On restriction of unitarizable representations of general linear groups and the non-generic local Gan-Gross-Prasad conjecture

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur-Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid-Minguez criterion from the p-adic setting.


Introduction
Let π be a smooth irreducible representation of the group GL n (F ), where F is a p-adic field. Let us consider GL n−1 (F ) as a subgroup of GL n (F ), embedded in a natural way, which in matrix form is described as GL n−1 (F ) 0 n−1,1 0 1,n−1 1 < GL n (F ) .
We study the branching laws that govern the decomposition of the restricted representation π| GL n−1 (F ) into irreducible representations of GL n−1 (F ). An approachable part of this question would be to describe the possible isomorphism classes of irreducible representations which appear as quotients of π| GL n−1 (F ) .
One fundamental result in that direction was achieved in [1], where it was shown that is at most one-dimensional, for all π ∈ Irr GL n (F ), σ ∈ Irr GL n−1 (F ). As a step forward, we are seeking for a meaningful description of the collection of pairs (π, σ), for which the latter morphism space is non-zero.
Date: August 13, 2018. An irreducible representation is said to be generic, if it can be produced on a Whittaker model of functions. It is a classical fact proved in [24] through the study of L-functions, that for every pair of generic representations (π, σ) as above, Hom GL n−1 (F ) (π, σ) = 0.
In later years, great efforts were focused on posing and proving analogous rules for (quasisplit) classical groups, in place of the general linear group [17,18,16,30]. The resulting branching laws became known as the local Gan-Gross-Prasad conjectures. In similarity with GL n , these laws for classical groups were always set to pertain the generic case, in the sense of generic Langlands parameters which parameterize irreducible representations.
Back in the GL n case, an effective answer for the general restriction problem is still largely considered unpractical. Yet, recently Gan-Gross-Prasad [15] have revisited this problem in an attempt to formulate branching laws that would extend beyond the generic case.
They stipulated a principle that clear combinatorial rules should describe the pairs (π, σ) with non-zero Hom GL n−1 (F ) (π, σ), when π and σ belong to a well-behaved class of representations. More precisely, they formulated a conjecture which concerns the case when both π and σ are in a subclass of unitarizable representations which is described by Arthur parameters.
An Arthur parameter for GL n (F ), in the definition of [15], stands for an admissible homomorphism φ : such that the image of W F is bounded, and the restriction of φ to each SL 2 (C) component is algebraic. Here W F stands for the Weil group of the field F . In particular, an Arthur parameter φ is a completely reducible representation, which decomposes as where {ψ i } are irreducible representations of W F with bounded image, and V d , for d ∈ Z >0 , denotes the unique isomorphism class of a d-dimensional irreducible algebraic representation of SL 2 (C).
Given an Arthur parameter φ, one can attach a L-parameter to it. Consequently, by the established local Langlands reciprocity this L-parameter gives rise to an irreducible representation π(φ) of GL n (F ). We will say that a representation which is constructed in this manner is of Arthur-type. Conjecture 1.1 ( [15]). Suppose that , are two Arthur parameters for GL n (F ), GL n−1 (F ), respectively.
Combining these results, we settle both directions for the case when at least one of π(φ 1 ), π(φ 2 ) is generic.
One motivation for the formulation of Conjecture 1.1 were the works of Clozel [12], Venkatesh [36] and Lapid-Rogawski [28] in the setting of unitary representations. They studied a restriction problem in the sense of direct integral decomposition, and showed that the Burger-Sarnak principle for automorphic representations implies some necessary combinatorial conditions to occur in a restriction.
Even though there is no immediate relation between the smooth and unitary problems, our Theorem 5.6 is consistent with [36, Proposition 2(1)] in a certain sense. Namely, the cases where I 1 = J 2 = ∅ hold in the statement of Conjecture 1.1 produce a refinement of the SL(2)-type condition required by the mentioned proposition from [36]. Further discussion of this relation can be found in [15].
1.1. Methods. We first make use of the classical Bernstein-Zelevinski filtration for the space of π| GL n−1 . This special feature of general linear groups allows for a translation (Proposition 5.4) of the restriction problem into questions on spaces of the form (1) Hom GL n−i (F ) (ν 1/2 ⊗ π (i) , (i−1) σ), where ν 1/2 is a certain character, and π → π (i) and π → (i) π are the Bernstein-Zelevinski derivative functors, which attach finite-length representations of a smaller rank group to a given irreducible representation. Thus, we are left with questions on morphism spaces inside a category of finite-length representations. Moreover, derivatives of Arthur-type representations are built out of (Bernstein-Zelevinski) products of derivatives of Speh representations. We call these derived Speh representations (which happen to be irreducible) quasi-Speh representations.
The reasoning portrayed thus far was employed already in [15] to tackle the problem and to prove some basic cases of Conjecture 1.1 (such as when π(φ 1 ), π(φ 2 ) are Speh representations themselves). Yet, the spaces (1) for the general case of Arthur-type representations were discovered to be substantially more intricate.
More precisely, the basic cases treated in [15] had the special feature that the relevant products of derivatives were all irreducible representations. In general, products of quasi-Speh representations are reducible. The sub-representation structure of such Bernstein-Zelevinski products is often a highly non-trivial issue, as evidenced by a batch of recent works ( [27,26,35,20]).
In the general case, one needs to obtain the information of which quotient and sub representations may morphisms in (1) factor through. To facilitate this interest, we transfer the problem into one about morphism spaces in other Lie-theoretic abelian categories. Namely, we move into the theories of affine Hecke algebras and quantum affine algebras of type A.
The passage from the category of smooth representations of GL n (F ) into module categories over affine Hecke algebras is done through the mechanism of Bernstein decomposition (Section 3.1). It is well known that each block of the former category can be described as modules over a complex algebra. The identification of these algebras with affine Hecke algebras was done in [7,6] through type theory, and independently in [21] through a more explicit approach.
Next, we are able to pass from modules over affine Hecke algberas to modules over quantum affine algebras by using the quantum affine Schur-Weyl duality functors (Section 6.3), developed by Chari-Pressley [11]. Since all functors involved turn out to be monoidal, in a suitable sense, we are finally left with morphism spaces between the quantum affine analogs of products of quasi-Speh representations.
The advantage of posing our problem in a language of quantum groups lies in a recent result of Hernandez [22] dealing with cyclic modules. It states that a product When transferring the notion of a cyclic products back to the p-adic setting through our sequence of functors, we end up (Propositions 6.1 and 6.3) with a notion of a product π 1 × · · · × π k of irreducible representations that has a unique irreducible quotient whose Zelevinski parameter is the sum of the Zelevinski parameters of π 1 , . . . , π k . This is precisely the notion that was studied by Lapid-Minguez in [27] for products of pairs of representations of the ladder class.
Since quasi-Speh representations belong to the ladder class, we are able to order them (Proposition 4.2) in such way that a combinatorial criterion from [27] would imply that π 1 × π 2 is "cyclic" whenever π 2 π 1 . Proposition 4.3 subsequently applies the result of Hernandez to claim essentially that the derivatives in (1) for our cases of interest can be decomposed into representations with expected unique irreducible quotients/subs. Such claim untangles the main difficulty behind one direction of Conjecture 1.1.
1.2. Paper structure. Section 2 surveys the basic tools needed to study the smooth representation theory of p-adic GL n , together with some necessary lemmas. In particular, we recall the Zelevinski multisegment parametrization, which is an essential tool in our work. We make note of the basic Proposition 2.3, whose statement has not appeared previously in the literature to the best of our knowledge. Section 2.5 recovers the necessary results from [27]. Section 3 portrays the categorical passage from representations of GL n (F ) into those of affine Hecke algebras. Proposition 3.2 delves into the resulting correspondence between irreducible representations. The proof of the key Theorem 3.3 is delayed to Section 6.
Section 4 contains the gist of the combinatorial work in the class of quasi-Speh representations and their products, which is necessary for the proof of Conjecture 1.1.
In Section 5 we prove the main theorems discussed above. Finally, Section 6 surveys the necessary ingredients from the representations theory of U q ŝ l N with the aim of proving Theorem 3.3, which is shown to be essentially a translation of the main result of [22].
1.3. Acknowledgements. I would like to thank Wee Teck Gan, for sharing this problem with me and whose insights and optimism are an invaluable guide. Thanks are due to Erez Lapid for the continuing encouragement and useful conversations, to David Hernandez for sharing his results and expertise, to Dipendra Prasad for sharing his views on the problem and to Kei Yuen Chan and Gordan Savin for sharing their preprint and their views on similar themes.

2.
Background on representation theory of p-adic GL n Let F be a p-adic field. Unless explicitly stated, F will be fixed and omitted from our notation. We are interested in the representation theory of the groups G n := GL n (F ), for all n ≥ 1.
For a given n, let α = (n 1 , . . . , n r ) be a composition of n. We denote by M α the subgroup of G n isomorphic to G n 1 × · · · × G nr consisting of matrices which are diagonal by blocks of size n 1 , . . . , n r and by P α the subgroup of G n generated by M α and the upper unitriangular matrices. A standard parabolic subgroup of G n is a subgroup of the form P α and its standard Levi factor is M α .
For a p-adic group G, let R(G) be the category of smooth complex representations of G of finite length. Denote by Irr(G) the set of equivalence classes of irreducible objects in R(G). Denote by C(G) ⊆ Irr(G) the subset of irreducible supercuspidal representations.
We also write Irr = ∪ m≥0 Irr(G m ) and C = ∪ m≥1 C(G m ). For any n, let ν s = | det | s F , s ∈ C denote the family of one-dimensional representations of G n , where | · | F is the absolute value of F . For π ∈ R(G n ), we write ν s π = πν s := π ⊗ ν s ∈ R(G n ).
The map s → ν s is a group homomorphism from C to the group of characters of G n , whose kernel is 2πi log q F Z, where q F is the residue characteristic of F . The group of (unramified) characters {ν s : s ∈ C} of G n acts on C(G n ) by ρ → ρν s . For ρ ∈ C(G n ), we write o(ρ) for the (finite) order of the stabilizer of ρ for that action.
Given a set X, we write N(X) for the commutative monoid of maps from X to N = Z ≥0 with finite support. We will sometimes treat an element A ∈ N(X) as a finite subset of X, by writing x ∈ A for a given x ∈ X, in case A(x) > 0.
There is a natural embedding X → N(X) which sends an element to its indicator function. Thus, we will often treat elements of X as elements in N(X).

2.1.
Multisegments. The elements of Irr are classified by multisegments in the following manner.
A segment ∆ = [a, b] ρ is a formal object defined by a triple ([ρ], a, b), where ρ ∈ C and a ≤ b are two integers, up to the equivalence [a, b] It is also useful to refer to the empty segment given in the form [a, a − 1] ρ , for any ρ ∈ C and integer a.
Let Seg denote the collection of all segments. Elements of Mult := N(Seg) are called multisegments.
The Zelevinski classification [37] defines a bijection Similarly, the Langlands (quotient) classification, which describes irreducible representations of any reductive p-adic group, can be stated for the case of the groups {G n } ∞ n=1 , as another 1 bijection L : Mult → Irr . The relation between both bijections Z and L is further discussed in Section 3.2. For a segment ∆ = [a, b] ρ and s ∈ C, we write ∆ν s := [a, b] ρν s . We naturally extend this to an operation m → mν s on Mult. It is easy to check that, Z(mν s ) = Z(m)ν s and L(mν s ) = L(m)ν s hold.
The analogous statements remain true when replacing the Z bijection with L.
Given a supercuspidal representation ρ ∈ C, let Seg ρ denote the collection of segments of the form [a, b] ρν s , for some integers a, b and s ∈ C.
We will also let the field F vary for this part of the discussion, and write Seg F ρ , Mult F ρ , Mult F 0 for the corresponding objects, defined for an arbitrary p-adic field F . Given ρ 1 ∈ C(GL n 1 (F 1 )) and ρ 2 ∈ C(GL n 2 (F 2 )) with q which naturally extends to an isomorphism of monoids φ ρ 1 ,ρ 2 :

2.2.
Gelfand-Kazhdan involution. The outer automorphism g → (g t ) −1 on G n , gives an involutive auto-equivalence η of the category R(G n ). Let us write η for the composition of η with the operation of taking the contragredient (smooth dual) representation. The resulting involution η is a contragredient functor with satisfies the special property η(π) ∼ = π, for all π ∈ Irr, as shown by a classical result of Gelfand-Kazhdan.
The existence of η also gives the following well-known corollary.
Recall from the general theory of the Bernstein center, that any representation σ ∈ R(G n ) splits uniquely to the form σ = ⊕ I σ I , where the sum goes over distinct multisets I ∈ N(C), so that supp(σ I ) = I.
Then, we have a natural identification Proof. Consider the parabolic induction functor i α : We are left to show that any morphism in the latter space is contained in the image of i α .

2.4.
Tadic classification of the unitary spectrum. Let Irr u ⊆ Irr be the subset of irreducible representations that are unitarizable, that is, those whose space can be equipped with a positive definite Hermitian form invariant under the group action. A classification of Irr u in terms of multisegments was achieved by Tadic in [34]. Let us briefly recall this classification.
It is easy to check that Irr u ∩ C = {ρ ∈ C : Re(ρ) = 0}. For each pair of integers a, b ∈ Z >0 and a supercuspidal ρ ∈ Irr u ∩ C, we define the Speh multisegment We then set π a,b ρ := L(m a,b ρ ) to be a (unitary 2 ) Speh representation. Let B ⊆ Mult denote the collection of Speh multisegments. Let us also define It is known ( [4]) that π 1 × π 2 is irreducible, for any π 1 , π 2 ∈ Irr u . Hence, the theorem above states that any π ∈ Irr u can be written in the form π = π 1 ×· · ·×π t , where Definition 2.6. 3 We say that π ∈ Irr u is of Arthur-type, if m i ∈ B, for all i = 1, . . . , t in the decomposition above.
In other words, Arthur-type representations are products of Speh representations.
Note, that proper Arthur-type representations built out of non-isomorphic elements of Irr u ∩ C will always have disjoint supercuspidal supports.
Given an Arthur-type representation π = L(m) and a real number It is easy to deduce from Theorem 2.5 that for every π ∈ Irr u there is a unique, up to order, factorization of the form where π 0 , π 1 , . . . , π k ∈ Irr u are of Arthur-type 4 and 0 < α 1 , . . . , α k < 1/2 are distinct real numbers.
2.5. Ladder representations and the Lapid-Minguez criterion. Speh representations are a special case of a class of irreducible representations known as ladder representations. A representation π ∈ Irr is called a proper 5 ladder representation, if it is given as π = L(∆ 1 + . . . + ∆ k ), for segments ∆ i , i = 1, . . . , k, satisfying Ladder representations were shown (e.g. [25,27,19]) to possess certain remarkable properties, which often make the ladder class more approachable for treatment of questions on general irreducible representations. Essentially the same class of representations was also studied in the literature under various names in various type A settings, such as calibrated affine Hecke algebra modules, snake modules for quantum affine algebras and homogeneous modules for KLR algberas.
Given a ladder representation π ∈ Irr and any representation σ ∈ Irr, it was shown [27, 5.15] that both π × σ and σ × π have a unique irreducible sub-representation. Lapid-Minguez have also devised an algorithm in [27] for computing the multisegment classifying that sub-representation. We recall in what follows one corollary of that algorithm.
Suppose, for that purpose, that Recall again that Z(m 1 +m 2 ) always appears as a sub-quotient in Z(m 1 )×Z(m 2 ). Among the results of [27] is a combinatorial criterion for determining when does Z(m 1 +m 2 ) actually appear as a sub-representation of Z(m 1 ) × Z(m 2 ).
Consider the set of indices I = {1, . . . , k 1 } × {1, . . . , k 2 }, and the following bipartite graph on the set of vertices I 1 ⊔ I 2 , where I 1 = I 2 = I (two copies of I). We say that elements (i 1 , j 1 ) ∈ I 1 and (i 2 , j 2 ) ∈ I 2 are in relation ( Consider the sets , if and only if, there is a matching function for the restricted bipartite graph (X m 1 ;m 2 , Y m 1 ;m 2 , ↔), i.e., there exists an injective function 2.6. Bernstein-Zelevinski derivatives. For given n 1 , n 2 , consider the subgroup U < G n 1 × G n 2 of upper unitriangular matrices in G n 2 . Let ψ be a non degenerate character of U.
We then have an obvious functor of taking co-invariants: The Bernstein-Zelevinski derivatives of [2] can be defined as functors R(G n ) → R(G n−i ) constructed by composing W with the Jacquet functor. More precisely, given π ∈ R(G n ), we set its i-th derivative to be The derivatives comply with a Leibniz rule, in the following sense.

Affine Hecke algebras
Given n ∈ Z >0 and q ∈ C, the root datum of GL n gives rise to the (extended) affine Hecke algebra H(n, q). In fact, we will not be using here the concrete algebraic structure of these algberas, but rather some abstract information on their categories of representations.
Yet, to avoid confusion let us recall a possible presentation of H(n, q): This is the complex algebra generated by T 1 , . . . , T n−1 and invertible y 1 , . . . , y n , subject to the relations We denote by M q n the category of finite-dimensional modules over the algebra H(n, q). For any n 1 , . . . , n t ∈ Z >0 , there is a natural embedding of algebras H(n 1 , q) ⊗ · · · ⊗ H(n t , q) ֒→ H(n 1 + . . . + n t , q) .
3.1. Equivalence to Bernstein blocks. Let ∼ be an equivalence relation on the elements of N(C) defined as follows: For A, B ∈ N(C), we say that A ∼ B, if there are representations ρ 1 , . . . , ρ t ∈ C and numbers s 1 , . . . , s t ∈ C, such that Note, that the number N A := n 1 + . . . + n t , where ρ i ∈ C(G n i ), for i = 1, . . . , t, is an invariant of the ∼-equivalence class of A (also known as the inertia class). Hence, we will write N Θ , where Θ denotes that equivalence class. Each inertia class Θ defines the Bernstein block R(Θ), which is the full sub-category of R(G N Θ ) consisting of representations π, such that supp(σ) belongs to Θ for all irreducible sub-quotients of π. The Bernstein decomposition [3] (in the case of GL n ) states that we have a decomposition of abelian categories where the product goes over all inertia classes Θ, for which N Θ = n.
We will call R(Θ) a simple block, if Θ = Θ(ρ, d) has a representative of the form d · ρ ∈ N(C), where ρ ∈ C and d ≥ 1 is an integer.
The clear consequence of the above is that for any ρ ∈ C, the irreducible representations appearing in the sequence of blocks {R(Θ(ρ, n))} ∞ n=0 are precisely those given by Z(Mult ρ ) or L(Mult ρ ).
For the trivial representation ν 0 of G 1 , we set Θ n = Θ(ν 0 , n), for all n ≥ 1. The simple block R(Θ n ) is called the principal (or Iwahori-invariant) block of R(G n ).
It was shown in [7] and [21] that for every simple block Θ = Θ(ρ, d), we have an equivalence of categories The equivalences {U Θ } are not canonical, yet they can be chosen in a way that is compatible with parabolic induction [32]. Namely, we are allowed to assume that holds, for all π 1 ∈ R(Θ(ρ, n 1 )) and π 2 ∈ R(Θ(ρ, n 2 )). The particular case of equivalences for principal blocks ∀n is in fact a classical theorem 6 of Borel [5] and Casselman.
We will fix a canonical choice of {U n } which is supplied by said theorem. Let us note that the irreducible representations of the algbera H(1, q F ) = C[y 1 , y −1 1 ] are naturally given by the variety C × . Note further that the irreducible representations in R(Θ 1 ) are given by the group of unramified characters {ν s : s ∈ C}. Under these identifications, the canonical equivalence U 1 takes ν s to the character given by q s F ∈ C × . We can use U n to push a parametrization of the irreducible representations of H(n, q F ) in terms of multisegments. Recall that the irreducible representations in R(Θ n ) are parameterized by Mult 0 , either through Z or L. Hence, we write bijectionŝ defined byẐ(m) = U nm (Z(m)), where Z(m) ∈ Irr(G nm ). Similarly,L is defined by L.
Remark 3.1. Note, thatẐ (orL) can in fact be described intrinsically (that is, without use of p-adic groups) as was done in [33]. In particular, these classifications are independent of the field F .
The following proposition shows that all equivalences {U Θ } preserve the Zelevinski and Langlands parametrizations of irreducible representations, in a natural sense.
Let E be a p-adic field with residue cardinality q E = q o(ρ) E stands for the trivial representation of GL 1 (E).

Let us writeẐ
: for the map as defined above (but for the field E in place of F ). Then, we can choose the collection {U Θ(ρ,n) } n , so that it satisfieŝ is defined to be the unique irreducible sub-representation of ζ(m) = Z(∆ 1 ) × · · · × Z(∆ k ) (for a prescribed ordering of the segments of m). In other words, we need to verify that U Θ(ρ,1) is allowed to be chosen so that for all s ∈ C, the representation ρν s in R(Θ(ρ, 1)) is sent to the character of H(1, q E ) = C[y 1 , y −1 1 ] given by y 1 → q s E . Let us recall the construction of U Θ(ρ,1) in [21]. Let V be the space of the representation ρ ∈ C(G m ). Then, G m acts on the space The functor U Θ(ρ,1) is then defined by taking π ∈ R(Θ(ρ, 1)) to U Θ(ρ,1) (π) := Hom Gm (W, π) , viewed as a H ∼ = H(1, q E )-module. For all s ∈ C, there is a projection of representations ψ s : ρ → ρν s given by ψ s (vt k ) = q ks F v. It is easy to verify that U Θ(ρ,1) (ρν s ) is a one-dimensional space spanned by ψ s | W .

3.2.
Consequences of a result of Hernandez. Section 6 deals with quantum affine algebras and the quantum affine Schur-Weyl duality functor. The following theorem on representations of affine Hecke algebras will be shown to be a manifestation of a theorem of Hernandez [22], when transferred through the duality functor.
We would like to extend the statement of the theorem above slightly. For that purpose let us recall that each algebra H(n, q) is equipped with the Iwahori-Matsumoto involutive automorphism θ n (see [29, I.6]. It gives rise to an involutive autoequivalence of M q n . In order to ease notation, we will simply write θ for all these involutions. It is known [29, Lemme I.7.1] 7 that the Iwahori-Matsumoto involution is compatible with the induction product, in the following sense. For all representations π 1 , . . . , π t in M q k 1 , . . . , M q kt , respectively, we have θ(π 1 × · · · × π t ) ∼ = θ(π t ) × · · · × θ(π 1 ) .
When restricting θ to irreducible representations, we obtain [29, Proposition I. 7.3] what is known as the Zelevinski involution in the context of p-adic groups, that is, for all m ∈ Mult 0 . Corollary 3.4. The statement of Theorem 3.3 remains valid, whenẐ is replaced withL. In addition, "quotient" may be replaced with "sub-representation".
Proof. It follows from an application of the functor θ and Proposition 2.2, which remains valid for representations of affine Hecke algebras through the equivalences {U n }.

Classes of irreducible representations
We would like to study certain classes of representations in Irr which naturally occur in the derivatives of unitarizable representations.

4.1.
Quasi-Speh representations. We will first deal with a subclass of ladder representations, which we will call quasi-Speh representations. These are parameterized by integers a, b ∈ Z >0 , c ∈ Z, so that 0 ≤ c ≤ a, and a representation ρ ∈ Irr u ∩ C. For such data, we define the multisegment We then set π a,b,c ρ := L(m a,b,c ρ ) to be a quasi-Speh representation. Note, that for a = c, these are the usual Speh representations π a,b ρ = π a,b,a ρ defined in Section 2.4. We also note that for b > 1 and all a, we have π a,b,0 ρ = π a,b−1 ρ ν −1/2 . The following identities make the class of quasi-Speh representation relevant to our discussion. See [25,Section 5.4] for the proof, which is attributed to Tadic. Proposition 4.1. Let ρ ∈ Irr u ∩ C be a given representations of G d . Let a, b ∈ Z >0 be given.
Then, the formula gives the derivatives of the Speh representation π a,b ρ . Let us define a preorder on the class of quasi-Speh representations. We will write For any given ρ ∈ Irr u ∩ C, the restriction of to the collection {π a,b,c ρ } a,b,c gives a total preorder. Proposition 4.2. If π 1 = L(m 1 ), π 2 = L(m 2 ) are two quasi-Speh representations, which satisfy π 2 π 1 , then π 1 ×π 2 has a unique irreducible quotient which is given by L(m 1 +m 2 ).
Recall the sets X = X m 1 ;m 2 , Y = Y m 1 ;m 2 and the relation ↔, as they were defined in Section 2.5.
We need to show there is an injective function f : X → Y , which satisfies x ↔ f (x), for all x ∈ X.
Having established that g((X ∩X) \ K) is always contained in Y , we can define f = g on (X ∩X) \ K. Injectivity is not interfered with the definition of f | K , because when x ∈ X satisfies x ↔ (i, i) for an index i, we clearly must have x ∈ K.
We are left with the task of extending f injectively to Moreover, the same argument shows that (i, 1) ∈X implies (i − 1, 1) ∈Ŷ . Hence, we can extend f by setting f (i, 1) = (i − 1, 1) without harming injectivity.

4.2.
Quasi-Arthur-type representations. We say that π ∈ Irr is quasi-Arthur-type, if it has the form and ρ ∈ Irr u ∩ C. The significance of quasi-Arthur-type representations appears through the following corollary of previous discussions. Then, the product π a 1 ,b 1 ,c 1 ρ × · · · × π a k ,b k ,c k ρ has a unique irreducible quotient, whose isomorphism class is given by the quasi-Arthurtype representation Proof. By Proposition 3.2 and the property (2), we can assume that we are dealing with finite dimensional representations of the corresponding affine Hecke algebras. The statement then follows from Proposition 4.2 combined with Corollary 3.4.
Let ρ ∈ Irr u ∩ C be given. We would like to define a similar involution on Mult ρ . Given ∆ = [a, b] ρν s ∈ Seg ρ , we set ∆ ! = [−b, −a] ρν −s , and extend it to an involution m → m ! , for all m ∈ Mult ρ .
When ρ ∼ = ρ ∨ , we clearly have m ! = m ∨ . Now, given m ∈ Mult ρ , we can write a unique decomposition m = m s + m a , with m s , m a ∈ Mult ρ , so that m ! s = m s and m s is the maximal multisegment with that property. Clearly, for a segment ∆ ∈ m a , we have ∆ ! ∈ m a . Lemma 4.4. Suppose that π is a quasi-Arthur-type representation.
Proof. We can assume that and for every segment ∆ ∈ n a , we have and m a,0 ρ ∨ , m a,−1 ρ ∨ are understood as empty multisegments. Clearly,

Proof. By Lemma 4.4 we have an equality of multisegments of the form
Let us further write a disjoint partition For a multisegment m = s∈K [α s , β s ] ρ ∈ Seg ρ , let us write a decomposition We obtain the identity A moment's reflection will show that such an identity forces a bijection t : ρ , for a subsetĨ ⊆ I ′ . This implies a bijection s : , for all i ∈Ĩ. The desired bijections u and d are easily constructed out of t and s after denoting 2 ), I 2 = t −1 (J 1 ) and I 3 = I 1 \Ĩ.

Main Theorems
We would like to determine for which pairs of representations π 1 , π 2 ∈ Irr u , such that π 1 ∈ Irr(G n ) and π 2 ∈ Irr(G n−1 ), the space is non-zero. Here we consider G n−1 as a subgroup of G n embedded in the corner, as described in the introduction section. Now, suppose that π 1 , π 2 are Arthur-type representations. It is easy to verify that they can be written uniquely (up to ordering) in the form Definition 5.1. We say that a pair (π 1 , π 2 ) of Arthur-type representations is in GGP position, if in terms of the decomposition above, there are disjoint partitions and bijections u : Theorem 5.2. Suppose that π 1 , π 2 ∈ Irr u are two Arthur-type representations, which satisfy Hom(ν 1/2 π (i) 1 , (j) π 2 ) = {0} , for some i, j.
Proof. Let us write π 1 ∼ = π and π 2 ∼ = π a 1 ,b 1 ρ 1 × · · · × π a l ,b l ρ l . Since the above products are independent of the order in which the factors are taken, we can assume that Then, by Propositions 2.9, 4.1 and the assumption on the non-vanishing homomorphism space, we deduce that for some integers c 1 , . . . , c k , c ′ 1 , . . . , c ′ l . By rearranging the factors in the product if necessary and using Corollary 2.4, we can decompose the above homomorphism space as In particular, the above decomposition shows that it is enough to consider the case where all ρ i 's and ρ ′ j 's are isomorphic. In other words, we are free to assume for the rest of the proof that π 1 , π 2 are of proper Arthur-type and that ρ i ∼ = ρ ′ j ∼ = ρ for all i, j and one fixed ρ ∈ C ∩ Irr u . Now, note that the given quasi-Speh representations satisfy π a k ,b k ,c k Hence, by Proposition 4.3, we have the quasi-Arthur type representation as the unique irreducible quotient of π By applying Lemma 2.1, we can deduce from the non-vanishing Hom space that σ 1 ν 1/2 ∼ = σ ∨ 2 . Finally, the statement follows from Corollary 4.5.
5.1. Bernstein-Zelevinski filtration. In order to study the morphism space above, we will make use of the analysis in [2] of restrictions of representations in Irr. Let us sketch the main ingredients used in that reference. The reader can also refer to [8,31] for very similar discussions. Recall the p-adic mirabolic groups {P n } situated within the inclusions G n−1 < P n < G n . In matrix form, P n is defined to be the subgroup of G n consisting of matrices whose bottom row is given by (0 . . . 0 1).
Bernstein-Zelevinski defined families of exact functors together with a set of identities between them. In particular, Φ − is left adjoint toΦ + , while Ψ − is left adjoint to Ψ + . In terms of these functors, for every 0 < i ≤ n, the i-th derivative of a representation π ∈ R(G n ) is given as Proposition 5.3. For any representation τ ∈ R(P n ) on a vector space V , there is a filtration of P n -representations as a P n -representation, for all 0 ≤ i < n.
Proposition 5.4. Let π 1 ∈ R(G n ) and π 2 ∈ R(G n−1 ) be given. Let {V i } be the filtration of π 1 | Pn as described in Proposition 5.3. Then, we have a natural identification of homomorphism spaces for all 0 ≤ i < n.
Proof. The proof of Proposition 5.4 will work for this case as well, after recalling that adjunctions of exact functors give natural identifications of Ext-spaces, as well as Homspaces. See e.g. [31, Proposition 2.2].

5.2.
Branching laws. We will first treat the Arthur-type case which was considered in Conjecture 1.1.
The above theorem can also be extended into a branching law governing the irreducible unitarizable quotients of a restriction of any unitarizable irreducible representation.
(2) For every i, j for which α i = β j holds, the pair (π i , σ j ) is in GGP position.

Quantum affine algebras
We would like to prove Theorem 3.3 by applying results from the representation theory of quantum affine algebras.
6.1. Setting. Let us recall parts of the theory of quantum affine algebras and their finitedimensional representations. We refer the reader to [10], [9,Chapter 12] , [14] for comprehensive study and discussions of these objects.
We are interested in the Hopf C-algebra A N,q = U q ŝ l N , which is defined for a fixed parameter q ∈ C × . Its definition involves generators and relations which depend on the Cartan matrix of the affine Lie algberaŝl N . We will refrain from stating the full definition, which can be easily found in the sources mentioned above, since our applications will not require it. Our assumption for the rest of this section will be that q is not a root of unity.
Recall ([10, Proposition 2.3]) that, as vector spaces, we have a triangular decomposition where U − , U 0 , U + are sub-algebras of A N,q . Also, recall that the definition of A N,q involves the invertible elements which generate a commutative co-commutative Hopf-sub-algebra K < A N,q . This subalgebra can be seen as the Cartan algebra (zero-part) in the triangular decomposition of the smaller quantum group U q (sl N ), which is naturally embedded in A N,q .
Let Ω be the set of complex characters λ of K, which satisfy λ(k i ) ∈ q Z , for all i = 1, . . . , N −1. Since K is co-commutative, Ω is an abelian group. Given a complex character P of the algebra U 0 , we will write ω(P ) to be its restriction to K.
We are further interested in the category C q N of type 1 representations of A N,q . It consists of all finite-dimensional representations V in which a certain specified central element c 1/2 ∈ U 0 acts trivially, and which comply with a weight space decomposition for the sub-algebra K, of the form where V λ is the λ-eigenspace of V .
6.2. Cyclic modules. For a representation V in C q N , we say that a vector v ∈ V is highest weight, if v is an eigenvector for the algebra U 0 and U + · v = 0. The character of U 0 by which it acts on a highest weight vector v is called the ℓ-weight of v (not to be confused with the previous weight decomposition by characters of the smaller algebra K).
Given an irreducible representation V in C q N , it is known there is a unique, up to scalar, highest weight vector v ∈ V . The ℓ-weight P of v characterizes the isomorphism class of V . We can write in this case V = V (P ). Let us write D N for the set of characters of U 0 which give rise to irreducible representations V (P ), P ∈ D N . The set D N can be described using what is known as Drinfeld polynomials.
The set D N also comes with a natural monoid structure, in the following sense. Given highest weight vectors v P ∈ V, v Q ∈ W of respective ℓ-weights P, Q ∈ D N , the vector v P ⊗ v Q ∈ V ⊗ W is also a highest weight vector of ℓ-weight P · Q ∈ D N .
Let V 1 , . . . , V k be irreducible representations in C N . Let v i ∈ V i , 1 ≤ i ≤ k be the corresponding highest weight vectors. Let W ⊆ V 1 ⊗ · · · ⊗ V k be the subrepresentation generated by the vector v 1 ⊗ · · · ⊗ v k .
We would like to reformulate the notion of cyclic tuples into a more categorical language.
Proposition 6.1. Let V = (V (P 1 ), . . . , V (P k )) be a tuple of irreducible representations in C q N , with P 1 , . . . , P k ∈ D N . The tuple V is cyclic, if and only if, the representation V (P 1 ) ⊗ · · · ⊗ V (P k ) has a unique irreducible quotient whose isomorphism class is given by V (P 1 · . . . · P k ).
Proof. For P ∈ D N , note that ω(P ) ∈ Ω by the assumptions on C q N . Recall ([13, Theorem 1.3(3)]) that dim V (P i ) w(P i ) = 1, and that for any λ ∈ Ω with V (P i ) λ = {0}, we have w(P i ) > λ in a certain partial order on Ω coming from the root system of sl N .
Conversely, suppose that M has an irreducible quotient Y ∼ = V (P 1 · . . . · P k ). Again, the one-dimensionality of M ω(P 1 ·...·P k ) requires m, and hence W , to project non-trivially on Y . Since Y is irreducible, we must have W = M. Theorem 6.2 (Hernandez [22]). Let V 1 , . . . , V k be irreducible representations in C N , such that for all 1 ≤ i < j ≤ k, the pair (V i , V j ) is cyclic. Then, the tuple (V 1 , . . . , V k ) is cyclic.
6.3. Quantum affine Schur-Weyl duality. Let q ∈ C × be fixed. For each k ≥ 1, N ≥ 2, Chari-Pressley [11] have defined a quantum affine Schur-Weyl duality functor F k,N : M q 2 k → C q N . Recall from Section 3 that M q 2 N stands for the category of finite-dimensional representations of the affine Hecke algebra H(k, q 2 ).
Let us recall some of the properties of these duality functors, as were studied in [11]. When k ≤ N, F k,N is a fully faithful functor. In other words, it serves as an embedding of the category M q 2 N into a certain full sub-category of C q N . In particular, in this case the functor preserves irreducibility of objects.
Proposition 6.3. [11,Theorem 7.6] For every N ≥ 2, there is a natural embedding of monoids ι N : D N → Mult 0 , for which F k P ,N (Ẑ(ι N (P ))) ∼ = V (P ) ∈ Irr(C q N ) holds, for all P ∈ D N with k P ≤ N, where k P is the integer for whichẐ(ι N (P )) ∈ Irr(M q F k P ). Moreover, for every 1 ≤ k ≤ N, the collection of multisegmentsẐ −1 (Irr(M q F k )) is contained in the image of ι N .
We finish by noting that the combination of Proposition 6.1, Theorem 6.2, Proposition 6.3 and (6) directly give the proof of Theorem 3.3.