Kazhdan–Lusztig Cells with Unequal Parameters

350 Pages·09.445 MB·English

by  Cédric Bonnafé

Algebra and Applications
Cédric Bonnafé
Kazhdan–Lusztig
Cells with
Unequal
Parameters
–
Kazhdan Lusztig Cells with Unequal Parameters
Algebra and Applications
Volume 24
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Université Paris Diderot, Paris, France
Alice Fialowski
Eötvös Loránd University, Budapest, Hungary
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University of Edinburgh, Edinburgh, UK
John Greenlees
Sheffield University Sheffield, UK
Gerhard Hiß
Aachen University, Aachen, Germany
Ieke Moerdijk
Radboud University Nijmegen, Nijmegen, The Netherlands
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Mina Teicher
Bar-Ilan University, Ramat Gan, Israel
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é é
C dric Bonnaf
–
Kazhdan Lusztig Cells
with Unequal Parameters
123
CédricBonnafé
Institut Montpelliérain Alexander
Grothendieck
CNRS–UniversitédeMontpellier
France
ISSN 1572-5553 ISSN 2192-2950 (electronic)
Algebra andApplications
ISBN978-3-319-70735-8 ISBN978-3-319-70736-5 (eBook)
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Preface
In their seminal paper on representations of Coxeter groups and Hecke algebras
[KaLu1], Kazhdan and Lusztig already had in mind lots of applications and con-
nections between their theory (now called Kazhdan–Lusztig theory) and several
problems in Lie theory: representations of Lie algebras and reductive groups, sin-
gularities of Schubert varieties, and also representations of Weyl groups in the
cohomologyofSpringerfibers.1TheywereinspiredbytheworksofJantzen[Jan],
Joseph [Jos1], [Jos2], and Vogan [Vog1], [Vog2], [Vog3]. The main objects
introducedintheirpaperaretheKazhdan–LusztigbasisofaHeckealgebra(whose
matrix inthestandard basisisgivenbythe Kazhdan–Lusztigpolynomials)andthe
partitions of the Coxeter groups into left, right, or two-sided cells. They state the
fundamentalKazhdan–Lusztigconjecture,whichrelatesthemultiplicitiesofsimple
modules of a semisimple complex Lie algebra in Verma modules to the Kazhdan–
Lusztig polynomials, and they also characterize rationally smooth Schubert vari-
eties in terms of Kazhdan–Lusztig polynomials.
In the decade following Kazhdan–Lusztig’s paper, the theory developed
incrediblyquickly,withtheproofoftheKazhdan–LusztigconjecturebyBrylinski–
Kashiwara[BrKa1],[BrKa2]andBeilinson–Bernstein[BeBe];theinterpretationof
Kazhdan–LusztigpolynomialsintermsoftheintersectioncohomologyofSchubert
varieties by Kazhdan and Lusztig [KaLu2]; the work of Lusztig on the represen-
tation theory offinite and p-adic reductive groups [Lus2], [Lus3], [Lus4], [Lus5],
[Lus8], [Lus11]; the works of several authors, such as Joseph [Jos3], Deodhar
[Deo1],Deodhar–Gabber–Kac[DeGaKa],Douglass[Dou],Lusztig[Lus1],[Lus7],
Shi [Shi1], Soergel [Soe1], Springer [Spr1], Xi [Xi1].
All these developments are concerned with the Hecke algebras with equal
parameters. In 1983, Lusztig [Lus9] generalized the construction of the Kazhdan–
Lusztig basis and the partition into cells in a framework that will be the one
1Eventhoughthislastpointisnotmentionedintheoriginalpaper[KaLu1],thisisexplainedin
[Lus22].
v
vi Preface
followed in this book: ðW;SÞ is a Coxeter system, A is a totally ordered abelian
group, and u:S!A[0 is a weight function (i.e., uðsÞ¼uðtÞ if s and t are
conjugate in W). The equal parameter case corresponds to the case where u is
constant. In the general case, the important property of Kazhdan–Lusztig polyno-
mials and structure constants to have nonnegative coefficients (which, in the equal
parameter case, follows from the geometric interpretation) is false, and the devel-
opmentofthetheorywasmoredifficultandsomuchslower.Letus,however,note
the works of Bremke [Bre1], [Bre2] and Xi [Xi2].
The interest in Kazhdan–Lusztig theory in the unequal parameter case was
boosted by the publication of Lusztig's book on the subject [Lus20]. In this book,
Lusztig
(cid:129) proposed a series of conjectures (denoted by (P1) up to (P15)) involving
Lusztig’s a-function, whose proof would allow the extension of some con-
structionsexistingonlyintheequalparametercase(especiallytheasymptotic
algebra);
(cid:129) recalled from [Lus14] how these conjectures follow from the positivity
properties in the equal parameter case;
(cid:129) proved these conjectures in the quasi-split case (see Chapter 15 for the
definition) for W finite2;
(cid:129) proposed a conjectural geometric interpretation of Kazhdan–Lusztig poly-
nomialsintermsofcharacter sheaves ondisconnected reductive groups(for
finite Weyl groups and particular weight functions);
(cid:129) proposedanotherconjecturelinkingthistheorytoHarish-Chandratheoryfor
unipotent representations of finite reductive groups (again for finite Weyl
groups and particular weight functions);
(cid:129) computed the constructible characters and the Lusztig families of all finite
Coxeter groups, for all choices of parameters.
This led to a new dynamics, where the central problems are to determine the
partitionsofWintocellsandtheinterpretationofconstructible characters:See,for
instance Geck [Gec2], [Gec3], [Gec4], [Gec5], [Gec6], Geck–Iancu [GeIa1],
Guilhot [Gui1], [Gui2], [Gui3], [Gui4], Leclerc–Miyachi [LeMi], Malle–Rouquier
[MaRo], Shi [Shi5], Shi–Yang [ShYa1], Bonnafé–Iancu [BoIa], and Bonnafé
[Bon1], [Bon2], [Bon3], [Bon4].
Last but not least, in their work on representations of rational Cherednik
algebrasatt¼0,GordonandMartinostatedtheconjecturethatKazhdan–Lusztig
families (defined using the partition into two-sided cells) coincide with Calogero–
Moser families of finite Coxeter groups (defined as the blocks of the restricted
Cherednik algebra), for any choices of parameters (see [Gor], [GoMa], [Mar], and
the works of Bellamy and Thiel [Bell], [BeTh]). This conjecture was extended to
2G.LusztighaspostedinJune2014anupdatedversionofthisbook:Seehttp://arxiv.org/abs/math/
0208154v2.Inthis version,heuses therecent results ofElias–Williamson [ElWi]to extendthe
resultsonthequasi-splitcasetoboundedCoxetergroups(itisconjecturedthatallCoxetergroups
arebounded,anditisprovedthataffineWeylgroupsarebounded).
Preface vii
constructible characters and the partition into cells by Rouquier and the author
[BoRo1], [BoRo2], [BoRo3]: This would provide a completely different interpre-
tationofcells.Evenifitisnottreatedinthisbook,thispossibleinterpretationisat
the heart of our motivation for studying Kazhdan–Lusztig theory with unequal
parameters (Chapter 18 about the action of the cactus group might be related to
these conjectures).
The present book aims to introduce the reader to Kazhdan–Lusztig theory with
unequalparameters,byusingonlyalgebraic/combinatorialmethods.Thegoalisto
go as far as possible using only these methods, by focusing on the general prop-
erties of the partition into cells (“general” in the sense that they are valid for all
Coxetersystems,andnotonlyforaclassofCoxetersystems,suchasfiniteoraffine
Weyl groups).3
We also tried as much as possible to provide proofs of the basic results of the
theory which are different (sometimes, just slightly different) from those one can
find in the literature. For instance, we try to avoid systematic induction arguments
involvingtheBruhatorder,asmostofthemcanbeviewedasparticularinstancesof
a general lemma about existence and uniqueness of an invariant basis of a module
underasemilinearinvolution(seeChapter2):Thisisaclassicalresultwhichisalso
used to construct canonical bases for representations of quantum groups.
Vailhauquès, France Cédric Bonnafé
October 2017
3There will be an exception to this: We will study the effect of multiplication by the longest
elementoncellswheneverWisfinite.
Acknowledgements
ItwouldbestatingtheobvioustosaythatIwouldhaveneverwrittenthisbookifI
had not been initiated to Kazhdan–Lusztig theory (especially the unequal param-
eters case) by Lacri Iancu and Meinolf Geck. Most of my knowledge in this field
owes to the discussions I had with them, which led to several common articles.
IwouldalsoliketothankwarmlyJérémieGuilhot,notonlyforprovidingthetype
G~ figures of Chapter 25, but also for explaining several applications of his
2
induction process to me (I could not reproduce all of them here).
IhavealsoincludedinChapter25picturesthatgivesomepartitionsintocellsof
some hyperbolic groups: They have been kindly provided by Paul Gunnells, and I
am happy to thank him here for his invaluable help. I also thank Mikhail
Belolipetskyforauthorizingmetoreproducethepictureofthehyperbolicpentagon
group.
I thank Ben Elias and Geordie Williamson for clarifying subtle points of the
theory of Soergel bimodules.
I also wish to thank Gunter Malle, Ulrich Thiel, and again Jérémie Guilhot for
their very efficient and careful reading of a preliminary version of this book.
ix
Contents
Part I Preliminaries
1 Preorders on Bases of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.A Preorders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.B Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.C Cell Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.D Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Preorders and Symmetrizing Forms . . . . . . . . . . . . . . . . . . . . . 8
2 Lusztig’s Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Invariant Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part II Coxeter Systems, Hecke Algebras
3 Coxeter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Characterizations of Coxeter Systems. . . . . . . . . . . . . . . . . . . . 19
3.2 First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Reflections, Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Braided Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Bruhat Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Parabolic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Longest Element, Left and Right Descent Set. . . . . . . . . . . . . . 30
3.8 More on Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.9 Conjugacy Classes of Involutions . . . . . . . . . . . . . . . . . . . . . . 32
3.10 Fixed Points under Automorphisms . . . . . . . . . . . . . . . . . . . . . 34
3.11 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.11.A Coxeter Matrix, Coxeter Graph . . . . . . . . . . . . . . . . . 36
3.11.B Reflection Representation. . . . . . . . . . . . . . . . . . . . . . 37
3.11.C Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
xi