Affine Hecke Algebras and Orthogonal Polynomials

187 Pages·2003·2.31 MB·english

by  I. G. Macdonald

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CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK
157 Affine Hecke Algebras and
Orthogonal Polynomials

I.G.Macdonald
QueenMary,UniversityofLondon
Affine Hecke Algebras
and
Orthogonal Polynomials
  
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Contents
Introduction pagevii
1 Affinerootsystems 1
1.1 Notationandterminology 1
1.2 Affinerootsystems 3
1.3 Classificationofaffinerootsystems 6
1.4 Duality 12
1.5 Labels 14
2 TheextendedaffineWeylgroup 17
2.1 Definitionandbasicproperties 17
2.2 ThelengthfunctiononW 19
2.3 TheBruhatorderonW 22
2.4 Theelementsu(λ(cid:2)),v(λ(cid:2)) 23
2.5 Thegroup(cid:5) 27
2.6 Convexity 29
2.7 Thepartialorderon L(cid:2) 31
2.8 Thefunctionsrk(cid:2),rk(cid:2) 34
3 Thebraidgroup 37
3.1 Definitionofthebraidgroup 37
3.2 TheelementsYλ(cid:2) 39
3.3 AnotherpresentationofB 42
3.4 Thedoublebraidgroup 45
3.5 Duality 47
3.6 Thecase R(cid:2) = R 49
3.7 Thecase R(cid:2) (cid:4)= R 52
v
vi Contents
4 TheaffineHeckealgebra 55
4.1 TheHeckealgebraofW 55
4.2 Lusztig’srelation 57
4.3 ThebasicrepresentationofH 62
4.4 Thebasicrepresentation,continued 69
4.5 Thebasicrepresentation,continued 73
4.6 TheoperatorsYλ(cid:2) 77
4.7 ThedoubleaffineHeckealgebra 81
5 Orthogonalpolynomials 85
5.1 Thescalarproduct 85
5.2 Thepolynomials Eλ 97
5.3 Thesymmetricpolynomials Pλ 102
5.4 TheH-modules Aλ 108
5.5 Symmetrizers 112
5.6 Intertwiners 117
5.7 Thepolynomials P(ε) 120
λ
5.8 Norms 125
5.9 Shiftoperators 137
5.10 Creationoperators 141
6 Therank1case 148
6.1 BraidgroupandHeckealgebra(type A ) 148
1
6.2 Thepolynomials E 150
m
6.3 Thesymmetricpolynomials P 155
m
6.4 BraidgroupandHeckealgebra(type(C∨,C )) 159
1 1
6.5 Thesymmetricpolynomials P 160
m
6.6 Thepolynomials E 166
m
Bibliography 170
Indexofnotation 173
Index 175
Introduction
Overthelastfifteenyearsorso,therehasemergedasatisfactoryandcoherent
theoryoforthogonalpolynomialsinseveralvariables,attachedtorootsystems,
anddependingontwoormoreparameters.Atthepresentstageofitsdevelop-
ment,itappearsthatanappropriateframeworkforitsstudyisprovidedbythe
notionofanaffinerootsystem:toeachirreducibleaffinerootsystemSthereare
associatedseveralfamiliesoforthogonalpolynomials(denotedbyEλ, Pλ,Qλ,
P(ε) inthisbook).Forexample,when S isthenon-reducedaffinerootsystem
λ
ofrank1denotedhereby(C1∨,C1),thepolynomials Pλ aretheAskey-Wilson
polynomials[A2]which,asiswell-known,includeasspecialorlimitingcases
alltheclassicalfamiliesoforthogonalpolynomialsinonevariable.
Ihavesurveyedelsewhere[M8]thevariousantecedentsofthistheory:sym-
metricfunctions,especiallySchurfunctionsandtheirgeneralizationssuchas
zonalpolynomialsandHall-Littlewoodfunctions[M6];zonalsphericalfunc-
tions on p-adic Lie groups [M1]; the Jacobi polynomials of Heckman and
Opdam attached to root systems [H2]; and the constant term conjectures of
Dyson,Andrewsetal.([D1],[A1],[M4],[M10]).ThelecturesofKirillov[K2]
also provide valuable background and form an excellent introduction to the
subject.
Thetitleofthismonographisthesameasthatofthelecture[M7].Thatreport,
for obvious reasons of time and space, gave only a cursory and incomplete
overviewofthetheory.Themodestaimofthepresentvolumeistofillinthe
gaps in that report and to provide a unified foundation for the theory in its
presentstate.
Thedecisiontotreatallaffinerootsystems,reducedornot,simultaneously
onthesamefootinghasresultedinanunavoidablycomplexsystemofnotation.
Inordertoformulateresultsuniformlyitisnecessarytoassociatetoeachaffine
root system S another affine root system S(cid:2) (which may or may not coincide
withS),andtoeachlabelling(§1.5)ofSaduallabellingofS(cid:2).
vii
viii Introduction
The prospective reader is expected to be familiar with the algebra and
geometry of (crystallographic) root systems and Weyl groups, as expounded
for example by Bourbaki in [B1]. Beyond that, the book is pretty well
self-contained.
We shall now survey briefly the various chapters and their contents. The
first four chapters are preparatory to Chapter5, which contains all the main
results. Chapter1 covers the basic properties of affine root systems and their
classification. Chapter2 is devoted to the extended affine Weyl group, and
collectsvariousnotionsandresultsthatwillbeneededlater.
Chapter 3 introduces the (Artin) braid group of an extended affine Weyl
group,andthedoublebraidgroup.Themainresultofthischapteristheduality
theorem(3.5.1);althoughitisfundamentaltothetheory,thereisatthistimeof
writingnocompleteproofintheliterature.Ihavetoconfessthattheproofgiven
hereofthedualitytheoremistheleastsatisfactoryfeatureofthebook,sinceit
consists in checking, in rather tedious detail, the necessary relations between
the generators. Fortunately, B. Ion [I1] has recently given a more conceptual
proofwhichavoidsthesecalculations.
ThesubjectofChapter4istheaffineHeckealgebraH,whichisadeforma-
tionofthegroupalgebraoftheextendedaffineWeylgroup.Weconstructthe
basicrepresentationofHin§4.3anddevelopitspropertiesinthesubsequent
sections.Finally,in§4.7weintroducethedoubleaffineHeckealgebraH˜,and
showthatthedualitytheoremforthedoublebraidgroupgivesrisetoaduality
theoremforH˜.
As stated above, Chapter5, on orthogonal polynomials, is the heart of the
book.Thescalarproductsareintroducedin§5.1,theorthogonalpolynomials
Eλin§5.2,thesymmetricorthogonalpolynomialsPλin§5.3,andtheirvariants
QλandPλ(ε)in§5.7.Themainresultsofthechapterarethesymmetrytheorems
(5.2.4) and (5.3.5); the specialization theorems (5.2.14) and (5.3.12); and the
normformulas(5.8.17)and(5.8.19),whichincludeasspecialcasesalmostall
theconstanttermconjecturesreferredtoearlier.
The final Chapter6 deals with the case where the affine root system S has
rank1.Hereeverythingcanbemadecompletelyexplicit.WhenSisoftypeA ,
1
thepolynomialsPλarethecontinuousq-ultraspherical(orRogers)polynomials,
and when S is of type (C∨,C ) they are the Askey-Wilson polynomials, as
1 1
mentionedabove.
The subject of this monograph has many connections with other parts of
mathematics and theoretical physics, such as (in no particular order) alge-
braic combinatorics, harmonic analysis, integrable quantum systems, quan-
tum groups and symmetric spaces, quantum statistical mechanics, deformed