Cohomology for quantum groups via the geometry of the nullcone

110 Pages·2014·1.017 MB·English

by  Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen

MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1077 (fourth of 5 numbers) • May 2014
Cohomology for Quantum Groups
via the Geometry of the Nullcone
Christopher P. Bendel
Daniel K. Nakano
Brian J. Parshall
Cornelius Pillen
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1077 (fourth of 5 numbers) • May 2014
Cohomology for Quantum Groups
via the Geometry of the Nullcone
Christopher P. Bendel
Daniel K. Nakano
Brian J. Parshall
Cornelius Pillen
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Bendel,ChristopherP.,1969-
Cohomology for quantum groups via the geometry of the nullcone / Christopher P. Bendel,
DanielK.Nakano,BrianJ.Parshall,CorneliusPillen.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 229,
number1077)
“May2014,volume229,number1077(fourthof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-9175-9(alk. paper)
1. Cohomology operations. 2. Algebraic topology. I. Nakano, Daniel K. (Daniel Ken),
1964- II. Parshall, Brian, 1945- III. Pillen, Cornelius, 1959- IV. American Mathematical
Society. V.Title.
QA169.B46 2014
512(cid:2).55–dc23 2013051269
DOI:http://dx.doi.org/10.1090/memo/1077
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Contents
Introduction vii
Chapter 1. Preliminaries and Statement of Results 1
1.1. Some preliminary notation 1
1.2. Main results 2
Chapter 2. Quantum Groups, Actions, and Cohomology 7
2.1. Listings 7
2.2. Quantum enveloping algebras 8
2.3. Connections with algebraic groups 10
2.4. Root vectors and PBW-basis 11
2.5. Levi and parabolic subalgebras 12
2.6. The subalgebra U (u ) 12
q J
2.7. Adjoint action 13
2.8. Finite dimensionality of cohomology groups 16
2.9. Spectral sequences and the Euler characteristic 17
2.10. Induction functors 20
Chapter 3. Computation of Φ and N(Φ ) 21
0 0
3.1. Subroot systems defined by weights 21
3.2. The case of the classical Lie algebras 23
3.3. The case of the exceptional Lie algebras 25
3.4. Standardizing Φ 26
0
3.5. Resolution of singularities 26
3.6. Normality of orbit closures 28
Chapter 4. Combinatorics and the Steinberg Module 31
4.1. Steinberg weights 31
4.2. Weights of Λ• 32
ζ,J
4.3. Multiplicity of the Steinberg module 33
4.4. Proof of Proposition 4.2.1 34
4.5. The weight δ 35
X
4.6. Types B ,C ,D 38
n n n
4.7. Type A 39
n
4.8. Type A with l dividing n+1 41
n
4.9. Exceptional Lie algebras 43
Chapter 5. The Cohomology Algebra H•(u (g),C) 49
ζ
5.1. Spectral sequences, I 49
5.2. Spectral sequences, II 50
iii
iv CONTENTS
5.3. An identification theorem 51
5.4. Spectral sequences, III 55
5.5. Proof of main result, Theorem 1.2.3, I 57
5.6. Spectral sequences, IV 57
5.7. Proof of the main result, Theorem 1.2.3, II 60
Chapter 6. Finite Generation 61
6.1. A finite generation result 61
6.2. Proof of part (a) of Theorem 1.2.4 62
6.3. Proof of part (b) of Theorem 1.2.4 62
Chapter 7. Comparison with Positive Characteristic 65
7.1. The setting 65
7.2. Assumptions 66
7.3. Consequences 66
7.4. Special cases 68
Chapter 8. Support Varieties over u for the Modules ∇ (λ) and Δ (λ) 71
ζ ζ ζ
8.1. Quantum support varieties 71
8.2. Lower bounds on the dimensions of support varieties 71
8.3. Support varieties of ∇ (λ): general results 72
ζ
8.4. Support varieties of Δ (λ) when l is good 73
ζ
8.5. A question of naturality of support varieties 74
8.6. The Constrictor Method I 75
8.7. The Constrictor Method II 75
8.8. Support varieties of ∇ (λ) when l is bad 76
ζ
8.9. E when 3|l 77
6
8.10. F when 3|l 77
4
8.11. E when 3|l 78
7
8.12. E when 3|l, 5|l 79
8
8.13. Support varieties of Δ (λ) when l is bad 80
ζ
Appendix A 81
A.1. Tables I 81
A.2. Tables II 84
Bibliography 89
Abstract
Let ζ be a complex (cid:5)th root of unity for an odd integer (cid:5) > 1. For any
complex simple Lie algebra g, let u = u (g) be the associated “small” quantum
ζ ζ
enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be
realizedasasubalgebraoftheLusztig(dividedpower)quantumenvelopingalgebra
U and as a quotient algebra of the De Concini–Kac quantum enveloping algebra
ζ
U . ItplaysanimportantroleintherepresentationtheoriesofbothU andU ina
ζ ζ ζ
way analogous to that played by the restricted enveloping algebra u of a reductive
group G in positive characteristic p with respect to its distribution and enveloping
algebras. In general, little is known about the representation theory of quantum
groups(resp.,algebraicgroups)whenl(resp.,p)issmallerthantheCoxeternumber
h of the underlying root system. For example, Lusztig’s conjecture concerning the
characters of the rational irreducible G-modules stipulates that p ≥ h. The main
result in this paper provides a surprisingly uniform answer for the cohomology
algebra H•(u ,C) of the small quantum group. When (cid:5) > h, this cohomology
ζ
algebra has been calculated by Ginzburg and Kumar [GK]. Our result requires
powerfultoolsfromcomplexgeometryandadetailedknowledgeofthegeometryof
thenullconeofg. Inthisway,themethodspointoutdifficultiespresentinobtaining
similarresultsfortherestrictedenvelopingalgebrauinsmallcharacteristics,though
theydoprovide someclarificationofknownresultstherealso. Finally, weestablish
•
that if M is a finite dimensional u -module, then H (u ,M) is a finitely generated
ζ ζ
H•(u ,C)-module, and we obtain new results on the theory of support varietiesfor
ζ
u .
ζ
ReceivedbytheeditorFebruary16,2011,and,inrevisedform,June15,2012.
ArticleelectronicallypublishedonOctober16,2013.
DOI:http://dx.doi.org/10.1090/memo/1077
2010 MathematicsSubjectClassification. Primary20G42,20G10;Secondary17B08.
Key wordsand phrases. Quantumgroups,cohomology,supportvarieties.
ResearchofthefirstauthorwassupportedinpartbyNSFgrantDMS-0400558.
ResearchofthesecondauthorwassupportedinpartbyNSFgrantDMS-1002135.
ResearchofthethirdauthorwassupportedinpartbyNSFgrantDMS-1001900.
(cid:3)c2013 American Mathematical Society
v

Introduction
Let g be a finite dimensional, restricted Lie algebra (as defined by Jacobson)
F
overanalgebraicallyclosedfieldF ofpositivecharacteristicp,withrestrictionmap
x (cid:5)→ x[p], x ∈ g . The restricted enveloping algebra u := u(g ) of g is a fi-
F F F
nite dimensional cocommutative Hopf algebra. In general, the cohomology algebra
•
H (u,F) is difficult to compute. However, Suslin-Friedlander-Bendel [SFB1,SFB2]
proved that, putting A := H2•(u,F) (the commutative subalgebra of the coho-
mology algebra concentrated in even degrees), the (algebraic) scheme SpecA is
homeomorphic to the closed subvariety N (g ) := {x ∈ g : x[p] = 0}. We call
1 F F
N (g ) the restricted nullcone of g ; it is a closed subvariety of the full nullcone
1 F F
N(g ) which consists of all [p]-nilpotent elements in g .
F F
When g is the Lie algebra of a reductive algebraic group G over F, the above
F
resultscanbeconsiderablysharpened. Forexample,ifp>h(theCoxeternumberof
G),thenH2•(u,F)∼=F[N (g )],thecoordinatealgebraofN (g )(cf. Friedlander-
1 F 1 F
Parshall[FP2]andAndersen-Jantzen[AJ]).Inaddition,theconditionp≥himplies
that N (g ) = N(g ). However, when p ≤ h, there is no known calculation
1 F F
of H•(u,F) (apart from some small rank cases). For all primes, N (g ) is an
1 F
irreducible variety [NPV, (6.3.1) Cor.], [UGA2, Thm. 4.2] and is the closure of a
G-orbit. These orbits have been determined in [CLNP], [UGA2, Thm. 4.2].
Now let g = gC be a complex simple Lie algebra, and let Uζ = Uζ(g) be the
quantum enveloping algebra (Lusztig form) associated to g at a primitive lth root
of unity ζ ∈ C. We regard U as an algebra over C obtained by base change
ζ
from the quantum enveloping algebra over the cyclotomic field Q(ζ). Here l > 1
is an odd integer, not divisible by 3 when g has type G . The representation
2
theory of U , when l = p, models or approximates the representation theory of
ζ
G. For example, part of the “Lusztig program” to determine the characters of the
irreducible G-modules amounts to showing, when l = p ≥ h, that the characters
of the irreducible G-modules having high weights in the so-called Jantzen region
coincide with the characters of the analogous irreducible modules for U . This
ζ
result has been proved by Andersen-Jantzen-Soergel [AJS] for all large p (no lower
bound known except for small rank). Recently, Fiebig [Fie], improving upon the
methods of [AJS], has provided a specific (large) bound on p for each root system
sufficient for the validity of the Lusztig character formula. The original conjecture
for p≥h remains open.
For all l, Lusztig [L1] has also introduced an analog, denoted u := u (g),
ζ ζ
of the restricted enveloping algebra u. Like u, u is a finite dimensional Hopf
ζ
algebra, though it is in general not cocommutative. It plays a role in the repre-
sentation theory of U much like that played by u in the representation theory
ζ
of G. For l > h, Ginzburg-Kumar [GK] have calculated the cohomology algebra
H•(u ,C). Byanexactanalogywiththecohomologyalgebraofofu,theyprovethat
ζ
vii
viii INTRODUCTION
H2•(u ,C) ∼= C[N(g)], the coordinate algebra of the nullcone N(g) consisting of
ζ
nilpotent elements in g. Recently, Arkhipov-Bezrukavnikov-Ginzburg [ABG, §1.4],
taking[GK]asamethodtopass fromtherepresentationtheoryofquantumgroups
to the geometry of the nullcone, provide a proof of Lusztig’s character formula for
U when l>h. These important connections have made the small quantum group
ζ
u an object of significant interest (cf. [ABBGM], [AG], [Be], [Lac1,Lac2]).
ζ
This paper presents new results on the cohomology of u . In particular, we
ζ
compute the cohomology algebra H•(u ,C) in most of the remaining cases when
ζ
l≤h. Our results are explicitlydescribed inSection1.2below. Weprove inChap-
ters 2–5 that H2•+1(u ,C) = 0, while H2•(u ,C) is isomorphic to, in most cases,
ζ ζ
the coordinate algebra of an explicitly described closed subvariety N(Φ ) of N(g)
0
(constructed in a similar way to the variety N (g ) discussed above). In Chap-
1 F
ter 2, we rigorously develop and present new results on the cohomology theory of
parabolic subalgebras for quantum groups. The application of powerful tools from
complex geometry represents at least one advantage that the quantum enveloping
algebrasituation (in characteristic zero) has over that in positive characteristic. In
Chapters 3 and 5, we demonstrate how the Grauert-Riemenschneider theorem and
thenormality ofcertainorbitclosuresinN(g), namely, thevarietiesN(Φ ), playa
0
vital role in carrying out our cohomology calculations. Some of the major input in
ourcalculationsoccursinChapter4, whereweanalyzethecombinatoricsinvolving
the multiplicity of the Steinberg module in certain cohomology modules related to
unipotent radicals of parabolic subalgebras.
In Chapter 6, we prove that R := H•(u ,C) is a finitely generated C-algebra.
ζ
•
Also,weshowthatifM isafinitedimensionalu -module,thenH (u ,M)isfinitely
ζ ζ
generatedoverR. Chapter7adaptssomeofourmethodstothecohomologyalgebra
•
H (u,F) in positive characteristic, making some ad hoc computations in [AJ] more
transparent. In particular, we can identify key vanishing results, known over C,
and as yet unproved in positive characteristic, which would be sufficient to extend
the cohomology calculations to positive characteristic p.
Finally, by building on results in Chapter 6, we define support varieties in
the quantum setting in Chapter 8 and exhibit some new calculations on support
varieties. Thetheoryofsupport varietiesfortherestrictedLiealgebrag attached
F
to a reductive group G in positive characteristic provides evidence for beautiful
connectionsbetweentherepresentationtheoryofGandthestructureandgeometry
of the restricted nullcone N (g ) ([NPV], [CLNP], [UGA1], [UGA2]). The results of
1 F
this paper strongly reinforce this expectation (as do those in [ABG] mentioned
above).
A number of applications and further developments which heavily depend on
the foundational results of this paper have arisen since this manuscript first ap-
peared as a preprint. Drupieski [Dr1,Dr2] has used the methods of this paper to
investigatethecohomologyandrepresentationtheoryforFrobenius-Lusztigkernels
ofquantumgroups. Moreover,FeldvossandWitherspoonhaveshownhowtoapply
our calculations to determine the representation type of the small quantum group
[FeW].
INTRODUCTION ix
Acknowledgements
TheauthorsacknowledgeLeonardChastkofsky,ChristopherDrupieski,William
Graham, Shrawan Kumar and Eric Sommers for several useful discussions at vari-
ous stagesof thisproject. Inparticular, Drupieski carefully went thoughthe entire
manuscript helping us to correct and improve many results. We also thank Jesper
Funch Thomsen for indicating how recent work of Christophersen can be used to
compute G -cohomology for the group E , and two undergraduate students Alloys
1 6
J. Samz and Sara Wood at the University of Wisconsin-Stout for their assistance
with the MAGMA computations used in this paper. The authors also would like
to also express their appreciation to the referees for their suggestions.
Finally, the first, second, and last authors acknowledge the hospitality and
support of the Department of Mathematics at the University of Virginia in Spring
2006 and Spring 2007 during work on this project.