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ROBINSON-SCHENSTED-KNUTH CORRESPONDENCE IN THEREPRESENTATION THEORY OF THE GENERAL LINEAR GROUP

OVER A NON-ARCHIMEDEAN LOCAL FIELD(WITH AN APPENDIX BY MARK SHIMOZONO)

MAXIM GUREVICH AND EREZ LAPID

Abstract. We construct new “standard modules” for the representations of general lin-ear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky’s multisegments.

Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrixpolynomial rings, indexed by bitableaux. Hence, our main result provides a link betweenthe dual canonical basis (coming from quantum groups) and the DRS basis.

Contents

1. Introduction 12. Operations on multisegments 53. Commutativity of algorithms 114. Representation theoretic applications 155. Enhanced multisegments 186. Odds and ends 22Appendix A. Multisegments, MW, and RSK

by Mark Shimozono 24References 32

1. Introduction

The Zelevinsky classification ([32]) is one of the cornerstones of the representation theoryof reductive groups over a non-Archimedean local field F . It classifies the equivalenceclasses of the irreducible (complex, smooth) representations of the general linear groups ofall ranks over F in terms of multisegments, which are essentially a combinatorial object.

The irreducible representations may all be obtained as socles of the so-called standardmodules (that are also indexed by multisegments). The standard modules are the par-abolic induction of certain irreducible, essentially square-integrable representations, andconstitute a basis for the Grothendieck group. The situation is analogous to that in cat-egory O where the Verma modules play the role of standard modules. This is in fact not

Date: June 19, 2020.1

2 MAXIM GUREVICH AND EREZ LAPID

a coincidence. It is explained by the Arakawa–Suzuki functors [1] which provide a linkbetween category O of type A and representations of GLn(F ), n ≥ 0. See [2, 11, 16] formore details.

In this paper we present a new class of RSK-standard modules, that are parabolicallyinduced from ladder representations. Its construction relies on an application of the well-known Robinson–Schensted–Knuth (RSK) correspondence. The new class is again in bi-jection with irreducible representations, in such a way that each irreducible representationis realized as a subrepresentation (and conjecturally, the socle) of the corresponding RSK-standard module.

Ladder representations are a class of irreducible representations with particularly niceproperties [2,10,13,14]. In particular, parabolic induction from two ladder representationsis well understood.

Let us describe the new construction in more detail. Roughly speaking, a multisegmentm is a collection of pairs of integers [ai, bi], i = 1, . . . , n, with ai ≤ bi. Denote by Z(m) theirreducible representation of GLN(F ) corresponding to m, as defined by Zelevinsky.

Now, the RSK correspondence attaches to m a pair of semistandard Young tableauxof the same shape of total size n. For our purposes it will be more convenient to use amodified version of RSK, m 7→ (Pm, Qm) where Pm and Qm are inverted Young tableaux (ofthe same shape, of total size n). By an inverted Young tableau we mean that the entriesin each row are strictly decreasing and the entries in each column are weakly decreasing,unlike the usual convention in which the entries in each row are weakly increasing and theentries in each column are strictly increasing. As in the classical case, Pm is filled by theai’s and Qm by the bi’s.

Suppose that the pair (Pm, Qm) is given by

Pm =c1,1 c1,2 . . . c1,λ2 . . . c1,λ1

c2,1 c2,2 . . . c2,λ2

......

...

ck,1 . . . ck,λk

Qm =d1,1 d1,2 . . . d1,λ2 . . . d1,λ1

d2,1 d2,2 . . . d2,λ2

......

...

dk,1 . . . dk,λk

To each row of the resulting shape we attach a ladder representation by setting

πi = Z([ci,1, di,1] + . . .+ [ci,λi, di,λi

]), i = 1, . . . , k .

In other words, πi is the irreducible representation whose defining multisegment is encodedin the pair of i-th rows of the tableaux (Pm, Qm).

The RSK-standard module attached to m is now defined as

Λ(m) = πk × · · · × π1 ,

where the (Bernstein–Zelevinsky) product denotes (normalized) parabolic induction.An intriguing angle on the new class of modules comes from combinatorial invariant

theory. Let R be the direct sum of Grothendieck groups of smooth finite-length repre-sentations, of all GLn(F ), n ≥ 0. The Bernstein–Zelevinsky product makes R into a

RSK FOR GLn OVER LOCAL FIELDS 3

commutative ring and Zelevinsky showed that R is the polynomial ring over Z freely gen-erated by the classes of irreducible, essentially square-integrable representations. In otherwords, the standard modules form the monomial basis forR (as a Z-module). On the otherhand, in the context of invariant theory, Doubilet, Rota, and Stein constructed a basis forpolynomial matrix rings, parameterized by pairs of tableaux, consisting of products of mi-nors [7] (see also [6,28]. As will be explained in §6.3, the RSK-standard modules categorifythe DRS basis for matrix polynomial rings. This phenomenon follows from the paramountdeterminantal property [14] of ladder representations. The construction of Λ(m) as a prod-uct of ladders parallels the recipe of DRS for obtaining all basis elements as products ofminors.

Our main result is the following.

Theorem 1.1. For every multisegment m, Z(m) occurs as a subrepresentation of Λ(m).More precisely,

Z(m) ∼= soc(soc(. . . soc(soc(πk × πk−1)× πk−2)× . . . × π2)× π1) .

Here, soc(τ) stands for the socle (i.e., the maximal semisimple subrepresentation) of arepresentation τ .

We expect (Conjecture 6.1) that as in Zelevinsky’s case, soc(Λ(m)) is itself irreducible(hence isomorphic to Z(m)) and occurs with multiplicity one in the Jordan–Holder sequenceof Λ(m).

Note that the parameter k = k(m), i.e. the number of rows in the tableaux Pm, Qm, isthe width of the multisegment, as defined and studied in [10]. In particular, it was shownin [ibid.] that k is the minimal number of ladder representations whose product containsZ(m) as a subquotient. In that respect, the RSK-standard modules possess a minimalityproperty.

The case of Theorem 1.1 with k(m) = 2 and certain regularity conditions was previouslyshown in [9] using quantum shuffle methods and equivalences to module categories of quiverHecke algebras. In fact, the papers [9,10] were the point of departure of the present work.

Coming back to the categorification context, irreducible representations are known togive the (dual) canonical basis (a la Lusztig or Kashiwara) of R (see [18], for example).Thus, Theorem 1.1 gives a certain link between the DRS basis and the canonical basis.Similar links were explored in [19] by means of quantum matrix rings. Our results andexpectations open a new perspective on those links in a categorial setting, i.e. it is thesubmodule structure which is used to characterize relations between those two bases.

A natural question which arises is what can be said about the other irreducible con-stituents of Λ(m). Based on empirical evidence, we conjecture that for any irreduciblesubquotient Z(n) of Λ(m) other than Z(m), the RSK-data (Pn, Qn) is strictly smaller than(Pm, Qm) with respect to the product order of the domination order of inverted Youngtableaux (see §6.2). Once again, this would be analogous to the situation for Zelevinskyclassification, where the pertinent partial order on multisegments (as originally defined in[32]) is closely related to the Bruhat order on the symmetric group.

In particular, this would give a positive answer to the question which still remains open,of whether the classes of RSK-standard modules indeed form a Z-basis for (an appropriate

4 MAXIM GUREVICH AND EREZ LAPID

subgroup of) R. Certain related questions can be answered (Proposition 6.5) using thebasis theorem of [6].

Recall that the classical RSK correspondence admits several (equivalent) implementa-tions. For our purposes, it is best to use Knuth’s algorithm [12, §4] which constructs theYoung tableaux row by row, rather than the earlier (and perhaps more commonly used)Robinson-Schensted insertion/bumping algorithm which fills them box by box. (A pictorialapproach to Knuth’s algorithm was given by Viennot [29].)

A key part of the proof of Theorem 1.1 is an intriguing relation between the Knuthalgorithm and the description of the socles of certain induced representations due to thesecond-named author and Mınguez [15]. In turn, this description is closely related tothe Mœglin–Waldspurger algorithm for the Zelevinsky involution [20, §II.2]. In fact, themain new combinatorial input (Corollary 3.4), which is interesting in its own right, is thatroughly speaking, under certain conditions the Knuth algorithm commutes with the firststep of the Mœglin–Waldspurger algorithm.

In an appendix by Mark Shimozono, further illuminating relations between those twocombinatorial algorithms are presented. The slightly different point of view of the ap-pendix suggests that some of our questions may be resolved in future work through thecombinatorial theory of tableaux and crystals.

The connection between RSK and the Mœglin–Waldspurger algorithm is further elu-cidated by considering enhanced multisegments, that is, “multisegments with dummies”.More precisely, starting with a (genuine) multisegment m = ∆1 + · · ·+∆k we add empty(dummy) segments of the form [a, a− 1], a ∈ Z. This will not affect Z(m) but will changethe output of RSK. In Theorem 5.1, to any irreducible representation π = Z(m) we concocta whole slew of “standard modules” admitting π as a subrepresentation (and conjecturally,as the socle) by adding dummy segments to m at will.

Let na, a ∈ Z be the number of dummy segments of the form [a, a− 1]. There are twoextreme cases, as stated in Proposition 5.3. If na ≪ na+1 for all a in the support of m thenthe standard module will be the one defined by Zelevinsky (taking the product of Z(∆i)in a suitable order). On the other hand if na ≫ na+1 for all a in the support of m, then wewill get the standard module in the sense of Langlands. In other words, the output of RSKessentially coincides (up to dummy segments) with the Mœglin–Waldspurger algorithm.Thus, we get a family of standard modules interpolating between the standard modules ofZelevinsky and Langlands.

Both the Zelevinsky classification and the RSK correspondence admit geometric inter-pretations (starting with [30, 31] for the former, [22, 25, 27] for the latter). However, weare not aware of a geometric interpretation of the abovementioned partial order definedthrough RSK. It would be interesting to find a geometric interpretation of the results andconjectures presented here.

Finally, it would be interesting to know whether other type A representation categories,such as those of quiver Hecke algebras (through the Brundan-Kleshchev isomorphisms[4]), quantum affine algebras (through quantum affine Schur-Weyl duality [5]) or eventhe classical category O (through the Arakawa-Suzuki functors [1, 11]), can be used toreinterpret the results of this paper.

RSK FOR GLn OVER LOCAL FIELDS 5

Part of this work was carried out while the authors participated in the month-longactivity “On the Langlands Program: Endoscopy and Beyond” at the Institute for Math-ematical Sciences of the National University of Singapore, Dec. 2018 – Jan. 2019. Thefirst-named author and Mark Shimozono took part in the thematic trimester program onrepresentation theory in the Institut Henri Poincare, Paris, Jan. 2020, where the fruitfulconditions of the venue lead to the addition of Appendix A. We would like to thank theIMS, the IHP and the organizers of the two programs for their warm hospitality.

2. Operations on multisegments

2.1. Multisegments. A segment ∆ (of length b− a+ 1) is a subset of Z of the form

[a, b] = {n ∈ Z : a ≤ n ≤ b}

for some integers a ≤ b. We will write b(∆) = a and e(∆) = b. We also write←

∆ =[a− 1, b− 1] and −∆ = [a + 1, b]. Note that if a = b, then −∆ is the empty set.

We denote by Seg the set of all segments.Given ∆1, ∆2 ∈ Seg, we write ∆1 ≪ ∆2 if b(∆1) < b(∆2) and e(∆1) < e(∆2). This is a

strict partial order on Seg.A multisegment is a multiset of segments, i.e., a formal finite sum m =

∑i∈I ∆i where

{∆i ∈ Seg}i∈I are segments.If m 6= 0, we will write minm = mini∈I b(∆i).The size |m| of a multisegment m is by definition, the sum

∑i∈I |∆i| of the cardinalities

of its segments.For any set X , we denote by N(X) the ordered commutative monoid of finite multisets

on X , that is, finitely supported functions from X to N = Z≥0. We view elements of N(X)as formal finite (possibly empty) sums of elements of X .

From this point of view, we denote byM := N(Seg) the ordered monoid of multisegments.If m′ ≤ m we say that m′ is a sub-multisegment of m.

A ladder is a nonzero multisegment of the form∑k

i=1∆i, for segments ∆i, where ∆i+1 ≪∆i for i = 1, . . . , k − 1.

We will write Lad ⊆ M for the collection of ladders.

2.2. Mœglin–Waldspurger involution. Let us recall the Mœglin–Waldspurger algo-rithm from [20, §II.2]. More precisely, we will adopt a mirror variation on [loc. cit.], inwhich minimal segments are taken in place of maximal ones.1

For any multisegment 0 6= m =∑

i∈I ∆i ∈ M we define a segment ∆◦(m) and a multi-segment m† as follows.

Let i1 ∈ I be such that b(∆i1) = minm and e(∆i1) is minimal. If there is no i ∈ Isuch that ∆i1 ≪ ∆i and b(∆i) = b(∆i1) + 1, then we set ∆◦(m) := [minm,minm] ∈ Seg.Otherwise, let i2 ∈ I be such that ∆i1 ≪ ∆i2 , b(∆i2) = b(∆i1) + 1 and e(∆i2) is minimalwith respect to these properties. Continuing this way, we define an integer k > 0 andindices i1, . . . , ik ∈ I, such that

1This is justified by the fact that the Mœglin–Waldspurger algorithm is compatible with contragredient.

6 MAXIM GUREVICH AND EREZ LAPID

(1) For all j < k, ∆ij ≪ ∆ij+1, b(∆ij+1

) = b(∆ij ) + 1, and e(∆ij+1) is minimal with

respect to these properties.(2) There does not exist i ∈ I such that ∆ik ≪ ∆i and b(∆i) = b(∆ik) + 1.

We set ∆◦(m) := [b(∆i1), b(∆ik)] = [minm,minm+ k − 1] ∈ Seg.We call I∗ = {i1, . . . , ik} ⊆ I a set of leading indices of m. It is not unique, but the

set {∆i1 , . . . ,∆ik} depends only on m. We also write i∗min = i1, i∗max = ik and denote by

i 7→ i+ : I∗ \ {i∗max} → I∗ \ {i∗min} the bijection ij 7→ ij+1. The inverse will be denoted byi 7→ i−.

We set m† :=∑

i∈I ∆∗i ∈ M (discarding the summands that are empty sets), where

(1) ∆∗i =

{−∆i i ∈ I∗,

∆i otherwise.

We denote the resulting map m 7→ (m†,∆◦(m)) by

MW : M \ {0} → M× Seg .

We say that m is non-degenerate if ∆∗i 6= ∅, for all i ∈ I. Equivalently, ∆i 6= [minm,minm]for all i ∈ I.

Applying the map MW repeatedly we obtain the Mœglin–Waldspurger involution2 m 7→m# onM which is the combinatorial counterpart of the Zelevinsky involution [32, §9]. Moreprecisely, m# is defined recursively by

0# = 0, m# = ∆◦(m) + (m†)#.

In particular, the map MW is injective.The subset Lad is preserved under m 7→ m#.

2.3. Modified RSK correspondence for multisegments. Let n ≥ 0 be an integer andλ = (λ1, . . . , λk) a partition of n, i.e. λ1 ≥ · · · ≥ λk > 0 are integers and λ1 + · · ·+ λk = n.The partition λ gives rise to a Young diagram of size n. The conjugate partition λt := µ =(µ1, . . . , µl) is given by l = λ1, µj = max{i : λi ≥ j}.

A semistandard Young tableau of shape λ is a filling of a Young diagram of shape λ byintegers, such that the entries along each row are weakly increasing and the entries downeach column are strictly increasing.

The classical Robinson–Schensted–Knuth (RSK) correspondence is a bijection betweenN(Z×Z) (i.e., multisets of pairs of integers) and pairs of semistandard Young tableaux ofthe same shape. We refer to [8, §4] or [26, §7.11-13] for standard references on RSK.

We will consider a slight modification of RSK where semistandard Young tableaux arereplaced by inverted Young tableaux. By definition, an inverted Young tableau of shape λ isa filling of the Young diagram of λ by integers, i.e. a double sequence zi,j ∈ Z, i = 1, . . . , k,j = 1, . . . , λi, that satisfies

zi,1 > · · · > zi,λi, ∀i = 1, . . . , k ,

z1,j ≥ · · · ≥ zµj ,j, ∀j = 1, . . . , l .

2 The fact that m 7→ m# is an involution is not obvious from the definition.

RSK FOR GLn OVER LOCAL FIELDS 7

We denote by T the set of pairs of inverted Young tableaux (P,Q) of the same shape.(See [22] for a similar nonstandard convention. The appendix of [8] also discusses closelyrelated variants of RSK.)

Thus, the modified RSK correspondence is a bijection

RSK′ : N(Z× Z) −→ T .

It can be defined using a modification of the Schensted insertion/bumping algorithm wherewe replace strict inequalities by weak inequalities in the opposite direction and vice versa.It is advantageous, however, to use a modification of the Knuth algorithm which we willrecall below. We remark that if RSK′(

∑i∈I(ai, bi)) = (P,Q), the ai’s and the bi’s comprise

the entries of the tableaux P and Q, respectively.We may identify Seg as a subset of Z×Z by ∆ 7→ (b(∆), e(∆)). Hence, we may identify

M with a subset of N(Z × Z). Thus, for any multisegment m ∈ M we may consider thepair of inverted Young tableaux

(Pm, Qm) := RSK′(m) ∈ T .

In what follows we will only consider the restriction of RSK′ to M.

2.3.1. Ladders and tableaux. First, we would like to be able to describe certain elementsof T in terms of ladders.

Let l1 =∑k

i=1∆i and l2 =∑k′

i=1∆′i be two ladders with ∆i+1 ≪ ∆i, i = 1, . . . , k−1 and

∆′i+1 ≪ ∆′i, i = 1, . . . , k′ − 1.

We say that l2 is dominant with respect to l1, if k′ ≥ k and

←

∆i ≪ ∆′i for all i = 1, . . . , k.We say that the pair (l2, l1) is permissible if l2 is dominant with respect to l1 and for all

i = 1, . . . , k and j = 1, . . . , k′ such that←

∆i ≪ ∆′j and either j = k′ or←

∆i 6≪ ∆′j+1 (and inparticular, j ≥ i), we have e(∆r) ≥ b(∆′j−i+r) for all r = 1, . . . , i.

Let us write L′ for the collection of tuples (l1, . . . , lm) ∈ Ladm, for some m, such that liis dominant with respect to li+1, for all 1 ≤ i < m.

We may think of L′ as a subset of T as follows. Given l = (l1, . . . , lm) ∈ L′ whereli =

∑ni

j=1∆ij with ∆i

j+1 ≪ ∆ij we construct

(P (l), Q(l)) ∈ T

by letting the (i, j)-th entry in P (l) (resp. Q(l)) be b(∆ij) (resp. e(∆i

j)). The map l 7→(P (l), Q(l)) is an injection of L′ into T . Its image is the set of pairs (P,Q) of tableaux ofthe same shape such that Pi,j ≤ Qi,j for all entries of the tableaux.

Finally, we denote by L ⊆ L′ the subset consisting of tuples (l1, . . . , lm) ∈ L′ such that(li, lj) is permissible for all 1 ≤ i < j ≤ m.

2.3.2. The Knuth implementation. Let us fix a multisegment 0 6= m =∑

i∈I ∆i ∈ M. Wewill explicate (Pm, Qm) introducing some terminology for multisegments.

We define the depth function d = dm : I → Z≥0 by

d(i) = max{j : ∃i0 = i, i1, . . . , ij ∈ I such that ∆ir ≪ ∆ir+1, r = 0, . . . , j − 1} .

We let d = d(m) = maxi∈I d(i) be the depth of m.

8 MAXIM GUREVICH AND EREZ LAPID

It is clear that if d(i) = d(j), then either ∆i = ∆j or ∆i ) ∆j or ∆i ( ∆j .

Lemma 2.1.

(1) For any i ∈ I and 0 ≤ k < d(i) there exists i′ ∈ I such that ∆i ≪ ∆i′ and d(i′) = k.(2) Let i1, i2 ∈ I be such that d(i1) = d(i2) and ∆i2 ⊆ ∆i1. Then for any i ∈ I such

that ∆i2 ⊆ ∆i ⊆ ∆i1 and d(i) > d(i1), there exists j ∈ I such that ∆i2 ⊆ ∆j ⊆ ∆i1,d(j) = d(i1) and ∆i ≪ ∆j.

Proof. The first part is clear. To prove the second part we argue by induction on d(i)−d(i1).Let i′ be such that ∆i ≪ ∆i′ and d(i′) = d(i)−1. Then, b(∆i1) ≤ b(∆i) < b(∆i′). However,we cannot have ∆i1 ≪ ∆i′ since otherwise d(i1) ≥ d(i′) + 1 = d(i). Therefore ∆i′ ⊆ ∆i1 .Similarly, we cannot have ∆i2 ≪ ∆i′ and therefore ∆i′ ⊇ ∆i2 . If d(i

′) = d(i1) we are done.Otherwise, we apply the induction hypothesis to i′. �

For any k = 0, . . . , d, choose an admissible enumeration {ik1, . . . , ikl } of d−1(k), namely

such that ∆ik1⊇ . . . ⊇ ∆ik

l, and set jk := ikl .

We will say that ikr is a distinguished index (with respect to the enumeration) if eitherr = l or ∆ikr+1

6= ∆ikr.

Let σ be the permutation of the index set I, whose cycle decomposition is given by{(ik1, . . . , i

kl )}k=0,...,d.

Define

∆′i = [b(∆i), e(∆σ(i))] ∈ Seg, i ∈ I,(2a)

I♮ = {j0, . . . , jd}, I ′ = I \ I♮(2b)

l(m) =∑

i∈I♮

∆′i, m′ =∑

i∈I′

∆′i ∈ M .(2c)

For reference, we will write i∨ = σ(i), for all i ∈ I.Note that for any i ∈ I♮ we have

(3) b(∆′i) = maxj∈I:d(j)=d(i)

b(∆j), e(∆′i) = maxj∈I:d(j)=d(i)

e(∆j) .

Thus, it follows from Lemma 2.1(1) that l(m) is a ladder.

Lemma 2.2. Suppose that i, i′ ∈ I are such that ∆i∨ ( ∆i′ ( ∆i. Then, dm(i′) < dm(i).

Proof. Assume on the contrary that dm(i′) ≥ dm(i). By Lemma 2.1(2), there is j ∈ I, for

which dm(j) = dm(i), ∆i∨ ⊆ ∆j ⊆ ∆i, and either i′ = j or ∆i′ ≪ ∆j . In both cases, it isclear that ∆i∨ ( ∆j ( ∆i. This contradicts the definition of i∨. �

Lemma 2.3. For all i ∈ I ′ we have dm′(i) ≤ dm(i).

Proof. Note that when i, j ∈ I ′ and dm(i) = dm(j), either ∆′i = ∆′j or ∆

′i ( ∆′j or ∆

′i ) ∆′j .

Suppose that ∆′jk ≪ . . . ≪ ∆′j1 ≪ ∆′j0 for given indices j0, . . . , jk ∈ I ′. We need to showthat k ≤ dm(jk).

By the remark above dm(jc+1) 6= dm(jc) for all c < k. Suppose on the contrary thatdm(jc+1) < dm(jc) for some c. Since b(∆jc) = b(∆′jc) > b(∆′jc+1

) = b(∆jc+1) we necessarily

RSK FOR GLn OVER LOCAL FIELDS 9

have ∆jc ( ∆jc+1. Also, by definition of m′, there exists i1 ∈ I such that dm(i1) = dm(jc+1)and e(∆i1) = e(∆′jc+1

) < e(∆′jc) ≤ e(∆jc). Thus, ∆i1 ( ∆jc for otherwise ∆i1 ≪ ∆jc anddm(jc) < dm(i1) = dm(jc+1) contradicting our assumption. In particular, by Lemma 2.1(2),there exists i2 ∈ I such that dm(i2) = dm(jc+1), ∆jc ≪ ∆i2 and ∆i2 ( ∆jc+1. Once again,by the definition of m′, we necessarily have e(∆′jc+1

) ≥ e(∆i2). Hence, e(∆jc) ≥ e(∆′jc) >e(∆′jc+1

) ≥ e(∆i2) in contradiction to the condition ∆jc ≪ ∆i2 .We conclude that dm(j0) < dm(j1) < . . . < dm(jk), which implies that k ≤ dm(jk) as

required. �

We define a map

K : M \ {0} → Lad×M, K(m) = (l(m),m′) .

It is clear that K(m) is well defined (i.e., does not depend on the choice of admissibleenumerations of the fibers of d). We call l(m) the highest ladder of m and m′ the derivedmultisegment of m.

Define recursivelyRSK : M → L′

byRSK(0) = ∅, RSK(m) = (l(m),RSK(m′)), m 6= 0.

Adapting the discussion of [12, §4] to our conventions we obtain that

RSK′(m) = (P (RSK(m)), Q(RSK(m))).

2.3.3. On the image of RSK. The fact that we consider only multisegments rather thanmultisets of arbitrary pairs of integers means that there are restrictions on the image ofRSK. This situation is explained in Appendix §A, where a full characterization of theimage of (a certain variant of) RSK is given by means of key tableaux in the sense ofLascoux and Schutzenberger, which has links to the theory of Demazure crystals.

A partial characterization from a somewhat different point of view will be given here,and will be relevant for the core of our arguments. A combined understanding of boththese approaches may be beneficial for a better understanding of the entire theme of thispaper. We leave it for future endeavors.

Let m be a multisegment and l a ladder. We say that l is dominant with respect tom, if l is dominant with respect to any ladder sub-multisegment of m. We say that thepair (l,m) is permissible if (l, l′) is permissible (and in particular, dominant) for any laddersub-multisegment l′ of m. Denote by A ⊆ Lad×M the set of permissible pairs.

Proposition 2.4. The map K defines a bijection

K : M \ {0} → A.

Moreover, the image of the map RSK is contained in L.

Remark 2.5. The map RSK is not onto L. For instance, ([3, 3]+[1, 2], [2, 3], [1, 2]) is not inthe image of RSK since RSK([1, 3]+ [2, 2]) = ([2, 3], [1, 2]) but ([3, 3]+ [1, 2], [1, 3]+ [2, 2])is not permissible.

10 MAXIM GUREVICH AND EREZ LAPID

However, if we restrict RSK to the set of multisegments m =∑

i∈I ∆i such that b(∆i) ≤e(∆j) for all i, j ∈ I, then the image consists of the subset of L′ consisting of the tuples(l1, . . . , lm) ∈ L′ such that b(∆) ≤ e(∆′) for any ∆ and ∆′ which occur in

∑li. We refer

to this as the saturated case.

Proof. Fix 0 6= m ∈ M as before.

It follows from Lemma 2.1(1) and equations (3) that←

∆j ≪ ∆′i for any i ∈ I♮ and j ∈ I

such that d(j) ≥ d(i). In particular,←

∆′j ≪ ∆′i if in addition j ∈ I ′.It then follows by Lemma 2.3 that l(m) is dominant with respect to m′.Note that e(∆j) ≥ b(∆i) for all i, j ∈ I with d(i) = d(j). Hence, by (3) and the fact that

l(m) is a ladder, we see that e(∆′j) ≥ b(∆′i) for any i ∈ I♮ and j ∈ I ′ such that d(j) ≤ d(i).The fact that (l(m),m′) is permissible now follows again from Lemma 2.3. In conclusion,

K(m) ∈ A.To show that K is a bijection we describe the inverse K′ : A → M\{0} following [8, §4.2].Suppose that l =

∑j∈J ∆i, J = {1, . . . , m} is a ladder, such that ∆r+1 ≪ ∆r for all

r = 1, . . . , m−1. Let m =∑

i∈I ∆i be a multisegment (taking the index sets I, J as disjointsets), such that (l,m) is permissible. In particular, l is dominant with respect to m.

We define g = gm,l : I → J and f = fm,l : I → J by

g(i) = max{j ∈ J :←

∆i ≪ ∆j},

f(i) = min(g(i), {f(j)− 1 : j ∈ I,∆j ≪ ∆i}).

Equivalently,

f(i) = min{g(ik)− k : ∃i = i0, . . . , ik ∈ I such that ∆ir+1 ≪ ∆ir , r = 0, . . . , k − 1}.

By our assumption, f is well defined. Moreover, for any j ∈ J we may write the fiberYj = f−1(j) as Yj = {i1, . . . , ik} (possibly with k = 0) where ∆ir+1 ⊆ ∆ir , r = 1, . . . , k− 1,e(∆ik) ≥ b(∆j) ≥ b(∆ik), e(∆j) ≥ e(∆i1). (The condition e(∆ik) ≥ b(∆j) follows from thepermissibility of (l,m).)

Let σ be the permutation of I ∪ J whose cycles are (i1, . . . , ik, j) as we vary over j ∈ J .For any i ∈ I ∪ J , we set ∆′i = [b(∆σ(i)), e(∆i)]. Note that we still have ∆′ir+1

⊆ ∆′ir ,r = 1, . . . , k − 1 and also ∆′i1 ⊆ ∆′j.

Finally, set K′(l,m) =∑

i∈I∪J ∆′i. It is easy to see that K′ is the inverse of K.

It remains to prove the last statement of the proposition regarding the map RSK. Tothat end, we need to show that for any pair (l,m) ∈ A with m 6= 0 we have (l, l(m)), (l,m′) ∈A as well.

Suppose that (l,m) ∈ A. Assume that {0, . . . , n}∩I = ∅ and write l =∑n

i=0∆i. Suppose

that←

∆′i ≪ ∆j for some i ∈ I♮ and j = 0, . . . , n with j maximal. Let i1, i2 ∈ I be such

that d(i1) = d(i2) = d(i) and ∆′i = [b(∆i1), e(∆i2)]. Clearly←

∆i1 ≪ ∆j and←

∆i2 ≪ ∆j .

Suppose that j 6= n. Since←

∆′i 6≪ ∆j+1 we cannot have both←

∆i1 ≪ ∆j+1 and←

∆i2 ≪ ∆j+1.

Let i′ ∈ {i1, i2} be such that←

∆i′ 6≪ ∆j+1. If j = n then take i′ = i1 or i2 – it makes sodifference. In either case, there exist k0, . . . , ks = i′ with s = d(i) such that ∆kr ≪ ∆kr−1,

RSK FOR GLn OVER LOCAL FIELDS 11

r = 1, . . . , s and d(∆kr) = r for all r. Since (l,m) is permissible, for any r ≤ s we havee(∆kr) ≥ b(∆j−s+r). This implies that e(∆′k) ≥ b(∆j−s+r) where k ∈ I♮ is such thatd(k) = r. It follows that (l, l(m)) is permissible.

Next, suppose that i0, . . . , ik ∈ I ′ with ∆′ir ≪ ∆′ir−1, r = 1, . . . , k. We claim that

there exist j0, . . . , jk such that ∆′jr ≪ ∆′jr−1, r = 1, . . . , k d(jr) = d(ir) for all r and

e(∆jr) ≤ e(∆′ir) for all r and e(∆′ik) = e(∆jk).We construct jr by descending induction on r. Let jk be such that e(∆′ik) = e(∆jk)

and d(ik) = d(jk). Suppose that jr was constructed and r ≥ 0. Let j′ be such thate(∆j′) = e(∆′ir−1

) and d(j′) = d(ir−1). Then e(∆j′) > e(∆′ir) ≥ e(∆jr). If ∆jr ≪ ∆j′, takejr−1 = j′. Otherwise, ∆j′ ⊇ ∆jr . Take jr−1 such that d(jr) = d(j′) and ∆jr ≪ ∆jr−1. Thennecessarily ∆jr−1 ⊆ ∆j′ and hence e(∆j′) ≥ e(∆jr−1) as required.

Now let j ∈ {0, . . . , n} be the maximal index such that ∆jk ≪ ∆j . By the permissibilityof (l,m) we have e(∆jr) ≥ b(∆j+r−k) for all r. On the other hand, ∆′ik ≪ ∆j and therefore,if j′ ∈ {0, . . . , n} is the maximal index such that ∆′ik ≪ ∆j′ , then j′ ≥ j. Therefore,e(∆′ir) ≥ e(∆jr) ≥ b(∆j+r−k) ≥ b(∆j′+r−k). It follows that (l,m

′) is permissible. �

2.3.4. An inductive description. We finish our discussion on the RSK algorithm with thefollowing lemma, which allows for inductive arguments in certain cases.

Lemma 2.6. Assume that m 6= 0 and min l(m) = minm. Let d = d(m) be the depth of mand let i0 ∈ d−1m (d) be such that ∆i0 ⊇ ∆i for all i ∈ d−1m (d). Then

(1) d−1m (d) = {i ∈ I : b(∆i) = minm, ∆i0 ⊇ ∆i} .(2) ∆i0 ≤ l(m).(3)

∑i∈d−1

m(d)\{i0}

∆i ≤ m′. (It is of course not excluded that d−1m (d) = {i0}.)

(4)

K(m−

∑

i∈d−1m (d)

∆i

)=

(l(m)−∆i0 ,m

′ −∑

i∈d−1m (d)\{i0}

∆i

).

Proof. Let ∆ be the segment in l(m) with b(∆) = min l(m). By equation (3), b(∆i) = minmfor any i ∈ d−1m (d). Similarly, e(∆) = e(∆i0) and hence, ∆ = ∆i0 .

The equality in (1) becomes obvious from the fact that dm(i) ≥ dm(i0) for such i ∈ I.The rest of the lemma follows from the description of K(m). �

3. Commutativity of algorithms

We turn now to study the relations between the Mœglin–Waldspurger algorithm andRSK.

Let m =∑

i∈I ∆i ∈ M ba a multisegment, with notation as before. We will consider themore involved case which is not covered by Lemma 2.6, namely, when minm < min l(m).

Lemma 3.1. Suppose that m is non-degenerate (see §2.2) and let I∗ be a set of leadingindices for m. (See §2.2.) Then

(1) For every i ∈ I we have dm(i) ≤ dm†(i) ≤ dm(i) + 1.

12 MAXIM GUREVICH AND EREZ LAPID

(2) We write

I♭ = {i ∈ I : ∃i′ ∈ I∗ such that b(∆i) = b(∆i′),∆i ( ∆i′ and dm(i) = dm(i′)} .

Then, I∗ and I♭ are disjoint. The equality dm†(i) = dm(i) + 1 holds if and only ifi ∈ I♭. Moreover, if i ∈ I♭ and i′ ∈ I∗ are such that b(∆i) = b(∆i′), ∆i ( ∆i′ anddm(i) = dm(i

′), then i′ 6= i∗min and dm†(i) = dm(i′−).

(3) d(m) = d(m†).(4) dm is injective on I∗.

Remark 3.2. In general, it is not true that dm(i−) = dm(i) + 1 for all i ∈ I∗ \ {i∗min}.

Proof. Let i ∈ I. To show that dm(i) ≤ dm†(i) suppose that ∆i1 ≪ . . . ≪ ∆ik for somei1, . . . , ik ∈ I with i1 = i. We claim that we can choose these indices so that wheneverir ∈ I∗ for some r, either r = k or b(∆ir+1) > b(∆ir) + 1 or b(∆ir+1) = b(∆ir) + 1 andir+1 ∈ I∗. Indeed, whenever ir ∈ I∗ with r < k and b(∆ir+1) = b(∆ir)+1 with ir+1 /∈ I∗ wemay replace ir+1 by the leading index i such that b(∆i) = b(∆ir+1). Iterating this processwe will get the required property. With this extra property we have ∆∗i1 ≪ . . . ≪ ∆∗ik .Thus dm(i) ≤ dm†(i).

It is clear that I♭ ∩ I∗ = ∅ and that if i ∈ I♭ then dm†(i) > dm(i). Indeed, if i′ ∈ I∗ issuch that b(∆i) = b(∆i′), ∆i ( ∆i′ and dm(i) = dm(i

′) then ∆∗i = ∆i ≪ ∆∗i′ and thereforedm†(i) > dm†(i′) ≥ dm(i

′) = dm(i). Moreover, it is clear from the definition of i∗min thati′ 6= i∗min and since i′ ∈ I∗ we have ∆i′− ⊇ ∆i. We claim that dm(i

′) + 1 = dm(i′−). Clearly,

dm(i′−) > dm(i

′). If dm(i′−) > dm(i

′) + 1 then there exists j ∈ I such that ∆i′− ≪ ∆j

and dm(j) > dm(i′). If b(∆j) = b(∆i′) then ∆j ⊇ ∆i′ since i′ ∈ I∗ and we would get a

contradiction. Otherwise, ∆i ≪ ∆j and again we get a contradiction.It remains to show that dm†(i) ≤ dm(i) + 1 for all i ∈ I with equality only if i ∈ I♭. We

prove this by descending induction on e(∆i).The statement is trivial if dm†(i) = 0. Suppose that dm†(i) > 0. Then there exists i1 ∈ I

such that dm†(i) = dm†(i1)+1 and ∆∗i ≪ ∆∗i1 . If ∆i 6≪ ∆i1 then i1 ∈ I∗, b(∆i) = b(∆i1) and∆i ( ∆i1 . Our claim follows in this case since dm(i1) ≤ dm(i) and by induction hypothesisdm†(i1) = dm(i1).

Assume therefore that ∆i ≪ ∆i1 . Then dm(i) ≥ dm(i1)+1. It follows from the inductionhypothesis that

dm†(i) = dm†(i1) + 1 ≤ dm(i1) + 2 ≤ dm(i) + 1.

Assume that dm†(i) = dm(i) + 1. Then, again by induction hypothesis i1 ∈ I♭, i.e. thereexists i2 ∈ I∗ such that b(∆i2) = b(∆i1), ∆i1 ( ∆i2 and dm(i1) = dm(i2). In particulari2 6= i∗min. Let i3 = i−2 ∈ I∗. Since i1 /∈ I∗ we must have e(∆i3) ≥ e(∆i1). Hence ∆∗i ≪ ∆∗i3and therefore dm(i3) ≤ dm†(i) − 1 = dm(i). (Note that dm(i3) = dm†(i3) by inductionhypothesis.) On the other hand, dm(i3) ≥ dm(i2) + 1 = dm(i1) + 1 = dm†(i1) = dm†(i) − 1.Thus, dm(i3) = dm(i). Also, b(∆i3) = b(∆i) for otherwise ∆i ≪ ∆i3 and then dm(i) ≥dm(i3) + 1, in contradiction to what we just proved.

Finally, the last part of the lemma is evident. �

Proposition 3.3. Suppose that m 6= 0 and minm < min l(m). Then

RSK FOR GLn OVER LOCAL FIELDS 13

(1) For any i ∈ I∗, there exists j ∈ I such that b(∆j) > b(∆i) and dm(i) = dm(j). Inparticular, m is non-degenerate.

(2) For any i ∈ I∗, let i# ∈ I be the distinguished index (see §2.3.2) such that b(∆i) =b(∆i#), dm(i) = dm(i

#) and e(∆i#) is minimal with respect to these properties.Then,

I♯ = {i# : i ∈ I∗}

is a set of leading indices for m′. In particular, ∆◦(m) = ∆◦(m′).(3) Assume (as we may) that I∗ consists of distinguished indices. In particular, i = i#

whenever ∆i = ∆i# , for i ∈ I∗. Let τ be the permutation of the index set I definedby τ(i) = i for all i /∈ I∗ ∪ I♯, τ(i) = i− for all i ∈ I∗ \ I♯, and

τ(i#) =

{(i+)# if i 6= i∗max and (i+)# 6= i+,

i otherwise

for all i ∈ I∗. Then for any i ∈ I,

(∆∗τ(i))′ =

{−∆′i i ∈ I♯,

∆′i otherwise.

Proof. (1) Assume on the contrary that i ∈ I∗ and b(∆j) ≤ b(∆i) whenever dm(j) =dm(i). Assume further that b(∆i) is minimal with respect to this property.If i = i∗min, then dm(i) = d(m), because we cannot have ∆j ≪ ∆i′, for any

j ∈ I and i′ ∈ d−1m (dm(i)). However, in this case we will get a contradiction to theassumption that minm < min l(m).Suppose that i 6= i∗min and let j ∈ I be any index such that dm(j) = dm(i

−). Thendm(j) > dm(i) and therefore there exists i′ ∈ I such that ∆j ≪ ∆i′ and dm(i

′) =dm(i). By our assumption b(∆j) < b(∆i′) ≤ b(∆i) and hence b(∆j) ≤ b(∆i−). Weget a contradiction to the minimality of i.The non-degeneracy part is clear, since if b(∆i) = e(∆i) had been satisfied for

some i ∈ I∗, and j ∈ I had been such that b(∆j) > b(∆i), then ∆i ≪ ∆j wouldhave implied dm(i) > dm(j).

(2) It follows from part (1) that I♯ ∩ I♮ = ∅. In particular, e(∆(i#)∨) ≤ e(∆i#), for all

i ∈ I∗. Also, since i# is distinguished, its defining property imposes

(4) b(∆i#) < b(∆(i#)∨) .

Clearly, (i∗min)# = i∗min ∈ I♯. We first claim that ∆′i∗min

is the shortest segment of m′

which begins at minm.Suppose on the contrary that this is not the case. Then, ∆′i ( ∆′i∗min

⊆ ∆i∗minfor

some i ∈ I with b(∆i) = minm. Thus, e(∆i∨) = e(∆′i) means ∆i∨ ( ∆i∗min. On

the other hand, by the defining property of i∗min, ∆i∗min( ∆i (inequality because of

∆′i 6= ∆′i∗min). Yet, because of b(∆i) = b(∆i∗min

) we have dm(i) ≤ dm(i∗min), which now

contradicts Lemma 2.2.Now, let i ∈ I∗ with i 6= i∗max be fixed. To ease the notation, set j = i#, j′ =

(i+)#, k = j∨, k′ = (j′)∨ ∈ I.

14 MAXIM GUREVICH AND EREZ LAPID

We have ∆i ≪ ∆i+ and therefore

dm(i) = dm(j) = dm(k) > dm(i+) = dm(j

′) = dm(k′) .

By (4), b(∆k) ≥ b(∆j) + 1 = b(∆j′). Thus, ∆k ( ∆j′, since otherwise we wouldhave dm(j

′) ≥ dm(k). So, by Lemma 2.2, we cannot have ∆k′ ⊆ ∆k. The depthinequality also forbids the condition ∆k′ ≪ ∆k. Hence, we must have e(∆k) ≤e(∆k′). Now, an equality e(∆k) = e(∆k′) together with the implied containment∆k ⊆ ∆k′ would again contradict the depth inequality. Summing up, e(∆k) <e(∆k′), which means ∆′j ≪ ∆′j′.Next, we prove that with i, j, k, j′, k′ ∈ I as before, there does not exist a segment

∆ of m′ such that ∆′j ≪ ∆, b(∆) = b(∆′j) + 1(= b(∆i) + 1) and ∆ ( ∆′j′.Suppose otherwise. Then such a segment satisfies ∆ = [b(∆l), e(∆l∨)], for an

index l ∈ I ′. By the assumptions, b(∆l) = b(∆i) + 1, e(∆k) < e(∆l∨) < e(∆k′) ande(∆l∨) ≤ e(∆l).In particular, ∆k ( ∆l. Now, either ∆j ≪ ∆l or ∆l ( ∆j . By applying Lemma

2.2 in the latter case, we obtain dm(l) < dm(j) in both cases.If ∆i ≪ ∆l, we set m = l. Otherwise, by Lemma 2.1(1), there is m ∈ I, such

that ∆i ≪ ∆m and dm(m) = dm(l). In that case, e(∆l) ≤ e(∆i) < e(∆m) forcesb(∆m) = b(∆i) + 1.By the definition of i+ we have ∆i+ ⊆ ∆m, which implies that dm(i

+) ≥ dm(m).On the other hand, we have either ∆l ≪ ∆k′ or ∆l∨ ≪ ∆k′ or ∆l∨ ( ∆k′ ( ∆l.

In all three case, with Lemma 2.2 for the latter, we reach a contradiction to thedepth inequality.Finally, set jmax = (i∗max)

#. We are left to show that there is no segment ∆ of m′

such that ∆′jmax≪ ∆ and b(∆) = b(∆i∗max

) + 1(= b(∆′jmax) + 1).

Assume the contrary. Then, arguing like before, we obtain m ∈ I with ∆i ≪ ∆m

and b(∆m) = b(∆i∗max) + 1. This contradicts the defining property of i∗max.

(3) First note that it follows from Lemma 3.1 that (∆∗i )′ = ∆′i, for all i ∈ I with

dm(i) /∈ dm(I∗).

For any i ∈ I∗, let

Ji = {j ∈ I : dm(j) = dm(i), b(∆j) = b(∆i) and ∆j ( ∆i}.

Thus, Ji = ∅ if and only if i = i# (since both are distinguished). For conveniencewe set Ji+ = ∅ when i = i∗max. By Lemma 3.1, we have

d−1m† (t) = d−1m (t) ∪ Ji+ \ Ji ,

for t = dm(i).Let {i1, . . . , il} be the admissible enumeration of d−1m (t). Then the indices of Ji

(if non-empty) appear as a contiguous block (in decreasing order of e(∆j), endingwith i#) right after the occurrence of i (since i is distinguished). Upon removingthe indices of Ji (if any) and inserting instead the indices of Ji+ next to i (again, indecreasing order of e(∆j), ending with (i+)#) we obtain an admissible enumeration{i′1, . . . , i

′l′} of d−1

m† (t) (with respect to m† =∑

i∈I ∆∗i ).

RSK FOR GLn OVER LOCAL FIELDS 15

It follows that

(∆∗i )′ =

{∆′i+ if Ji+ 6= ∅,−∆′

i#otherwise,

while if Ji+ 6= ∅, then(∆∗(i+)#)

′ = −∆′i# .

For all i′ ∈ d−1m† (t) \ {i, (i+)#}, we have (∆∗i′)

′ = ∆′i′. In particular, il = i′l′ and(∆∗il)

′ = ∆′il.The proposition follows. �

Corollary 3.4. For any 0 6= m ∈ M with minm < min l(m) we have

(K × Id)(MW(m)) = (Id×MW)(K(m)) .

In other words, l(m) = l(m†), ∆◦(m) = ∆◦(m′) and (m†)′ = (m′)†.

Proof. It follows from Proposition 3.3(1) that I∗ ∩ I♮ = I♯ ∩ I♮ = ∅. Hence, l(m) = l(m†).The rest of Proposition 3.3 shows that ∆◦(m) = ∆◦(m′) and (m†)′ = (m′)†. �

4. Representation theoretic applications

4.1. Basics. For the rest of the paper we fix a non-archimedean local field F with normal-ized absolute value |·| and consider representations of the general linear groups GLn(F ),n ≥ 0. All representations are implicitly assumed to be complex and smooth.

For any segment ∆ = [a, b] ∈ Seg, we write Z(∆) and L(∆) for the character |det|a+b2 of

GLb−a+1(F ) and the Steinberg representation of GLb−a+1(F ) twisted by |det|a+b2 , respec-

tively.Normalized parabolic induction will be denoted by ×. More precisely, if πi are represen-

tations of GLni(F ), i = 1, . . . , k and n = n1 + · · ·+ nk, we write

π1 × · · · × πk = IndGLn(F )Pn1,...,nk

(F ) π1 ⊗ · · · ⊗ πk ,

where Pn1,...,nkis the parabolic subgroup of GLn consisting of upper block triangular matri-

ces with block sizes n1, . . . , nk and π1⊗· · ·⊗πk is considered as a representation of Pn1,...,nk

via the pull-back from GLn1(F )× · · · ×GLnk(F ).

Given a multisegment m ∈ M, we can write it (in possibly several ways) as m =∑k

i=1∆i,where for any i < j, we have ∆i 6≪ ∆j . Then the representations

Z(m) = soc(Z(∆1)× · · · × Z(∆k)) ,

L(m) = soc(L(∆k)× · · · × L(∆1)) ,

are both irreducible and, up to equivalence, depend only on m. (We recall that soc standsfor the socle, i.e., the sum of all irreducible subrepresentations.) The Mœglin–Waldspurgerinvolution switches the two, namely

L(m) ∼= Z(m#), Z(m) ∼= L(m#).

In fact, the map Z(m) 7→ L(m) is a specialization of a functorial duality on the categoryof admissible smooth representations of GLn ([3, 24]).

16 MAXIM GUREVICH AND EREZ LAPID

Remark 4.1. More generally, we can fix a (not necessarily unitary) irreducible supercuspidalrepresentation ρ of GLd(F ) and consider the irreducible representations

Z([a, b]ρ) = soc(ρ |det|a × · · · × ρ |det|b), L([a, b]ρ) = soc(ρ |det|b × · · · × ρ |det|a)

of GL(b−a+1)d(F ). (When ρ is the trivial character of GL1(F ) = F ∗ this coincides withthe previous notation.) We can then define Z(mρ) and L(mρ) for any multisegment m

as before. Theorem 4.3 below and its proof will hold without change. Given irreduciblesupercuspidal representations ρ1, . . . , ρk of GLdi(F ) such that ρi 6≃ ρj |det|

r for all i 6= j andr ∈ Z, and any multisegments m1, . . . ,mk, the representation Z((m1)ρ1)× · · · × Z((mk)ρk)is irreducible. Moreover, by Zelevinsky classification, any irreducible representation ofGLn(F ) can be written uniquely in this form (up to permuting the factors) [32]. A similarstatement holds for L(m). Therefore, for all practical purposes it is enough to deal with asingle ρ. For concreteness we take ρ to be the trivial character of F ∗, but as was pointedout above this is essentially immaterial.

4.2. Suppose that l is a ladder. Then, for any irreducible representation τ of GLn(F ),each of the representations soc(Z(l)× τ) and soc(τ × Z(l)) is irreducible and occurs withmultiplicity one in the Jordan–Holder sequence of Z(l)× τ [15]. Thus, for any m ∈ M andl ∈ Lad, there is a multisegment soc(m, l) ∈ M, such that

soc(Z(m)× Z(l)) ∼= Z(soc(m, l)).

A simple recursive algorithm for the computation of soc(m, l), which relies on the Mœglin–Waldspurger algorithm, is given in [ibid.].3 We recall the result (using the notation of§2.2).

Proposition 4.2. ([15, Proposition 6.15] and [16, Lemma 3.16]) Let 0 6= m ∈ M andl ∈ Lad.

(1) Suppose that min l ≤ minm. Let ∆ be the unique segment in l for which b(∆) =min l. Then,

soc(m, l) = soc(m− n, l−∆) + n+∆

where upon writing m =∑

i∈I ∆i,

n =∑

i∈I:b(∆i)=b(∆) and e(∆i)≤e(∆)

∆i ≤ m .

(2) Suppose that minm < min l. Then, soc(m, l) is characterized by the condition

MW(soc(m, l)) =(soc(m†, l),∆◦(m)

).

We remark that we also have

soc(L(l)× L(m)) ∼= L(soc(m, l)).

This can be proved by either repeating the argument of [15] or using the functorial prop-erties of the Zelevinsky involution.

3Strictly speaking, we have to pass to the contragredient.

RSK FOR GLn OVER LOCAL FIELDS 17

4.3. Main result. We use the notation of §2.3.Theorem 4.3.

(1) For any 0 6= m ∈ M we have

soc(m′, l(m)) = m, soc(L(l(m))× L(m′)) = L(m).

(2) For any (l,m) ∈ A we have

soc(m, l) = K−1(l,m), soc(L(l)× L(m)) = L(K−1(l,m)).

(3) Given 0 6= m ∈ M write RSK(m) = (l1, . . . , lk) ∈ L and define recursively

πk = Z(lk), πi = soc(πi+1 × Z(li)), i = k − 1, . . . , 1 .

Then, π1∼= Z(m).

Similarly, letting

π′k = L(lk), π′i = soc(L(li)× π′i+1), i = k − 1, . . . , 1 ,

we have π′1∼= L(m).

In particular, Z(m) (resp. L(m)) occurs as a sub-representation of

(5) Λ(m) := Z(lk)× · · · × Z(l1)(resp. Λ′(m) := L(l1)× · · · × L(lk)

).

Proof. First note that the second part is merely a reformulation of the first part and thelast part follows directly from the first two parts. Thus, it suffices to prove the first part.Moreover, by the remark following Proposition 4.2, it is enough to prove the assertion

soc(m′, l(m)) = m.

We argue by induction on |m|, using Proposition 4.2. The base of the induction is trivial.Suppose that 0 6= m =

∑i∈I ∆i ∈ M. We separate into cases.

Suppose first that min l(m) = minm. Let i0 ∈ I be as in Lemma 2.6 and set

n =∑

i∈I:dm(i)=d(m)

∆i ≤ m .

Then, by Lemma 2.6,

K(m− n) =(l(m)−∆i0 , m′ − (n−∆i0)

).

Since |m− n| < |m|, the induction hypothesis now implies that

m− n = soc(m′ − (n−∆i0), l(m)−∆i0

).

It follows from the first part of Proposition 4.2 and Lemma 2.6(1) that soc(m′, l(m)) = m.Suppose now that minm < min l(m). By the second part of Proposition 4.2 and Corollary

3.4, we have

MW(soc(m′, l(m))) =(soc

((m′)†, l(m)

),∆◦(m′)

)

=(soc

((m†)′, l(m†)

),∆◦(m)

).

Yet, since |m†| < |m|, the induction hypothesis implies that the last expression is nothingbut MW(m). The result follows from the injectivity of the map MW. �

18 MAXIM GUREVICH AND EREZ LAPID

Remark 4.4. In [10], the width invariant k = k(m) was defined for every m =∑

i∈I ∆i ∈ M

to be the maximal number of distinct indices i1, . . . , ik ∈ I for which ∆ir+1 ⊆ ∆ir for allr = 1, . . . , k − 1. By standard properties of the RSK correspondence, k(m) is the numberof rows in the tableaux of RSK′(m).

It was shown in [10] that if there exist l1, . . . , ll ∈ Lad such that Z(m) appears as asubquotient of Z(l1) × · · · × Z(ll), then l ≥ k(m). This underlines a minimality propertyof Λ(m).

Remark 4.5. For m =∑

i∈I [ai, bi] let m∨ =

∑i∈I [−bi,−ai] so that Z(m∨) is the contragre-

dient of Z(m). Then, RSK(m∨) is related to RSK(m) by the Schutzenberger involution(modified to our conventions).

We also point out that(m#)∨ = (m∨)#

since Z(m)∨ = Z(m∨) and L(m∨) = L(m)∨.

It would be interesting to extend the second part of Theorem 4.3 to an arbitrary pair ofa ladder l and a multisegment m.

5. Enhanced multisegments

The results of the previous section stay put if we slightly extend the notion of multiseg-ments to include empty segments of the form [a + 1, a], a ∈ Z.

Since the proofs are essentially the same, we will only state the results in the modifiedcase, indicating the points where the proofs have to be adjusted.

5.1. Extended definitions. For an integer a, we will write δa = (a + 1, a) ∈ Z × Z andconsider it as a dummy segment. As with segments, we write b(δa) = a + 1 and e(δa) = a.We write

Seg = Seg ∪ {δa : a ∈ Z}

for the set of enhanced segments.

The relation ≪ is defined on Seg in the same fashion as it was defined on Seg. By our

convention, for ∆1,∆2 ∈ Seg, we write ∆1 ⊆ ∆2, if b(∆2) ≤ b(∆1) and e(∆1) ≤ e(∆2).Next, we embed the monoid N(Z) into N(Z×Z) by assigning to each a = a1+ . . .+ak ∈

N(Z) the dummy multisegment ∂a = δa1 + . . . + δak ∈ N(Z × Z). We will often refer todummy multisegments as elements of N(Z) by implicitly referring to this embedding. Wewrite supp ∂a for the underlying subset {a1, . . . , ak} of Z (without multiplicity).

Given a dummy multisegment ∂ and an integer a ∈ Z, we let n∂(a) denote the multiplicityof δa in ∂.

We will call M = N(Seg) the set of enhanced multisegments. Again taking N(Z) as thecollection of dummy multisegments, we naturally obtain an identification

M = M× N(Z) .

Using this presentation we will denote an enhanced multisegment as m = (m, ∂(m)) ∈ M,where m is the underlying (genuine) multisegment and ∂(m) is a dummy multisegment.

RSK FOR GLn OVER LOCAL FIELDS 19

We may view M as a sub-monoid of N(Z×Z), by taking m 7→ m+ ∂(m). In particular,

RSK′(m) = (Pm, Qm) ∈ T is well-defined, for all m ∈ M.Since the relation ≪ and the dominance condition remain well-defined for enhanced

segments, we may also define Lad ⊆ M and L′ ⊆⋃

m Ladmanalogously to Lad and L′.

The results of §2.3.2 all remain valid for enhanced multisegments and, as a result, givean extension of the previously defined map K to

K : M \ {0} → Lad× M, K(m) = (l(m),m′) .

Similarly, we define RSK : M → L′ recursively and finally obtain

RSK′(m) = (P (RSK(m)), Q(RSK(m))) ,

for all m ∈ M.

5.2. Enhanced results. For any enhanced multisegment m = m+ ∂(m) we write

Z(m) = Z(m), L(m) = L(m).

Similarly, if l is an enhanced ladder and m is an enhanced multisegment we write

soc(l,m) = soc(l,m).

In the newly defined terms, Theorem 4.3 can now be extended to the following statement.

Theorem 5.1. For any enhanced multisegment m ∈ M with non-zero m, we have

soc (m′, l(m)) = m, soc (L (l(m))× L (m′)) = L(m).

The point is that in general, m′ 6= m′ and l(m) 6= l(m). Hence, Theorem 5.1 is a genuineextension of Theorem 4.3.

For m ∈ M with a non-zero m, let us write RSK(m) = (l1, . . . , lk) ∈ L′. We then formthe representation

Λ(m) := Z(lk)× · · · × Z(l1) .

Viewed differently, we may start from a choice of a multisegment 0 6= n ∈ M and an

auxiliary choice of a dummy multisegment ∂ ∈ N(Z). Taking (n, ∂) ∈ M, we defineΛ∂(n) := Λ(n, ∂).

Corollary 5.2. For any 0 6= n ∈ M and ∂ ∈ N(Z), the irreducible representation Z(n)occurs as a sub-representation of Λ∂(n).

The case ∂ = 0 amounts to Theorem 4.3.

5.2.1. On the proof of Theorem 5.1. Essentially the same arguments that appeared in theprevious sections for multisegments, continue to hold for the case of enhanced multiseg-ments. In this manner, the proof of Theorem 5.1 follows the same lines as that of Theorem4.3. Let us give some details on the necessary adjustments.

Lemma 2.6 holds for m ∈ M, with the tweaked assumption min l(m) = minm, and

d = d(m) being the essential depth, that is,

d(m) = maxi∈I

{dm(i) : ∆i ∈ Seg} .

20 MAXIM GUREVICH AND EREZ LAPID

The commutativity of the two algorithms which is the culmination of §3 requires us toextend MW to a map

(6) MW : M \ N(Z) → M× Seg MW(m) = (∆◦(m),m†)

as follows. Given m =∑

i∈I ∆i ∈ M with m 6= 0, we set ∆◦(m) = ∆◦(m) and definem† =

∑i∈I ∆

∗i as in the multisegment case, but without discarding empty segments. In

other words, if ∆∗i = −∆i according to the original algorithm and ∆i = [a, a], we now set

∆∗i = δa ∈ Seg instead of discarding it. Note that by our convention the leading exponentsI∗ are taken from m (not from the dummy part). Iterating this procedure will give rise tom# together with a dummy multisegment.

With this algorithm in mind, practically all arguments of §3 remain valid, with thenotable exception of Lemma 3.1(2), whose statement should be slightly adjusted. We notethat our refined definition of MW allows us to drop the non-degeneracy assumptions in

this section. Finally, Corollary 3.4 holds for all m ∈ M with minm < min l(m).The arguments of §4 can now be applied in the enhanced multisegment setting, with m

substituting m in the required places. In the concluding part of the proof of Theorem 5.1we use the observations that (m′)† = (m′)† = (m†)′.

5.3. Relation to standard modules. While our newly discovered family {Λ(m)}m∈M ofRSK-standard modules satisfies some intriguing properties (see next section for a furtherdiscussion), Corollary 5.2 shows that each choice of a dummy multisegment ∂ ∈ N(Z), givesrise to yet another family {Λ∂(m)}m∈M of representations. Fixing m ∈ M we will showthat in a suitable sense, varying ∂ allows us to interpolate between the standard modulesa la Zelevinsky and Langlands.

Recall that for a given a multisegment m =∑k

i=1∆i ∈ M, Zelevinsky constructed thestandard module

ζ(m) = Z(∆1)× · · · × Z(∆k) ,

where ∆i’s are enumerated so that for any i < j, we have ∆i 6≪ ∆j . Similarly,

λ(m) = L(∆k)× · · · × L(∆1) ,

will stand for the standard module in the sense of Langlands.4

Recall further the involution m 7→ m# on M (see §2.2), for which Z(m) ∼= L(m#) andL(m) ∼= Z(m#). In particular, for each m ∈ M, Z(m) = soc(ζ(m)) = soc(λ(m#)).

Note that viewing a segment ∆ = [a, b] ∈ Seg as a singleton multisegment, we have∆# = [a, a] + [a+ 1, a+ 1] + . . .+ [b, b] ∈ Lad.

Proposition 5.3. Let m =∑

i∈I ∆i ∈ M be a multisegment consisting of r segments, andlet ∂ ∈ N(Z) be a dummy multisegment.

(1) Suppose that the following two conditions are satisfied.

4Note that we reversed the order of factors from the usual convention in order to obtain L(m) as asubrepresentation, rather than a quotient.

RSK FOR GLn OVER LOCAL FIELDS 21

(a) Whenever e(∆j) < e(∆i) we have

n∂(e(∆i)) ≥ n∂(e(∆j)) + #{j′ : e(∆′j) = e(∆j)}.

(b) For any i ∈ I and t such that t < e(∆i) we have n∂(t) ≤ n∂(e(∆i)).Then, Λ∂(m) ∼= ζ(m).

(2) Suppose that n∂(i) ≥ n∂(i+ 1) + r, for all minm ≤ i < maxm (when m 6= 0).Then, Λ∂(m) ∼= λ(m#).

The first part follows from the following elementary lemma by induction.

Lemma 5.4. Let 0 6= m ∈ M be an enhanced multisegment, with m =∑

i∈I ∆i.

(1) Suppose that n∂(m)(e(∆i)) > 0, for all i ∈ I. Then,(a) dm(i) depends only on e(∆i).(b) l(m) =

∑a∈supp ∂(m) δa.

(c) m′ = m− l(m).(2) Suppose that m 6= 0 and let j be the minimum of e(∆i), i ∈ I. Let ∆ be the segment

in m such that e(∆) = j and b(∆) is maximal with respect to this property.Suppose that n∂(m)(j

′) = 0, for all j′ ≤ j, but n∂(m)(e(∆i)) > 0, for all i such thate(∆i) > j. Then,(a) dm(i) depends only on e(∆i).(b) l(m) = ∆ +

∑a∈supp ∂(m) δa.

(c) m′ = m− l(m).

�

For the second part of Proposition 5.3, we recall the relation (m#)∨ = (m∨)# for multi-segments (see Remark 4.5). This means that instead of (6) we may consider the map

m 7→ (m†,∆◦(m)) ,

where m† := ((m∨)†)∨ and ∆◦(m) := (∆◦(m∨))∨. Applying it iteratively will give rise tom#, together with a dummy multisegment. (In fact, for genuine multisegments, this wasthe original description in [20].)

Hence, the second part of Proposition 5.3 follows by a simple induction from the followingelementary lemma whose proof is an easy exercise (cf. [20, Remarques II.2]).

Lemma 5.5. Let m ∈ M be an enhanced multisegment with 0 6= m =∑

i∈I ∆i.Suppose that n∂(m)(j) = 0 for all j ≥ maxm, but n∂(m)(j) > 0 for all minm ≤ j < maxm.

Then,

(1) For any i ∈ I, dm(i) = maxm − e(∆i), if there exist i0, . . . , ir = i ∈ I withr = maxm − e(∆i) such that ∆i = ∆ir ≪ . . . ≪ ∆i0 ; otherwise dm(i) = maxm −e(∆i)− 1.

(2) l(m) = ∆◦(m)# +∑

a:n∂(m)(a)>0,a+1<b(∆◦(m)) δa.

(3) m′ = m† −∑

a:n∂(m)(a)>0 δa.

�

22 MAXIM GUREVICH AND EREZ LAPID

Remark 5.6. While Proposition 5.3 (which is a simple combinatorial statement) and The-orem 5.2 give a new perspective on the Mœglin–Waldpusrger involution, they do not givea genuinely new proof of the relation Z(m) = L(m#), which is one of the main results of[20]. The reason is that the techniques and ideas of [ibid.] are used in the results of [15]and hence indirectly in the proof of Theorem 5.2.

6. Odds and ends

6.1. Socle irreducibility. It is natural to ask whether as in the case of the traditionalstandard modules, the newly constructed class of RSK-standard modules possesses theproperty of having a unique irreducible sub-representation. Although we are currentlyunable to prove this in general, we expect that this is indeed the case, namely

Conjecture 6.1. Let m be a multisegment and let Λ(m) be as in (5). Then soc(Λ(m)) isirreducible, hence (by Theorem 4.3), Z(m) ∼= soc(Λ(m)).

A weaker form of this conjecture would be the following

Conjecture 6.2. Suppose that RSK(m) = (l1, . . . , lk). Define recursively π′1 = Z(l1),π′i = soc(Z(li)× π′i−1), i = 2, . . . , k. Then Z(m) ∼= π′k.

Note that Conjecture 6.2 is not a formal consequence of Theorem 4.3 since in general itis not true that soc(soc(π1×π2)×π3) ≃ soc(π1×soc(π2×π3)), even if πi are supercuspidal.For instance, we can take π1 = π3 to be the trivial character of F

∗ and π2 to be the absolutevalue on F ∗.

It is tempting to attempt to prove Conjecture 6.2 by the same method as Theorem 4.3.Suppose that RSK(m) = (l1, . . . , lk). Call lk the lowest ladder of m and write V (m) =(lk,

′m) where ′m = RSK−1(l1, . . . , lk−1) (the fact that it is well defined may follow fromarguments in the spirit of §A). We need to show that

Z(m) = soc(Z(lk)× Z(′m)) .

As before, it is natural to use the recipe of [15, §6.3]. The simple case is when max lk =maxm. Assume that max lk < maxm. In this case we have to show that

(7) V (m†′

) = (lk, (′m)†

′

)

where m†′

is the analogue of m† for the end points.

6.2. Triangular structure. For any integer r and an inverted Young tableau Y , let Y≥rbe the part of Y consisting of the entries that are bigger than or equal to r. Clearly, Y≥r isalso an inverted Young tableau (of possibly smaller size). Recall the dominance order onthe set of Young diagrams defined by

(λ1, . . . , λk) ≺ (λ′1, . . . , λ′k′) if k ≤ k′ and

j∑

i=1

λi ≥

j∑

i=1

λ′j for all j = 1, . . . , k.

RSK FOR GLn OVER LOCAL FIELDS 23

(We will only compare Young diagrams of the same size. In this case ≺ encodes the closurerelation of unipotent orbits in GLn(C), parameterized by partitions via the Jordan normalform.) Define a partial order on the set of inverted Young tableaux by

Y ≤ Y ′ if sh(Y≥r) ≺ sh(Y ′≥r) for all r ∈ Z ,

where sh(X) is the shape of X , i.e., its underlying Young diagram. (We will only compareinverted Young tableaux whose entries coincide as multisets.) The product partial order onT induces a partial order on L (which will be denoted by ≤), according to the identificationsof §2.3.1.

Conjecture 6.3. Let m be a multisegment and let Λ(m) be as in (5). In the Grothendieckgroup we have

[Λ(m)] = [Z(m)] +

l∑

i=1

[Z(ni)] ,

where RSK(ni) < RSK(m) for all i = 1, . . . , l.5

We verified this conjecture by computer calculation for all multisegments consisting ofn segments with n ≤ 8. The computation involves writing Z(lk)× · · · × Z(l1) in terms ofstandard modules ([14]) and decomposing standard modules into irreducible representa-tions – the multiplicities are given by the value at 1 of Kazhdan–Lusztig polynomials withrespect to the symmetric group Sn.

We do not know whether the partial order on M (or on the symmetric group for thatmatter) given by m1 ≤ m2 ⇐⇒ RSK(m1) ≤ RSK(m2) has already been considered inthe literature. Likewise, we are so far unaware of a befitting geometric interpretation or asimpler combinatorial description of this partial order.

Let Rn, n ≥ 0 be the Grothendieck groups of GLn(F ), n ≥ 0 and let R′ be the subgroupof ⊕n≥0Rn generated by [Z(m)], m ∈ M.

Conjecture 6.3 would imply the following weaker conjecture.

Conjecture 6.4. The classes of RSK-standard representations

Λ(m), m ∈ M,

form a Z-basis for R′.

6.3. Relation to the DRS basis. Recall that R′, equipped with the parabolic induc-tion product, becomes a commutative ring, which is freely generated by the variables{[Z(∆)]}∆∈Seg. With these generators, the classes of standard representations become themonomial basis for R′ (as a Z-module).

Let a, b ⊆ Z be two given subsets of integers. Let Ma,b be the collection of multisegmentsm, for which the begin points (resp. end points) of all segments comprising m belong toa (resp. b). Let Ra,b be the subgroup of R′ generated by [Z(m)], m ∈ Ma,b. In other

words, Ra,b is the subring of R′ generated by the classes of Z([a, b]) where a ∈ a and b ∈ b

and a ≤ b. By the properties of RSK′, we have [Λ(m)] ∈ Ra,b, for all m ∈ Ma,b. Clearly,

5The ni’s are not necessarily distinct. Also, not all n’s with RSK(n) < RSK(m) necessarily occur.

24 MAXIM GUREVICH AND EREZ LAPID

Conjecture 6.4 is equivalent to the statement that Λ(m), m ∈ Ma,b, is a Z-basis for Ra,b,

for all a, b.Now, assume that a, b are such that, a ≤ b, for every a ∈ a and b ∈ b (the so-called

saturated case – cf. Remark 2.5). Then, Ra,b can be identified with the coordinate ring of

the space of matrices of size |a| × |b|. Bases for such polynomial rings were considered in[6,7] in the context of combinatorial invariant theory. Our class of RSK-standard modulesinside the rings Ra,b becomes precisely the bitableaux basis considered in [ibid.]. Thus,with the above identifications in place, we obtain a weak form of Conjecture 6.4.

Proposition 6.5. (cf. [6, 7, 28]) For a, b ⊆ Z in the saturated case, the classes

[Λ(m)], m ∈ Ma,b,

form a Z-basis for Ra,b.

Proof. Under our assumptions, RSK gives a bijection between Ma,b and the set of pairs

of tableaux (P,Q) in T , for which P (resp. Q) is filled by numbers from a (resp. b) (seeRemark 2.5). Thus, it is enough to prove that the set of elements

B ={[Z(l1)] · . . . · [Z(lm)] : (l1, . . . , lm) ∈ L′, li ∈ Ma,b

}

gives a basis for Ra,b.

For a ladder l =∑k

i=1[ai, bi], we have the following determinantal identity [14]

[Z(l)] =∑

σ∈Sk

ǫ(σ)

k∏

i=1

[Z([ai, bσ(i)])] ,

as elements of R′. Here, ǫ(σ) is the sign of the permutation σ.Identifying [Z([ai, bj ])] with variables in the polynomial ring Ra,b, we see that our con-

struction of B coincides with the construction of the basis in [6, §4]. �

Appendix A. Multisegments, MW, and RSK

by Mark Shimozono6

This appendix contains two main combinatorial results. The first result, Proposition A.9,describes the effect of the mapMW on the cRSK tableau pair. The second result, TheoremA.13, is a characterization of the image of the set of multisegments under RSK expressed interms of the right and left key tableau construction of Lascoux and Schutzenberger, appliedto the RSK tableau pair. The complete proofs will appear in a separate publication.

In this appendix, to conform with the bulk of the combinatorics literature, semistandardYoung tableaux are used; in contrast, in the main part of this article, inverted Youngtableaux are used; see Remark A.2.

6Department of Mathematics, Virginia Polytechnic Institute and State University;mshimo@math.vt.edu

RSK FOR GLn OVER LOCAL FIELDS 25

A.1. Partitions and skew shapes. Let Y be Young’s lattice of partitions. The diagramD(λ) of λ ∈ Y is the set D(λ) = {(i, j) ∈ Z2

>0 | 1 ≤ j ≤ λi}. The (i, j)-th box is in thei-th row and j-th column with matrix-style indexing. The transpose or conjugate partitionλt ∈ Y is defined by (i, j) ∈ D(λ) if and only if (j, i) ∈ D(λt). The notation µ ⊆ λ meansD(µ) ⊆ D(λ). If µ ⊆ λ let λ/µ := D(λ)\D(µ); this is called a skew shape. The skew shaperefers to this set difference of boxes, and can be achieved by various pairs (λ, µ). A skewshape is normal (resp. antinormal) if it has a unique northwestmost (resp. southeastmost)corner. A skew shape is a horizontal (resp. vertical) strip if it has at most one box in eachcolumn (resp. row). Let H (resp. V) denote the set of horizontal (resp. vertical) stripsand let Hr (resp. Vr) be the set of horizontal (resp. vertical) strips of size r.

A.2. Skew tableaux. In this section a semistandard tableau of shape λ/µ is a functionT : λ/µ → Z>0 which is weakly increasing within rows going from left to right and strictlydecreasing from bottom to top. Let Tabλ/µ denote the set of semistandard tableaux ofshape λ/µ. A tableau (resp. antitableau) is a semistandard skew tableau of normal (resp.antinormal) shape. More precisely, by a skew tableau we mean a translation equivalenceclass of such maps. The (row-reading) word of a skew tableau T is defined by word(T ) =· · ·u(2)u(1) where u(i) is the word obtained by reading the i-th row of T from left to right.Using the row reading word a skew tableau may be regarded as a word. A tableau (resp.antitableau) word is a word of the form word(T ) where T is a tableau (resp. antitableau).

A.3. Knuth equivalence. Let ≡ be Knuth’s equivalence relation on the set of words A∗

with symbols in the totally ordered set A = {1 < 2 < · · · < n} [12]. It is the transitiveclosure of relations uxzyv ≡ uzxyv for x ≤ y < z and uyxzv ≡ uyzxv for x < y ≤ z wherex, y, z ∈ A and u, v ∈ A∗.

Proposition A.1. [12] Every Knuth class in A∗ contains a unique tableau word and aunique antitableau word.

Denote by P(u) the unique tableau whose word is Knuth equivalent to u. This is theSchensted P -tableau. For a tableau T denote by Tց the unique antitableau whose wordis Knuth equivalent to the word of T . Then P(u)ց is the unique antitableau word that isKnuth equivalent to u.

Remark A.2. (1) An inverted Young tableau of §2.3 is the same thing as an antitableauexcept written a different way: they are mapped to each other by reflection acrossa northeast to southwest antidiagonal. See further §A.6.

(2) Passing from a tableau T to its antitableau Tց is a nontrivial computation. Giventhe word of Tց, taking its reverse complement with respect to some ambient al-phabet containing all appearing entries, directly gives the word of a tableau knownas the Schutzenberger involution of T .

A.4. Pieri bijections. A row word (resp. column word) is a weakly increasing (resp.strictly decreasing) word. Let Row (resp. Col) denote the set of row (resp. column) wordsand Rowr (resp. Colr) those of length r.

26 MAXIM GUREVICH AND EREZ LAPID

Proposition A.3. [17, Theoreme 2.10] Let µ ∈ Y and r ∈ Z≥0. There are bijections

Rowr × Tabµ∼=

⊔

λ∈Yλ/µ∈Hr

Tabλ (u, T ) 7→ P(uT )(8)

Colr × Tabµ∼=

⊔

λ∈Yλ/µ∈Vr

Tabλ (u, T ) 7→ P(uT )(9)

Tabµ × Rowr∼=

⊔

λ∈Yλ/µ∈Hr

Tabλ (T, u) 7→ P(Tu)(10)

Tabµ × Colr ∼=⊔

λ∈Yλ/µ∈Vr

Tabλ (T, u) 7→ P(Tu).(11)

Example A.4. We illustrate how to compute the inverse of the bijection (9). Let λ =(4, 3, 3, 2, 1) and µ = (3, 2, 2, 1, 1). Let U be the unique skew standard tableau of shapeλ/µ ∈ V, in which the boxes are labeled in increasing value from top to bottom. Picturedbelow is a tableauX of shape λ. We perform successive Schensted reverse column insertionsstarting at the boxes of U starting with the bottommost. Each reverse insertion producesan additional single entry factored to the left: the bumping path along which entries shiftto the left, is depicted in green.

U = 123

4

X = 1 2 3 32 3 44 4 55 66

≡ 1 2 3 32 3 44 4 556

6

≡ 1 2 3 32 3 44 556

46

≡ 1 2 3 33 44 556

246

≡ 2 3 33 44 556

1246

= uT.

Example A.5. We illustrate how to compute the inverse of the bijection (11). We use thesame λ, µ, and U as before but a different tableau Y . We perform successive Schenstedreverse row insertions starting at the boxes of U starting with the bottommost. Eachreverse insertion produces an additional single entry factored to the right: the bumpingpath along which entries shift upwards, is depicted in green.

Y = 1 1 1 22 2 23 3 34 65

≡1

1 1 2 22 2 33 3 645

≡

12

1 1 2 32 2 63 345

≡

123

1 1 2 62 23 345

≡

1236

1 1 22 23 345

= Tu.

A.5. Column RSK correspondence. Let Row∗ be the set of infinite sequences of rowwords u• = (· · · , u(2), u(1)) with u(i) ∈ Row, only finitely many of which are nonempty.

RSK FOR GLn OVER LOCAL FIELDS 27

Given u• ∈ Row∗ let P0 = ∅ be the empty tableau and for j ≥ 1 let Pj = P(u(j)Pj−1).Let P(u•) = P(· · ·u(2)u(1)), the Schensted P -tableau of the juxtaposition of the row wordsu(i). Note that shape(Pj)/shape(Pj−1) ∈ H by Proposition A.3 (8). This sequence ofhorizontal strips defines a semistandard tableau Q(u•) of the same shape as P(u•) in whichthe boxes of the horizontal strip shape(Pj)/shape(Pj−1) are filled with j’s for all j. Thisyields a bijection we denote by cRSK, the column insertion Robinson-Schensted-Knuthcorrespondence:

Row∗⊔

λ

Tabλ × Tabλ

u• (P(u•),Q(u•))

✲cRSK

✲

(12)

Example A.6. Let u• = (· · · , 56, 34, 233, 122, 12, 1). We have P0 empty,

P1 = 1 P2 =1 12 P3 =

1 1 1 22 2

P4 =1 1 1 22 2 23 3

P5 =

1 1 1 22 2 23 3 34

P(u•) = P6 =

1 1 1 22 2 23 3 34 65

Q(u•) =

1 2 3 32 3 44 4 55 66

A.6. Comparison with multisegment notation. Without loss of generality in thisdiscussion we only consider segments which are intervals of positive integers. Define theinjective map ι : M → Row∗ such that ι(m) = u• = (· · ·u(2), u(1)) where the word u(j)

contains the value i the same number of times the multisegment [i, j] appears in m. Theimage of ι is the set of flagged tuples of row words, those such that all values appearing inu(i) are less than or equal to i.

Let րւ be the reflection across an antidiagonal. There is an involutive bijection inv :Tabλ → ITabλ′ from the set of semistandard tableaux of shape λ, to the set ITabλ′ ofinverted tableaux of the conjugate or transposed shape λ′, defined by inv(T ) =րւ (Tց).That is, first take the antitableau and then flip.

1 1 42 3 → 1 1

2 3 4 →4 13 12

28 MAXIM GUREVICH AND EREZ LAPID

The following diagram commutes:

M

⊔

λ

ITabλ × ITabλ

Row∗⊔

λ

Tabλ × Tabλ

✲RSK′

❄

ι

❄

inv×inv

✲

cRSK

Let RSK′(m) = (Pm, Qm) and K(m) = (l(m),m′). Then

• l(m) is the ladder [a1, b1] + [a2, b2] + · · ·+ [ar, br] where a1a2 · · · ar and b1b2 · · · br arethe first rows of Pm and Qm respectively.

• m′ is uniquely specified by the condition that Pm′ and Qm′ are obtained from Pm

and Qm by removing their respective first rows.

Let ι(m) = u• so that by the commutativity of the above diagram, P(u•) = inv(Pm)and Q(u•) = inv(Qm). Consider the map Row∗ → Lad×Row∗ mapping u• to (l(u•), (u•)′)commuting the diagram

M Lad×M

Row∗ Lad×Row∗

✲K

❄

ι

❄

Id×ι

✲

By definition, the ladder l(u•) is obtained by the pair of column words coming from therightmost rows of the antitableaux P(u•)ց and Q(u•)ց. Rather than computing the fullantitableaux there is a more efficient way to compute l(u•) and (u•)′. Let P(u•) 7→ (T, u)be the application of the inverse of the bijection 11 for λ = shape(P(u•)) and µ obtained byremoving the first column of λ. Similarly Q(u•) 7→ (U, v). Then (u, v) is the pair of columnwords defining l(u•). Moreover (u•)′ is uniquely defined by the property that P((u•)′) = Tand Q((u•)′) = U .

Remark A.7. There is a version of K which, instead of producing the last columns of theantitableaux of P(u•) and Q(u•), produces the first columns of P(u•) and Q(u•).

A.7. The map MW and cRSK. Let u• ∈ Row∗ be upper triangular and nonempty withcorresponding multisegment m. Let u•† ∈ Row∗ correspond to m† and let ∆◦(u•) = ∆◦(m).

Given a nonempty word u and m ∈ Z>0, let Um(u) be the word obtained by scanning ufrom right to left, selecting the rightmost copy of m in u, then the next m+1 to its left, thenext m+ 2 to its left, and so on, until the left end of u is encountered, and then replacingeach selected letter i by i+ 1. It can be shown that Um can be defined on Tabλ/µ via the

row reading word and on and on Row∗ via the map u• 7→ · · ·u(2)u(1). Given u• ∈ Row∗

RSK FOR GLn OVER LOCAL FIELDS 29

nonempty, let m = min(u•) be the minimum value among the words in u•. Denote by

k = k(u•) ∈ Z≥0 the maximum index such that m + j ∈ u(m+j) for all 0 ≤ j < k. Define

U(u•) ∈ Row∗ by removing a copy of m+ j from u(m+j) for all 0 ≤ j < k.We briefly define the algorithm MW on u• and describe its affect on the cRSK tableau

pair. Let min(u•) be the minimum value among the words in u•. We first consider U(u•)

with m = min(u•) and k = k(u•). Then u•† = Um+k(u•). Let k = km(u

•) be the sum of kand the number of values incremented by Um+k. By definition ∆◦(u•) = [m,m+ k − 1].

Example A.8. With u• as in Example A.6 we have U(u•) = (· · · , 56, 34, 233, 122, 1, ∅),

m = 1, k = 2, u•† = U3(U(u•)) = (· · · , 66, 35, 234, 122, 1, ∅) and k = 5.

Given a tableau T and a subtableau S, both of partition shape, let T − S be theskew tableau obtained by removing S from T . For a ≤ b let Cb

a be the column wordb(b− 1) · · · (a+ 1)a.

Proposition A.9. Let ∅ 6= u• ∈ Row∗, m = min(u•), (P,Q) = (P(u•),Q(u•)), m =min(P ), and (P †, Q†) = (P(u•†),Q(u•†)).

(1) k(u•) is the maximum k ∈ Z≥0 such that C = Cm+k−1m is contained in the first

column of Q.(2) The value k in the definition of u•† equals the maximum k such that Cm+k−1

m iscontained in the first column of P .

(3) We have

Q† = P(Q− C).(13)

In particular shape(Q)/shape(Q†) ∈ Vk.(4) Let (T, c) ∈ Tabshape(Q†) × Colk be the unique pair of Proposition A.3 such that

P(Tc) = P . Then T = P(U(u•)), c = C, and P † = Um+k(T ).

Example A.10. With u• as in Example A.6 (P †, Q†) are given in Figure 1, computed as inProposition A.9.

A.8. Key tableaux. Say that a tableau is a key tableau if it has the property that any ofits columns, viewed as a subset, contains any column to its right.

Consider a word with symbols in the set {1, 2, . . . , n}. The weight of a word u is thesequence wt(u) = (m1(u), m2(u), . . . , mn(u)) ∈ Zn

≥0 where mj(u) is the multiplicity of theentry j in u.

Given β ∈ Zn≥0 let β+ be the unique partition in the orbit Sn · β and let wβ ∈ Sn be

the shortest element such that wβ(β+) = β. The key tableau Kβ of weight β is the unique

tableau of shape β+ and weight β. Given a fixed partition λ the orbit Sn · λ is a poset:α ≤ β if wα ≤ wβ in Bruhat order.

A.9. Left and right keys; image of RSK′. Recall horizontal and vertical strips from§A.1. Say that a vertical (resp. horizontal) strip is λ-removable if it has the form λ/µ forsome µ.

30 MAXIM GUREVICH AND EREZ LAPID

P =

1 1 1 22 2 23 3 34 65

Q =

1 2 3 32 3 44 4 55 66

T ⊗ C =

1 1 22 2 33 34 65

⊗ 12 Q† = P(Q− C) =

2 3 33 4 44 55 66

P † =

1 1 22 2 43 35 66

Figure 1. The computation of Q† and P † from Q and P

Say that a vertical strip of size r is grounded if it contains a box in each of the firstr rows. It is obvious that for a given r a partition has at most one grounded removablevertical strip of size r. This strip exists if and only if λ has a column of size r.

Let P be a tableau of shape λ. Let L = λ1 be the number of columns of λ. For eachcolumn size r = λt

j of λ, let µ ⊆ λ be the unique partition such that λ/µ ∈ Vr is grounded.By Proposition A.3 there is a unique pair (u, T ) ∈ Colr ×Tabµ (resp. (T, u) ∈ Tabµ×Colrsuch that P(uT ) = P (resp. P(Tu) = P ). Denote the word u by c−j (P ) (resp. c+j (P )).

Lemma A.11. Considering column words as subsets, we have c−j (P ) ⊇ c−j+1(P ) and

c+j (P ) ⊇ c+j+1(P ) for all j.

The left key K−(P ) (resp. right key K+(P )) of P is by definition the tableau of the sameshape as P whose j-th column is given by the column word c−j (P ) (resp. c+j (P )) for all j.By Lemma A.11 K−(P ) and K+(P ) are key tableaux and in particular semistandard.

Example A.12. For the tableaux (P,Q) = (P(u•),Q(u•)) from Example A.6 the factoriza-tions defining c+J (P ) and c−j (Q) are given as follows. See Example A.5 for the method ofproducing the 4-box right column factor from P and Example A.4 for how to obtain the4-box left column factor from Q.

P =

1 1 1 22 2 23 3 34 65

≡

1 1 22 23 35

·

12346

≡

1 1 22 23 345

·

1236

≡

1 1 22 23 34 65

·123≡

1 1 12 2 23 3 34 65

· 2

K+(P ) =

1 1 1 22 2 23 3 34 66

wt(K+(P )) = (3, 4, 3, 1, 0, 2)

RSK FOR GLn OVER LOCAL FIELDS 31

Q =

1 2 3 32 3 44 4 55 66

≡

12456

·

2 3 33 44 56

≡

1246

·

2 3 33 44 556

≡124·

2 3 33 44 55 66

≡ 2 ·

1 2 33 3 44 4 55 66

K−(Q) =

1 1 1 22 2 24 4 45 66

wt(K−(Q)) = (3, 4, 0, 3, 1, 2).

For α = wt(K+(P )) = (3, 4, 3, 1, 0, 2) we have wα = s1s5s4 and for β = wt(K−(Q)) =(3, 4, 0, 3, 1, 2) we have wβ = s1s3s5s4s5. We see that wα ≤ wβ.

Theorem A.13. Let u• ∈ Row∗. Then u• is flagged (in the image of a multisegment underι) if and only if wt(K+(P(u•))) ≤ wt(K−(Q(u•))).

This theorem can be proved using the ideas of [21] combined with the combinatorics ofthe crystal graphs of Demazure modules in highest weight modules for Uq(sln).

Remark A.14. A similar condition works for enhanced multisegments; the flagged conditionfor u• is changed to requiring that all values in u(i) be at most i + 1, and that instead ofBruhat-comparing the weights α and β of the right key of P(u•) and the left key of Q(u•),one uses the weight (0, β1, β2, . . . ) instead of β.

A.10. Theorem A.13 in terms closer to inverted tableaux. We wish to state thecondition of Theorem A.13 in terms of the inverted tableau pair (Pm, Qm). The first trans-formation is to apply րւ so instead the condition is on a pair of antitableaux. It sufficesto define left and right key for an antitableau. By definition the left key and right keyapply to a tableau. Since each Knuth class contains a unique tableau word and antitableauword, it makes sense to talk about the left and right keys of an antitableau. This sufficesto describe the required condition implicitly.

However, since the map between a tableau and its antitableau is nontrivial, we wish togive a more direct way to compute the left and right key, directly from the antitableau.For this it suffices to compute the column factors c±j of an antitableau.

We will do this by example, using anti-analogues of the methods given in Examples A.4and A.5.

Example A.15. Consider the tableau X of shape λ = (4, 3, 3, 2, 1) from Example A.4.Suppose instead of X we are given its antitableau. It is pictured in Figure 2.

Let us compute the column c−2 (Xց), which has size 4 since λt

2 = 4. Let µ ⊆ λ be theunique partition such that λ/µ is the grounded λ-removable vertical strip of size 4. Wemake a skew tableau U of the anti-shape of λ/µ filled with the numbers 1 through 4 fromtop to bottom.

We now perform a sequence of successive internal row insertions on Xց at the boxesof U starting with the box labeled 1 (cf. [23]). The bumping paths, which push entriesdownward, are in green. Example A.4 produced a factorization X ≡ uT : the currentcomputation produces the factorization Xց ≡ uTց.

32 MAXIM GUREVICH AND EREZ LAPID

U =

123

4 Xց =

23 3

1 4 43 5 5

2 4 6 6 ≡

23

1 3 43 4 5

2 4 5 66

≡

23

3 41 4 5

2 3 5 646

≡

23

3 44 5

1 3 5 6246

≡

23

3 44 5

3 5 61246

= uTց.

Figure 2. Computing the left key of an antitableau

Y ց = 11 2

1 2 32 3 4

2 3 5 6

≡1

12

1 2 32 3 4

2 3 5 6

≡

12

12

1 32 3 4

2 3 5 6

≡

123

12

1 32 4

2 3 5 6

≡

1236

12

1 32 4

2 3 5

= Tցu.

Figure 3. Computing the right key of an antitableau

Example A.16. Example A.5 computed a factorization Y ≡ Tu. We illustrate a factoriza-tion Y ց ≡ Tցu. The antitableau Y ց is pictured in Figure 3. Using the same tableau Uas in Example A.15 we apply successive internal column insertions on Y ց at the boxes ofU in the same order as before. The bumping paths push to the right.

References

[1] Tomoyuki Arakawa and Takeshi Suzuki, Duality between sln(C) and the degenerate affine Hecke

algebra, J. Algebra 209 (1998), no. 1, 288–304. MR1652134[2] Dan Barbasch and Dan Ciubotaru, Ladder representations of GL(n,Qp), Representations of reductive

groups, 2015, pp. 117–137. MR3495794[3] Joseph Bernstein, Roman Bezrukavnikov, and David Kazhdan, Deligne-Lusztig duality and wonderful

compactification, Selecta Math. (N.S.) 24 (2018), no. 1, 7–20. MR3769724[4] Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-

Lauda algebras, Invent. Math. 178 (2009), no. 3, 451–484. MR2551762[5] Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J.

Math. 174 (1996), no. 2, 295–326. MR1405590[6] J. Desarmenien, Joseph P. S. Kung, and Gian-Carlo Rota, Invariant theory, Young bitableaux, and

combinatorics, Advances in Math. 27 (1978), no. 1, 63–92. MR0485944[7] Peter Doubilet, Gian-Carlo Rota, and Joel Stein, On the foundations of combinatorial theory. IX.

Combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216. MR0498650[8] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge Uni-

versity Press, Cambridge, 1997. With applications to representation theory and geometry. MR1464693(99f:05119)

[9] Maxim Gurevich, Quantum invariants for decomposition problems in type a rings of representations,2017. arXiv:1711.01721.

RSK FOR GLn OVER LOCAL FIELDS 33

[10] , Decomposition rules for the ring of representations of non-Archimedean GLn, Int. Math. Res.Not. IMRN (2019).

[11] Anthony Henderson, Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of

type A, Represent. Theory 11 (2007), 95–121 (electronic). MR2320806[12] Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970),

709–727. MR0272654[13] Arno Kret and Erez Lapid, Jacquet modules of ladder representations, C. R. Math. Acad. Sci. Paris

350 (2012), no. 21-22, 937–940. MR2996769[14] Erez Lapid and Alberto Mınguez, On a determinantal formula of Tadic, Amer. J. Math. 136 (2014),

no. 1, 111–142. MR3163355[15] , On parabolic induction on inner forms of the general linear group over a non-archimedean

local field, Selecta Math. (N.S.) 22 (2016), no. 4, 2347–2400. MR3573961[16] Erez Lapid and Alberto Mınguez, Geometric conditions for �-irreducibility of certain representations

of the general linear group over a non-archimedean local field, Adv. Math. 339 (2018), 113–190.MR3866895

[17] Alain Lascoux and Marcel-P. Schutzenberger, Le monoıde plaxique, Noncommutative structures inalgebra and geometric combinatorics (Naples, 1978), 1981, pp. 129–156. MR646486 (83g:20016)

[18] Bernard Leclerc, Maxim Nazarov, and Jean-Yves Thibon, Induced representations of affine Hecke al-

gebras and canonical bases of quantum groups, Studies in memory of Issai Schur (Chevaleret/Rehovot,2000), 2003, pp. 115–153. MR1985725

[19] Bernard Leclerc and Jean-Yves Thibon, The Robinson-Schensted correspondence, crystal bases, and

the quantum straightening at q = 0, Electron. J. Combin. 3 (1996), no. 2, Research Paper 11, approx.24 pp. (electronic). The Foata Festschrift. MR1392496

[20] C. Mœ glin and J.-L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math. 372 (1986),136–177. MR863522

[21] Victor Reiner and Mark Shimozono, Key polynomials and a flagged Littlewood-Richardson rule, J.Combin. Theory Ser. A 70 (1995), no. 1, 107–143. MR1324004

[22] Daniele Rosso, Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags,Canad. J. Math. 64 (2012), no. 5, 1090–1121. MR2979579

[23] Bruce E. Sagan and Richard P. Stanley, Robinson-Schensted algorithms for skew tableaux, J. Combin.Theory Ser. A 55 (1990), no. 2, 161–193. MR1075706

[24] Peter Schneider and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building,

Inst. Hautes Etudes Sci. Publ. Math. 85 (1997), 97–191. MR1471867 (98m:22023)[25] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics,

vol. 946, Springer-Verlag, Berlin, 1982. MR672610 (84a:14024)[26] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics,

vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota andappendix 1 by Sergey Fomin. MR1676282 (2000k:05026)

[27] Robert Steinberg, An occurrence of the Robinson-Schensted correspondence, J. Algebra 113 (1988),no. 2, 523–528. MR929778 (89c:05006)

[28] Richard G. Swan, On the straightening law for minors of a matrix, 2016. arXiv:1605.06696.[29] G. Viennot, Une forme geometrique de la correspondance de Robinson-Schensted (1977), 29–58. Lec-

ture Notes in Math., Vol. 579. MR0470059[30] A. V. Zelevinskiı, The p-adic analogue of the Kazhdan-Lusztig conjecture, Funktsional. Anal. i

Prilozhen. 15 (1981), no. 2, 9–21, 96. MR617466 (84g:22039)[31] , Two remarks on graded nilpotent classes, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 199–200.

MR783619 (86e:14027)[32] A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations

of GL(n), Ann. Sci. Ecole Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR584084 (83g:22012)

34 MAXIM GUREVICH AND EREZ LAPID

Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel.

E-mail address : maxg@technion.ac.il

Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

E-mail address : erez.m.lapid@gmail.com