Homological branching law for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$: projectivity and indecomposability

This paper studies homological properties of irreducible representations restricted from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. We establish the following: (1) classify irreducible smooth representations of $\mathrm{GL}_{n+1}(F)$ which are projective when restricted to $\mathrm{GL}_n(F)$; (2) prove that each Bernstein component of an irreducible smooth representations of $\mathrm{GL}_{n+1}(F)$, restricted to $\mathrm{GL}_n(F)$, is indecomposable. In appendixes, we study some aspects of Speh representations, and in particular we give an explicit formulae of Ext-groups between Speh representations in terms of symmetric group representations and discuss the related Ext-brancing law.


Introduction
Let F be a non-Archimedean local field. Let G n = GL n (F ). Let Alg(G n ) be the category of smooth representations of G n . This paper is a sequel of [CS18b] in studying homological properties of representations in Alg(G n+1 ) restricted to G n , which is originally motivated from the study of D. Prasad in his ICM proceeding [Pr18]. In [CS18b], we show that for generic representations π and π ′ of G n+1 and G n respectively, the higher Ext-groups Ext i Gn (π, π ′ ) = 0, for i ≥ 1, which was previously conjectured in [Pr18]. This result gives a hope that there is an explicit homological branching law, generalizing the multiplicity one theorem [AGRS10], [SZ12] and the local Gan-Gross-Prasad conjecture [GGP12].
The main techniques in [CS18b] are utilizing some Hecke algebra structure developed in [CS19,CS18] and simultaneously applying left and right Bernstein-Zelevinsky derivatives, based on the classical approach of using Bernstein-Zelevinsky filtration on representations of G n+1 restricted to G n [Pr93,Pr18]. We shall extend these methods further, in combination of other things, to obtain new results in this paper.
In [CS18b], we showed that an essentially square-integrable representation π of G n+1 is projective when restricted to G n . However, those representations do not account for all irreducible representations whose restriction is projective. The first goal of the paper is to classify such representations: Theorem 1.1. Let π be an irreducible smooth representation of G n+1 . Then π| Gn is projective if and only if (1) π is essentially square integrable, or (2) n + 1 is even, and π ∼ = ρ 1 × ρ 2 for some cuspidal representations ρ 1 , ρ 2 of G (n+1)/2 .
There are also recent studies of the projectivity under restriction in other settings [APS17], [La17] and [CS18c]. A main step in our classification is to show that an irreducible smooth representation of G n+1 is projective restricted to G n if and only if π is generic and any irreducible quotient of π| Gn is generic, which turns the projectivity into a Hombranching problem. This is a consequence of two things: (1) the Euler-Poincaré pairing formula of D. Prasad [Pr18] and (2) the Hecke algebra argument used in [CS18b] by G. Savin and the author. Roughly speaking, (1) is used to show nonprojectivity while (2) is used to show projectivity.
The second part of the paper studies indecomposability of a restricted representation. It is clear that an irreducible representation (except one-dimensional ones) restricted from G n+1 to G n cannot be indecomposable as it has more than one nonzero Bernstein component. However, the Hecke algebra realization in [CS18b,CS19] of the projective representations in Theorem 1.1 immediately implies that each Bernstein component of those restricted representation is indecomposable. This is a motivation of our study in general case, and precisely we prove: Theorem 1.2. Let π be an irreducible representation of G n+1 . Then each Bernstein component of π| Gn is indecomposable.
For a mirabolic subgroup M n of G n+1 , it is known [Ze80] that π| Mn is indecomposable for an irreducible representation π of G n+1 . The approach in [Ze80] uses the Bernstein-Zelevinsky filtration of π to M n and that the bottom piece of the filtration is irreducible. We prove that the bottom piece is indecomposable as a G n -module, and then make use of left and right derivatives, developed and used to prove main results in [CS18b]. The key fact is that left and right derivatives of an irreducible representation are asymmetric. We now make more precise the meaning of 'asymmetric'. We say that an integer i is the level of an irreducible representation π if the left derivative π (i) (and hence the right derivative (i) π) is the highest derivative of π. Theorem 1.3. Let π be an irreducible smooth representation of G n . Let ν(g) = |det(g)| F . Suppose i is not the level of π. Then ν 1/2 · π (i) and ν −1/2 · (i) π have no isomorphic irreducible quotients whenever ν 1/2 · π (i) and ν −1/2 · (i) π are non-zero.
Computing the structure of a derivative of an arbitrary representation is a difficult question in general. Our approach is to try to approximate the information of derivatives of irreducible ones by some parabolically induced modules, whose derivatives can be computed via geometric lemma. On the other hand, the Speh representations behave more symmetrically for left and right derivatives, which motivates our proof to involve Speh representations.
In Appendix A, we demonstrate a way to compute Ext-groups by a Koszul-type resolution constructed in [Ch16] and explain the Ext-branching problem. Hom -branching law for Arthur parameter representations, which includes Speh representations, is recently studied in [GGP] and [Gu18]. In Appendix B, we explain how an irreducible representation appears as the unique submodule of the product of Speh modules.
Section 2 studies derivatives of generic representations, which simplify some computations for Theorem 1.1. The results also give some guiding examples in the study of this paper and [CS18b].
1.1. Acknowledgements. This article is a part of the project in studying Extbranching laws, and the author would like to thank Gordan Savin for a number of helpful discussions. Part of the idea on indecomposability for restriction was developed during the participation in the program of 'On the Langlands Program: Endoscopy and Beyond' at IMS of National University of Singapore in January 2019. The author would like to thank the organizers for their warm hospitality.

Bernstein-Zelevinsky derivatives of generic representations
Denote a(∆) = ν a ρ and b(∆) = ν b ρ. The relative length of ∆ is defined as b − a + 1 and the absolute length of ∆ is defined as l(b − a + 1). We can truncate ∆ form each side to obtain two segments of absolute length r(b − a): Moreover, if we perform the truncation k-times, the resulting segments will be denoted by (kl) ∆ and ∆ (kl) (We remark that the convention here is different from the previous paper [CS18b] for convenience later). If i is not an integer divisible by l, then we set ∆ (i) and (i) ∆ to be empty sets. We also denote For a singleton segment [ρ, ρ], we abbreviate as [ρ]. For a representation ρ of G l , define n(ρ) = l.
Let U n be the group of unipotent upper triangular matrices in G n . For i ≤ n, let P i be the parabolic subgroup of G n containing the block diagonal matrices diag(g 1 , g 2 ) (g 1 ∈ G i , g 2 ∈ G n−i ) and the upper triangular matrices. Let P i = M i N i with the Levi M i and the unipotent N i . Let N − i be the opposite unipotent subgroup.
Let ν : G n → C given by ν(g) = |det(g)| F . Let Let R − n−i be the transpose of R n−i . We shall use Ind for normalized induction and ind for normalized induction with compact support.
For a smooth representation π of G n , define π (i) and (i) π to be the left and right Bernstein-Zelevinsky derivatives of π as in [CS18b]. To recall it, let ψ i be a character on U i given by ψ i (u) = ψ(u 1,2 + . . . + u i−1,i ), where ψ is a nondegenerate character on F . Following [CS18b], define π (i) to be the left adjoint functor of Ind Gn R n−i π ⊠ ψ i . Let θ n : G n → G n given by θ n (g) = g −T , the inverse transpose on g. Define the left derivative The level of an admissible representation of π is the largest integer i such that π (i) = 0 and π (j) = 0 for all j > i. By using [Ze80] and (2.1), if i is the level of π, then (i) π = 0 and (j) π = 0 for all j > i. When i is the level for π, we shall call π (i) and (i) π to be the highest left and right derivative of π respectively, where we usually drop the term of left and right if no confusion.
We shall often use the following lemma: Proof. By definitions, λ(m) = ζ(m) and hence has unique submodule and quotient. Since all segments in m are singletons, the submodule is generic [Ze80].
It is known that λ(m) always has a generic representation as the unique submodule. We shall prove a slightly stronger statement (using the Zelevinsky theory).
Proposition 2.3. Let m be a multisegment. Then λ(m) can be embedded to a full principle series. In particular, λ(m) has a unique simple submodule and moreover, the submodule is generic.
Proof. Let ρ be a cuspidal representation in the cuspidal support of λ(m) such that for any cuspidal representation ρ ′ in the cuspidal support λ(m), ρ does not precede ρ ′ . Let ∆ be a segment in m with the shortest relative length among all segments By definition of λ(m), we have that On the other hand, we have that Thus we have that The above isomorphism follows from the Zelevinsky theory and our choice of ∆. Now λ(m \ {∆} + ∆ − ) embeds to a full principle series λ ′ by induction. Thus this gives that b(∆) × λ(m \ {∆} + ∆ − ) embeds to b(∆) × λ ′ , which is also a principle series, and so does λ(m) by (2.2).
The second assertion follows from Lemma 2.2.
Proof. This is almost the same as the proof of [CS18b, Lemma 2.2]. More precisely, it follows for an irreducible G n -representation π, and the fact that θ n−i (τ ) ∼ = τ ∨ for any irreducible G n−i -representation τ .
Proposition 2.5. Let π be an irreducible smooth representation of G n+1 . The socle and cosocle of π (i) (and (i) π) are multiplicity-free.
Proof. Let π 0 be an irreducible quotient of π (i) . Let π 1 be a cuspidal representation of G i−1 which is not a unramified twist of a cuspidal representation in the cuspidal support of π 0 . Then Ext j G n+1−k (ν 1/2 · π (k) , (k−1) (π 0 × π 1 )) = 0 for all j and k < i. A long exact sequence argument using Bernstein-Zelevinsky filtration gives: ≤ dim Hom Gn (π, π 0 × π 1 ). Now the last dimension is at most one by [AGRS10] and so is the first dimension. This implies the cosocle statement by Lemma 2.1 and the socle statement follows from Lemma 2.4.
A smooth representation π of G n is called generic if π (n) = 0. The Zelevinsky classification of irreducible generic representations is in [Ze80], that is St(m) is generic if and only if any two segments in m are unlinked. With Proposition 2.5, the following result essentially gives a combinatorial description on socle and cosocle of the derivatives of a generic representation.
Corollary 2.6. Let π be an irreducible generic representation of G n+1 . Then any simple quotient and submodule of π (i) (resp. (i) π) is generic.
Proof. By Lemma 2.4, it suffices to prove the statement for quotient. Let m = {∆ 1 , . . . , ∆ r } be the Zelevinsky segment m such that Since any two segments in m are unlinked, we can label in any order and so we shall assume that for i < j, b(∆ j ) does not precede b(∆ i ). Then geometric lemma produces a filtration on π (i) whose successive subquotient is isomorphic to If π ′ is a simple quotient of π (i) , then π ′ is a simple quotient of one successive subquotient in the filtration, or in other words is a simple submodule of λ(m ′ ) for a multisegment m ′ . Now the result follows from Proposition 2.3.
Remark 2.7. One can formulate the corresponding statement of Proposition 2.5 for affine Hecke algebra level using a sign module in [CS19]. Then it might be interesting to ask for an analogue result for affine Hecke algebra over fields of positive characteristics.
Here we give a consequence to branching law: Corollary 2.8. Let π be a generic irreducible representation of G n+1 . Let π ′ be an irreducible smooth representation of G n and let m be a Zelevinsky multisegment with π ′ ∼ = m . If Hom Gn (π, m ) = 0, then each segment in m has relative length at most 2.
Let π 0 = m . By using the Bernstein-Zelevinsky filtration, Hom Gn (π, π 0 ) = 0 implies that we have (i−1) π 0 ֒→ (i−1) ζ(m) and so (i−1) ζ(m) has a generic composition factor. On the other hand, geometric lemma gives that (i−1) π 0 admits a filtration whose successive quotients are isomorphic to (i 1 ) ∆ 1 × . . . × (ir) ∆ r where i 1 , . . . , i r run for all sums equal to i − 1. Then at least one such quotient is non-degenerate and so in that quotient, all (i k ) ∆ k are cuspidal. Following from the derivatives on ∆ , ∆ can have at most of relative length 2.
Corollary 2.9. Let π be an irreducible generic representation of G n+1 . Then the projections of π (i) and (i) π to any cuspidal support component have unique simple quotient and submodule. In particular, the projections of π (i) and (i) π to any cuspidal support component are indecomposable.
Proof. For a fixed cuspdial support, there is an unique irreducible smooth generic representation. Now the result follows from Proposition 2.5 and Corollary 2.6.
Lemma 3.2. Let π be an irreducible G n+1 -representation. If π (i) has a non-generic irreducible submodule or quotient, then there exists a non-generic representation π ′ of G n such that Hom Gn (π, π ′ ) = 0. The statement still holds if we replace π (i) by (i) π.
Proof. By Lemma 2.4, it suffices just to consider that π (i) has a non-generic irreducible quotient, say λ. Now let where τ is a cuspidal representation such that τ is not an unramified twist of a cuspidal representation appearing in a segment in m. Here m is a multisegment with π ∼ = m . Now for j < n(τ ) since τ is in the cuspidal support of (j−1) π ′ whenever it is nonzero while τ is never in the cuspidal support of ν 1/2 · π (j) . Moreover, (n(τ )−1) π ′ has a simple quotient isomorphic to ν 1/2 λ. This checks the Hom and Ext conditions in Lemma 2.1 and hence proves the lemma. The proof for (i) π is almost identical with switching left and right derivatives in suitable places.
We can formulate the conditions (i) and (ii) combinatorially as follows. Let π ∼ = St(m) for a multisegment m = {∆ 1 , . . . , ∆ r }. Then (i) is equivalent to r = 1; and (ii) is equivalent to that r = 2, and ∆ 1 and ∆ 2 are not linked, and the relative lengths of ∆ 1 and ∆ 2 are both 1. We also call those m to be of restricted projective type.
Lemma 3.5. Let π be an irreducible generic representation of G n+1 , which is not restricted-projective. Then there exists an irreducible non-generic representation π ′ of G n such that Hom Gn (π, π ′ ) = 0.
Proof. It suffices to construct an irreducible non-generic representation π ′ satisfying the Hom and Ext properties in Lemma 2.1.
Case 1: r ≥ 3; or when r = 2, each segment has relative length at least 2; or when r = 2, ∆ 1 ∩ ∆ 2 = ∅. We choose a segment ∆ ′ in m with the shortest absolute length. Now we choose a maximal segment ∆ in m with the property that ∆ ′ ⊂ ∆. By genericity, ν −1 a(∆) / ∈ ∆ k for any ∆ k ∈ m. Let where τ is a cuspidal representation so that St(m ′ ) is a representation of G n and τ is not an unramified twist of any cuspidal representation appearing in a segment of m. To make sense of the construction, it needs the choices and the assumptions on this case. Let k = n(a(∆)), l = n(τ ). Let Now as for i < k + l + 1, either ν −1/2 a(∆) or τ appears in the cuspidal support of (i−1) π, but not in that of ν 1/2 π (i) , = 0 and all j. By Corollary 2.6, ν 1/2 · π (k+l+1) has a simple generic quotient isomorphic to ν 1/2 St( − ∆). On the other hand, (k+l) π ′ can be computed as follows: Here the non-zeroness comes from the fact that St(m ′ ) is the unique quotient of λ(m ′ ), and the isomorphism follows from Frobenius reciprocity. Since taking the derivative is an exact functor, we have that St(ν 1/2 · − ∆) is a subrepresentation of (k+l) St(m ′ ). Thus we have Case 2: r = 2 with ∆ 1 ∩ ∆ 2 = ∅ and one segment having relative length 1 (and not both having relative length 1 by the definition of restricted projective type). By switching the labeling on segments if necessary, we assume that ∆ 1 ⊂ ∆ 2 . Let p and let l be the absolute and relative length of ∆ 2 respectively. Let where τ is a cuspidal representation of G k (here k is possibly zero) so that St(m ′ ) is a G n -representation. Note that π ′ is non-generic. By Corollary 2.6 and geometric lemma, a simple quotient ν 1/2 ·π (p) is isomorphic to ν 1/2 a(∆ 1 ). Similar computation as in (3.4) gives that a simple module of (p−1) π ′ is isomorphic to ν 1/2 a(∆ 1 ). This implies the non-vanishing Hom between those two G n+1−p -representations.
Proof. When π is essentially square-integrable, it is proved in [CS18b]. We now assume that π is in the case (2) of Definition 3.4. It is equivalent to prove the condition (2) in Theorem 3.3. Let π ′ ∈ Irr(G n ) with Hom Gn (π, π ′ ) = 0. We have to show that π ′ is generic. Note that the only non-zero derivative of π (i) can occur when i = n + 1 and n+1 2 .
Theorem 3.7. Let π be an irreducible G n+1 -representation. Then π| Gn is projective if and only if π is restricted-projective in Definition 3.4.
Proof. The if direction is proved in Lemma 3.6. The only if direction follows from Lemma 3.5 and Theorem 3.1.
One advantage for such classification is that those restricted representations admit a more explicit realization as shown in [CS18b]: Theorem 3.8. Let π, π ′ be irreducible smooth representations of G n+1 . If π and π ′ are restricted projective, then π| Gn ∼ = π ′ | Gn . In particular, π| Gn is isomorphic to the Gelfand-Graev representation ind Gn Un ψ n . Proof. We have shown that π and π ′ have to be generic. Then we apply [CS18b, Corollary 5.5 and Theorem 5.6].
Definition 4.1. The affine Hecke algebra H l (q) of type A is an associative algebra over C generated by θ 1 , . . . , θ l and T w (w ∈ S l ) satisfying the relations: (1) θ i θ j = θ j θ i ; (2) T s k θ k − θ k+1 T s k = (q − 1)θ k , where q is a certain prime power and s k is the transposition between the numbers k and k + 1; Let A l (q) be the subalgebra generated by θ 1 , . . . , θ l . Let H W,l (q) be the subalgebra generated by T s 1 , . . . , T s l−1 . Let sgn be the 1-dimensional H W,l (q)-module characterized by T s k acting by −1.

Bernstein decomposition asserts that
where R(G n ) is the category of smooth G n -representations. Let B(G n ) is the set of inertial equivalence classes of G n and let R s (G n ) be the full subcategory of R(G n ) associated to s. For a smooth representation π of G n , define π s to be the projection of π to the component R s (G n ).
For each s ∈ B(G n ), [BK93] and [BK00] associate with a compact group K s and a finite-dimensional representation τ of K s , such that the convolution algebra is isomorphic to the product H n 1 (q 1 ) ⊗ . . . ⊗ H nr (q r ) of affine Hecke algebra of type A, denoted by H s . For a smooth representation π of G n , the algebra H(K s , τ ) acts naturally on the space Hom Ks (τ, π) ∼ = (τ ∨ ⊗ π) Ks . This defines an equivalence of categories: Proposition 4.3. For any s ∈ B(G n ), for any two submodules π 1 , π 2 in (ind Gn Un ψ n ) s , the intersection of π 1 and π 2 is non-zero. In particular, (ind Gn Un ψ n ) s is indecomposable.
Proof. This follows from the fact that Π s | As is isomorphic to A s and any two A ssubmodules of A s has non-zero intersection.
Remark 4.4. We give a proof for indecomposability for Π s more directly as below, which argument can be applied to other other connected quasisplit reductive groups G. For any s ∈ R(G), [BH03] showed that Hom G (Π s , Π s ) is isomorphic to the Bernstein center Z s of R s (G) [BH03]. Since R s (G) is an indecomposable category, we also have Z s is indecomposable as Z s -module. This implies that End G (Π s ) is indecomposable as Z s -module and hence Π s is indecomposable in B s (G).

Preserving indecomposability of Bernstein-Zelevinsky induction.
Lemma 4.5. Let π be an irreducible smooth representation of G n . Then Hom Gn (π, ind Gn Un ψ) = 0. Proof. Let s be the type such that π is an object in B s (G n ). Now π s is an irreducible finite-dimensional H s -module, but there is no finite-dimensional submodule for (ind Gn Un ψ n ) s ∼ = H s ⊗ H W,s sgn s (as there is no finite-dimensional submodule of A s as A s -module). Hence the Hom space is zero.
Lemma 4.6. Let P = LN be the parabolic subgroup containing upper triangular matrices and block-diagonal matrices diag(g 1 , . . . , Proof. Let w be a permutation matrix in G n . Then w(N ) ∩ U n contains a unipotent subgroup {I n + tu k,k+1 : t ∈ F } for some k if and only if w(N ) ⊂ U − n . Here u k,k+1 is a matrix with (k, k + 1)-entry 1 and other entries 0. For any such w, it gives that P wB is the same open orbit in G n . Now the geometric lemma in [BZ77, Theorem 5.2] gives the lemma.
Theorem 4.8. Let π be an admissible indecomposable smooth representation of Since we did not prove the second adjointness of Bernstein-Zelevinsky induction ind Gn R n−i π ⊠ ψ i for all smooth (not necessarily admissible) representations, we shall not use at this point.
Proof. We first prove the following: Lemma 4.9. For an admissible indecomposable smooth representation π of G n−i , as endomorphism algebras, Geometric lemma gives that there exists a filtration on (π × ind G i U i ψ i ) N i such that successive quotients are isomorphic to where j + k = i. Here N j , P j (resp. N k , P k ) are subgroup of G n−i (resp. G i ), and for a G n−i−j × G j × G i−k × G k -representation π, we denote π w to be a G n−i−j × G j × G i−k × G k -representation whose action is given by (g 1 , g 2 , g 3 , g 4 ). π w v = (g 1 , g 3 , g 2 , g 4 ). π v.
Since π is admissible, we have a filtration on π N j by simple composition factors, and we denote those successive simple quotients of π N j by τ 1 ⊠τ ′ 1 , . . . , τ p ⊠τ ′ p [BZ76]. For notion simplicity, we set Π l = ind G l U l ψ l . This gives that ind Here the first isomorphism follows from the second adjointness, the second isomorphism follows from Lemma 4.6, and the last isomorphism uses that τ ′ q is irreducible (and so admissible).
The above isomorphisms imply that Hom G n−i ×G i (τ q × Π j ⊠ τ ′ q × Π k , π × Π i ) = 0 whenever j = 0. Thus (4.8) gives that for j = 0 To complete the proof of Lemma 4.9, it remains to show that the isomorphism is also compatible with the algebra structure. This follows from that the isomorphism Section 5], in particular, [BZ77, 5.5]). From Frobenius reciprocity, the isomorphism

Hence, it gives an element in
Now we prove the theorem. For each s ∈ B(G n ), if (ind Gn P i π 1 ⊠ π 2 ) s = 0, then there exists a Bernstein component (s 1 , s 2 ) such that (π 1 ) s 1 × (π 2 ) s 2 = (π 1 × π 2 ) s . Now combining with Lemma 4.9, we have that: where Z s is the Bernstein center of the category R s 2 (G i ). Now theorem follows from Lemma 4.8 since the right hand side has only 0 and 1 as idempotents if and only if the left hand side does.
Let M n be the mirabolic subgroup in G n+1 , that is the subgroup containing all matrices with the last row of the form (0, . . . , 0, 1). We have the following consequence restricted from M n to G n . Here G n is viewed as the subgroup of M n via the embedding g → diag(g, 1).
Corollary 4.10. Let π be an irreducible smooth representation of M n . Then for any s ∈ B(G n ), π s is indecomposable.
Proof. Suppose j(k) < i(k). By the definition of i(k), we have that Let ω be an indecomposable component of ν 1/2 · (r(k)) π such that (ind Gn R n−i+1 ω ⊠ ψ i−1 ) ∩ τ k = 0. Let π ′ be an irreducible quotient of ω. Let µ be a generic representation such that π ′ × µ ∈ B s (G n ) and for any cuspidal representation ρ in the cuspidal support of µ and any integer c, ν c ρ does not appear in ν 1/2 m and ν −1/2 m, where m is the multisegment for π = m . By comparing cuspidal supports, we have that for i < i(k), and any j Hom Gn (ind Gn R n−i(k)+1 ω ⊠ ψ i(k)−1 , π ′ × µ) = 0 By intersecting the filtration with τ k , we then have On the other hand, our construction on π ′ × µ give that for any i ≤ j(k), by comparing cuspidal supports. This implies that Hom Gn (τ k , π ′ × µ) = 0 since τ k embeds to π/ j(k) π. This gives a contradiction. One similarly proves that j(k) < i(k) is impossible. Thus i(k) = j(k).

4.5.
Indecomposability of restricting an irreducible representation. We now prove our main result: Theorem 4.12. Let π be an irreducible representation of G n+1 . Then for each s ∈ R(G n ), π s is indecomposable whenever it is nonzero.
Theorem 4.13. Let π be an irreducible representation of G n+1 . If i is not the level for π, then ν 1/2 · π (i) and ν −1/2 · (i) π does not have isomorphic irreducible quotient whenever the two derivatives are not zero.
The proof of Theorem 4.13 will be carried out in Section 5. Note that the converse of the above theorem is also true, which follows directly from the well-known highest derivative due to Zelevinsky [Ze80, Theorem 8.1].

Asymmetric property of left and right derivatives
We are going to prove Theorem 4.13 in this section. The idea lies in two simple cases: The first one is a generic representation. Since an irreducible generic representation is isomorphic to λ(m) ∼ = St(m) for a Zelevinsky multisegment m, a simple counting on cuspidal support on derivatives can show Theorem 4.13 for that case. The second one is an irreducible representation whose Zelevinsky multisegment has segments with relative length strictly greater than 1. In such case, one can narrow down the possibility of irreducible submodule via the embedding m (i) ֒→ ζ(m) (i) and (i) m ֒→ (i) ζ(m), and use geometric lemma to compute the submodules of derivatives of ζ(m) (i) and (i) ζ(m). The combination of these two cases seems to require some extra work. We shall use the notations and terminology for Speh representations in the appendices.

5.2.
Proof of Theorem 4.13. By Lemma 2.6, it suffices to prove the same statement for submodules of the derivatives. Let m be the Zelevinsky multisegment with π ∼ = m . We shall assume that the cuspidal representations in each segment of m is an unramified twist of a fixed cuspidal representation ρ. We shall prove that Theorem 4.13 for such π. The general case follows from this by writing an irreducible representation as a product of irreducible representations of such specific form.
Since taking derivative is an exact functor, π (i) embeds to ζ(m) (i) and so does π ′ .
We write m as the sum of Speh mulitsegments For each m k , write m k = a(m k , ∆ k ) and define b(m k ) = b(∆ k ). We shall label m k in the way that b(m k ) does not precede b(m l ) for k < l. Furthermore, the labelling satisfies the property that (♦) for any m p and p < q, m p + ∆ is not a Speh multisegment for any ∆ ∈ m q . (♦♦) if m p ∩ m q = ∅ and p ≤ q, then m q ⊂ m p . Let n 1 be the collection of all segments in m\(m ′ 1 +. . .+m ′ r ) such that any segment ∆ ′ in n 1 satisfies the property that (1) Define inductively that n k is the collection of all segments in m \ (m ′ 1 + . . . + m ′ k−1 + n 1 + . . . n k−1 ) such that any segment ∆ ′ in n k satisfies the property that By Lemma 7.4, we have a series of embedding: ֒→ζ(n 1 ) × m 1 × ζ(n 2 ) × m 2 × ζ(n 3 + . . . + n r + m 3 + m r + n r+1 ) ֒→ζ(n 1 ) × m 1 × ζ(n 2 + . . . + n r + m 2 + m r + n r+1 ) ֒→ζ(n 1 + . . . n r+1 + m 1 + . . . + m r ) = ζ(m) For simplicity, define, for k ≥ 0 Then for each k, we again have an embedding: As λ k is a product of representations, we again have a filtration on λ (i) k . This gives that π ′ embeds to a successive quotient of the filtration: Now we assume there exists a smallestk such that at least one of qk l is not equal to the level of m l . Now we shall denote such l by l * . We use similar strategy to further consider the filtrations on each n a by geometric lemma. For that we write n a = ∆ a,1 , . . . , ∆ a,r(a) and ok +1 = ∆k +1,1 , . . . , ∆k +1,r(k+1) .
We claim (*) that if ∆ a,b has an relative length at least L + 1, then p a,b = n(ρ). This indeed follows from Lemma 5.1(1), since π ′ is a composition factor of ζ(m ′ ) for some m ′ that ν 1/2 ( n 1 + . . . + nk + ok +1 ) ⊂ m ′ . Now from our choice ofk, we have that m l * (qk l * ) is not a Speh representation. We can write m l * = ν −x+1 ∆ * , . . . , ∆ * for a certain ∆ * with relative length L and some x. By (♦), ν∆ * / ∈ m l for any l > l * from our labelling on m l . Rephrasing the statement, we get the following statement: (**) ν 1/2 ∆ * / ∈ ν −1/2 m l for any l > l * . Now with (*), we have that π ′ is a composition factor of ζ(m ′′′ ) with m ′′′ contains all the segments ∆ − with ∆ in m that has relative length at least L + 1 and a segment ν 1/2 ∆ * , and we shall call the former segments (i.e. the segment in the form of ∆ − ) to be special for convenience.
We can apply the intersection-union process to obtain the Zelevinsky multisegment for π ′ from m ′′′ . However in each step of the process, any one of the two segments involved in the intersection-union cannot be special, otherwise, there exists l ≥ L + 1 such that the number of segments in the resulting multisegment with relative length l is more than the number of segments in m (i 1 ,...,ir) . Hence we obtain the following: (***) the number of segments ∆ in m (i 1 ,...,ir) such that ν 1/2 ∆ * ⊂ ∆ is at least equal to one plus the number special segments in m ′′′ satisfying the same properties Now we come to the final part of the proof. We now consider ν −1/2 · (i) π. Following the strategy for right derivatives, we have that for each k π ′ ֒→ ν −1/2 · (i) π ֒→ ν −1/2 · (i) λ k .

This gives that π
Again assume there exists a smallest k such that at least one of q k l is not equal to the level of m l .
We firstly consider the case that k ≥k. In this case we similarly have that where n a = (u a,1 ) ∆ a,1 , . . . , (u a,r(a) ) ∆ a,r(a) .
with u a,1 + . . . + u a,r(a) = u a and each u a,b = 0 or n(ρ). Since we assume that k ≥k, we have that (v l ) m l is a highest derivative and so is a Speh representation, and we can apply Lemma 7.4(1). Hence the unique subrepresentation of is isomorphic to where m l = (v l ) m l . Similar to (*) for right derivatives (but the proof could be easier here), we obtain the analogous statement for those n a . Now if such that ∆ = ν 1/2 ∆ * , then we must have that ∆ = ν −1/2 · − ∆ 0 or ν −1/2 · ∆ 0 for some segment ∆ 0 in m. However, by (**), the possibility ∆ = ν −1/2 ∆ 0 cannot happen. Thus we must have that ∆ is a special in the same sense as the discussion in right derivatives. This concludes the following: (****) The number of segments ∆ in ν −1/2 ( n 1 + m 1 + . . . + nk −1 + mk −1 + ok) with the property that ν 1/2 ∆ * ⊂ ∆ is equal to the number of special segments satisfying satisfying the same properties. Now the above statement contradicts to (***) since both Zelevinsky multisegments give an irreducible representation isomorphic to π ′ . Now the way to get contradiction in the case k ≥k is similar by interchanging the role of left and right derivatives. We remark that to prove the analogue of (**), one uses (⋄). And to obtain the similar isomorphism as (5.11), one needs to use Lemma 7.4(2). We can argue similarly to get an analogue of (***) and (****). Hence the only possibility that ν 1/2 · π (i) and so ν −1/2 · (i) π have an isomorphic irreducible quotient only if i is the level for π.

Another consequence.
Here is another consequence on the Hom-branching law in another direction: Corollary 5.2. Let π ′ be an irreducible smooth representation of G n . Let π be an irreducible smooth representation of G n+1 . Suppose π is not a 1-dimensional representation of G n+1 . Then Hom Gn (π ′ , π| Gn ) = 0.

Appendix A: Ext-groups for Speh representations
We studied the homological properties for generic representations in [CS18]. We shall discuss another interesting class of representations, namely the Speh representations, under restriction. We remark that in our context, we do not require Speh representations to be unitary. Speh modules will be used in the next section and we make a digression to discuss some related problems. Indeed, the problem of computing the restricted Ext-groups is partially reduced to computing the ordinary Ext-groups between Speh representations. We explain how one can compute such Ext-groups, and the way to do so is to pass to the setting of graded Hecke algebra modules, which makes use of a resolution constructed in [Ch16]. We define L(u(m, ∆)) to be the relative length of ∆.
We similarly define  such that a(m, ∆) is in R s (G n ). In [BK93], we have that R s (G n ) is equivalent to the category of representations of Hecke algebra H dm := H dm (q).
6.2. Ext-groups of Speh representations. Thus (4.6) gives that If u(m, d) and u(m ′ , d ′ ) have different cuspidal support, then for all i Thus we only have to consider that u(m, d) and u(m ′ , d ′ ) have the same cuspidal support. In H dm setting, that is to consider u(m, d) s and u(m ′ , d ′ ) s with the same central character of H md . Let Z be the center of H dm . Let J be the corresponding ideal in Z annihilating u(m, d) and u(m ′ , d ′ ). We need to pass to the graded Hecke algebra.
Definition 6.2. The graded Hecke algebra H dm is an associative algebra with unit generated by symbols t w (w ∈ S dm ) and x 1 , . . . , x dm satisfying the relations: There exists a corresponding ideal J in the center Z of H dm such that the category of finite-dimensional H dm -modules annihilated by some powers of J is equivalent to the category of finite-dimensional H dm -modules annihilated by some powers of J [Lu89] (also see [CS19, Theorem 6.2]). Let v(m, d) (resp. v(m ′ , d ′ )) be the corresponding H dm -module u(m, d) s (resp. u(m ′ , d ′ ) s ) under that equivalence of categories. Thus we also have: By tracing [BK93, Theorem 7.6.20] and Lusztig reduction [Lu89], we have that v(m, d) is isomorphic to an H dm -module which is irreducible restricted to C[S dm ]. Here C[S dm ] is generated by t s i (i = 1, . . . , dm − 1). According to [BM99,BC14], we shall denote such module by b(m, d), whose restriction to S dm -representation is isomorphic to the irreducible Specht module σ(m, d) of S dm corresponding to the partition (d, . . . , d), where d appears m-times. In particular, σ(m, 1) is the sign representation and σ(1, d) is the trivial representation.
(2) Case 2: l < b + m−1 2 . The argument is similar. Again suppose τ is a composition factor of a(m, ∆) × ∆ ′ . Using an argument similar to above, we have that the level of τ is either m + 1 or m. However, if the level of τ is m, then τ (m) would be a(m, ∆ − ) × ∆ ′ , which is irreducible by induction. Then it would imply that the Zelevinsky multisegment of τ (m) is m + 1 and contradicts that the number of segments for the Zelevinsky multisegment of the highest derivative of an irreducible representation π must be smaller than that for π. Hence, the level of τ must be m+1. Now repeating a similar argument as in (1) and using the induction, we obtain the statements. Proof. The statment is clear if ∆ is singleton. For the general case, we note by simple countng that the Zelevinsky multisegment of any composition factor must contain a segment ∆ 1 with b(∆ 1 ) ∼ = ν (m−1)/2 b(∆) and at least one segment ∆ 2 with b(∆ 2 ) ∼ = ν (m−1)/2 b(∆ − ). Then one proves the statement by a similar argument using highest derivative as in the previous lemma.  Proof. We shall label the segments m in the way that for i < j, (i) b(∆ i ) does not proceed b(∆ j ) and (ii) if b(∆ i ) = b(∆ j ), then a(∆ i ) does not proceed a(∆ j ). Let ∆ = ∆ 1 . Let k be the largest integer (k ≥ 0) such that ∆, ν −1 ∆, . . . , ν −k ∆ are segments in m. We claim that (7.14) m ֒→ m ′ × m \ m ′ and moreover m is the unique submodule of m ′ × m \ m ′ . By induction, the claim proves the lemma. We now prove the claim. We shall prove by induction that for i = 0, . . . , k, where m i = ∆, ν −1 ∆, . . . , ν −i ∆ . When i = 0 the statement is clear from the definition. Now suppose that we have the inductive statement for i. To prove the statement for i + 1. We now set n to be the collection of all segments ∆ ′ in m such that b(∆ ′ ) = b(∆), ν −1 b(∆), . . . , ν −i b(∆) and ν −i a(∆) proceeds a(∆ ′ ). We also setn to be the collection of all segments ∆ ′ in m such that b(∆ ′ ) = b(∆), ν −1 b(∆), . . . , ν −i b(∆) and a(∆ ′ ) precedes ν −i−1 a(∆). By using the Zelevinsky theory, we have that ζ(n) × ζ(n) ∼ = ζ(n + n) and ζ(m \ m i ) ֒→ ζ(n) × ζ(n) × ζ(m \ (n +n + m i ).
From our construction (and i = k), we have that ν −i−1 ∆ is a segment in m \ (n + n + m i ) and m \ (n +n + m i ) = ν −i−1 ∆ × ζ(m \ (n +n + m i+1 )) On the other hand The first and last isomorphism is by Lemma 7.3. The injectivity in forth line comes from the uniqueness of submodule in ζ(m i+1 ). The second isomorphism follows from [Ze80]. This proves (1) and (3). And (2), (4) and (5) follows from the inductive construction.
We shall need a variation which is more flexible in our application.