On a determinantal formula of Tadić

ON A DETERMINANTAL FORMULA OF TADI´ C EREZ LAPID AND ALBERTO M´ INGUEZ Abstract. We study a special class of irreducible representations of GLnover a local non-Archimedean field which we call ladder representations. This is a natural class in the admissible dual which contains the Speh representations. We show that the Tadi´c determinantal formula is valid for this class and analyze the standard modules pertaining to these representations. 1. Introduction Let Fbe a non-Archimedean locally compact field and, for every integer n≥0, set Gn= GLn(F). (with the convention that G0is the trivial group). Denote by Rnthe Grothendieck group of the category of smooth representations of Gnof finite length, and let R=⊕n≥0Rnwith the product structure defined by normalized parabolic induction. The commutative ring R(with the one-dimensional representation of G0as the identity element) was introduced by Zelevinsky, who showed that it is freely generated by the essentially square-integrable representations (of any Gn) [Zel80]. Hence, the monomials in these generators (the so-called standard modules) form a basis for R. Another natural basis for Ris given by the irreducible representations. The change of basis matrix is unitriangular (in an appropriate sense) and the coefficients can be expressed in terms of the Kazhdan-Lusztig polynomials. This fact was conjectured by Zelevinsky [Zel81] (and in a more precise form in [Zel85]) and proved in [CG97]. While in principle this solves the problem of decomposing a standard module into irreducible constituents, in practice the coefficients are very complicated. Nevertheless, in the case of a Speh representation, Tadi´c obtained a remarkable formula expressing its character as a linear combination of characters of standard modules with coefficients ±1 [Tad95]. Tadi´c’s ingenious argument transfers the problem to a question about complex groups. Later on, the argument was simplified by Chenevier and Renard [CR08] by a clever use of the Desnanot-Jacobi identity of determinants (also known as Dodgson’s rule of determinants). Both proofs rely on the determination of the composition series at the edge of a complementary series, and thus they heavily rely on unitarity. Our purpose in this note is to provide a different proof of Tadi´c’s formula which yields additional information on the structure of the Langlands quotient, and at the same time extends to a wide class of representations which we call ladder representations. (We caution Date: March 7, 2012. First named author partially supported by a grant from the Israel Science Foundation. Second named author partially supported by ANR-10-BLANC 0114, EPSRC grant EP/G001480/1, MTM2010-19298 and FEDER. 1 2 EREZ LAPID AND ALBERTO M´ INGUEZ that in the literature there is an unrelated, older notion of ladder representations for unitary groups.) The extra piece of information (part 1 of Theorem 1 below) is used in [FLO] to prove for this class of representations a conjecture of Jacquet about the existence of functionals invariant under a unitary group. Except for the Speh representations, ladder representations are not unitary. Instead of unitarity, we use Jacquet module technique. Unlike in other scenarios, it is necessary to study not only the semisimplification of the Jacquet module, but the finer structure of its submodules. Curiously, using our result one can turn the table and use the Desnanot-Jacobi determinantal identity to obtain the decomposition of certain induced spaces (including the ends of complementary series) as a corollary. To state our results, let us introduce some more notation. Denote by νthe character |det|on any Gn. (The nwill be implicit and hopefully clear from the context.) For any smooth representation πof Gnand a∈Rdenote by πνathe representation obtained from π by twisting it by the character νa. If π1, . . . , πrare smooth representations of Gn1, . . . , Gnr respectively, we will denote as usual by π1×π2× · · · × πr the representation of Gn1+n2+···+nrparabolically induced (normalized induction) from the representation π1⊗π2⊗ · · · ⊗ πrof the standard parabolic subgroup of Gn1+n2+···+nrof type (n1, . . . , nr) (with Levi subgroup Gn1× · · · × Gnr). Throughout we fix a positive integer dand an irreducible cuspidal (not necessarily unitary) representation σof Gd. By a segment [a, b] we mean a set of elements {a, a+1, . . . , b} where b≥aare integers. For any segment [a, b] we denote by ∆([a, b]) the unique irreducible quotient of σνa× · · · × σνb. It is an essentially square-integrable representation of GLd(b−a+1). The map [a, b]7→ ∆([a, b]) is a bijection between the segments and the essentially square-integrable representations whose cuspidal support is contained {σνi:i∈Z} [Zel80, §9.3]. We also write ∆([a, a −1]) = 1 (the one-dimensional representation of GL0) for any a∈Zand ∆([a, b]) = 0 if b<a−1. Let [a1, b1],...,[at, bt] be segments and let ∆i= ∆([ai, bi]). Assume that ∆idoes not precede ∆jfor any i<j,i.e., we do not have ai< aj≤bi+ 1 < bj+ 1 for i<j. Then the representation ∆1× · · · × ∆tadmits a unique irreducible quotient, the Langlands quotient, which we denote by L(∆1,...,∆t) (see for example [Rod82, Th´eor`eme 3]). Assume now that a1>· · · > atand b1>· · · > bt. In this case we say that ∆1,...,∆tis aladder and call L(∆1,...,∆t) a ladder representation. A particularly important subclass is the Speh representations where for every i= 1, . . . , t −1, ai+1 =ai−1 and bi+1 =bi−1. These representations comprise the building blocks for the unitary dual of Gn([Tad86]). Let Stbe the symmetric group on {1, . . . , t}. For any w∈Stlet Iw= ∆([aw(1), b1]) × · · ·×∆([aw(t), bt]). (This is 0 unless aw(i)≤bi+1 for all i.) In particular, for k= 1, . . . , t−1 let Kk=Isk⊆ IId where skis the transposition (k, k + 1). Theorem 1. Assume that ∆1,...,∆tis a ladder and let π= L(∆1,...,∆t). Then (i) The maximal proper submodule of ∆1× · · · × ∆tis Pt−1 k=1 Kk. ON A DETERMINANTAL FORMULA OF TADI ´ C 3 (ii) πsatisfies Tadi´c’s determinantal formula (1.1) π=X w∈St sgn wIw in R. As mentioned before, the formula (1.1) was proved by Tadi´c in the case of Speh representations [Tad95] and his proof was simplified in [CR08]. In the course of proving Theorem 1 we will analyze the Jacquet functor J(π) of πwith respect to the parabolic subgroup of type (d, . . . , d) and obtain a formula for its character. Remarkably, J(π) is the quotient of J(∆1×· · ·×∆t) by the sum of generalized eigenspaces (with respect to the torus action). We will introduce a certain directed graph E(π) whose vertex set consists of the segments ∆1,...,∆tdrawn sequentially in the plane, and the edges are horizontal and diagonal arrows. The character of J(π) will be expressed in terms of the possible vertex labellings of E(π) which are increasing with respect to the arrows. See Theorem 7 for the precise statement. In particular, the length of J(π) is equal, in the terminology of [Sta99, §7.10], to the number of standard Young Tableaux of the skew Young diagram (a1+ 1, . . . , at+t)/(b1+ 1, . . . , bt+t). Incidently, this can be computed by a well-known determinantal formula which is a consequence of the Jacobi-Trudi identity (cf. [Sta99, Corollary 7.16.3]). We may regard (1.1) as a p-adic analogue of the Jacobi-Trudi identity. We remark that for general irreducible representations of GLnthere is no known or conjectural simple description of their Jacquet modules in terms of the Langlands (or Zelevinsky) data. Theorem 1 has an interesting application which was pointed out to us by Marko Tadi´c and which is incorporated here with his kind permission. Namely, one can compute the full derivative (in the sense of Bernstein-Zelevinsky) of a ladder representation in terms of subordinate (in the sense of Zelevinsky) ladder representations. This expression had been conjectured by Tadi´c some time ago in the case of Speh representations [Tad87]. Interestingly enough, already in this case one has to use non-unitary ladder representations. In the last section we make some further comments about ladder representations and beyond. We first conjecture that roughly speaking, in the ladder case, the decomposition of ∆1×· · ·×∆tis quite uniform and does not depend in an essential way on the segments (except that some constituents may disappear, depending on the ordering of a1, b1, . . . , at, bt). Our second conjecture is a generalization of Theorem 1 part 1 to an arbitrary irreducible representation. This conjecture seems to be the first one addressing the fine structure of the Langlands quotient in the corresponding standard module. (Incidently, this conjecture, as well as the class of ladder representations, was inspired by analyzing functionals invariant under unitary groups – cf. [FLO].) We also provide a simple example of non-ladder irreducible representations which are not (fully) parabolically induced from representations of smaller GLn’s. To summarize, it seems that many results about Speh representations can be extended to ladder representations. Thus, the latter provide a suitable algebraic envelope for the 4 EREZ LAPID AND ALBERTO M´ INGUEZ former which avoids unitarity. We view the family of ladder representations as broad enough to include many interesting representations on the one hand and on the other hand sufficiently restricted so that one can approach their structure. Thus it provides a “litmus paper” for testing conjectures about representations of Gn(for instance Conjecture 2 alluded to above). See [BLM] for a follow-up about irreducibility questions. Finally, we remark that all the results and the proofs in this paper except for §5.5 carry over to inner forms of GLnwith minor changes. Acknowledgement. The authors would like to thank Shaun Stevens and the University of East Anglia for their hospitality. Also, the second named author wishes to thank the Hebrew University of Jerusalem for its hospitality. We also thank Ioan Badulescu and Laurent Clozel for useful discussions and Marko Tadi´c for his contribution in §5.5. Finally, we thank Ira Gessel for correspondence related to Proposition 10. 2. Notation and preliminaries 2.1. Throughout this article, we fix a non-Archimedean locally compact field F. Let G be the group of F-points of a connected reductive group defined over F, with the usual topology. We will only consider smooth representations of G, that is, representations such that the stabilizer of every vector is an open subgroup of G. Fix a minimal F-parabolic subgroup P0of Gand let M0be a Levi factor of P0defined over F. We denote by WGthe spherical Weyl group, defined by WG= NG(M0)/M0, where NG(M0) is the normalizer of M0in G. A parabolic subgroup Pof Gwill be called standard if it contains P0. Henceforth, the letter Pwill always denote a standard parabolic subgroup of Gwith an implicit standard Levi decomposition P=MU. Let (τ, V) be a representation of M, regarded as a representation of Pon which Uacts trivially. We denote by IM(τ) = IG M(τ) = IndG Pτ, the representation of Ginduced from τ. (We will always mean normalized induction.) We view IG Mas a functor. Its left adjoint, the Jacquet functor with respect to P, will be denoted by JM. For any representation π of G, we let jM=jM,π :π→JM(π) be the canonical projection. An irreducible representation πof Gis called cuspidal if it is not a composition factor of any representation of the form IG M(τ) with Pa proper parabolic subgroup of Gand τa representation of M. We denote by Irr G(resp. IrrcG) the set of equivalence classes of (resp. cuspidal) irreducible representations of G. For any π∈Irr Gthere exists, up to conjugacy, a unique pair (M, ρ) consisting of a Levi subgroup Mof Gand ρ∈IrrcMsuch that πis a composition factor of IG M(ρ). We call it the cuspidal support of πand write it supp(π). Let R(G) denote the Grothendieck group of the category of smooth representations of Gof finite length. The image of a representation πof Gfinite length in R(G) will be denoted by [π]. The Jacquet functor JMinduces a homomorphism JM:R(G)→R(M). Thus JM(π) is the semisimplification [JM(π)] of JM(π). ON A DETERMINANTAL FORMULA OF TADI ´ C 5 2.2. For any integer n≥0, set Gn= GLn(F). Let P0be the Borel subgroup of upper triangular matrices and let U0be its unipotent radical consisting of upper unitriangular matrices. The standard parabolic subgroups of Gare in bijection with compositions n= n1+· · · +nt. The corresponding standard Levi subgroup is the group of block diagonal invertible matrices with block sizes n1, . . . , nt. It is isomorphic to Gn1× · · · × Gnr. As in the introduction, if ρ1, . . . , ρrare representations of Gn1, . . . , Gnrrespectively, we will denote by (2.1) ρ1×ρ2× · · · × ρr=IGn M(ρ) the corresponding induced representation where ρis the representation ρ1⊗ρ2⊗· · ·⊗ρrof M. Here P=MU is the standard parabolic subgroup corresponding to the composition n=n1+· · · +nr. Given π∈Irr Gn, we can view the cuspidal support of πas the unique multi-set (i.e., set with multiplicities) (ρ1, . . . , ρr), ρi∈IrrcGni,n=n1+· · ·+nrsuch that πis a composition factor of ρ1×ρ2× · · · × ρr. 2.3. Let Rn=R(Gn) and R=⊕n≥0Rn. Then Ris a graded commutative ring with the product defined by (2.1). The identity element is the one-dimensional representation of G0. The natural ordering on Rwill be denoted by ≤. With respect to the addition R forms a lattice ordered group. We say that two non-negative elements of Rare disjoint if their meet is 0. We also write Irr = `n≥0Irr Gn. 2.4. Throughout the article, we fix a positive integer dand σ∈IrrcGd(not necessarily unitary). Write n=md. Let PP=MPUPbe the parabolic subgroup of type (d, . . . , d) in Gnwith MP≃Gd× · · · × Gd. For simplicity we write J=JP,j=jPand J=JP: R(Gn)→R(Gd× · · · × Gd) = R(Gd)⊗ · · · ⊗ R(Gd) (mtimes). Henceforth, we will only consider standard parabolic and Levi subgroups containing MP. Note that then Mis of type n1, . . . , ntwhere ddivides all ni’s. We write ni=dmi. 2.5. Denote by Zσthe set of cuspidal representations of the form σνi, for i∈Z. We denote by Irrσ⊆Irr the set of equivalence classes of irreducible representations of Gn(any n) whose cuspidal support is contained in Zσ. We denote by C=CGn σthe category of the finite length representations of Gnwith the property that all their irreducible subquotients belong to Irrσ. More generally, for any standard Levi M(containing MP) let CM=CM σbe the category of finite length representations (of M) with a similar property. We continue to write J(or JM) for the Jacquet functor with respect to PP∩M. 2.6. Let Wbe the Weyl group of Gn, identified with Snby Sn→W w7→ δi,w(j) and let WMbe the Weyl group of M. We identify WM\Wwith the set ˜ ΩMof left-WMreduced elements of W,i.e. the elements of Wof minimal length in their left WM-coset. 6 EREZ LAPID AND ALBERTO M´ INGUEZ Of particular importance will be the subset ΩMof ˜ ΩMdefined as follows. Let W0be the Weyl group of Gd× · · · × Gdand let W0be the set of reduced elements in W/W0which normalize MP. Note that W0is a subgroup of Wwhich we identify with Sm. We define ΩM= (W0∩WM)\W0identified with a subset of ˜ ΩM,i.e., ΩMis the set of elements of W0which are of minimal length in their left WM-coset. Explicitly, ΩM={w∈Sm:w−1(i)< w−1(i+ 1) ∀i6=m1, m1+m2, . . . , m1+· · · +mt−1}. In particular, |ΩM|=m m1m2... mt. We note that for any w∈ΩMwe have wUPw−1∩M=UP∩M and (2.2) wUPw−1∩P= (UP∩M)(wUPw−1∩U). Remark 1.ΩMis the set denoted by WM,MPin [BZ77, §2.11]. Note that ΩMnormalizes MP. Recall the Bruhat decomposition G=∪w∈WP0wU0and the relative Bruhat decomposition G=∪w∈˜ ΩMPwU0. We denote the Bruhat order on Wby ≤. Recall that w1≤w2if and only if P0w1U0⊆ P0w2U0in the p-adic topology of Gn. We refer to [BB05] for standard facts about the Bruhat order. The Bruhat order induces a partial order on ˜ ΩMand on ΩM. Once again, for w1, w2∈˜ ΩMwe have w1≤w2if and only if Pw1U0⊆Pw2U0. The Bruhat order gives rise to a topology on W. Namely, A⊆Wis open if and only if whenever w, w0∈Wand w≥w0∈A, we have w∈A. Equivalently, a subset A⊆W is open if and only if P0AU0is open in Gn. For instance, for any w∈Wthe subsets W≥w={w0∈W:w0≥w}and W>w ={w0∈W:w0> w}are open. Similarly for ˜ ΩM,≥win ˜ ΩM, etc.. (We topologize subsets of Wby the relative topology.) Note that W≥w=WM˜ ΩM,≥wfor any w∈˜ ΩM. 2.7. Let τ∈CMand let Π = IM(τ). For any open subset ωin ˜ ΩMconsider the PPinvariant subspace Πω= ΠM,ω := {ϕ∈IM(τ) : supp(ϕ)⊆PωU0}={ϕ∈IM(τ) : ϕGn\P ωU0≡0}. Let J(Π)M,ω be its image under j, which is a subrepresentation of J(Π). (We often omit Mif it’s clear from the context.) In particular for any w∈˜ ΩM, we can consider Π≥w, Π>w,J(Π)≥wand J(Π)>w. By [BZ77, §2.12], fixing a choice of Haar measures, the map ˜pw:ϕ7→ Z(UP∩w−1UPw)\UP jτ(ϕ(wu)) du ∈J(τ), ϕ ∈Π≥w ON A DETERMINANTAL FORMULA OF TADI ´ C 7 induces a surjective homomorphism pM,τ,w =pw:J(Π)≥w→(J(τ)wif w∈ΩM, 0 otherwise, whose kernel is J(Π)>w. (Note that ˜pwis well-defined because PwU0is closed in PW≥wU0.) Here J(τ)wis the vector space J(τ) with the twisted action of MPby w. Note that if β⊆τthen J(IM(β)) ⊆J(IM(τ)) and pM,β,w is the restriction of pM,τ,w. 2.8. Now let Lbe a standard Levi subgroup of M. Set ˜ ΩM L=˜ ΩL∩WMand ΩM L= ΩL∩WM. Recall that the map (w1, w2)7→ w1w2defines bijections ˜ ΩM Lט ΩM→˜ ΩLand ΩM L×ΩM→ΩL. Let %∈CLand τ=IM L(%), so that Π = IM(τ)≃˜ Π := IL(%). Let ιM Lthe equivalence of representations ιM L:˜ Π→Π defined by ιM Lϕ(g)=(h7→ ϕ(hg)). Correspondingly we have J(ιM L) : J(˜ Π) →J(Π). For any w∈ΩM Lwe may consider τ≥w and J(τ)L,≥w. The following result is probably well known. For convenience we include a proof. Proposition 2 (Compatibility with induction in stages).Under the above assumptions suppose that w=w1w2∈ΩLwhere w1∈ΩM L,w2∈ΩM. Then ιM L(˜ ΠL,≥w)⊆ΠM,≥w2 (2.3) p−1 M,τ,w2(J(τ)w2 ≥w1) = J(ιM L)(J(˜ Π)≥w)(2.4) pM L,%,w1◦pM,τ,w2◦J(ιM L) = pL,%,w on J(˜ Π)L,≥w.(2.5) Proof. Let Qbe the standard parabolic with Levi L. The relation (2.3) follows from the fact that QW≥wU0⊆PW≥w2U0. The inclusion ⊇of (2.4) follows from the relation (2.6) {m∈M:mw2U0∩QW≥wU06=∅} = (M∩Q)WM ≥w1(U0∩M). To prove (2.5) we first observe that w−1 1UPw1= (w−1 1UPw1∩M)Uand UP= (UP∩M)U, and hence by (2.2) (applied to w2) we have (2.7) w2UPw−1 2∩w−1 1UPw1= (w2UPw−1 2∩U)(UP∩w−1 1UPw1∩M) and (2.8) w2UPw−1 2∩UP= (w2UPw−1 2∩U)(UP∩M). Let ϕ∈˜ Π≥wand ϕ0=ιM Lϕ. Then ˜pL,w(ϕ) = Z(UP∩w−1UPw)\UP j%(ϕ(wu)) du =Z(UP∩w−1 2UPw2)\UPZ(UP∩w−1UPw)\(UP∩w−1 2UPw2) j%(ϕ(w1w2u1u2)) du1du2 =Z(UP∩w−1 2UPw2)\UPZ(w2UPw−1 2∩w−1 1UPw1)\(w2UPw−1 2∩UP) j%(ϕ(w1u1w2u2)) du1du2. 8 EREZ LAPID AND ALBERTO M´ INGUEZ Using (2.7) and (2.8) this equals Z(UP∩w−1 2UPw2)\UPZ(UP∩M∩w−1 1UPw1)\(UP∩M) j%(ϕ(w1u1w2u2)) du1du2 =Z(UP∩w−1 2UPw2)\UPZ(UP∩M∩w−1 1UPw1)\(UP∩M) j%(ϕ0(w2u2)(w1u1)) du1du2. Note that by (2.6) for any u2∈U0we have ϕ0(w2u2)∈τ≥w1. Hence we get Z(UP∩w−1 2UPw2)\UP ˜pM L,w1(ϕ0(w2u)) du =Z(UP∩w−1 2UPw2)\UP pM L,w1(jτ(ϕ0(w2u))) du =pM L,w1◦˜pM,w2(ϕ0). We conclude (2.5). Finally, we prove (2.4). We already know the inclusion ⊇. First note that P˜ ΩM,>w2U0= Q˜ ΩM L˜ ΩM,>w2U0and hence ΠM,>w2=ιM L(˜ ΠL,˜ ΩM L˜ ΩM,>w2). Thus, Ker pM,τ,w2=J(Π)M,>w2=J(ιM L)(J(˜ Π)L,˜ ΩM L(˜ ΩM)>w2)⊆J(ιM L)(J(˜ Π)L,≥w). Hence, to prove (2.4) it remains to show that (2.9) pM,τ,w2◦J(ιM L)(J(˜ Π)≥w) = J(τ)w2 ≥w1. We show this for all w1∈˜ ΩM Lby descending induction on `(w1). (Note that (2.5) holds trivially if w1∈˜ ΩM L\ΩM L.) For the longest element in ˜ ΩM L, the equality (2.9) follows from (2.5). For the induction step, (2.5) gives pM,τ,w2◦J(ιM L)(J(˜ Π)≥w) + J(τ)w2 >w1=J(τ)w2 ≥w1. On the other hand, by induction hypothesis we have pM,τ,w2◦J(ιM L)(J(˜ Π)>w) = J(τ)w2 >w1. The relation (2.9) follows.  Corollary 3. Under the same assumptions and notation suppose further that β∈CMis a subrepresentation of τ. Identify IM(β)(resp. J(IM(β))) with a subrepresentation of ˜ Π (resp. J(˜ Π)) via (ιM L)−1(resp. J(ιM L)−1). Let w=w1w2∈ΩLwhere w1∈ΩM L,w2∈ΩM. Then the image of J(IM(β)) ∩J(˜ Π)L,≥w→J(˜ Π)L,≥w/J(˜ Π)L,>w is isomorphic to the w2-twist of the image of J(β)∩J(τ)L,≥w1→J(τ)L,≥w1/J(τ)L,>w1. In particular if J(β) + J(τ)L,>w1⊇J(τ)L,≥w1 ON A DETERMINANTAL FORMULA OF TADI ´ C 9 then J(IM(β)) + J(˜ Π)L,>w1w2⊇J(˜ Π)L,≥w1w2. Indeed, the Corollary follows from the relation pL,w(J(ιM L)−1(J(IM(β))) ∩J(˜ Π)L,≥w)(2.5) =pM L,w1◦pM,w2(J(IM(β)) ∩J(ιM L)(J(˜ Π)L,≥w)) (2.4) =pM L,w1◦pM,w2(J(IM(β)) ∩p−1 M,w2(J(τ)w2 ≥w1)) = pM L,w1(J(β)w2∩J(τ)w2 ≥w1). 3. The graph E(π) 3.1. Ladder representations. As before, we fix a positive integer dand a cuspidal representation σof Gd. 3.1.1. By a segment [a, b] we mean a set of the form {a, a + 1, . . . , b}for some integers a≤b. For any segment [a, b] we denote by ∆([a, b]) the unique irreducible quotient of σνa× · · · × σνb: it is an essentially square-integrable representation of Gd(b−a+1). The map [a, b]7→ ∆([a, b]) is a bijection between the set of segments and the set of essentially square-integrable representations in Irrσ[Zel80, §9.3]. By definition, the length l(∆([a, b])) of ∆([a, b]) is b−a+1. For convenience, by a slight abuse of notation, we set ∆([a, a−1]) = 1 (the one-dimensional representation of G0) for any a∈Zand ∆([a, b]) = 0 (in R) for b < a −1. Let [a1, b1],...,[at, bt] be segments and let ∆i= ∆([ai, bi]) and m=P i l(∆i). Assume that for all i<j, ∆idoes not precede ∆j,i.e. we do not have ai< aj≤bi+ 1 < bj+ 1 for i<j. Then the representation I(∆1,...,∆t) := ∆1× · · · × ∆t admits a unique irreducible quotient, the Langlands quotient, which we denote by L(∆1,...,∆t) (see for example [Rod82, Th´eor`eme 3]). Any element of Irrσis of this form, for uniquely determined ∆1,...,∆t(up to permutation). More generally, by reordering ∆1,...,∆tif necessary, we define L(∆1,...,∆t) for any t-tuple. 3.1.2. Let [a1, b1],...,[at, bt] be segments and let ∆i= ∆([ai, bi]). We say that ∆1,...,∆t form a ladder if a1>· · · > atand b1>· · · > bt. In this case we term L(∆1,...,∆t) a ladder representation. A particularly important special case is the class of Speh representations for which ai+1 =ai−1 and bi+1 =bi−1 for all i= 1, . . . , t −1. These representations comprise the building blocks for the unitary dual of Gn([Tad86]). 3.2. Let [a1, b1],...,[at, bt] be segments, ∆i= ∆([ai, bi]) and m=P i l(∆i). Let π= L(∆1,...,∆t)∈Irr Gmd. We order the segments so that for all i<jwe have bi≥bjand if bi=bjthen ai≥aj. (In the sequel we only consider the case where b1>· · · > bt.) 16 EREZ LAPID AND ALBERTO M´ INGUEZ j0  ◦ ◦ ◦ ◦ ◦ ◦  G κj0   ◦ ◦ ◦ ◦ ◦  ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ G1◦ ◦ ◦ We define φ0=ι1(φ) : G→ {1, . . . , m}by φ0=φ◦κj0. Evidently λφ0=λφ. To check that φ0∈ M(G) we observe that by assumption on j0 φ0(j0,1) = φ(j0,1) < φ(j0−1,2) = φ0(j0−1,1) and (if j0≤b2) φ0(j0,2) = φ(j0,2) < φ(j0+ 1,1) < φ(j0,1) < φ(j0−1,1) = φ0(j0−1,2) where the first inequality follows from the maximality of j0. The other inequalities φ0(j, i)< φ0(j−1, i)j06=j∈[ai+ 1, bi] follow directly from the corresponding inequalities for φ. Finally, the inverse map M(G)\M(E(π)) → M(G1) is defined in exactly the same way. (The index j0is well-defined for φ∈ M(G)\ M(E(π)).) We infer that J(K) = XM(G1)=XM(G)\M(E(π)) =XM(G)−XM(E(π)) and J(π) = XM(E(π)). As was pointed out before, the first part of Theorem 7 in the case t= 2 now follows from the disjointness of XM(E(π)) and XM(G)−XM(E(π)). For future record we say that φ0as above is obtained from φby a flip-flop along the two rows. Of course, for this procedure we do not need to assume necessarily that φtakes values in {1, . . . , m}. Recall the bijection ψπ: ΩM−→ M(G) defined in (3.1). We further note that for any w∈ΩMsuch that ψπ(w)∈ι1(M(G1)) we have (4.3) J>w +K⊇J≥w. This follows from Lemma 6 because J≥w/J>w ≃σ[λψπ(w)], ψπ(w)/∈ M(E(π)) and K=JM(E(π)) by what we just proved. ON A DETERMINANTAL FORMULA OF TADI ´ C 17 4.4. The general case. Let Pk,k= 1, . . . , t −1 be the parabolic subgroup of type (n1, . . . , nk−1, nk+nk+1, nk+2, . . . , nt). Recall that, by definition, Kkis isomorphic to IMk(%k) where %k= ∆1⊗ · · · ⊗ ∆k−1⊗∆([ak+1, bk]) ×∆([ak, bk+1]) ⊗∆k+2 ⊗ · · · ⊗ ∆t. Let Gk=I(L(∆1,...,∆k−1,∆([ak+1, bk]),∆([ak, bk+1]),∆k+2,...,∆t)). (If ak> bk+1 + 1 then we set Gk=∅.) Using the map ι1defined above in the case t= 2 we define an injective map ιk:M(Gk)→ M(G) by flip-flop along rows k, k + 1. More precisely we define ιk(φ) = φ◦κk,j0for φ∈ M(Gk) where j0is the largest j∈[ak, bk+1 + 1] such that φ(j, k)< φ(j−1, k + 1) (with φ(ak−1, k + 1) = ∞) and κk,j0:G→Gkis the bijection κk,j0(j, i) = ((j, sk(i)) if j < j0, (j, i) otherwise. The map ιkis weight-preserving and therefore J(Kk) = J(IMk(%k)) = XM(Gk)=XMk(G) where Mk(G) is the image of ιk. We can characterize Mk(G) as the set of elements φin M(G) for which φ(j, k)< φ(j−1, k + 1) for some j∈[ak, bk+1 + 1]. It follows from the definition of M(E(π)) that ∪t−1 k=1Mk(G) is the complement of M(E(π)) in M(G). We conclude from Lemma 4 that J(Kk)⊆JM(E(π)) for all k, and hence J(K) = Pt k=1 J(Kk)⊆JM(E(π)). To conclude the first part of Theorem 7 it remains to show the following Lemma 8. J(K)⊇JM(E(π)). Proof. First note that the condition ψπ(w)∈ Mk(G) depends only on the right ΩMk-coset of w. Hence ψ−1 π(Mk(G)) = ωkΩMk where ωk= ΩMk M∩ψ−1 π(Mk(G)). Therefore, Ωπ(see §3.4) is the complement of ∪t−1 k=1ωkΩMk in ΩM. We will show by descending induction on `(w) that JM(E(π)) ∩J≥w⊆J(K) for all w∈ΩM. For the induction step (as well as for the base of the induction) we may assume by (4.1) that JM(E(π)) ∩J>w ⊆J(K). Recall that J≥w/J>w ≃σ[λψπ(w)]. Therefore, if w∈Ωπthen JM(E(π)) ∩J>w =JM(E(π)) ∩J≥wand the induction step follows trivially. On the other hand, if w /∈Ωπthen there exists ksuch that w=w1w2where w1∈ωkand w2∈ΩMk. By (4.3) and Corollary 3 applied with L=M,M=Mk,%=δand β=%kwe infer that J(Kk) + J>w =J(IMk(%k)) + J>w ⊇J≥w. Hence J≥w∩JM(E(π)) ⊆J(Kk)∩JM(E(π)) +J>w ∩JM(E(π)) ⊆J(K) + J>w ∩JM(E(π)) 18 EREZ LAPID AND ALBERTO M´ INGUEZ and we can apply the induction hypothesis to conclude that J≥w∩JM(E(π)) ⊆J(K) as required.  It remains to show the second part of Theorem 7. Let M: ∆1× · · · × ∆t→∆t× · · · × ∆1be the “longest” intertwining operator. It is known that the image of Mis π(or equivalently, Ker M=L). Since Jis a functor we have a map J(M) : J→J(∆t× · · · × ∆1). Lemma 9. J(M)factors through an injective map on JM(E(π)). Therefore J(π)≃ JM(E(π)). Proof. It is clear that for each k= 1, . . . , t−1, Mfactors through the intertwining operator obtained by switching ∆k−1and ∆k. Hence, Ker M⊇Kkfor all kand consequently Ker M⊇K. Thus, Ker J(M)⊇J(K) and by the first part of the Theorem proved above it follows that J(M) factors through JM(E(π)). It remains to show that J(M) is injective on I=JM(G)\M(E(π)). Let si1. . . silbe a reduced decomposition of the longest element in St. Correspondingly we decompose M=Ml◦ · · · ◦ M1 where Mjis the intertwining operator Mj: ∆wj−1(1) × · · · × ∆wj−1(t)→∆wj(1) × · · · × ∆wj(t) where wj=si1. . . sij. We will show that for all j= 1, . . . , l J(Mj) is injective on the image of Iunder J(Mj−1◦ · · · ◦ M1), i.e. (4.4) Ker J(Mj)∩J(Mj−1◦ · · · ◦ M1)(I) = 0. Recall that wj−1(ij)< wj−1(ij+ 1) = wj(ij) and Ker Mjis equivalent to ∆wj(1) ×· · ·×∆wj(ij−1) ×∆([awj−1(ij), bwj(ij)])×∆([awj(ij), bwj−1(ij)])×∆wj(ij+2) ×· · ·×∆wj(t) which is equal to I(wj−1(ij),wj(ij)) in R. It follows from (4.2) that [Ker Mj]≤[K], and hence [Ker J(Mj)] = J(Ker Mj)≤J(K). Since [I] and J(K) are disjoint, we deduce (4.4).  This concludes the proof of Theorem 7. 5. Application to Tadi´ c’s formula We continue to use the notation and assumptions of the previous section. Our goal in this section is to prove the second part of Theorem 1, namely Tadi´c’s determinantal formula (1.1) π=X w∈St sgn wIw. Let (5.1) S◦ t:= {w∈St:aw(i)≤bi+ 1, i = 1, . . . , t}. We note that only elements of S◦ tcontribute to the sum on the right-hand side of (1.1). ON A DETERMINANTAL FORMULA OF TADI ´ C 19 Given w∈Stwe have w∈S◦ tif and only if w(i)≥rifor all iwhere ri= min{j:aj≤ bi+ 1})≤i. Thus |S◦ t|=Qt i=1(i+ 1 −ri). Indeed, r1≤ · · · ≤ rt; there are t+ 1 −rt possibilities for w(t), t−rt−1possibilities for w(t−1) given w(t), etc. Remark 4.In light of Remark 3 we can view (1.1) as a p-adic analogue of the Jacobi-Trudi identity. 5.1. To simplify notations, let E=E(L(∆1,...,∆t)) and, for any w∈S0 t, let Iw= I(L(∆([aw(1), b1]),...,∆([aw(t), bt]))). Recall that M(E)⊆ M(IId). Set M0={(φ, w) : w∈S◦ t, φ ∈ M(Iw)}\{(φ, Id) : φ∈ M(E)}. Proposition 10. There exists an involution ∗on M0such that if (φ, w)∗= (φ∗, w∗)then sgn(w∗) = −sgn(w)and λφ=λφ∗. (In the case t= 2 this involution was used in section 4.3 above.) Before proving the proposition, it will be convenient to introduce the following convention. Given a graph Iand φ∈ M(I) we extend φto Z2by setting φ(j, i) =          ∞j < ai, i = 1, . . . , t, −∞ j > bi, i = 1, . . . , t, ∞i < 1, −∞ i > t. Example 4. In example 3, φstd extends to ∞∞∞∞ ∞ 5 ◦4 ◦ oo3 ◦ oo2 ◦ oo1 ◦ oo−∞ ∞∞∞11 ◦10 ◦ oo9 ◦ oo8 ◦ oo7 ◦ oo6 ◦ oo−∞ −∞ 15 ◦14 ◦ oo13 ◦ oo12 ◦ oo−∞ −∞ −∞ −∞ −∞ −∞ −∞ Proof. The statement and the proof are closely related to the Gessel-Viennot Lemma (cf. [Sta97, Theorem 2.7.1]). Let (φ, w)∈ M0. Define Y=Y(φ,w)={(j, i)∈Z2:φ(j, i)< φ(j−1, i + 1)}. Note that by our convention we have (aw(i+1), i)∈Ywhenever w(i)> w(i+1). Thus, Y6= ∅. Also, we have 1 ≤i < t and j∈[aw(i), bi+1 + 1] for all (j, i)∈Y. Let (j0, i0) = (jφ 0, iφ 0) be the maximal element in Ywith respect to the lexicographic order (from left to right). Clearly j0∈[amin(w(i0),w(i0+1)), bi0+1 + 1]. Let si0be the transposition (i0, i0+ 1), set w∗=wsi0and define φ∗(j, i) = (φ(j, si0(i)) j < j0, φ(j, i) otherwise. 20 EREZ LAPID AND ALBERTO M´ INGUEZ By definition, it is clear that λφ∗=λφand sgn(w∗) = −sgn(w). Let us check that w∗∈S0 t. It is clear that aw∗(i)≤bi+ 1 for i6=i0, i0+ 1. Also aw∗(i0)=aw(i0+1) ≤j0≤bi0+1 + 1 < bi0+ 1 and aw∗(i0+1) =aw(i0)≤j0≤bi0+1 + 1. Thus, w∗∈S0 t. Next, we show that φ∗∈ M(Iw∗). It is clear that for all 1 ≤i≤twe have φ∗(j, i) = ∞ if and only if j < aw∗(i)and φ∗(j, i) = −∞ if and only if j > bi. We also need to show that for all 1 ≤i≤tand j∈[aw∗(i)+ 1, bi] we have φ∗(j−1, i)> φ∗(j, i).This follows from the similar property of φif either i6=i0, i0+ 1 or j6=j0. For the remaining two cases we have φ∗(j0−1, i0) = φ(j0−1, i0+ 1) > φ(j0, i0) = φ∗(j0, i0) since (j0, i0)∈Y, and φ∗(j0−1, i0+ 1) = φ(j0−1, i0)> φ(j0, i0)> φ(j0+ 1, i0)> φ(j0, i0+ 1) = φ∗(j0, i0+ 1) where the last inequality follows from the maximality of (j0, i0). Finally we verify that (φ∗, w∗)∗= (φ, w). Let (j0, i0) = (jφ 0, iφ 0), (j∗ 0, i∗ 0) = (jφ∗ 0, iφ∗ 0), Y= Y(φ,w)and Y∗=Y(φ∗,w∗). Observe that φ∗(j0, i0) = φ(j0, i0)< φ(j0−1, i0) = φ∗(j0−1, i0+1), so that (j0, i0)∈Y∗. Suppose that (j, i)>(j0, i0) in the lexicographic order. Then φ∗(j, i) = φ(j, i) (since j≥j0) and φ∗(j−1, i+1) = φ(j−1, i+1) (since j≥j0and if j=j0 then i>i0). Thus, φ∗(j, i)> φ∗(j−1, i + 1) and (j, i)/∈Y∗. Thus, (j0, i0) is the maximal element in Y∗, so that (j∗ 0, i∗ 0)=(j0, i0). It readily follows that (φ∗, w∗)∗= (φ, w).  Corollary 11. We have an equality in the Grothendieck group: XM(E(L(∆1,...,∆t))) =X w∈St sgn(w)XM(Iw). Equivalently (by (3.2)), (5.2) J(L(∆1,...,∆t)) = X w∈St sgn(w)J(Iw). Proof. First note that only w∈S0 tcontributes. With the notation as in the previous proposition, we have X w∈S0 t sgn(w)XM(Iw)=X w∈S0 t sgn(w)X φ∈M(Iw) σ[λφ] =XM(E)+X (φ,w)∈M0 sgn(w)σ[λφ]. But, applying the involution, we have P(φ,w)∈M0sgn(w)σ[λφ] = 0. The corollary follows.  ON A DETERMINANTAL FORMULA OF TADI ´ C 21 5.2. Proof of (1.1).It is known that L(∆1,...,∆t) = X w∈S◦ t cwIw for some integers cw. Applying Jwe obtain J(L(∆1,...,∆t)) = X w∈S◦ t cwJ(Iw). Combined with (5.2), we infer that X w∈S◦ t sgn(w)J(Iw) = X w∈S◦ t cwJ(Iw). On the other hand by [Zel80, §6.9], the elements J(Iw), w∈S◦ tare free over Zin R. We conclude that cw= sgn(w) for all w. This concludes the proof of Theorem 1. 5.3. Note that if we define the matrix (mi,j)i,j=1,...,t with coefficients in Rby mi,j = ∆([ai, bj]) then the identity (1.1) reads L(∆1,...,∆t) = det (mi,j). Thus, by [CR08, Lemma 6.1], we immediately get the following result. Corollary 12. Let ∆i= ∆([ai, bi]),i= 1, . . . , t be a ladder. Let π= L(∆1,...,∆t−1)×L(∆2,...,∆t), π1= L(∆1,...,∆t)×L(∆2,...,∆t−1), π2= L(∆([a1, b2]),...,∆([at−1, bt])) ×L(∆([a2, b1]),...,∆([at, bt−1])). Then in the Grothendieck group we have π=π1+π2. For Speh representations see [Tad06]. Remark 5.Note that π2= 0 if ai> bi+1 + 1 for some i= 1, . . . , t −1. In a subsequent paper [BLM] we show that π=π1is irreducible in this case. Otherwise we show that π1 and π2are irreducible, so that πhas length 2, and π1(resp. π2) is the unique irreducible quotient (resp. subrepresentation) of π. 5.4. We can rephrase Theorem 1 in terms of Zelevinsky classification [Zel80]. For any segment [a, b] we write Z([a, b]) = L(σνb, . . . , σνa). Thus Z([a, b]) is the unique irreducible subrepresentation of σνa× · · · × σνb. As usual we write Z([a, a −1]) = 1 and Z([a, b]) = 0 if b < a −1. More generally, for a multisegment [a1, b1],...,[at, bt] such that for any i < j [ai, bi] does not precede [aj, bj] we write Z([a1, b1],...,[at, bt]) for the unique irreducible subrepresentation of Z([a1, b1]) × · · · × Z([at, bt]). The Zelevinsky involution takes L(∆([a1, b1]),...,∆([at, bt])) to Z([a1, b1],...,[bt, at]). Recall that the class of ladder representations is invariant under Zelevinsky involution. Therefore, if a1>· · · > atand b1>· · · > btthen Z([a1, b1],...,[at, bt]) is a ladder representation and conversely any ladder representation can be expressed this way. By applying Zelevinsky involution on (1.1) we obtain 22 EREZ LAPID AND ALBERTO M´ INGUEZ Corollary 13. Suppose that a1>· · · > atand b1>· · · > bt. Then we have Z([a1, b1]) × · · · × Z([at, bt]) = X w∈S◦ t sgn w Z([aw(1), b1]) × · · · × Z([aw(t), bt]) = det(Z([ai, bj]))1≤i,j≤t in R. 5.5. We conclude this section with another application, due to Marko Tadi´c, for the computation of the (full) derivative of a ladder representation. This formula had been conjectured by Tadi´c for Speh representations [Tad87]. The interesting point is that even for Speh representations one needs to use non-unitary ladder representations. We are grateful to Marko Tadi´c for kindly allowing us to include this application here. Recall that for any finite length representation πof Gnwe can consider its full derivative D(π) which is a sequence of finite length representations of Gi,i= 0, . . . , n (see [BZ77, §4] for definition and basic properties). The functor Don ⊕∞ n=0CGninduces a ring homomorphism of the Grothendieck group (which is a subring of R). We have D(Z([a, b])) = Z([a, b]) + Z([a, b −1]) ([Zel80, Theorem 3.5]). Theorem 14 (Tadi´c).Suppose that a1>· · · > atand b1>· · · > bt. Then the full derivative D(Z([a1, b1],...,[at, bt])) of Z([a1, b1],...,[at, bt]) is given by the direct sum of Z([a1, b0 1],...,[at, b0 t]) where (b0 1, . . . , b0 t)range over all sequences such that bi−b0 i∈ {0,1}for all iand b0 1<· · · < b0 t. In other words, (5.3) D(Z([a1, b1],...,[at, bt])) = ⊕m0Z(m0) where the sum is over all multisegments m0which are subordinate to [a1, b1],...,[an, bn](in the sense of [Zel80, §7.4]) and which form a ladder. Proof. Using Corollary 13 and the fact that Dis a ring homomorphism we have D(Z([a1, b1],...,[at, bt])) = D(det(Z([ai, bj]))1≤i,j≤t) = det(D(Z([ai, bj])))1≤i,j≤t = det(Z([ai, bj]) + Z([ai, bj−1]))1≤i,j≤t in R. By the multi-linearity of the determinant we get X 1,...,t∈{0,1} det(Z([ai, bj−j]))1≤i,j≤t. Let b0 j=bj−jso that b0 j≥b0 j+1,j= 1, . . . , t −1. If b0 j=b0 j+1 for some jthen the corresponding determinant in the sum above vanishes, since the matrix contains two identical columns. Therefore we remain with X ∀j bj−b0 j∈{0,1},b0 1<···<b0 t det(Z([ai, b0 j]))1≤i,j≤t. ON A DETERMINANTAL FORMULA OF TADI ´ C 23 Once again using Corollary 13 we obtain D(Z([a1, b1],...,[at, bt])) = X ∀j bj−b0 j∈{0,1},b0 1<···<b0 t Z([a1, b0 1],...,[at, b0 t]) in R. Now observe that the cuspidal supports of all representations in the above sum are different. Therefore, their infinitesimal characters (in the sense of Bernstein center) are all different. Hence, D(Z([a1, b1],...,[at, bt])) = M ∀j bj−b0 j∈{0,1},b0 1<···<b0 t Z([a1, b0 1],...,[at, b0 t]) as an isomorphism of representations (of the various GLn’s).  Note the similarity between this argument and the standard proof of branching laws for the unitary group or the symmetric group (e.g., [Bum04, ch. 42, 44]). We can rephrase Theorem 14 in terms of Langlands classification as follows. If ∆1,...,∆t form a ladder with ∆i= ∆([ai, bi]) then D(L(∆1,...,∆t)) = XL(∆([a0 1, b1]),...,∆([a0 t, bt])) where the sum is over a0 i∈[ai, ai−1−1], i= 1, . . . , t with the convention that a0=∞. (In particular, a0 1>· · · > a0 n.) This can be proved by applying the Mœglin-Waldspurger algorithm to both sides of (5.3). Alternatively, we can follow the argument of Theorem 14 using the fact that D([a, b]) = Pa0≥a∆([a0, b]) ([Zel80, Proposition 9.6]) – of course only a0≤b+ 1 contribute. We will obtain (5.4) D(L(∆1,...,∆t)) = X a0 i≥ai,i=1,...,t det ∆([a0 i, bj]). If the a0 jare not distinct then the corresponding summand vanishes. The summands for which a0 i≥ai−1for some i > 1 cancel in pairs. Indeed, on this set we can define an involution by switching a0 i0and a0 i0−1where i0>1 is the largest index such that a0 i0≥a0 i0−1. This involution negates the corresponding summand. We remain with the a0 isuch that a0 i< ai−1as required. 6. Odds and ends We will list several questions and conjectures arising from the results above. 6.1. Decomposition of standard modules. It is natural to ask whether in the case of a ladder representations L(∆1,...,∆t) the decomposition of ∆1× · · · × ∆tin R, or more generally of Mw:= Iwfor any w∈S◦ t(see (5.1)), is any simpler than in the general case. As was pointed out by Tadi´c, MId is not multiplicity free in general for t > 3 [Tad95, §6]. Let Lw= L(∆([aw(1), b1]),...,∆([aw(t), bt])). Recall that Lw6= 0 if and only if w∈S◦ t. Note that for w, w0∈S◦ twe have w0≥wif and only if [aw0(1), b1],...,[aw0(t), bt]≤[aw(1), b1],...,[aw(t), bt] 24 EREZ LAPID AND ALBERTO M´ INGUEZ in the partial order on multi-segments introduced by Zelevinsky in [Zel80, §7]. Therefore we have Mw=X w0≥w cw,w0Lw0 for some coefficients cw,w0∈Z≥0. We have cw,w = 1 for all w∈S◦ t. Lemma 15. There exists a unique w0 0∈S◦ tsuch that (6.1) S◦ t={w∈St:w≤w0 0}. Moreover, w0 0avoids 312 pattern, i.e. there does not exist a triple i < j < k such that w0 0(k)< w0 0(i)< w0 0(j). Finally Lw0 0is generic. Proof. We first note that the set S◦ tis closed from below in St(i.e., it is closed in the Bruhat order topology of St– see §2.6). Indeed, suppose that w∈S◦ tand w(i)> w(j) for some i < j. Let sbe the transposition (i, j). We verify that w0:= ws ∈S◦ t. If k6=i, j then aw0(k)=aw(k)≤bk+ 1. On the other hand, aw0(i)=aw(j)≤bj+ 1 < bi+ 1 and aw0(j)=aw(i)< aw(j)≤bj+ 1. Thus w0∈S◦ tand S◦ tis closed from below as claimed. It remains to show that S◦ thas a unique maximal element w0 0. We define w0 0by recursion as follows. Suppose that w0 0(t), . . . , w0 0(i+ 1) were defined for some 1 ≤i≤t. Then define w0 0(i) to be the minimal index j6=w0 0(t), . . . , w0 0(i+1) such that aj≤bi+1. (Such jexists because ai, . . . , at≤bi+ 1.) By definition, w0 0∈S◦ t. To see that w0 0is the unique maximal element of S◦ tsuppose that w∈S◦ twith w6=w0 0. Then we claim that wis not maximal in S◦ t. Indeed, let ibe the maximal index such that w(i)6=w0 0(i). Then by definition of w0 0we necessarily have w(i)> w0 0(i). Let sbe the transposition (i, j) with j=w−1(w0 0(i)). Note that j < i since w(k) = w0(k) for all k > i by assumption. Also, w(j) = w0 0(i)< w(i) so that ws > w. On the other hand ws ∈S◦ tbecause aws(k)=aw(k)≤bk+ 1 for all k6=i, j while aws(i)=aw(j)=aw0 0(i)≤bi+ 1 and aws(j)=aw(i)≤bi+ 1 ≤bj+ 1. We conclude (6.1). Let us prove now that w0 0avoids the pattern 312. Assume on the contrary that it does not. Then w0 0(k)< w0 0(i)< w0 0(j) for some triple i < j < k. It follows that aw0 0(i)< aw0 0(k)≤bk+ 1 < bj+ 1. However, by the choice of w0 0(j) we would then have w0 0(j)≤w0 0(i) in contradiction. Finally, to show that Lw0 0is generic assume on the contrary that [aw0 0(j), bj] precedes [aw0 0(i), bi] for some i<j. Then w0 0(i)< w0 0(j) and aw0 0(i)≤bj+ 1 in contradiction with the minimality of w0 0(j).  Remark 6.One can show conversely that any w∈Stwhich avoids 312 pattern can be realized as w0 0for some ladder of rank t. These permutations are exactly the inverses of the stack sortable permutations. Their number is known to be the Catalan number Ct([Sta99, p. 224]). We conclude that cw,w0>0 for all w≤w0≤w0 0and cw,w0 0= 1 for all w∈S◦ t. Set cw,w0= 0 if w6≤ w0. Let c0 w,w0be the inverse matrix of (cw,w0)w,w0∈S◦ t. Then c0 w,w0has integer ON A DETERMINANTAL FORMULA OF TADI ´ C 25 entries, c0 w,w = 1 for all w∈S◦ tand c0 w,w0= 0 if w6≤ w0. We have Lw=X w0≥w c0 w,w0Mw0. Conjecture 1. We have cw,w0=Pw,w0(1) where Pw,w0are the Kazhdan-Lusztig polynomials. Equivalently, (6.2) c0 w,w0= (−1)l(w)−l(w0)Pw0w0,w0w(1). In other words, the relation between the Mw’s and Lw’s is analogous to the relation between Verma modules and simple highest weight modules in the category O(cf. [Hum08]). Note that the relation (1.1) is the case w= Id of (6.2). Also, note that Pw,w0 0≡1 for all w≤w0 0because w0 0avoids the pattern 312 [LS90]. This is consistent with the fact that cw,w0 0= 1. In principle, it should be possible to check whether Conjecture 1 is in accordance with Zelevinsky’s conjectures [Zel85] proved in [CG97]. However, this is not straightforward since the Kazhdan-Lusztig polynomials appearing in [Zel85] are pertaining to the much bigger symmetric group Smwhere we recall that m=Pt i=1(bi+ 1 −ai). We will not pursue this question any further here. At any rate, we checked that Conjecture 1 holds for t= 3, in which case cw,w0= 1 for all w≤w0, and t= 4, in which case cw,w0=         0w06≥ w, 2w0= (1,3)(2,4) and w≤(2,3), 2w0= (1,4) and w≤(1,2)(3,4), 1 otherwise. (See [Tad95, §6] for a special case.) Another interesting problem in connection with ladder representations would be to determine (the semisimplification of) all Jacquet modules, not only the minimal one. This seems to be unknown even for Speh representations. As was pointed out to us by Arno Kret, knowing it in this case would already have interesting consequences. 6.2. Imprimitive representations. We say that π∈Irr Gnis (parabolically) imprimitive if it is not (fully) induced from a proper parabolic subgroup. It is known that any π∈Irr can be expressed as the product of imprimitive representations in a unique way, up to reordering. In other words, the imprimitive representations are, roughly speaking, the prime elements of Irr. Thus, it is desirable to characterize imprimitive representations in terms of their Zelevinsky (or Langlands) data. In general, by [Zel80, Proposition 8.4], if L(∆1,...,∆t) is not imprimitive then there exists a non-trivial partition {i1, . . . , ir}t{j1, . . . , jt−r}={1, . . . , t}, such that (6.3) L(∆1,...,∆t)≃L(∆i1,...,∆ir)×L(∆j1,...,∆jt−r). In the case of ladder representations we can easily describe the imprimitive ones. More precisely, we say that a ladder is proper if ai≤bi+1 + 1 (i.e., ∆i+1 precedes ∆i) for all i= 1, . . . , t −1. Then we have