⊠ S x − ) k ′ +2 , η ′ ) if k ′ < m x +1 − , k ′ ≡ m x +1 mod 2 , ( k ′ − m x +1 + m x − , b, φ, η ) if k ′ ≥ m x +1 − . (b) If κ = δ x +1 and b > , then ( a k ′ , b k ′ , φ k ′ , η k ′ ) is equal to ( a , b, φ ′ + ( ρ ⊠ S x +1 ) k ′ − ( ρ ⊠ S x − ) k ′ , η ′ ) if k ′ < m x +1 , k ′ ≡ m x +1 mod 2 , ( a , b − , φ ′ + ( ρ ⊠ S x +1 ) k ′ +1 − ( ρ ⊠ S x − ) k ′ − , η ′ ) if k ′ < m x +1 , k ′ m x +1 mod 2 , ( k ′ − m x +1 + m x − , b, φ, η ) if k ′ ≥ m x +1 . Similarly, if κ = δ x +1 and b = 0 , then ( a k ′ , b k ′ , φ k ′ , η k ′ ) is equal to ( ( a , , φ ′ + ( ρ ⊠ S x +1 ) k ′ − ( ρ ⊠ S x − ) k ′ , η ′ ) if k ′ < m x +1 , ( k ′ − m x +1 + m x − , , φ, η ) if k ′ ≥ m x +1 . (3) Suppose that m x − = 0 . If b > , then ( a k ′ , b k ′ , φ k ′ , η k ′ ) is equal to (0 , b, φ ′ + ( ρ ⊠ S x +1 ) k ′ − ( ρ ⊠ S x − ) k ′ , η ′ b ) if k ′ < m x +1 , k ′ ≡ m x +1 mod 2 , (0 , b − , φ ′ + ( ρ ⊠ S x +1 ) k ′ +1 − ( ρ ⊠ S x − ) k ′ − , η ′ b ) if k ′ < m x +1 , k ′ m x +1 mod 2 , ( k ′ − m x +1 , b, φ, η ) if k ′ ≥ m x +1 . Similarly, if b = 0 , then ( a k ′ , b k ′ , φ k ′ , η k ′ ) is equal to ( (0 , , φ ′ + ( ρ ⊠ S x +1 ) k ′ − ( ρ ⊠ S x − ) k ′ , η ′ b ) if k ′ < m x +1 , ( k ′ − m x +1 , , φ, η ) if k ′ ≥ m x +1 . Examples of the Aubert duals. Let π be an irreducible representation of G n . ByTheorem 2.13 and Proposition 3.8, if one could compute M + ρ ∨ ( π ) for all ρ , one would obtain theLanglands data of the Aubert dual ˆ π of π almost explicitly. In this subsection, we give somespecific examples of explicit calculations of M + ρ ∨ ( π ) and ˆ π for some irreducible representations π of G n = Sp n ( F ). When φ = S d ⊕ · · · ⊕ S d t ∈ Φ( G ) and η i = η ( S d i ), where d , . . . , d t areall odd, we write π ( φ, η ) = π ( d η , . . . , d η t t ) . We also write = GL ( F ) , and ∆[ x, y ] = ∆ [ x, y ].First, let us consider π (1 + , − , + , + , + , + , − ) ∈ Irr(Sp ( F )). Then M + (cid:0) π (1 + , − , + , + , + , + , − ) (cid:1) = (cid:2) (2 , M + (cid:0) L (∆[1 , − π (1 + , − , − , − , − )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , M + (cid:0) L (∆[1 , − π (1 + , − , − , − , − )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , , (1 , M + (cid:0) L (∆[0 , − π (1 + , + , + , − , − )) (cid:1)(cid:3) (1) = (cid:2) (2 , , (3 , , (1 , , (2 , M + (cid:0) L (∆[0 , − π (1 + , + , + , − , − )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , , (1 , , (2 , , (0 , M + (cid:0) L (∆[ − , − π (1 + , − , − )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , , (1 , , (2 , , (0 , , (1 , M + (cid:0) L (∆[ − , − , ∆[0 , − π (1 + )) (cid:1)(cid:3) (2) = (cid:2) (2 , , (3 , , (1 , , (2 , , (0 , , (1 , , ( − , M + (cid:0) L ( | · | − , ∆[0 , − π (1 + )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , , (1 , , (2 , , (0 , , (1 , , ( − , , (0 , M + (cid:0) L ( | · | − , | · | − ; π (1 + )) (cid:1)(cid:3) = (cid:2) (2 , , (3 , , (1 , , (2 , , (0 , , (1 , , ( − , , (0 , , ( − , , ( − , π (1 + ) (cid:3) . The equations (1) and (2) are non-trivial. We explain these equations.(1) We compute the highest derivative D ( k ) |·| ( L (∆[0 , − π (1 + , + , + , − , − ))) by apply-ing Jantzen’s algorithm recalled in § L (∆[0 , − π (1 + , + , + , − , − )) ֒ → ∆[0 , − ⋊ L ( | · | − ; π (1 + , + , + , − , − )) . By Corollary 4.2 (2)-(d), D (2) |·| ( L ( | · | − ; π (1 + , + , + , − , − ))) = L ( | · | − ; π (1 + , + , + , − , − )) . Therefore, Jantzen’s Claim 3 says that D (2) |·| ( L (∆[0 , − π (1 + , + , + , − , − ))) = L (∆[0 , − π (1 + , + , + , − , − ))is the highest derivative.(2) We have to show that Jac |·| ( L (∆[ − , − , ∆[0 , − π (1 + ))) = 0. It follows from R |·| − ( L (∆[ − , − , ∆[0 , − M − (cid:0) ˆ π (1 + , − , + , + , + , + , − ) (cid:1) = (cid:2) ( − , , ( − , , ( − , , ( − , , (0 , , ( − , , (1 , , (0 , , (2 , , (1 , π (1 + ) (cid:3) . By Theorem 2.13 and Proposition 3.8, we haveˆ π (1 + , − , + , + , + , + , − ) ֒ → ∆[ − , − × ( | · | − ) × ∆[ − , − × ( | · | − ) × ∆[0 , − ⋊ π (1 ǫ , ǫ , ǫ , ǫ , + )for some sign ǫ ∈ {±} . Since π (1 ǫ , ǫ , + ) = D (1) |·| − ◦ D (2) |·| ◦ D (1) |·| − ◦ D (3) |·| − ◦ D (1) |·| − ◦ D (3) |·| − (ˆ π (1 + , − , + , + , + , + , − ))up to multiplicity, we must haveˆ π (1 ǫ , ǫ , + ) = D (1) |·| ◦ D (2) |·| ◦ D (1) |·| ◦ D (3) |·| ◦ D (1) |·| ◦ D (3) |·| ( π (1 + , − , + , + , + , + , − )) N AN ALGORITHM TO COMPUTE DERIVATIVES 29 = L (∆[ − , − , ∆[0 , − π (1 + )) . To determine ǫ ∈ {±} , we compute the Aubert duals of π (1 + , + , + ) and π (1 − , − , + ).When ǫ = +, we have M + (cid:0) π (1 + , + , + ) (cid:1) = (cid:2) (2 , M + (cid:0) π (1 + , + , + ) (cid:1)(cid:3) = (cid:2) (2 , , (1 , M + (cid:0) π (1 + , + , + ) (cid:1)(cid:3) = (cid:2) (2 , , (1 , , (0 , π (1 + ) (cid:3) . Hence ˆ π (1 + , + , + ) ֒ → | · | − × ( | · | − ) ⋊ π (1 + , + , + ) . When ǫ = − , we have M + (cid:0) π (1 − , − , + ) (cid:1) = (cid:2) (1 , M + (cid:0) π (1 − , − , + ) (cid:1)(cid:3) = (cid:2) (1 , , (2 , M + (cid:0) π (1 − , − , + ) (cid:1)(cid:3) = (cid:2) (1 , , (2 , , (0 , M + (cid:0) π (3 + ) (cid:1)(cid:3) = (cid:2) (1 , , (2 , , (0 , , (1 , π (1 + ) (cid:3) . Hence ˆ π (1 − , − , + ) ֒ → ∆[ − , − × ∆[0 , − ⋊ π (1 + ) . Therefore, the correct sign is ǫ = − .This method does not always determine ˆ π . For example, let us consider π ǫ = L (∆[0 , − π (1 ǫ , ǫ , + )) ∈ Irr(Sp ( F )) for a sign ǫ ∈ {±} . Then for any ǫ ∈ {±} , we have M + ( π ǫ ) = (cid:2) (0 , , (1 , , (2 , , ( − , π (1 + ) (cid:3) . Using Theorem 2.13 and Proposition 3.8, this implies that ˆ π ǫ = π ˆ ǫ for some ˆ ǫ ∈ {±} . However,the correspondence ǫ ˆ ǫ is not determined.5. Proof of Theorem 4.1 In this section, we prove Theorem 4.1.5.1. The case a = 0 . First, we consider the case a = 0. Proof of Theorem 4.1 when a = 0 . Let π = L (∆ ρ [ x − , − x ] b ; π ( φ, η )). The assertions (1)and (4) follow from Example 3.12 and Proposition 3.13. For (2) and (3), we note that D ( k ) ρ |·| x ( π ) = 0 = ⇒ k ≤ m by Lemma 3.4.To determine D ( m ) ρ |·| x ( π ), we will use Mœglin’s construction. Write ψ = φ + ρ ⊠ ( S a − t ′ + · · · + S a − )+ ( ρ ⊠ S x − ) m ′ + ( ρ ⊠ S x ⊠ S ) b ′ + ( ρ ⊠ S x +1 ) m + ρ ⊠ ( S a + · · · + S a t ) , where a − t ′ ≤ · · · ≤ a − < x − x + 1 < a ≤ · · · ≤ a t , and φ ρ ⊠ S d for any d > A -parameter of the form ψ > = φ + ρ ⊠ ( S a − t ′ + · · · + S a − ) + ( ρ ⊠ S x − ) m ′ + ( ρ ⊠ S x ⊠ S ) b ′ + ρ ⊠ ( S y +1 + · · · + S y m +1 )+ ρ ⊠ ( S a ′ + · · · + S a ′ t ) , where 2 y i + 1 ≡ a ′ i ≡ x + 1 mod 2 such that 2 x + 1 < y + 1 < · · · < y m + 1 < a ′ < · · · < a ′ t .When π = π ( ψ, l, η, η ), we set π > = π ( ψ > , l, η, η ). Then Mœglin’s construction says that π = J ◦ J ( π > ), where J = Jac ρ |·| ym ,...,ρ |·| x +1 ◦ · · · ◦ Jac ρ |·| y ,...,ρ |·| x +1 ,J = Jac ρ |·| a ′ t − ,...,ρ |·| at +12 ◦ · · · ◦ Jac ρ |·| a ′ − ,...,ρ |·| a . Set π ′ = J ◦ J ′ ( π > ) with J ′ = Jac ρ |·| ym ,...,ρ |·| x ◦ · · · ◦ Jac ρ |·| y ,...,ρ |·| x . Then π ′ = π < ψ ( ψ ′ , l, η, η ), where ψ ′ = φ + ρ ⊠ ( S d − t ′ + · · · + S d − )+ ( ρ ⊠ S x − ) m ′ + ( ρ ⊠ S x ⊠ S ) b ′ + ( ρ ⊠ S x − ) m + ρ ⊠ ( S d + · · · + S d t ) . However, the order < ψ for ( ρ ⊠ S x − ) m ′ + m and ( ρ ⊠ S x ⊠ S ) b ′ is ρ ⊠ S x − < ψ · · · < ψ ρ ⊠ S x − | {z } m ′ < ψ ρ ⊠ S x ⊠ S < ψ ρ ⊠ S x − < ψ · · · < ψ ρ ⊠ S x − | {z } m . To change the order so that ρ ⊠ S x − < ψ ′ ρ ⊠ S x ⊠ S for all ρ ⊠ S x − and ρ ⊠ S x ⊠ S , weuse a result of Xu [24, Theorem 6.1]. By this theorem, we see that π ′ = π ( ψ ′ , l ′ , η ′ , η ), where l ′ and η ′ are given in Theorem 4.1 (3). In particular, π ′ = 0 by Example 3.12 and Proposition3.13.Now, by [22, Corollary 5.4, Lemma 5.7] and Lemma 3.4, one can write π > ֒ → m × i =1 ∆ ρ [ y i , x ] ⋊ J ′ ( π > ) . Hence J ( π > ) ֒ → ( ρ | · | x ) m ⋊ J ′ ( π > ) so that we conclude that π ֒ → ( ρ | · | x ) m ⋊ π ′ . Since D ( m ) ρ |·| x ( π ) is irreducible or zero by Proposition 2.7, we see that D ( m ) ρ |·| x ( π ) = π ′ . (cid:3) The case x > . In this subsection, we will prove the following proposition. Proposition 5.1. Assume that x > . Consider π = L (( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b ; π ( φ, η )) and π = L (∆ ρ [ x − , − x ] b ; π ( φ, η )) . Set l = ( max { a − m x − + 1 , } if m x − m x +1 = 0 , δ x +1 = κ, max { a − m x − , } otherwise . If D ( k ) ρ |·| x ( π ) and D ( k ) ρ |·| x ( π ) are the highest derivatives, then k − k = l . N AN ALGORITHM TO COMPUTE DERIVATIVES 31 Assume this proposition for a moment. If l = 0 so that k = k , since x ≥ 1, we have π ֒ → ( ρ | · | − x ) a ⋊ π ֒ → ( ρ | · | x ) k × ( ρ | · | − x ) a ⋊ D ( k ) ρ |·| x ( π ) . Hence we have a non-zero map D ( k ) ρ |·| x ( π ) → ( ρ | · | − x ) a ⋊ D ( k ) ρ |·| x ( π ) . Since D ( k ) ρ |·| x ( π ) is irreducible, this map must be an injection. If l > 0, then π ֒ → ( ρ | · | − x ) l ⋊ L (( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ, η )) . Since D ( k ) ρ |·| x ( L (( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ, η ))) is the highest derivative, we must have D ( k ) ρ |·| x ( π ) = D ( k ) ρ |·| x ( L (( ρ | · | − x ) a − l , ∆ ρ [ x − , − x ] b ; π ( φ, η ))) ֒ → ( ρ | · | − x ) a − l ⋊ D ( k ) ρ |·| x ( π ) . Therefore, Proposition 5.1 implies Theorem 4.1 (for x > Proof of Proposition 5.1. We note that x − > 0. As explained in [8, § π = D ( α ) ρ |·| x − ( π ), π = D ( β ) ρ |·| x ( π ) and π = D ( γ ) ρ |·| x − ( π ) are the highest derivatives, then Jac ρ |·| x ( π ) = 0 and π ֒ → ( ρ |·| x ) max { β − α − γ, } × ( ρ |·| x − ) max { α − β + γ, } × L ( ρ |·| x − , ρ |·| x ) min { α,β − γ } × ∆ ρ [ x, x − γ ⋊ π . Hence we see that D ( k ) ρ |·| x ( π ) is the highest derivative with k = max { β − α − γ, } + γ. We compute α, β, γ by a case-by-case consideration using Jantzen’s algorithm ( § m x − = 0;(2) δ x +1 = 0, δ x − = 0 and m x − > δ x +1 = 1 and δ x − = 0;(4) δ x +1 = 0, δ x − = 1 and m x − ≡ δ x +1 = 1, m x +1 ≡ δ x − = 1 and m x − ≡ δ x +1 = 1, m x +1 ≡ δ x − = 1 and m x − ≡ δ x +1 = 0, δ x − = 1 and m x − ≡ δ x +1 = 1, δ x − = 1 and m x − ≡ δ x +1 = 1, δ x − = 1 and m x − ≡ φ = φ − ( ρ ⊠ S x − ) m x − + ( ρ ⊠ S x − ) m x − − , we see that π = D ( b + m x − − ρ |·| x − ( π ) = L (cid:16) ( ρ | · | − x ) a , ∆ ρ [ x − , − x ] b , ∆ ρ [ x − , − ( x − π ( φ , η ) (cid:17) is the highest derivative, so that α = b + m x − − 1. Note that π ֒ → L ( ρ | · | − x , ∆ ρ [ x − , − ( x − × ∆ ρ [ x − , − ( x − b × ( ρ | · | − x ) a + b − ⋊ π ( φ , η ) if a > 0. By [8, Proposition 3.4 (1)], with φ = φ − ( ρ ⊠ S x +1 ) m x +1 + ( ρ ⊠ S x − ) m x +1 , wesee that π = D ( a + b − m x +1 ) ρ |·| x ( π ) = L ( ρ | · | − x , ∆ ρ [ x − , − ( x − b +1 ; π ( φ , η ))is the highest derivative, so that β = a + b − m x +1 . By Corollary 4.2 for a = 0, we have γ = k = (cid:26) m x +1 if b ≡ ,m x +1 − b ≡ , Therefore, k − k = max { a − m x +1 − κ + 1 , } , where κ ∈ { , } such that κ ≡ b mod 2.In every case, a similar argument implies that k = k + l with l being in the assertion. (cid:3) The case x = 1 . In this subsection, we prove: Proposition 5.2. Proposition 5.1 holds even when x = 1 . As explained after Proposition 5.1, this proposition also implies Theorem 4.1 for x = 1.Note that Jantzen’s strategy to determine k from α, β, γ cannot be applied to the case x = 1.Instead of this, we use the following proposition. Proposition 5.3. Let φ ∈ Φ gp ( G ) and η ∈ c S φ . Suppose that ρ is self-dual of the same typeas φ . Set δ = (cid:26) if φ ⊃ ρ ⊠ ( S + S ) and η ( ρ ⊠ S ) = η ( ρ ⊠ S ) , otherwise . Consider π = L (( ρ |·| − ) a ; π ( φ, η )) . Then ρ |·| − ⋊ π is irreducible if and only if a ≥ m φ ( ρ ) − δ .Proof. Write m = m φ ( ρ ) and m = m φ ( ρ ⊠ S ). By Theorem 3.9, we may assume that a > a ≤ m , by Example 3.5, we have π ∈ Π ψ with ψ = φ − ρ a + ( ρ ⊠ S ⊠ S ) a . Also, if δ = 1 and a ≤ m − 1, by Proposition 3.14, we have π ∈ Π ψ with ψ = φ − ρ a + ( ρ ⊠ S ⊠ S ) a − ( ρ + ρ ⊠ S ) + ρ ⊠ S ⊠ S . In particular, when a ≤ m − δ , if D ( k ) ρ |·| ( π ) = 0, then k ≤ m − δ by Lemma 3.4. Since π ֒ → ( ρ | · | − ) a ⋊ π ( φ, η ) ֒ → ( ρ | · | ) m − δ × ( ρ | · | − ) a ⋊ D ( m − δ ) ρ |·| ( π ( φ, η )) , We have D ( m − δ ) ρ |·| ( π ) = 0. Since it is irreducible, we have D ( m − δ ) ρ |·| ( π ) ֒ → ( ρ | · | − ) a ⋊ D ( m − δ ) ρ |·| ( π ( φ, η )) . This implies the reducibility of ρ | · | − ⋊ π when a < m − δ .Suppose that a = m − δ > 0. We prove the irreducibility of ρ | · | − ⋊ π by inductionon a . Suppose that ρ | · | − ⋊ π is reducible and choose an irreducible quotient σ . Notethat D ( a ) ρ |·| − ( σ ) = 0. By [7, Proposition 4.1], σ = L (( ρ | · | − ) a ; π ( φ , η )) or σ = L (( ρ | ·| − ) a , ∆ ρ [0 , − π ( φ , η )) with φ = φ − ρ + ρ ⊠ S , φ = φ − ρ . N AN ALGORITHM TO COMPUTE DERIVATIVES 33 We consider the former case σ = L (( ρ | · | − ) a ; π ( φ , η )) with φ = φ − ρ + ρ ⊠ S . Note that m φ ( ρ ) = m − ≥ δ . By induction hypothesis, σ = ρ | · | − ⋊ L (( ρ | · | − ) a − ; π ( φ , η )) is anirreducible induction. In particular, D ( m +2 − δ ) ρ |·| ( σ ) = 0. Hence D ( m +2 − δ ) ρ |·| ( ρ | · | − ⋊ π ) = 0,which implies that D ( m +1 − δ ) ρ |·| ( π ) = 0. This contradicts Lemma 3.4. Now, we consider thelatter case σ = L (( ρ | · | − ) a , ∆ ρ [0 , − π ( φ , η )) with φ = φ − ρ so that m ≥ 2. Note thatJac ρ ( σ ) is nonzero and irreducible (up to multiplicity) since D (2) ρ ( π ) = 0 by Lemma 3.4. If σ ′ is the unique irreducible component of Jac ρ ( σ ), then σ ′ = L (( ρ | · | − ) a +1 ; π ( φ , η )) . Recall that a + 1 = m + 1 − δ . By induction hypothesis, σ ′ = ( ( ρ | · | − ) ⋊ L (( ρ | · | − ) m − − δ ; π ( φ , η )) if m > , ( ρ | · | − ) − δ ⋊ L (( ρ | · | − ) m − ; π ( φ , η )) if m = 2is an irreducible induction. Hence we have D ( m +3 − δ ) ρ |·| ( σ ′ ) = 0. Therefore we have D ( m +3 − δ ) ρ |·| (Jac ρ ( ρ | · | − ⋊ π )) = 0= ⇒ D ( m +2 − δ ) ρ |·| ( ρ | · | − ⋊ π ) = 0= ⇒ D ( m +1 − δ ) ρ |·| ( π ) = 0 , which contradicts Lemma 3.4.The case where a > m − δ follows from the case a = m − δ . This completes the proof. (cid:3) Now we prove Proposition 5.2 Proof of Proposition 5.2. Recall that π = L (( ρ |·| − ) a , ∆ ρ [0 , − b ; π ( φ, η )) and π = L (∆ ρ [0 , − b ; π ( φ, η )),and that D ( k ) ρ |·| ( π ) and D ( k ) ρ |·| ( π ) are the highest derivatives. Set l = ( max { a − m + 1 , } if m m = 0 , δ = κ, max { a − m , } otherwise . We will prove that k − k = l .Note that π ֒ → ( ρ | · | − ) l ⋊ L (( ρ | · | − ) a − l , ∆ ρ [0 , − b ; π ( φ, η )) . By applying Example 3.5 and Proposition 3.14 to L (( ρ | · | − ) a − l , ∆ ρ [0 , − b ; π ( φ, η )), we seethat D ( k ) ρ |·| ( L (( ρ | · | − ) a − l , ∆ ρ [0 , − b ; π ( φ, η ))) is the highest derivative. Hence we have k ≤ k + l . Also, we see that if l = 0, then k = k .Assume that l > 0. Since π is a unique irreducible subrepresentation of( ρ | · | − ) a × ∆ ρ [0 , − b ⋊ π ( φ, η ) ∼ = ∆ ρ [0 , − b × ( ρ | · | − ) a ⋊ π ( φ, η ) , by Proposition 5.3, we see that π ֒ → ∆ ρ [0 , − b × ( ρ | · | − ) a − m + δ ⋊ L (( ρ | · | − ) m − δ ; π ( φ, η )) ֒ → ρ b × ( ρ | · | − ) b + a − m + δ ⋊ L (( ρ | · | − ) m − δ ; π ( φ, η )) ∼ = ρ b × ( ρ | · | ) b + a − m + δ ⋊ L (( ρ | · | − ) m − δ ; π ( φ, η )) ֒ → ρ b × ( ρ | · | ) b + a − m + m ⋊ D ( m − δ ) ρ |·| ( L (( ρ | · | − ) m − δ ; π ( φ, η ))) . This implies that k ≥ a − m + m . Since k + l = a − m + m , we conclude that k = k + l . (cid:3) References [1] J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups . AmericanMathematical Society Colloquium Publications, . American Mathematical Society, Providence, RI , 2013.xviii+590 pp.[2] H. Atobe, Jacquet modules and local Langlands correspondence . Invent. Math . (2020), no. 3, 831–871.[3] A.-M. Aubert, Dualit´e dans le groupe de Grothendieck de la cat´egorie des repr´esentations lisses de longueurfinie d’un groupe r´eductif p -adique . Trans. Amer. Math. Soc . (1995), no. 6, 2179–2189 and Erratum .ibid. (1996), 4687–4690.[4] W. T. Gan and S. Takeda, A proof of the Howe duality conjecture . J. Amer. Math. Soc . (2016), no. 2,473–493.[5] C. Jantzen, Jacquet modules of p-adic general linear groups . Represent. Theory (2007), 45–83.[6] C. Jantzen, Tempered representations for classical p -adic groups . Manuscripta Math . (2014), no. 3-4,319–387.[7] C. Jantzen, Jacquet modules and irrreducibility of induced representations for classical p -adic groups . Manuscripta Math . (2018), no. 1-2, 23–55.[8] C. Jantzen, Duality for classical p -adic groups: the half-integral case . Represent. Theory (2018), 160–201.[9] T. Konno, A note on the Langlands classification and irreducibility of induced representations of p -adicgroups . Kyushu J. Math . (2003), no. 2, 383–409.[10] A. Kret and E. Lapid, Jacquet modules of ladder representations . C. R. Math. Acad. Sci. Paris (2012),no. 21-22, 937–940.[11] E. Lapid and A. M´ınguez, On a determinantal formula of Tadi´c . Amer. J. Math . (2014), no. 1,111–142.[12] E. Lapid and A. M´ınguez, On parabolic induction on inner forms of the general linear group over anon-archimedean local field . Sel. Math. New Ser . , 2347–2400 (2016).[13] E. Lapid and M. Tadi´c, Some results on reducibility of parabolic induction for classical groups . Preprint2017. Available at: .[14] A. M´ınguez, Correspondance de Howe explicite: paires duales de type II . Ann. Sci. ´Ec. Norm. Sup´er. (4) (2008), no. 5, 717–741.[15] A. M´ınguez, Sur l’irr´eductibilit´e d’une induite parabolique . J. Reine Angew. Math . (2009), 107–131.[16] C. Mœglin, Sur certains paquets d’Arthur et involution d’Aubert–Schneider–Stuhler g´en´eralis´ee . Repre-sent. Theory (2006), 86–129.[17] C. Mœglin, Paquets d’Arthur discrets pour un groupe classique p -adique . Automorphic forms and L -functions II . Local aspects , 179–257, Contemp. Math., , Israel Math. Conf. Proc., Amer. Math. Soc.,Providence, RI , 2009.[18] C. Mœglin, Comparaison des param`etres de Langlands et des exposants `a l’int´erieur d’un paquet d’Arthur . J. Lie Theory (2009), no. 4, 797–840.[19] C. Mœglin, Multiplicit´e dans les paquets d’Arthur aux places p -adiques . On certain L -functions , 333–374,Clay Math. Proc., , Amer. Math. Soc., Providence, RI , 2011.[20] C. Mœglin and J.-L. Waldspurger, Le spectre r´esiduel de GL( n ). Ann. Sci. ´Ecole Norm. Sup. (4) (1989),no. 4, 605–674.[21] M. Tadi´c, Structure arising from induction and Jacquet modules of representations of classical p -adicgroups . J. Algebra (1995), no. 1, 1–33.[22] B. Xu, On the cuspidal support of discrete series for p -adic quasisplit Sp ( N ) and SO ( N ). ManuscriptaMath . (2017), no. 3-4, 441–502.[23] B. Xu, On Mœglin’s parametrization of Arthur packets for p -adic quasisplit Sp( N ) and SO( N ). Canad. J.Math . (2017), no. 4, 890–960.[24] B. Xu, A combinatorial solution to Mœglin’s parametrization of Arthur packets for p -adic quasisplit Sp( N ) and O( N ). J. Inst. Math. Jussieu (2019). http://dx.doi.org/10.1017/S1474748019000409 . N AN ALGORITHM TO COMPUTE DERIVATIVES 35 [25] A. V. Zelevinsky, Induced representations of reductive p -adic groups. II. On irreducible representations of GL( n ). Ann. Sci. ´Ecole Norm. Sup . (4) (1980), no. 2, 165–210. Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido,060-0810, Japan E-mail address ::