On the product functor on inner forms of the general linear group over a non-Archimedean local field

Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $\sigma$ of $G_n$. Lapid-M\'inguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $\pi \boxtimes \sigma$ when $\pi$ is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some $G_m$, on which the parabolic induction functor $\tau \mapsto \tau \times \sigma$ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-H\"older sequence of the big derivative.


Introduction
Let F be a non-Archimedean local field and let D be a finite-dimensional F -central division algebra.Let G n = GL n (D) be the general linear group over D. Let Alg(G n ) be the category of smooth representations of G n over C. The parabolic induction is an important tool in constructing representations and plays a central role in the Zelevinsky classification of irreducible representations of GL n (F ) [Ze80].Recently, Aizenbud-Lapid and Lapid-Mínguez [LM16, LM19, LM20, AL22, LM22] extensively study the irreducibility of parabolic inductions, with rich connections to combinatorics and geometry.
This paper focuses on some homological aspects of parabolic inductions.The main purpose is to elaborate some observations and results in [Ch22], which we use functorial properties of parabolic inductions for proving the local non-tempered Gan-Gross-Prasad conjecture [GGP20].Our main result addresses the remark in [Ch22, Section 9.2].
We first explain the main object-the product functor.We denote by × the normalized parabolic induction.For a fixed irreducible representation π and a full subcategory A of Alg(G n ), define × π,A : A → Alg(G n+k ), given by × π,A (ω) = π × ω.Here we regard × π,A as a functor such that for a morphism f : π ′ → π ′′ in A, × π,A (f ) = Id π × f , the one induced from the parabolic induction (see Section 3.1 for more precise descriptions).
Some general results about the product functor with respect to smooth duals and cohomological duals are given in Section 3.
We briefly recall the Zelevinsky theory [Ze80] for D = F , see Section 2.1 for more notations.A segment takes the form [a, b] ρ for a supercuspidal representation ρ of some G m and a, b ∈ C with b − a ∈ Z ≥0 .Zelevinsky [Ze80] associates each segment ∆ with a representation ∆ , called a segment representation.A multisegment is a multiset of segments.Let Mult be the set of multisegments.For m ∈ Mult, let m be the associated Zelevinsky module [Ze80].
The irreducibility of the parabolic induction is extensively studied in [LM16].A first question is that for a given irreducible representation π of G n , how one can find another irreducible representation π ′ of G m such that π × π ′ is also irreducible.One way to do so is via 'building from the (basic) segment case'.The precise meaning is as follows.Set Then, for any n ∈ M π , n × π is irreducible [LM16, Proposition 6.1].The converse is not true in general i.e. if n × π is irreducible, it is not necessary that n ∈ M π .
We write m 1 ≤ Z m 2 if m 1 is obtained from m 2 by a sequence of intersection-union operations (see Section 2.2).Our observation is that the set M π is closed under intersectionunion operations in the following sense: Theorem 1.1.(=Theorem 4.1) Let π be an irreducible representation of G n .For n ∈ M π , if n ′ is another multisegment with n ′ ≤ Z n, then n ′ ∈ M π .
Our proof for Theorem 1.1 uses properties from intertwining operators on -irreducible representations.Another possible approach for proving Theorem 1.1 is to use the combinatorial criteria of Lapid-Mínguez in [LM16, Proposition 5.12].
We now set A π = Alg π (G n ) to be the full subcategory of Alg(G n ) whose objects are of finite length and have all simple composition factors isomorphic to m for some m ∈ M π .The significance of Theorem 1.1 is that one can obtain plenty examples of extensions from the set M π and so A π is not semisimple in most of cases.Indeed, those extensions are preserved under × π,Aπ , shown in Proposition 5.7 and Theorem 9.2.This in turn implies our main result: Theorem 1.2.(=Theorem 10.2) Let π be an irreducible representation of G n .Then × π,Aπ is a fully-faithful functor.
[Ch22] deals with the case that π is a Speh representation and A is some subcategory coming from the irreducibility of the product between a cuspidal representation and π.
A key new ingredient in the proof of Theorem 1.2 is a construction of extensions between two irreducible representations.This differs from the approach used in [Ch22], although we also need a basic case (when π is also a segment representation) from [Ch22].The main idea comes from a study of first extensions in the graded Hecke algebra case in [Ch18].Roughly speaking, those extensions for two non-isomorphic representations come from Zelevinsky standard modules, and those for two isomorphic representations reduce to the tempered case.However, we remark that we do not have a concrete classification for indecomposable modules of length 2.
For the self-extension case, we actually have more general statement: Theorem 1.3.(=Theorem 9.2) Let π 1 and π 2 be irreducible representations of G k and G l respectively such that π 1 × π 2 is still irreducible.Suppose λ is an indecomposable representation of length 2 with both simple composition factors isomorphic to π 2 .Then π 1 × λ is also indecomposable.
Perhaps an interesting point of Theorem 1.3 is that the parabolic induction does not preserve indecomposability in general.In other words, some non-trivial self-extensions can be trivialized under parabolic inductions (see Remark 9.3).
Theorem 1.3 concerns about indecomposable modules of length 2. Our proof relies on some constructions of those modules.One important ingredient is analogous properties in the affine highest weight category introduced by Kleshchev [Kl15] (also see [Ka17]), see the proofs in Section 5. Roughly speaking such ingredient reduces to the computations of Ext-groups for tempered modules.Such Ext-groups are now better understood due to the work on discrete series by Silberger, Meyer, Opdam-Solleveld [Si79, Me06, OS09] using analytic methods and by [Ch16] using algebraic methods; and more general case [OS13] via R-groups.We also refer the reader to [Ch18] for more discussions.
Recent articles [LM16, LM19, LM20, AL22, LM22] study the conditions of irreducibility for more general multisegments.In particular, when one of the multisegments arises from a so-called -irreducible representation, there are some precise conjectures connecting to the geometry of nilpotent orbits due to Geiß-Leclerc-Schröer and Lapid-Mínguez [GLS11,LM19,LM20].Thus one may hope for a version of Theorem 1.2 for replacing the segment case by other interesting classes of representations such as Speh, ladder or even -irreducible representations.One main problem goes back to understand the analog of the set M π in Theorem 1.1 and so Alg π (G n ) in Theorem 1.2.In [GL21], Gurevich-Lapid introduce a new class of representations parabolically induced from ladder representations, and so it is natural to ask if the extensions arising from those standard representations can be used to define a suitable analogue of Alg π (G n ).
We now consider a Jacquet functor version of above discussions.For an irreducible representation π of G k and for an admissible representation τ of G n , define where N − is the opposite unipotent radical of the standard parabolic subgroup in to the first factor.We shall call such D π to be a big derivative, and D π (τ ) has a natural G n−k -module structure (also see Definition 8.1).
The big derivative D π is the right adjoint functor of the product functor, if we consider the functors are for the category of all smooth representations.However, this is not entirely correct if we restrict the functor to the full subcategory Alg π (G n ) defined above.Nevertheless, there are some interesting cases that D π forms an adjoint functor for × π,Alg π (Gn) .For example, the case considered in [Ch22] works.In those cases, we could deduce that the big derivative is irreducible (see Lemma 11.3 for a precise statement).This is consequently applied to prove: Theorem 1.4.(c.f.Theorem 12.7) Suppose D = F .Let π be a generic irreducible representation of G k .Let τ be any irreducible representation of G n such that D π (τ ) = 0. Then D π (τ ) satisfies the socle irreducible property (i.e.D π (τ ) has unique simple submodule and such simple submodule appears with multiplicity one in the Jordan-Hölder sequence of D π (τ )).
When π is a cuspidal representation, the analogous statement for Theorem 1.4 for the affine Hecke algebra of type A is shown in the work of  by exploiting the explicit structure of a principal series.
The irreducibility part of the socle in Theorem 1.4 is shown by Kang-Kashiwara-Kim-Oh [KKKO15] (also see [AL22]) in a greater generality on -irreducible representations.For the irreducibility part of the socle, some variants of more specific cases using Gelfand-Kazhdan method are also shown by Aizenbud-Gourevitch [AG12].
Our emphasis on Theorem 1.4 is on the application of the product functor, which gives a basic case in Proposition 11.5.An advantage of this method is that one does not have to compute some internal structures of some modules.Hence, it has a higher potential for other applications such as the one in [Ch22].We shall also show how to extend the socle irreducible result to the case of generic representations (i.e.full version of Theorem 1.4) in the appendix.
As an analog of the problem of studying the irreducibility of parabolic inductions, one may ask the irreducibility of big derivatives.The product functor provides a technique on such problem as shown in the article while Theorem 1.4 provides some concrete examples.
A more classical viewpoint on studying parabolic inductions and Jacquet functors is on the Grothendieck group of the category of smooth representations of GL n (F )'s, in which those functors give a Hopf algebra structure [Ze80,Ze81,Ta90,Ta95,HM08].We hope this work could emphasis on some interesting higher structures associated to parabolic inductions and Jacquet functors.For a supercuspidal representation ρ of G n , let s ρ be the unique value in We also consider the empty set as a segment and also set [a, a − 1] ρ = ∅.The absolute length l abs ([a, b] A multisegment is a multiset of non-empty segments, and we also consider the empty set as a multisegment.For a multisegment m = {∆ 1 , . . ., ∆ k }, define l abs (m) = l abs (∆ 1 ) + . . .+ l abs (∆ k ), called the length of m.
We introduce the following notations: ) of all the smooth G n -representations of finite length; • let Alg f be the set of smooth representations of some G n (in other words, it is the set of all objects in Alg f (G n ) for some n); • Seg n = the set of segments of absolute length n; and Seg = ∪ n Seg n ; • Mult n = the set of multisegments of length n; and Mult = ∪ n Mult n ; • for π ∈ Alg f , JH(π) = the set of simple composition factors in π (i.e.multiplicities are not counted); • for π ∈ Irr, we write csupp(π) = {σ 1 , . . ., σ r } to be the unique multiset of supercuspidal representations such that π is a composition factor of σ 1 × . . .× σ r ; • for π ∈ Alg f , let soc(π) be the socle (i.e.maximal semisimple submodule) of π and let cosoc(π) be the cosocle (i.e.maximal semisimple quotient) of π.Since we are working on representations over C, we shall not distinguish cuspidal representations and supercuspidal representations.
2.2.More notations for segments and multisegments.For m ∈ Mult, two segments ∆ 1 and ∆ 2 in m are said to be linked if ∆ 1 ∪ ∆ 2 is still a segment and A multisegment m is said to be generic if any two segments in m are not linked.
As in [Ze80], for m, n ∈ Mult n , we say that m is obtained by an intersection-union process if there are two linked segments Here + and − represent the union and substraction as multisets.Write m < Z n if m can be obtained by a sequence of intersection-union operations from n, and write m ≤ Z n if m = n or m < Z n.
that there is a unique simple submodule in ζ(m), which will be denoted m .It is independent of a choice of a labeling above.There is a one-to-one correspondence The Zelevinsky classification is due to Zelevinsky [Ze80] when D = F and due to Mínguez-Sécherre [MS13,MS14] when D is general.
On the other hand, λ(m) has unique simple quotient, denoted by St(m).This gives another one-to-one correspondence The above correspondence in the form due to Langlands is known for D = F in [Ze80] and the general case follows from the local Jacquet-Langlands correspondence due to Deligne-Kazhdan-Vignéras [DKV84] for the zero characteristic and Badulescu [Ba02] for positive characteristics.Such classification also has significance in the unitary dual problem, see work of Tadić, Sécherre, Badulescu-Henniart-Lemaire-Sécherre [Ta90, Sc09, BHLS10].
2.4.Parabolic inductions and Jacquet functors.For non-negative integers n 1 , . . ., n r with n 1 + . . .+ n r = n, let P n1,...,nr be the parabolic subgroup containing the subgroup G n1 × . . .× G nr as block diagonal matrices and all upper triangular matrices.We shall call P n1,...,nr to be a standard parabolic subgroup.(Note that when n i is zero, G ni is regarded as the trivial group and we may simply drop the term.We include such case for the convenience of notations later.)Let N n1,...,nr be the unipotent radical of P n1,...,nr , and let N − n1,...,nr be the unipotent radical of the parabolic subgroup opposite to P n1,...,nr .For π 1 ∈ Alg(G n1 ) and π 2 ∈ Alg(G n2 ), define π 1 × π 2 to be where δ is the modular character of P n1,n2 .The G n -action on π 1 ×π 2 is the right translation on those functions i.e. for f ∈ π 1 × π 2 , Here we consider π 1 ⊠ π 2 as a P n1,n2 -representation by the inflation.We shall simply call π 1 × π 2 to be a product.The product is indeed an associative operation and so there is no ambiguity in defining π 1 × . . .× π r .
For a parabolic subgroup P of G n with the Levi decomposition LN , define the Jacquet functor, as a L-representation: where δ P is the modular character of P .Both parabolic inductions and Jacquet functors are exact functors.For n 1 +. ..+n r = n, the parabolic induction π → Ind Gn Pn 1 ,...,nr π has the Jacquet functor π → π Nn 1 ,...,nr as its left adjoint functor, and has the opposite Jacquet functor π → π N − n 1 ,...,nr as its right adjoint functor.
Following [LM19], for π ∈ Irr, π is said to be -irreducible if π × π is irreducible.Let Irr be the set of -irreducible representations.In the content of quantum affine algebras, it is called real modules, see e.g.work of Hernandez-Leclerc and Kang-Kashiwara-Kim-Oh [HL13,KKKO15].A particular class of -irreducible representations is those St(m) for a generic multisegment m ∈ Mult.

Geometric lemma. For
The geometric lemma [BZ77] gives that (π 1 × π 2 ) Ni admits a filtration, whose successive subquotients are of the form: Moreover, the bottom layer in the filtration of (π 1 × π 2 ) Ni is when i 1 = min {n 1 , i} and the top layer in the filtration of (π 1 × π 2 ) Ni is when i 2 = min {n 2 , i}.
We sometimes use the formulas implicitly in computing layers involving the geometric lemma.
3. Some generalities of the product functor 3.1.Product functor.Let A be a full Serre subcategory of Alg(G n ).For a fixed irreducible representation π of G k , we define the product functor and, for a morphism f from τ to τ ′ in A and F ∈ π × τ (under the realization in Section 2.4), Note that we do not assume × π,A preserves simple objects at this point.
Proof.Exactness follows from that the parabolic induction is an exact functor.The faithfulness then follows from that the functor sends a non-zero object to a non-zero object.
3.2.Smooth dual functor.In this section, we specify to D = F .Let θ : G n → G n given by θ(g) = g −t , the transpose inverse of g.This induces a covariant auto-equivalence for Alg(G n ), still denoted by θ.
One main example is the category Alg π (G n ), as shown in Theorem 4.1 later.Define = θ • ∨ and so it is also a contravariant functor.
Then × π,A is fully-faithful if and only if × π,A is fully-faithful.
Proof.We only prove the if direction, and the only if direction can be proved similarly.We have the following isomorphisms: The first and last isomorphisms follow from taking the duals (and the representations are admissible).The second isomorphism follows from the compatibility between taking duals and parabolic inductions [MW86, Page 173].The third isomorphism follows from a result of Gelfand-Kazhdan [GK75, Theorem 2] (see [BZ76,Theorem 7.3]).The fourth isomorphism follows from the if direction.

Cohomological dual functor.
Let H(G n ) be the space of compactly supported smooth C-valued functions on G n , viewed as a G n -representation with the action given by: We first recall the following result of Bernstein: and Alg(G n2 ) respectively.Let π 1 and π 2 be finitely-generated objects in R 1 and R 2 respectively.Let R be the unique Bernstein component containing the object We remark that the switch in the terms on the RHS comes from switching the induction between a standard parabolic subgroup and its opposite one.
Corollary 3.5.Let R be a Bernstein component of Alg(G n ) and let R f be the full subcategory of R of all objects of finite lengths.Let A be a full Serre subcategory of R f .Let π ∈ Irr.This gives a full subcategory D(A) whose objects are D(π) for objects π in A and morphisms D(f ) for morphisms f in A. Then × π,A is a fully-faithful functor if and only if × D(π),D(A) is a fully-faithful functor.Here × D(π),D(A) is defined in Proposition 3.3.
Proof.This follows from Theorem 3.4 and that D is a fully-faithful contravariant functor.

Product with a segment representation and intersection-union process
Recall that for π ∈ Irr, M π is the set of all multisegments n such that for any segment ∆ in n, ∆ × π is irreducible.
We say that π ∈ Alg f is SI or socle irreducible if soc(π) is irreducible and occurs with multiplicity one in the Jordan-Hölder sequence of π.
Proof.For π 1 , π 2 ∈ Alg f , let R π1,π2 be the normalized non-zero intertwining operator from π 1 × π 2 to π 2 × π 1 (see [LM19, Section 2]).By the transitivity of ≤ Z , we reduce to the case that m ∈ M π is of two linked segments.Now, fix an arbitrary π ∈ Irr, and let and is an isomorphism: By switching the labelling if necessary, we also have: Again, by switching labelling if necessary, we may and shall assume that Here ] and [LM16, Lemma 6.17] for general) and the SI property of We now consider two cases separately: (1) τ 1 ∼ = τ .Then τ 1 appears in the submodule of ∆ 1 ∪ ∆ 2 + ∆ 1 ∩ ∆ 2 × π and so appears in the submodule of ∆ 2 × ∆ 1 × π.Using (4.3), we also have that In other words, the socle and cosocle of π The proof is similar to the previous case, but we consider quotients rather than submodules and use (4.4) rather than (4.3).Proof.This follows from definitions.

Some explicit criteria for a multisegment in M
5.2.Indecomposability.We remark that an analogous result holds for other connected reductive groups with replacing the Zelevinsky classification by the Langlands classification (also see [Ch18]).For the Langlands classification version, we remark that there is also an analogous statement for branching laws [Ch23], with the generic case conjectured by D.
Prasad [Pr18] and proved in [CS21] by Savin and the author.
Proof.We shall prove by an induction on the sum of the numbers of segments in m and n.When both m and n are empty sets, there is nothing to prove. Let Now, by relabelling if necessary (using Lemma 5.1), we may assume that ∆ 1 is a shortest segment in m with b(∆ 1 ) ∼ = ρ, and similarly assume that ∆ ′ 1 is a shortest segment in n with b(∆ ′ 1 ) ∼ = ρ.We now consider the following three cases: Then, Frobenius reciprocity gives that: ).Now one analyzes the layers from the geometric lemma on the term ( ∆ r × . . .× ∆ 1 )) N n−l abs (∆ ′ 1 ) (also see Section 2.6).One sees that no layer has the same cuspidal support as , and so this gives such desired Ext-vanishing by the standard argument on an action of the Bernstein center.
One applies Frobenius reciprocity to give that: Now again analysing layers in the geometric lemma on ) N l abs (∆ 1 ) (see Section 2.6 again), one can compare cuspidal supports to give Ext-vanishing.
Then we apply the Frobenius reciprocity as (*).Then, again we compute the layers from the geometric lemma on the term . Then, a cuspidal support consideration on the G l abs (∆1) factor in G l abs (∆1) ×G n−l abs (∆1) gives that only possible layers contributing a non-zero Ext-group take the form: . Now, by the Künneth formula, The latter term is zero by the induction, and so such layer will also give vanishing Ext-groups.Now, since all layers in the geometric lemma give vanishing Ext-groups, we again have that Ext Proof.The basic case is that when all the segments in m are unlinked.In such case, either ζ(n) does not have the same cuspidal support as m ; or n is not generic.That case then follows from Lemma 5.4.
We now consider that some segments in m are unlinked.Then it admits a short exact sequence: where ω is the kernel of the surjection.Then, the Zelevinsky theory [Ze80, Theorem 7.1] implies that any simple composition factor of ω has the associated multisegment m ′ with m ′ ≤ Z m.Thus we still have m ′ ≤ Z n.Inductively on ≤ Z (the basic case explained above), we have that: Ext i Gn (ω, ζ(n)) = 0 for all i.Thus a long exact sequence argument gives that . Now the former one is zero by Lemma 5.4 and so the latter one is also zero.
Proof.Let π be the simple quotient of λ and let π ′ be the simple submodule of λ.We consider the following three cases: • Case 1: π < Z π ′ .Let p be the multisegment such that π ′ ∼ = p .We have the following short exact sequence: Then applying Hom Gn (., ζ(p)), we have the following long exact sequence: . By Lemma 5.5, the first and last terms are zero, and the third term has onedimensional.Thus the unique map from λ to ζ(p) is still non-zero when restricting to π ′ .Since π ′ is the unique simple module, the map must then be an embedding.
• Case 2: π ′ < Z π.In such case, we consider λ ∨ , which has simple submodule π ′∨ and simple quotient π ∨ .We still have that π ′∨ < Z π ∨ .Now the argument in Case 1 gives the embedding λ ∨ ֒→ ζ(p) for some multisegment p. • Case 3: π ′ and π are not ≤ Z -comparable.It suffices to prove that Ext 1 Gn (π, π ′ ) = 0 i.e. such indecomposable λ does not happen.To this end, let p and p ′ be the multisegments such that π ∼ = p and π ′ ∼ = p ′ .We consider the following short exact sequences: where ω is the cokernel of the first injection.Then, a long exact sequence argument with Lemma 5.5 gives Gn ( p , p ′ ).The former one is zero since any simple composition factor ω ′ in ω also satisfies p ≤ Z ω ′ .Thus the latter Ext is also zero.

Define
Alg π (G n ) to be the full subcategory of Alg f (G n ) of objects, all of whose simple composition factors are isomorphic to m for some m ∈ M π .In other words, Alg π (G n ) is the full Serre subcategory generated by simple objects of the form m for m in M π .
Proposition 5.7.Let λ be a representation of G n of length 2. Suppose λ is indecomposable.Suppose the two simple composition factors of λ are not isomorphic and both are in Alg π (G n ).Then π × λ is still an indecomposable representation of length 2.
Proof.By Proposition 5.2(1), we have that π × λ has length 2. To show the indecomposability, it suffices to show that π × λ has either unique simple quotient or unique simple submodule.Let π 1 be the simple quotient of λ and let π 2 be the simple submodule of λ.Let m be the multisegment associated to π.
According to the proof of Lemma 5.6, we must have one of the following two cases: • Case (1): π 1 < Z π 2 .In such case, there exists an embedding, by Lemma 5.6, for some multisegment p.Thus we also have an embedding: But the latter module embeds to ζ(m+p) by Proposition 5.2(3), which has a unique submodule.Thus, π × λ also has unique submodule and so is indecomposable.
has unique simple submodule.We have the embedding, by (the proof of) Lemma 5.6 again: for some q ∈ Mult.We now consider the following embeddings: where the first embedding follows from (5.5) and the second embedding follows from Proposition 5.2 and Lemma 5.3.6.Some results involving the geometric lemma 6.1.A counting problem.In order to give a favour of using the geometric lemma below, let us first consider the following lemma involving some counting arguments.We first define some notions.
For m ∈ Mult and ∆ ∈ Seg, let where k is the multiplicity of ∆ in m.In particular, m ∆ is a submultisegment of m.For example, if For two segments ∆, ∆ ′ , we write ∆ < b ∆ ′ if either one of the following conditions holds: By abuse of notations, we write A = ∪ i ∆ + i as a multiset of cuspidal representations.Let k be the number of segments in m ∆ .If Proof.Note that there are k copies of b(∆) in ∪ k j=1 ∆.Hence, we must also have k copies of b(∆) in A. Thus, we must have k-copies of ∆ i in m b=b(∆) such that ∆ + i = ∆ i .We write such k segments as ∆ i1 , . . ., ∆ i k .Now, recall that ∆ is ≤ b -maximal from our choice, and so if one of ∆ ij = ∆, then ∆ ij contains the cuspidal representation ν −1 ρ a(∆).Thus it is impossible.Hence, all ∆ ij = ∆.Then a simple count gives that A = ∪ k j=1 ∆.
We now study some applications on the above Lemma 6.1.For notational simplicity, for m ∈ Mult, we set This coincides with the notion ζ(m) in Section 3.2 when Let n 1 = l abs (m 1 ), n 2 = l abs (m 2 ), i 1 = n 1 − l abs ((m 1 ) ∆ ) and let i 2 = n 2 − l abs ((m 2 ) ∆ ).Let i = i 1 + i 2 .We now consider the filtration for ( ζ(m 1 ) × ζ(m 2 )) Ni from the geometric lemma in Section 2.5.The only layer from that filtration, which has the same cuspidal support as m ∆ ⊠ m − m ∆ , takes the form where Proof.The problem on the layer can be transferred to the counting problem by using the Jacquet functor computations in Section 2.6.Then the lemma follows from Lemma 6.1.
This implies Thus, any simple composition factor in π Ni appears as a composition factor in: Then csupp(τ 1 ) ∪ csupp(τ 3 ) (union as a multiset) has k number of b(∆).Now we suppose some b(∆) come from csupp(τ 3 ) to obtain a contradiction.In such case, τ 3 ⊠ τ 4 also appears in ζ(m − m ∆ ) N i ′′ .But the latter term can be computed from the geometric lemma again.One sees that if csupp(τ 3 ) contains b(∆), then it contains all the cuspidal representation in a segment We have concluded that all b(∆) in csupp( m ∆ ) arises from csupp(τ 1 ).Then, we must have that τ 1 = m ∆ and i ′ = i and i ′′ = 0.This shows that the only layer in the geometric lemma giving the desired module is m ∆ ⊠ m − m ∆ .This shows the lemma.

A refined computation.
Lemma 6.4.We use the notations in Lemma 6.2.We consider the filtration for ( m 1 × m 2 ) Ni from the geometric lemma.The only layer from that filtration, which has the same cuspidal support as m ∆ ⊠ m − m ∆ , takes the form Proof.Note that the geometric lemma is functorial and so the first assertion follows from the corresponding one in Lemma 6.2.The second assertion then follows from Lemma 6.3.
In the following applications, we shall need two modifications.One is to use Lemma 6.4 repeatedly while another one is to replace m 2 with an indecomposable module of length 2. We shall avoid notation complications to give such precise statements and the meaning will become clearer when one sees the required statements in the following proofs.

Constructing self-extensions
For π in Irr(G n ), we first show that self-extensions of π can be constructed via selfextensions of its associated Zelevinsky standard module.Then we study self-extensions of Zelevinsky standard modules and show it can be reduced to a tempered case via a categorical equivalence in Corollary 7.5.
m)) Thus, combining with the above isomorphism, We remark that the injection in Lemma 7.1 is not an isomorphism in general.For example, if one takes m We now explain Lemma 7.1 in module language via the Yoneda extension interpretation ([Ma95, Ch III Theorem 9.1], also see [Ma95, Section 6, Pages 71 and 83]).We can interpret an element in Ext 1 Gn ( m , m ) as a short exact sequence: By using Lemma 7.1, there exist short exact sequences such that the following diagram commutes: / / 0 Since the leftmost and rightmost vertical maps are injections, the middle vertical maps are also injective.In other words, we obtain: Lemma 7.2.Let π be an indecomposable representation of G n of length two with both irreducible composition factors isomorphic to m for some m ∈ Mult n .Then there exists an indecomposable representation π ′′ of G n which • admits a short exact sequence:

Extensions between two ζ(m).
Let ∆ 1 , . . ., ∆ r be all the distinct segments such that m ∆i = ∅ and label in the way that Lemma 7.3.For m ∈ Mult n , and for any i, Let n 1 = l abs (m b=ρ ′ ) and let n = l abs (m).Now, ), which follows from first applying Frobenius reciprocity and then using the geometric lemma and Lemma 6.1 to single out the only layer that has the same cuspidal support as ζ(m ∆ ) ⊠ ζ(m − m ∆ ).We now repeat similar process for ζ(m − m ∆ ).
We now focus on i = 1 in Lemma 7.3 to study first extensions.We now have the following: Suppose each of π 1 and π 2 admits a filtration with successive subquotients isomorphic to ζ(m).Let M = G(m) and let P be the standard parabolic subgroup containing M .Then (1) for each i = 1, 2, there exists an admissible M -representation τ i which admits a filtration with successive subquotients isomorphic to [m] such that π i ∼ = Ind Gn P τ i , and (2) Proof.Let n = l abs (m).Let P = M N be the Levi decomposition.We first consider (π i ) N .Let τ i be the projection of (π i ) N to the component that has the same cuspidal support as [m].By repeated use of Lemma 6.2 (under the situation that m 2 in Lemma 6.2 is empty), we have that τ i admits a filtration whose successive subquotients are isomorphic to [m].Then, applying Frobenius reciprocity, we have Hom G(m) ((π i ) N , τ i ) ∼ = Hom Gn (π i , Ind Gn P τ i ) and so we obtain a map f in Hom Gn (π i , Ind Gn P τ i ) corresponding to the surjection Claim: f is an isomorphism.Proof of the claim: If f is not an isomorphism, then by counting the number of composition factors, we must have an embedding m ι ֒→ π i such that f • ι = 0.However, via the functoriality of Forbenius reciprocity, we also have a composition of maps is zero.However, this is not possible since the multiplicity of [m] in (π i ) N agrees with that in τ via the construction above.Now the claim gives that π i ∼ = Ind Gn P τ i and this proves (1).We now prove (2).The if direction is clear.For the only if direction, let f : Ind Gn P τ 1 → Ind Gn P τ 2 be the isomorphism.Then the corresponding map, denoted f , under Frobenius reciprocity is given by: f (x) = f (x)(1), where 1 is the evaluation at the identity by viewing f (x) as a function in Ind Gn P τ 2 ; and x is any representative in (π 1 ) N .Since f is an isomorphism, the map f is surjective.Thus the multiplicity of [m] in τ 1 must be at least that in τ 2 .Similarly, we can obtain that the multiplicity of [m] in τ 2 must be at least that in τ 1 .Since the two multiplicities agree, f restricted to τ 1 in (π 1 ) N must be an isomorphism.Proof.Using Proposition 7.4 (and its notations), one can write π i = Ind Gn P τ i for some τ i in C 1 .It remains to see that the induction functor in the previous proposition also defines an isomorphism on the morphism spaces.The induced map is injective since the parabolic induction sends any non-zero objects to non-zero objects.Now it follows from Frobenius reciprocity, The last isomorphism follows from the proof of Proposition 7.4 that τ i is the component of (π i ) N that has the same cuspidal support as τ i .Now the finite-dimensionality of the Hom spaces implies that the injection is also an isomorphism, as desired.
Corollary 7.6.We use the notations in Corollary 7.5.Let τ be an object in C 2 and let π be the corresponding representation under the equivalence in Corollary 7.5.Then By the equivalence of categories, we have an embedding: Let l = dim Hom Gn ( m , π).Suppose l > k.Then, we have an embedding: l times m ⊕ . . .⊕ m ֒→ π = Ind Gn P τ.Now since the Jacquet functor is exact, we have that: This then gives a contradiction.Hence, we must have that l = k.

Big derivatives
In this section, we introduce the notion of big derivatives and describe some basic properties. where By applying the element This gives the following isomorphism: We shall use the identification frequently in Section 11.
We similarly define the left version as: where π We remark that D σ and D ′ σ are left-exact, but not exact.
We only prove results for D and results for D ′ can be formulated and proved similarly (c.f.Section 3.2).When π is -irreducible, D π (τ ) is either zero or has unique simple submodule [KKKO15].If D π (τ ) = 0, we shall denote by D π (τ ) the unique submodule.Proof.For the given condition, we have that σ 1 × . . .× σ s is still irreducible for s ≤ r.
Thus it reduces to r = 2. Let n 1 = deg(σ 1 ) and n 2 = deg(σ 2 ).We have: for any where the second, forth and fifth isomorphisms follow from Frobenius reciprocity, the first, third and last ones follow from the adjointness of the functors.We remark that the forth isomorphism also uses taking parabolic inductions in stages.Now the natural isomorphism between the two derivatives follows from the Yoneda lemma.
Let I σ1 (τ ) be the unique simple submodule of τ × σ 1 .The unique submodule must factor through the embedding: Then inductively, we have that 8.3.The segment case.We now consider a special case of the product functor and we recall the following result shown in [Ch22].For ∆ ∈ Seg, let C = C ∆ be the full subcategory of Alg f (G n ) whose objects π satisfy the property that for any simple composition factor τ in π and any σ ∈ csupp(τ ), σ ∈ ∆ (c.f.[Ch22, Section 9.1]).Let k = l abs (∆).The product functor Proof.This is a special case of [Ch22, Theorem 9.1].
Corollary 8.5.Let ∆ ∈ Seg.Let m be a multisegment with all segments equal to ∆.Let n be a submultisegment of m.
One may also compare the above two statements with Lemma 11.3 and Remark 11.4 below.9. Indecomposability under product functor: isomorphic cases 9.1.Indecomposibility.The following result is well-known (see [Ta90, Proposition 2.3]), but we shall use some similar computations in the proof of Theorem 9.2 and so we sketch some main steps in the following proof.
Let s i = l abs ((m 1 ) ∆i ) and let t i = l abs ((m 2 ) ∆i ) for i = 1, . . ., r.Let and let Let n 1 = l abs (m 1 ) and let n 2 = l abs (m 2 ).Note that We now apply the Frobenius reciprocity: Then, by Lemma 6.4, a possible layer that can contribute to the non-zero Hom is where Indeed, this is the only possible layer by Lemma 6.4.
In such layer (*), by Lemma 6.4 again, the only direct summand that can (possibly) contribute the Hom via Frobenius reciprocity is

Now we inductively have that Hom
and so Künneth's formula gives (9.8).
The idea of proving the following theorem is that one first enlarges to some modules close to standard modules.In particular, one uses Lemma 7.2 for a module of length 2. Then one applies Frobenius reciprocity and does some analysis as in the proof of Lemma 9.1.Theorem 9.2.Let π 1 , π 2 ∈ Irr.Suppose π 1 × π 2 is irreducible.Let π be a representation of length 2 with the two simple composition factors isomorphic to π 2 .Then π is indecomposable if and only if π 1 × π is indecomposable.
Proof.Let m 1 and m 2 be multisegments such that Note that the if direction is clear.We now prove the only if direction.By Lemma 7.2 and taking the dual, there exists π ′′ ∈ Alg f such that • π ′′ admits a short exact sequence To prove the latter one, it suffices to show that Now we apply some similar strategy as the proof of Lemma 9.1.Let n = m 1 + m 2 .Let ∆ 1 , . . ., ∆ r be all the distinct segments such that n ∆i = ∅, and . Now Frobenius reciprocity gives that: The analysis in the proof of Lemma 9.1 gives that the only possible layer contributing a non-zero Hom in (*) (via the geometric lemma on (π 1 × π ′′ ) N ) is of the form: , where • N ′ is the unipotent radical corresponding to the partition (s 1 , . . ., s r ) and N ′′ is the unipotent radical corresponding to the partition (t 1 , . . ., t r ); . . .× G sr × G tr (by permutating the factors in an obvious way); • P is the standard parabolic subgroup in G ′ containing G s1 × G t1 × . . .× G sr × G tr .Thus, we have that: ).Indeed, as in Lemma 9.1, which uses Lemma 6.4 inductively, the only component in (Ind G ′ P ((π 1 ) N ′ ⊠ (π ′′ ) N ′′ ) φ that can contribute to the nonzero Hom is: ).As the functors described in the proof of Corollary 7.5, τ satisfies Ind Gn 2 P * τ ∼ = π ′′ , where P * is the standard parabolic subgroup containing G t1 × . . .× G tr , and Corollary 7.5 implies that τ is indecomposable and of length 2 with both factors isomorphic to [m 2 ].In particular, τ has a unique simple quotient.Now we return to compute the latter Hom of (**).Let In such case, applying the second adjointness, such Hom is equal to where N − is the unipotent radical in P t s1,t1 × . . .× P t sr ,tr ⊂ G u1 × . . .× G ur .By using Hom-tensoring adjointness, the previous Hom turns to: where we regard [n] as a G s1 × . . .× G sr -representation via the embedding: ).
Let σ i = (m 1 ) ∆i .Finally, using Künneth formula on (***) and combining with (*), we have that: and so, by Corollary 8.5, , where n ′ i = (n) ∆i − (m 1 ) ∆i .Now, as discussed above τ has only unique simple quotient and we so have that the Hom space has dimension 1, as desired.Thus, we have Hom Remark 9.3.In general, the parabolic induction does not preserve self-extensions.For example, we consider π = [0] .Let τ = (π × π) N1 .Then 10. Fully-faithfulness of the product functor 10.1.A criteria for proving fully-faithfulness.We recall the following criteria for proving fully-faithfulness: Lemma 10.1.[Ch22, Lemma A.1] Let A and B be abelian Schurian k-categories, where k is a field.Let F : A → B be an additive exact functor.Suppose the following holds: (1) any object of A has finite length; (2) for any simple objects X and Y in A, the induced map of Then F is a fully-faithful functor.
The original statement of [Ch22, Lemma A.1] is stated in a slightly different way, but the proof still applies.
We remark that elements in Ext 1 A (X, Y ) can be interpreted as short exact sequences in Yoneda extensions [Ma95], and the addition corresponds to the Baer sum.Then the exact functor F : A → B sends a short exact sequence to a short exact sequence and so induces a map from Ext 1 A (X, Y ) to Ext 1 B (F (X), F (Y )) above.
11. Application on the SI property for big derivatives 11.1.More notations on derivatives.Recall that the big derivative is defined in Definition 8.1.For ∆ ∈ Seg, set D ∆ = D St(∆) .
For π ∈ Irr, let D ∆ (π) be the unique submodule of For a generic multisegment m, we similarly set D m = D St(m) .For π ∈ Irr, we similarly define D m (π) as the unique submodule of D m (π) if D m (π) = 0 and define D m (π) = 0 otherwise.The uniqueness of the simple submodule in D ∆ (π) and D m (π) follows from [LM16] and [KKKO15].
Set i = l abs (m).With a slight reformulation from above, we also have that: 11.2.η-invariants and ∆-reduced representations.We shall first discuss more results on derivatives.Define ǫ ∆ (π) to be the largest non-negative integer k such that and a ≤ c.Define mx(π, ∆) to be the multisegment that contains exactly the ∆-saturared segments ∆ ′ with the multiplicity ǫ ∆ ′ (π).We shall call π to be ∆-reduced if mx(π, ∆) = ∅.
We give two useful properties related to the η-invariant, which will also be used in the appendix.Those properties are also useful in the study of the Bernstein-Zelevinsky derivatives One may further consider the indecomposable component τ in m N1 which contains [0, 1] ⊠ [1] as the submodule.It is shown by (some variants of) [Ch21, Corollary 2.9] (also see [Ch22+]) that τ is the direct summand with all the simple composition factors in m N1 with the same cuspidal support as [0, 1] ⊠ [1] .Thus we have the following relation: 11.4.An application.We give an application on studying how to embed some Jacquet modules into some layers arising from the geometric lemma.The study of how to do such embedding will be used in [Ch22+b,Ch22+c] for studying commutations of some derivatives and integrals.An alternate way to see Proposition 11.7 is that if mx(π, ∆) contains more than one segment, then τ cannot be written into the form ω ′ ⊠ St(∆) for some ω ′ ∈ Alg f since this otherwise will imply D ∆ (π) ∼ = ω ′ and so τ ∼ = D ∆ (π) ⊠ St(∆) giving a contradiction.This consequently gives: Corollary 11.8.Let π = St(n) for some generic n ∈ Mult.Let ∆ be a segment such that D ∆ (π) = 0. Let ω be a representation of finite length such that π ֒→ ω × St(∆).
12. Appendix: SI property of big derivatives for generic representations It is interesting to generalize Proposition 11.5 to a larger family of big derivatives.We shall explain how to extend to generic representations in this appendix.
Claim 1: n is generic.Proof of Claim 1: It follows form the choice of ∆ * that there is no segment ∆ in m such that ∆ ∆ * .Then it is direct to check from the genericity of m that n is also generic.
We shall use Claim 1 later.We now consider some other multisegments: We see that the LHS is ∆-reduced (since D t−k ∆+∆ ′ +o (τ × St(t)) = 0 for a ∆-saturated segment ∆ ′ ) and so is the RHS.Now it follows from Lemma 12.3 that we also have mx(D p • D o ′ (τ ), ∆) = ∅.
Claim 1 shows that D o ′ +p (τ ) is SI by induction.With Claim 4, the SI property of D o ′ +p (τ ) × St(t ′ ) now follows from [LM22, Lemma 7.1] and so we also have the SI property for D o+p (τ × St(t)) by Claim 3. Since π ֒→ τ × St(t), we now also have the SI property of D σ (π).
Remark 12.8.We remark that [KKKO15, Corollary 3.7] shows that there is a unique simple submodule of D σ (π) for σ ∈ Irr and π ∈ Irr.Indeed, for the special case of generic representations, it also follows from [Ch21, Proposition 2.5] (also see [Ch22+]), using some inputs from branching laws.

1. 1 .
Acknowledgment.The author would like to thank Erez Lapid for drawing attention to [LM22, Section 7].The author would like to thank Max Gurevich for his inputs on Section 4 and discussions on [GL21].This project is supported in part by National Key Research and Development Program of China (Grant No. 2020YFA0713200), the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No: 17305223) and NSFC grant for Excellent Young Scholar (Project No.: 12322120).The author would like to thank referees' comments on improving expositions of the article, and thank one of the referees for an input in the proof of Theorem 4.1 that leads to a simplification.The author declares that he has no conflict of interest.2. Notations 2.1.Basic notations.Let F be a non-Archimedean local field and let D be a finitedimensional F -central division algebra.Let G n = GL n (D), the general linear group over D. The group G 0 is viewed as the trivial group.For g ∈ G n , let ν(g) = |Nrd(g)| F , where Nrd : G n → F × is the reduced norm and |.| F is the absolute value on F .All the representations we consider are smooth over C and we usually omit the adjectives 'smooth' and 'over C'.For a representation π of G n , we write deg(π) for n.We shall usually not distinguish representations in the same isomorphism class.

Lemma 6. 3 .
Let m ∈ Mult.Let ∆ be a ≤ b -maximal element.Let i = l abs (m) − l abs (m ∆ ).Then the direct summand in m Ni with same cuspidal support as m ∆ ⊠ m − m ∆ is actually isomorphic to m ∆ ⊠ m − m ∆ .Proof.Let π = m .Then, from standard results of the Zelevinsky classification [Ze80],

7. 1 .
Constructing extensions from ζ(m).Let m ∈ Mult.Let π = m .In this subsection, we explain how to construct extensions between two copies of m from extensions of two copies of ζ(m).One may compare with the study in [Ch18, Section 3].We first show that one can do that by showing Lemma 7.1 and then reinterpret the result via the Yoneda extension lemma.Lemma 7.1.Let m ∈ Mult n .Then we have a natural embedding Ext 1 Gn ( m , m ) ֒→ Ext 1 Gn ( m , ζ(m)) ∼ = Ext 1 Gn (ζ(m), ζ(m)).Proof.We have 0 → m ֒→ ζ(m) → K → 0, where K is the cokernel of the first embedding.Now, by Lemma 5.4, we have that, for all i, Ext i Gn (K, ζ(m)) = 0. Thus a standard long exact sequence gives that Ext i Gn ( m , ζ(m)) ∼ = Ext i Gn (ζ(m), ζ(m)).(7.6)Long exact sequence now gives that

Corollary 7. 5 .
Let m ∈ Mult n .Let C 1 be the full subcategory of Alg f (G n ) whose objects admit a finite filtration with successive subquotients isomorphic to ζ(m).Let C 2 be the full subcategory of Alg f (G(m)) whose objects admit a finite filtration with successive subquotients isomorphic to [m].There is a categorical equivalence between C 1 and C 2 .Here Alg f (G(m)) is the category of smooth representations of finite length of G(m).