Computing the Lusztig-Vogan bijection

Computing the Lusztig-Vogan Bijection

by

David B Rush

"

S.B., Massachusetts Institute of Technology (2013)

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Computing the Lusztig-Vogan Bijection

by

David B Rush

Submitted to the Department of Mathematics on August 8, 2017, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

Let G be a connected complex reductive algebraic group with Lie algebra g. The Lusztig-Vogan bijection relates two bases for the bounded derived category of G-equivariant coherent sheaves on the nilpotent cone 11 of g. One basis is indexed by

A+, the set of dominant weights of G, and the other by Q, the set of pairs ((9, 8)

consisting of a nilpotent orbit (9 C n and an irreducible G-equivariant vector bundle

(S -÷ (9. The existence of the Lusztig-Vogan bijection -/: Q M A+ was proven by Bezrukavnikov, and an algorithm computing -y in type A was given by Achar. Herein we present a combinatorial description of - in type A that subsumes and dramatically simplifies Achar's algorithm.

Thesis Supervisor: David A. Vogan Jr.

Acknowledgments

... I thence

Invoke thy aide to my adventrous song, That with no middle flight intends to soare Above th' Aonian Mount, while it persues Things unattempted yet in prose or rhime.

John Milton, Paradise Lost (1667)

Pursuing a Ph.D. is an epic undertaking. The originality of my poetry and prose is for others to judge. For myself, I can only be humbled by the support that sustained me throughout the journey.

David A. Vogan Jr. was the best advisor I could have ever imagined. When he first described the Lusztig-Vogan bijection to me, I was immediately taken, but I would scarcely have believed that I would one day have something to say about it.

My gratitude to David - for suggesting the problem and offering invaluable help and encouragement along the way to a solution - cannot be overstated.

Thanks are also due to Roman Bezrukavnikov and George Lusztig for serving on my thesis committee and offering thoughtful remarks. George and David were first to conjecture the existence of the Lusztig-Vogan bijection, and Roman was first to prove it. Knowing that my work enriched their understanding of a phenomenon they discovered is an honor that will last me a lifetime.

I also want to thank my undergraduate research advisors, Victor Reiner and David

Speyer. The projects I tackled while in college - on minuscule posets and plethysm coefficients - were not only rewarding in their own right, but, more so than any classwork, convinced me I could contribute to the mathematical enterprise.

During my undergraduate years at MIT, several faculty members were generous enough to share their perspectives on mathematics with me. I want to thank in particular Steven Kleiman and Michael Sipser.

I made it to MIT because the spark of interest I showed for mathematics as a

child was kindled into a flame of passion. Robert Arrigo and Zuming Feng were not only masterful teachers; they were inspirational mentors.

Most important: my parents Andrew and Susie Rush; my brother Jonathan Rush; my roommates Alexander Remorov, XiaoLin Shi, and Hunter Spink, and all my friends inside and outside academe. The day may come when I've forgotten the math

I knew in graduate school, but I'll still remember the times we spent together.

Finally, Emily, who made my soul sing. My mind had but to follow.

My late grandfather John W. Chun emigrated from Korea and fell in love with my

grandmother and American literature. He earned a doctorate in English at the Ohio State University, becoming the first member of my family to be awarded a Ph.D. This dissertation is dedicated to his memory, fondly and with reverence.

Throughout my graduate studies, I was supported by the US National Science Foundation Graduate Research Fellowship Program.

Overview

In 1989, Lusztig concluded his landmark four-part study of cells in affine Weyl groups

[9, 10, 11, 121 with an almost offhand remark:

"... we obtain a (conjectural) bijection between Xdom and the set of pairs

(u, p), (up to G-conjugacy) with u E G unipotent and p an irreducible

representation of ZG(u)."

By Xdom, Lusztig meant the set of dominant weights of a connected complex

reductive algebraic group G. (We refer to this set as A+.) We denote by Q the set of pairs (C, V), where C C G is a unipotent conjugacy class and V is an irreducible

representation of the centralizer ZG(u) for u E C, which is uniquely determined by C

up to inner isomorphism.

So elementary an assertion was Lusztig's claim of a bijection between A+ and Q that its emergence from so deep an opus was in retrospect an obvious indication that the close connection between the sets in question transcends the setting in which it was first glimpsed.

Indeed, Vogan's work on associated varieties [141 led him to the same supposition only two years later. Let g denote the Lie algebra of G, and let )1* denote the nilpotent cone of the dual space g*. Fixing a compact real form K of G with Lie algebra t, let C be the category of finitely generated (S(g/t), K)-modules for which each prime ideal in the support corresponds under the Nullstellensatz to a subvariety of (g/t)* C g* contained in n*. In 1991, Vogan [14] showed that Q - in an alternate incarnation as the set of pairs (0, V), where (9 C 17* is a coadjoint orbit and V is an irreducible

group Ko(Q). That A+ also indexes such a basis pointed to an uncharted bijection. Further evidence for the existence of what has come to be known as the Lusztig-Vogan bijection was uncovered by Ostrik [13], who was first to consider Q and A+ in the context in which the conjecture was ultimately confirmed by Bezrukavnikov [3]

-that of the equivariant K-theory of the nilpotent cone of g. Let 1? denote this nilpotent cone. Ostrik examined (G x C*)-equivariant coherent sheaves on fl. Subsequently, Bezrukavnikov examined G-equivariant coherent sheaves on n and proved Lusztig and Vogan's claim.

Let D := Db(CohG(fl)) be the bounded derived category of G-equivariant coherent sheaves on n. Bezrukavnikov [31 showed not only that Q and A+ both index bases for the Grothendieck group Ko(O), but also that there exists a bijection 7: Q -+

A+ uniquely characterized by the following property: For any total order < on A+

compatible with the root order, if < is imposed on Q via -1, then the change-of-basis matrix is upper triangular.

In his proof, Bezrukavnikov did not construct '}. Instead, he exhibited a t-structure

on D, the heart of which is a quasi-hereditary category with irreducible objects in-dexed by Q and costandard objects inin-dexed by A+. This entailed the existence of -Y, but left open the question of how -y is computed. 1

In his 2001 doctoral thesis [1], Achar set G := GL(C) and formulated algorithms to compute inverse maps Q -* A+ and A+ -+ Q that yield an upper triangular change

of basis in Ko(E). Then, in a follow-up article [2], he showed that his calculations carry over to Ko(O) and therefore that his bijection agrees with Bezrukavnikov's.

Achar's algorithm for -y- is elegant and simple. Unfortunately, his algorithm for y is a series of nested while loops, set to terminate upon reaching a configuration satisfying a list of conditions. Progress is tracked by a six-part monovariant, which is whittled down as the algorithm runs. Achar [1, 2] proved that his algorithm halts on every input after finitely many steps. But it does not directly describe the image of a given pair ((, V) C Q.

'In type A, the existence of the Lusztig-Vogan bijection also follows from Xi's work on the based ring of the affine Weyl group [151, in which he proved a more general conjecture of Lusztig [12].

In this article, we present an algorithm that directly describes the terminal con-figuration returned by Achar's algorithm on an input in Q, bypassing all of Achar's while loops and obviating the need for an accompanying monovariant. The upshot is a combinatorial algorithm to compute y for G = GL,(C) that encompasses and

expedites Achar's algorithm and holds the prospect of extension to other classical groups.2

2

In section 7, _{we present an analogous algorithm which we conjecture to compute -y for even}

Contents

1 Introduction 19

1.1 Sheaves on the nilpotent cone . . . . 19

1.2 The nilpotent cone of gl . . . . 22

1.3 Sheaves on the nilpotent cone of g1 . . . . 27

1.4 The Lusztig-Vogan bijection for

GL

1.5 O utline . . . . 32

2 The Algorithm, Integer-Sequences Version 35 2.1 O verview . . . . 35

2.2 Functions . . . . 37

2.3 The algorithm . . . . 39

2.4 Exam ples . . . . 40

3 Weight Diagrams 49 4 The Algorithm, Weight-Diagrams Version 55 4.1 The algorithm . . . . 55

4.2 P roperties . . . . 62

5 Proof of Theorem 4.2.5 67 6 Proof of Theorem 1.5.1 83 7 The Symplectic Group 109 7.1 The nilpotent cone of SP2n . . . . 109

7.1.1 Even orbits, odd parts . . . .

7.1.2 Even orbits, even parts . . . .

7.2 Weights and representations . . . .

7.2.1 Preliminaries on symplectic groups

7.2.2 Even orbits, odd parts . . . .

7.2.3 Preliminaries on orthogonal groups 7.2.4 Even orbits, even parts . . . .

7.3 The algorithms . . . .

7.3.1 Even orbits, odd parts . . . .

7.3.2 Even orbits, even parts . . . .

. . . . 111 . . . . 112 . . . . 113 . . . . 113 . . . . 115 .. .... 117 . . . . 121 . . . . 123 . . . . 123 . . . . 126

List of Figures

1-1 The Young diagram of a, colored by rows . . . . 25

1-2 The Young diagram of a, partitioned by distinct parts . . . . 25

1-3 The Young diagram of a, colored by columns . . . . 26

1-4 The Young diagram of a, partitioned by distinct parts and colored by colum ns . . . . 26

2-1 The Young diagram of [2,1] . . . . 41

2-2 The Young diagram of [4,3, 1] . . . . 44

3-1 A diagram pair of shape-class [4, 3, 2, 1, 1] . . . . 51

4-1 A diagram after steps 1 and 2 . . . . 58

4-2 The diagram pair obtained from the first branch . . . . 60

4-3 The diagram pair obtained from the recursion tree . . . . 60

4-4 The diagram pairs obtained from the first and second branches . . . . 61

connected complex reductive algebraic group

Lie algebra of G

nilpotent cone of g

bounded derived category of G-equivariant coherent sheaves on 71

nilpotent element nilpotent orbit of X stabilizer of X

pair consisting of nilpotent orbit Ox and irreducible Gx-representation V

intersection cohomology complex associated to (Ox, V) equivalence classes of pairs (Ox, V)

complex associated to weight A via Springer resolution weight lattice of G

dominant weights of G Lusztig-Vogan bijection

complex associated to weight A via T*(G/P) - 0

partition associated to X

distinct parts of oz with multiplicities reductive part of Gx

conjugate partition to a

parabolic subgroup associated to X Levi factor of Px

Index of Notation

x

'D

X

Gx

(Ox, V)

[a* .

7

x9a X, V(v('),-.,v(t)) [vi, .... ,vi] Ga Gred V(a,v) Pa La AZ WAJ WA Wa Pa W dom(p) QC, A a, %t(a, v) A(a, v) J(a, v) dom(t)

p(a,

v,

C

p

Levi subgroup of Lx containing Gd representative element of Ox

Ox

irreducible GLa,-representation with highest weight v(t)

y"(1) 0 . .. Z V(t)

integer sequence Gxo

Ga-representation arising from

v(v(l).,(m))

Px Lx,,

dominant weights of La

irreducible GLa*-representation with highest weight A7

WvA Z... MWAs

Weyl group of La

half-sum of positive roots of L, Weyl group of G

unique dominant weight of G in W-orbit of p dominant integer sequences with respect to a

dominant weights p of L, such that V(a'v) occurs in decomposition of W" as direct sum of irreducible Gred-representations

integer-sequences version of algorithm Achar's algorithm

weight-diagrams version of algorithm

rearrangement of entries of t in weakly decreasing order candidate-sum function

candidate-ceiling and -floor functions strict weak order

1.2 1.3 1.3 1.3

1.3

1.3

1.3

1.5

1.5

V1 (a, v, P)

V1_(OZ,

X

Ii

#(X,i)

9(a, t)(i)

(a,/3)

Fv

ranking-by-ceilings and -floors functions permutation

column-ceilings and -floors functions

iterative integer-sequences version of algorithm weight diagrams of shape-class a

weight diagram

zth entry from top in Jth column of X

map D, -+ Da, diagram pair map D, -+ Q,,

map Da A map D, A+

weight diagrams with f rows

entry of X in ith row and Jth column row-survival function

number of branches

number of rows surviving into xth branch number of boxes in ith row of X

sum of entries in ith row of X column-reduction operation diagram-concatenation operation row-partition function symplectic partition irreducible SP2-representation pv(l) M ... Z Fv(rn) Sp SI,

Gx-representation arising from Fp(v"'v('n)_Sp irreducible Ob-representation

Gx-representation arising from F(/"v()" algorithm for even orbits, odd parts

2.2 2.2 2.2 2.2 3 3 3 3 3 3 3 3 4.1 4.1 4.1 4.1 4.1 4.2 4.2 5 5 6 7.1 7.2 7.2 7.2 7.2 7.2 7.2 7.3

ranking-by-zero function column-zero function

algorithm for even orbits, even parts unspooling functions .Ro(a, v) 'Vo(a, v, a) 9F (V) 7.3 7.3 7.3 7.3

Chapter 1

Introduction

1.1

Sheaves on the nilpotent cone

Let G be a connected complex reductive algebraic group with Lie algebra 0. An element X E g is nilpotent if X E [g, g] and the endomorphism ad X: g -- g is nilpotent. The nilpotent cone 17 comprises the nilpotent elements of g. Since Ti is a subvariety of g (cf. Jantzen [61, section 6.2), we may consider the bounded derived category D := Db(CohG(Tl)) of G-equivariant coherent sheaves on T7.

Let X

c

Write Gx for the stabilizer of X in G. To an irreducible representation V of Gx corresponds the G-equivariant vector bundle

E(gx ,v) G x Gx V -* OX

with projection given by (g, v) + Ad(g)(X). Its sheaf of sections S(ox,v) is a G-equivariant coherent sheaf on Ox. To arrive at an object in the derived category D, we build the complex S(OxV)[I dim Ox] consisting of 8(ox,v) concentrated in degree

- dim Ox. Then we set

where

jV,

Let Qpre be the set of pairs {(Ox, V)}xEn consisting of a nilpotent orbit Ox

and an irreducible representation V of the stabilizer Gx. We assign an equivalence relation to Qpre by stipulating that (Ox, V) - (Oy, W) if there exists g E G and

an isomorphism of vector spaces r: V -- W such that Ad(g)X = Y and the group

isomorphism Ad(g): Gx -+ Gy manifests 7r as an isomorphism of Gx-representations. Note that (Ox, V) ~ (0y, W) implies Ox = (y and E(ox,v) ~ E(oy,w). Thus,

the map associating the intersection cohomology complex IC(gx,v) in 0 to the equiv-alence class of (Ox, V) in Qpre is well-defined. Set Q := Qp/ ~. Then Q indexes the

family of complexes {IC(0x,V)}(Ox,V)EQ. (The notation (Ox, V)

C

On the other hand, weights of G also give rise to complexes in D. To see this, let B be a Borel subgroup of G, and fix a maximal torus T C B. A weight A E Hom(T, C*) is a character of T, from which we obtain a one-dimensional representation CA of B

by stipulating that the unipotent radical of B act trivially. Then

LA := GxB CA - G/B

is a G-equivariant line bundle on the flag variety G/B. Its sheaf of sections A is a G-equivariant coherent sheaf on G/B which may be pulled back to the cotangent

bundle T*(G/B) along the projection p: T*(G/B) -+ G/B.

From the Springer resolution of singularities 7r: T*(G/B) -+ 72, we obtain the direct image functor 7r., and then the total derived functor R7r*. We set

AA := R7Tr*p*XA 0.

Let A := Hom(T, C*) be the weight lattice of G, and let A+

c

dominant weights with respect to B. The family of complexes {AA}AEA+ is sufficient to generate the Grothendieck group Ko(v), so it is this family which we compare to

Theorem 1.1.1 (Bezrukavnikov [3]). The Grothendieck group KO(D) is a free abelian

group for which both the sets {[IC(oxv)]}(ox,v)cQ and {[AA]}AEA+ form bases. There exists a unique bijection -y: Q -+ A+ such that

[IC(,y)] E span{[AA : A < y(Ox, V)},

where the partial order on the weights is the root order, viz., the transitive closure of the relations v < w for all v, w

c

B.

Furthermore, the coefficient of [A7((ox,v)] in the expansion of [IC(oxvy) is k1.

The association of the complex A, to the weight A evinces a more general con-struction of objects in 0 that is instrumental in identifying the bijection 'y. Let P D B be a parabolic subgroup, and let Up be its unipotent radical. Denote the Lie algebra of Up by up. The unique nilpotent orbit ( for which 0

n

subset of up is called the Richardson orbit of P, and there exists a canonical map 7r: T*(G/P) - 0 analogous to the Springer resolution.

Let L be the Levi factor of P that contains T. From a weight A E A dominant

with respect to the Borel subgroup BL := B n L of L, we obtain an irreducible L-representation WA with highest weight A, which we may regard as a P-L-representation

by stipulating that Up act trivially. Then

MsA := G xp WA G/P

is a G-equivariant vector bundle on G/P. Pulling back its sheaf of sections MhA to

the cotangent bundle T*(G/P) along the canonical projection p: T*(G/P) -+ G/P, and then pushing the result forward onto 7, we end up with the complex

A:= R7rp*MTA E 0.

Note that the Richardson orbit of B is the regular nilpotent orbit (9reg, uniquely characterized by the property (greg = 11. The Levi factor of B containing T is T itself.

Thus, for all A E A, the complex AB is defined and coincides with AA, meaning that the above construction specializes to that of {AA}AEA, as we claimed.

1.2

_{The nilpotent cone of gfn}

Henceforward we set G := GLs(C). Then g =gr(C). Let X c g be nilpotent.

The existence of the Jordan canonical form implies the existence of positive integers a, > . - aj summing to n and vectors vI,.. . , vf such that

C = span{Xjvi :1 <i < ,0 j < ai - 1}

and X'ivi = 0 for all i (cf. Jantzen [61, section 1.1).

Express the partition a :=[a... , a] in the form [k", ... , ka- ], where ki > ... >

km are the distinct parts of a and at is the multiplicity of kt for all 1 < t < m. Let V be the at-dimensional vector space spanned by the set {vi : ai = kt}. Define a map

px: GL(VI) x - - -x GL(V) -+ Gx

by px(gi, . . ., g.)(Xivi) : Xigtvi for vi E Vt.

Note that ypx is injective. Let Ged be the image of px, and let Rx be the unipo-tent radical of Gx. From Jantzen [61, sections 3.8-3.10, we see that Gd is reductive and Gx = GrdRx. Since Rx acts trivially in any irreducible Gx-representation, specifying an irreducible representation of Gx is equivalent to specifying an irre-ducible representation of GXd, which means specifying an irreducible representation

of GLai...,a. := GLai x ... x

GLa.-Let a* = [a*,..., a*] be the conjugate partition to a, where s := a1. For all

1

j

ai >

j},

Define subgroups Lx c Px C G by

and

Lx :={g E G : g (V(j)) = V(j) for all 1

<j

Since Px is the stabilizer of the partial flag

{o} C 01) c .. -c V(s) = Cn

it follows immediately that Px C G is a parabolic subgroup and Lx c Px is a Levi factor. Furthermore, the Richardson orbit of Px is none other than (x (cf. Jantzen

[6], section 4.9). For general G, this implies that the connected component of the

identity in Gx is contained in Px. In our case G = GL, the conclusion is stronger:

Gx

c

The claim Gx c Px follows from the observation that Vj) is the kernel of Xi for all 1 <

j

Since XktV

C

m kt

C = X

t=1 j=1

is a refinement of the decomposition C = = V(j). Set

L := {g E : g(XkV) - Xktj- for all 1 < t < m, 1 j < kt}.

Then Lx C Lx, and the inclusion Gid c Lrf follows directly from the definition of

Let Xx be the isomorphism

In kt

-+ fJ

fi

given by g '-4 ji(gXk,-1V, .... gv), and let Ox be the isomorphism

Lx -+ GL(V(1)) x ... x GL(V(s))

given by g -+ (glv(1), ...

,gv(s))-From the analysis above, we may conclude that the composition

bxpx: GLai.am

factors as the composition

m

Xxx: GLai,...,a, - (GLat )kt

t=1

(which coincides with the product, over all 1 < t < m, of the diagonal embeddings GLat + (GLat)k,), followed by the product, over all 1 < j < s, of the inclusions

GLat a GL, . This description of Oxyox allows us to detect the appearance of certain [IC(oxv)] classes in the expansion on the Q-basis of a complex arising via

the resolution T*(G/Px) -÷ (x.

Example 1.2.1. Set n:= 11. Then G = GL1 1. Set

0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 X :0 0 1 0 0 0 0 1 0 0 0 0

The partition encoding the sizes of the Jordan blocks of X is a = [4, 3, 2, 1, 1]. The Young diagram of a is depicted in Figure 1-1. Each Jordan block of X corresponds to a row of a.

Figure 1-1: The Young diagram of a, colored by rows

We may express a in the form [4', 31, 21, 12], where 4 > 3 > 2 > 1 are the distinct parts of a. Then G d is the image under the isomorphism cpx of

GL1 x GL1 x GL1 x GL2

-Each factor of the preimage corresponds to a distinct part of a (cf. Figure 1-2).

Figure 1-2: The Young diagram of a, partitioned by distinct parts

The conjugate partition a* is [5, 3, 2, 1]. The isomorphism ox maps Lx onto

GL5 x GL3 x GL2 x GL1.

Each factor of the image corresponds to a column of a (cf. Figure 1-3).

The group Lif lies inside Lx and contains Gd. The isomorphism xX maps Lif isomorphically onto

)1-Figure 1-3: The Young diagram of a, colored by columns

Each factor of the image corresponds to an ordered pair consisting of a distinct part of a and a column of a (cf. Figure 1-4).

Figure 1-4: The Young diagram of a, partitioned by distinct parts and colored by columns

The composition

)xpx: GL₁,₁,₁,₂ - GL₅ ,3,2,1

factors as the product of diagonal embeddings

Xx~ox: GL1,1,1,2 -÷ (GL1)4 x (GL1)3 x (GL1)2 x (GL2) 1, followed by the product of the inclusions

1.3

_{Sheaves on the nilpotent cone of gf,}

Let el, . .. , en, be the standard basis for C". From the nilpotent orbit Ox, we choose the representative element X,,

c

ei H-+ 0

for all 1< i < a* and

~-2+

for all2j s 1

Example 1.3.1. Maintain the notation of Example 1.2.1. Then

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0

To see that X,, E Ox, let g E G be given by Xi-jvi F- eat+... +a;1+i, and observe

that Xe = gXg- 1. Thus, N =

Uc.

For each factor GLat of GLai...,am, we identify the weight lattice with the char-acter lattice Zat of the maximal torus (C*)a, of invertible diagonal matrices, and we

assign the partial order induced by the Borel subgroup of invertible upper triangular matrices. Then the isomorphism classes of irreducible GLal.am-representations are indexed by m-tuples of integer sequences (v(1),... , v(m)) such that v(t) is a domi-nant weight of GLa, for all 1 < t < m. The m-tuple (v(1),.. , v(m)) corresponds to the representation

v(v(l).--V(M)) __ VVM .. M)VM

where VU() denotes the irreducible GLagrepresentation with highest weight v(t). We say that an integer sequence v = [/i,... , ve) is dominant with respect to a if ai = ai+1 implies vi > vi+1 . Note that the dominance condition holds precisely when v is the concatenation of an m-tuple of dominant weights (v(1), .. ., v(m)). For such v, we denote by V(cv) the representation of G, := Gx, (or of Gred := Gre, depending on context) arising from the representation VO"'),-v(m)) of GL,.,am. Via the association

(a, v) -

((9,

we construe Q as consisting of pairs of integer sequences (a, v) such that a

[a1 . , ael is a partition of n and v = [vl,. . ., vj] is dominant with respect to a. Let B C G be the Borel subgroup of invertible upper triangular matrices, and let T C B be the maximal torus of invertible diagonal matrices. The weight lattice

A = Hom(T, C*) 2 Z' comprises length-n integer sequences A = [A,,... , An]. Those

weights A E A which are weakly decreasing are dominant with respect to B and belong to A+.

Set P: Px, and L, := Lx,. Then

Pa = g E G : ge+--+;1+ E span{e+,..., e+---+;

and

Lc =

{g

lattice of La. Given a weight A E A, let Ai be its restriction to the factor GLa*

of La GLa,.. This realizes A as the concatenation of the s-tuple of integer sequences (A,... ,A). If Ai is weakly decreasing for all 1

j

with respect to the Borel subgroup B, := BL,, in which case A belongs to A+, the set of dominant weights of La with respect to Ba. For A

E

the irreducible GLa*-representation with highest weight Ai, and we set

W11:= WA M .. w- A

which indeed has highest weight A.

We rely on the complexes A := Ap- associated to weights A E A+ to interpolate between the Q- and A+-bases for Ko(O). Weights of La are also weights of G, so it is reasonable to expect that the expansion of [Ac] on the A+-basis be easy to compute. On the other hand, representations of La restrict to representations of Gd, and it turns out that this relationship lifts to the corresponding objects in D. The following results of Achar [21 encapsulate these statements formally.

Lemma 1.3.2 (Achar [21). Let (a,v) E Q, and let A E A+. Suppose that V(a'7) occurs

in the decomposition of the La-representation WA as a direct sum of irreducible G.da representations. Then, when [Ac] is expanded on the Q-basis for Ko(D), the coefficient

of [ICa,v1 is nonzero.

Lemma 1.3.3 (Achar [21). Let Wa be the Weyl group of La, and let pa be the half-sum

of the positive roots of La. For all A

c

[Ace =] (-1)w[AA+pQ.wpJ.

wEW,

Let W be the Weyl group of G, and, for all p

c

dominant weight in the W-orbit of p. When [A,,] is expanded on the A+-basis for Ko(O), the coefficient of [AA1 is zero unless A < dom(p) (cf. Achar [2], Proposition 2.2). Thus, if p E A+, it follows from Lemma 1.3.3 that [A'] E {span[AA] : A < dom(p + 2

Let Q, be the set of all dominant integer sequences v/ with respect to a. Given v E Q,, set

A

On input (a, v), our algorithm finds a weight p E A+ such that

I|p+

+

The intuition behind this approach is straightforward. For all p

C

expansion of [Ac] on the -basis takes the form

[A"]

=

(v(av),

where < denotes the dominance order on partitions of n. On the other hand, the expansion of [Ac] on the A+-basis takes the form

[Ac] = [Adom(jL+2p,)] + CA [AA].

A<dom(p+2pa)

We compare the equations. There is a single maximal-weight term in the right-hand side of the second equation. It follows that there is a single maximal-weight term in the expansion of the right-hand side of the first equation on the A+-basis.

By Theorem 1.1.1, the maximal weight must be '-(a, v) or among the sets {'Y(a, v) :

v 4 1/} and {y(B, ) : 3 < a}. In the former case, we may conclude immediately that

'y(a, v) = dom(p + 2p,). It turns out that mandating the minimality of |p + 2pafj suffices to preclude the latter possibility.

1.4

The Lusztig-Vogan bijection for GL

2

Set n := 2. Then G = GL2. The weight lattice A comprises ordered pairs [Al., A21 E

Z2, and A+ = {[A, A2] E Z2: A, > A2

}.

The variety n c g is the zero locus of the determinant polynomial. Each matrix 'This follows from Claim 2.3.1 in [1], except that y is defined differently. In [2], Theorem 8.10, Achar shows that the bijection -y constructed in [11 coincides with the bijection in Theorem 1.1.1.

0 1

of rank 1 in g is similar to ,so Tn is the union of

0 0 the G-orbit of

0

0

0

0

(the zero orbit) and

0

I

0

To the zero orbit corresponds the partition [1,1]. Hence

Q[1,1 =

{[v, v

Note that G _[1,1=1 = Lg,,

_~

and A,] = f[pl, A21 E Z2

: P1 1> P2}

For all [,P, P2] E A+, the irreducible L [,,]-representation Wl1'J12l is isomorphic

as a G ed-representation to V([ll][111,A2). Thus, for all [vi, v2 E Qp, ,

A , = [iv2}

Our algorithm sets [PI, P21 := [v1, v21.

On the Q-basis, [A' ] expands as

A ' =~j [-[CQ1,11,[V,1 )].

Since W,] = W 6 2 and p[] = [1, -j], it follows that [A ] expands on

the A+-basis as

A'l = -

[A

Hence y([l, 11, [v, v2]) = [vi + 1, v2 - 1] = dom([pi, [21 + 2p[,]), which confirms

that the output is correct.

We turn our attention to the regular orbit, to which corresponds the partition [2]. Recall that G,d a GL, and L[_[2] = 2] 2 GL, x GLj. Hence

Q[2] = {[V1] E ZI and A+ = {[PI, P2] E Z2

Furthermore, the composition @xpx of the isomorphisms

SOx[

GL1. For all [l, P21 E A+, the irreducible L[2]-representation W[I1P2] is isomorphic as a G ed -representation to VQ2,[Pl+A21.

Thus, for all [v1] E Q[21

A = {[[2,1p21 E A l:] + A2 = V1}.

Our algorithm sets

[P1, p2] = On the Q-basis, [A 21 ] expands as

IA [2 IC([21,[,11)] + C[17,21 [IC([1,1,[ 1, 21)

-lest2[2]

Since W[2] is trivial and P[2] = [0,01, it follows that [A [ expands on the A+-basis as

A l=A

From our analysis above, we know that -y([l, 11, [1, 21) = [&l + 1, '2 - 1], so there

cannot exist [l1, 21 E Q[1,1] such that Y([1, 1], [6, 2l) =[F"1, L"11 I Hence y([2], [vi]) = [F"j1, ["#J] = dom([/1i, P2] + 2P[2]).

2

1.5

Outline

The cynosure of this article is the integer-sequences version of our algorithm, which admits as input a pair (a, v) E Q and yields as output a weight %(a, v) E A+. The output, which consists of a weight of each factor of LQ, is obtained recursively: The weight of the first factor GLa; is computed; then the input is adjusted accordingly, and the algorithm is called on the residual input to determine the weight of each of the remaining factors.

The algorithm design is guided by the objective of locating Q(a, v) in A+, and 2

keeping

II2t(a,

I1%2(a, v) + 2p.II = min{jIp + 2paII : p c A+,}.

Our main theorem is the following.

Theorem 1.5.1. Let (a, v) E Q. Then -y(a, v) = dom(Qt(a, v)

+

We prove the main theorem via a combinatorial apparatus introduced by Achar

[1, 2] - weight diagrams. A weight diagram X of shape-class a encodes several integer sequences, including a weight h(X)

c

outputs a weight diagram A(a, v) of shape-class a such that hA(a, v) E A+, and

||hA(a, v) + 2pc|| is minimal (cf. [2], Corollary 8.9). Achar's conclusion (cf. [2],

Theorem 8.10) is that Theorem 1.5.1 holds with hA(a, v) in place of Q(a, v).

The minimality of I|hA(a, v)+2p, I is basic to Achar's algorithm, which maintains a candidate output X at each step, and performs only manipulations that do not increase IIhX + 2p, 11. In contrast, 2t(a, v) is computed one entry at a time. The

minimality of 12t(a, v)

+

with the main theorem, by comparison of our algorithm with Achar's.

To connect 2[ to A, we introduce a third algorithm A, built with the same tools as %, but configured to output weight diagrams rather than integer sequences.3 For all (a, v)

c

The integer sequences version bears little ostensible resemblance to Achar's algo-rithm. However, despite the architectural differences, the relationship between the weight diagrams version and Achar's algorithm is impossible to miss: A(a, v) always exactly matches A(a, v). Hence hA(a, v) = hA(a, v). The main theorem follows immediately.

In summary, the algorithm 2t is a bee-line for computing -y, akin to an ansatz.

'A actually outputs pairs of weight diagrams, so what we refer to in the introduction as A (a, v) is denoted in the body by piM(a, v).

Heuristically, we presume the main theorem holds "because" Q(a, v/) E A+, such that I12(a, v)

+

The rest of this article is organized as follows. In Chapter 2, we present the integer-sequences version of our algorithm, along with several example calculations. We devote Chapters 3-6 to a proof of the main theorem.

In Chapter 3, we define weight diagrams. A weight diagram of shape-class a encodes an element each of Qa, A+, and A+, and we give a correct proof of Proposition 4.4 in Achar [2] regarding the relations between the corresponding objects in D.

In Chapter 4, we present the weight-diagrams version of our algorithm and delin-eate its basic properties. In Chapter 5, we follow up on our work in Chapter 4 with a proof that Q(a, v) = hA (a, v).

In Chapter 6, we state Achar's criteria for a weight diagram to be distinguished. Then we prove that A outputs a distinguished diagram on any input. As we explain, this implies that the diagrams A(a, v) and A(a, 1/) are identical for all (a, V) E Q.

Finally, in Chapter 7, we fulfill our promise to (attempt to) reformulate our algo-rithm for other classical groups. Focusing on G = Sp2,(C), we discuss the structure

of the nilpotent cone ii of g = s2n, and, for each even orbit Ox c Ti, we find a

parabolic subgroup Px for which the Richardson orbit is Ox. Then we present a conjectural analogue of 2t for computing -y on pairs (Ox, V) E Q such that Ox is even.

Chapter 2

The Algorithm, Integer-Sequences

Version

2.1

Overview

Fix a partition a = [ai, ... , ac] with conjugate partition a* = [*, . .., *]. Given an integer sequence t of any length, let dom(t) be the sequence obtained by rearranging the entries of t in weakly decreasing order. (This is consistent with the notation of section 1.3, for dom(t) e Wt

n

c

Let v E Q,. On input (a, v), our algorithm outputs an integer sequence p of length n satisfying the following conditions:

" p is the concatenation of an s-tuple of weakly decreasing integer sequences (P, 1 . . ,Ps) such that pi is of length a* for all 1 < j < s;

" There exists a collection of integers {vi} 1 i<f such that 1 J Qi

1

i 4' + +. I Vi'Cai

for all 1< i e and piL == dom([vi,, . . . , v*,j]) for all 1 <j <.

Recall that the former condition indicates P E A+. The latter condition implies

Although we could construct a collection {vi,} 1 i5 such that vi = vi,1+- - +vi,as

for all i and obtain p as a by-product (by setting p' : dom([vi,j,. . . , va*,!]) for all

j),

Remark 2.1.1. Were we seeking to minimize |p, it would suffice to choose, for all i, integers vi1,... , vi,i E

{

}

1<j<aj

induce the output p.

However, our task is to minimize ||p + 2p,11, in which case we cannot confine each vij to the set

{

straight-1<j<ai

away, and learning the order (from greatest to least) of the entries in each sequence

[vi,,., vaj post hoc, risks needlessly inflating

S

Sa* - -a* 2

dom([v1,j, . . . , va,,]) + 2 2 2 ' ' 2 Il |p + 2pal 12

j=1

But how can we know what the order among the integers vij, . . . , v,* will be before their values are assigned? Our answer is simply to stipulate the order, and pick values pursuant thereto - by deciding o-, then pi, and setting [vij, ... , vag, ] :=

The algorithm runs by recursion. Roughly: c- is determined via a ranking func-tion, which compares candidate sums, each measuring how the addition of 2pa to p

might affect a subset of the collection {Vi} 1<;< , subject to a hypothesis about oa.

After a' is settled, the corresponding candidate sums are tweaked (under the aegis of a column function) to compute pl. Then pl is "subtracted off," and the algorithm is called on the residual input v', defined by v := vi - p l, returning p2 ...*

'See section 2.4 for an example in which there exists i, j such that vij must not belong to

2.2

Functions

Describing the algorithm explicitly requires us to introduce formally several prelimi-nary functions.

Given a pair of integer sequences

(a, V) = ([ai, .. , ai], [Vi, ,

v)

an integer i E {1, .1.., f}, and an ordered pair of disjoint sets (Ia, Ib) satisfying Ia UIb =

{1,... ,

\

C(a, ,i, Ia, Ib) : 1i -

E

+

jCIa jEIb

We also define the candidate-ceiling and candidate-floor functions C_ 1 and C1

from the candidate-sum function:

e-1I(a,

a ,1

C(a,

V,

C1 (C, 1/, z, I l,

Ib):-We proceed to define the functions employed in determining a-. Let 9(f) be the set of strict weak orders p on the set {1,. . . ,

}.2

ranking-by-floors algorithms R_1 and R1 compute functions Nf x Z x 9(f) -+ &j iteratively

over f steps.

To illustrate, say R_1(a, v, p) = -. On the ith step of the ranking-by-ceilings algorithm, o- 1(1),... , c- 1(i - 1) have already been determined. Set

Ji := {-(-

1)}

{1,

Let Ti7) c Jj be the set comprising the maximal elements of the restriction of p to Jj.

2

A strict weak order is a strict partial order such that incomparability is an equivalence relation; it amounts to a total order on the equivalence classes.

Then a-1(i) is designated the numerically minimal

j

c

(e_1 (a, I, ,

J, j

{j}),

is lexicographically maximal.

Analogously, say Zi(a, v, p) = a. On the zth step of the ranking-by-floors

algo-rithm, 9-1(f),..., oa(C - i -+ 2) have already been determined. Set

Ji := { 1(e), ... , a-1

(e - i+ 2)} and Jj := {1, ... , } \ Ji.

Let Mi c Jj[ be the set comprising the minimal elements of the restriction of p to Jj.

Then a--1 ( - i + 1) is designated the numerically maximal I E Mi among those for

which

(el1(a ,

ji,

\{jJi),

- aj,

is lexicographically minimal.

Remark 2.2.1. A permutation a E (3 induces a total order p, on the set {1,...,7}

given by {o-1(1) > - > a--1 (f)}. If l 1(a, v, p) = a, then p, is compatible with p.

Equipped with the ranking-by-ceilings and ranking-by-floors algorithms, we are ready to define the functions employed in computing pl: the column-ceilings and

column-floors algorithms U-1 and U1. Both algorithms are iterative with

e

compute functions Ne x Z x 6 - Zom, where Zom c Z' denotes the subset of

weakly decreasing sequences.

Say U-1(a, v, a-) = [t1,..., Le]. On the zth step of the column=ceilings algorithm, ti,..., ti_ 1 have already been determined. Then

ti

: = (?_,(a, , a-,(1), a--{,

...

, i - 1}, o{i

+ 1,...

, })

-

+2i - 1

unless the right-hand side is greater than ti_ 1, in which case t :=ti1.

algorithm, Le, .. . , f-i+2 have already been determined. Then

+: (a, V, e- 1( - i + 1), o-{1, ..

,

unless the right-hand side is less than Qe-i+2, in which case tei+1 :=-i+2.

2.3

The algorithm

We assemble these constituent functions into a recursive algorithm % that computes a map

Y x e x 9() x { 1} -+ Z , where yn, denotes the set of partitions of n with f parts.

On input (a, vi, p, c), the algorithm sets

el = Ie~a vp).

Next it sets

p1 :'U(a, 1, u1)

Then it determines the residual input. It sets

a' : [a , . . . , It defines v' by

v: - p 1ig)

for all 1 < i < a*.

It defines p' to be the restriction of p to {1,... , a2}, refined by the relations i >

j

for all 1 i,

j

Finally, the algorithm prepends pl to %(a', v', p', -e) and returns the result. Let 0 E 9(f) be the empty order. Whenever we write %(a, v), we refer to

!Qt (a, V,

0,

The use of recursion makes our instructions for computing Q(a, v) succinct. At

the cost of a bit of clarity, we can rephrase the instructions to use iteration, and thereby delineate every step in the computation.

Consider the algorithm 2

iter: Yf,e x ZI -_+ Zn defined as follows.

On input (a, v), it starts by setting a' := av, p : = 1 0, ( := RI(al, v1, p1), and p1 := 7-_(al , al).

Then, for 2

< j

* It sets a2 := [a*,. , a*

* It defines vi by vi :=v-3 for allli

" It defines p2 to be the restriction of pia to {1, .. , ac}, refined by the relations

i > i' for all 1 5 i, i' <" such that piil0() > 1W)l,) (unless the refinement is not a strict weak order, in which case p :=p-1

* It sets oa := IR_ (as, vi, p);

" _{It sets p := t(_1)j(a ,ViO 3).}

Finally, it returns the concatenation of (p2,...,

It should be clear that 2iter(a, v) agrees with Q(a, v), so we consider our definition

of Qfiter an alternate definition of a (which comes in handy in proving Theorem 4.2.5).

2.4

Examples

We study three examples. First, to illustrate the workings of the ranking functions, we consider the orbit (9[2,1]. Given v E Q[2,1], the algorithm makes exactly one meaningful

comparison - _{to determine whether o- is the trivial or nontrivial permutation in 62.} Second, to underscore the advantages of our approach, we consider an input pair (a, v) for which there exists only one collection {v,3} 15 s such that vij, E

{F1[

j

Last, we revisit the orbit O[4,3,2,1,1] featured in Example 1.2.1 and compute % on the input pair ([4, 3, 2, 1, 1], [15, 14, 9, 4, 4]), taken from Achar's thesis [1]. We also discuss the computation of t

triv, in which an iteration of the ranking procedure involves a nontrivial strict weak order; in particular, p3 is nonempty.

Example 2.4.1. Set a := [2, 1]. Then a* = _{[2, 1]. Reading Gd and LQ off the Young} diagram of a (cf. Figure 2-1), we see that Gre d [2,1] = GL, x GL1 and L[2,1] GL2 x GL1.

Figure 2-1: The Young diagram of [2, 1]

Note that

[2,= {[vi, v2] E Z2} and A, = {[A,, A2, A3] : A, > A2

Let v = [v1 , v2] E Q[2,1]. On input (a, v), the algorithm computes a' := R-1(a, v, 0).

Since a, > a2, the triple

(C_1(a, v, 1, 0, {2}), a1, v1)

is lexicographically greater than the triple

(C-

if and only if

C-1 (a, v, 1, 0, {2}) > C_1 (a, v, 2, 0, {1}). (2.4.1)

of the ranking-by-ceilings algorithm, (oa)~1(2) C {1, 2} \ {(- 1) 1(1)}, so a-' is the identity in

e

Evaluating the candidate ceilings, we find:

?-1(v, v,1,0, {2}) = _1 ([2, 1], [v1, v2], 1,0, {2}) =

FV

C-1

(a,

+

We treat each case separately.

1. Suppose vi 2v2.

The algorithm computes p' := U_ 1(a, v, -1). By definition,

S= C-1(, , 1, 0, {2}) - 1 =

F

Since

C_

v,

and F111 > v2, it follows that

Pi C1(a, v, 2, {1}, 0) + 1.

Hence

S= _1(, , f2,{1}, 0) + 1 = v2

Then the algorithm sets a' := [1]; it defines v' by p- - F i

11

+ 11

C

(a,

L

'

The algorithm computes p1 : l1-(a, v, crl). By definition,

S=

Since

i

+

2

and v2 ['1], it follows that

>

e-

{2}, 0) +

Hence

p2 = C-J(a, v, 1, {2}, 0)+1 =

Then the algorithm sets a' := [11; it defines v' by

- V Vi 11 Vi - 1 2 2 ' and it sets p' := 0

c

F

+1

Clearly, t(a', v', p', 1) = C1(', v1', 1, 0, 0) = v' = Hence We conclude that Qt([2, 1], [V1, v2]) = vi > 2v2 vi < 2v2- 1

Since P[2,1] = [1, - , 0], assuming Theorem 1.5.1 holds, we find

-y(2, 1], [1/1, Example 2.4.2. Set a := [3,2,2, 1]. [['+ ] , ["- ]lJ , v2 - 1] [V2+ 1, ["1[,11 [ ] -1J Then a* = [4, 3, 1]. > 2v2 < 2v2 - 1

Reading Gred and L, off the diagram of a (cf.

L, 2 GL4 x GL3 x GL1.

Figure 2-2), we see that G re ' GL1 x GL2 x GL1 and

I

Figure 2-2: The Young diagram of [4,3, 11

Note that Q =

{E

=

2w-J

I

L

Set v := [15, 8, 8, 4] G Q,. On input (a, v), the algorithm computes a := IR_1(a, v, 0) = 1234. Next it computes PI := 'L_(a, V,o1) = [4, 4,4,4]. Then it sets and p':= 0 C 9(3). a' := [2, 1, 1c, o[11, 4 4],

To finish off, it computes

% (a', v', p', 1) = [4, 4, 4, 7].

Thus,

%(a, v) = [4,4,4,4,4,4, 4,7].

Since

Pa = 2' -2'

17 0, -1,

01

assuming Theorem 1.5.1 holds, we find

(a, 1/) = [7, 7, 6, 5,4, 3, 2, 1]. Note that - = 5 and al 7/6) 7,1) V/A = =)4. a2 a3 a4

Therefore, if {vi} 1 i 4 C Z is a collection such that vij

c

{

z,3, then

/1,1 V1,2 = V1,3 =5

[i]H} for all