Dimension data of closed subgroups and algebraic vector bundles on punctured affine spaces and ...

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Dimension data of closed
subgroups and algebraic vector
bundles on punctured affine
spaces and smooth quadrics
Doctoral Thesis
Author(s):
Yu, Jun
Publication date:
2013
Permanent link:
https://doi.org/10.3929/ethz-a-009911467
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In Copyright - Non-Commercial Use Permitted
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DISS. ETH NO. 20748
Dimension data of closed subgroups and algebraic vector
bundles on punctured affine spaces and smooth quadrics
A dissertation submitted to
ETH Zurich
for the degree of
Doctor of Sciences
presented by
Jun Yu
MPhil Mathematics HKUST
born September 18, 1982
citizen of China
accepted on the recommendation of
Prof. Dr. Brent Doran (examiner)
Prof. Dr. Gisbert Wu¨stholz
Prof. Dr. Jiu-Kang Yu
2013

2
Acknowledgments
I would like to express my profound gratitude to Professor Brent Doran for all of his
support during this PhD project. In particular, I benefited a lot from communications
with him on geometric invariant theory and motive theory. Many thanks are due to
Professor Gisbert Wu¨stholz and Professor Jiu-Kang Yu for kindly accepting to be co-
examiners and for reading my thesis. I thank the Swiss National Science Foundation for
the financial support of this PhD project.
Special thanks go to Jiu-Kang Yu and Jinpeng An for the collaboration in a joint
paper. I am grateful to Professor Michael Larsen and Professor Richard Pink for their
pioneering work on dimension data and to Professor Geoffrey Horrocks for his seminal
work on algebraic vector bundles.
I would like to thank my colleagues in ETH and friends in Zurich for sharing their
knowledge with me and for a lot of help on various issues.
Finally, I deeply thank my parents and my sister for all of their support and love over
many years.
3
Summary
Thisthesiscontainstwoparts. Thefirstisaboutthedimensiondataofclosedsubgroups
ofcompactLiegroups. Thesecondisaboutthebehaviorofalgebraicvectorbundlesunder
the pull-back functor associated with a map p : Y = SL /SL −→ X = An+1\{0}.
n n+1 n n
The purpose of Chapter 1 is to study Lie primitive and Lie quasi-primitive subgroups
of compact Lie groups. We prove a generalization of a theorem of Borel and Serre. A
lemma proved in this chapter is important for the proof of the main theorem of Chapter
2.
The purpose of Chapter 2 is to prove a rigidity result for dimension data, which implies
that the topological space of dimension data of closed subgroups of a given compact Lie
group is sequentially compact. We also give some consequences of this rigidity result to
other rigidity and finiteness questions.
InChapter3,wesolvethedimensiondatumproblemandthelineardependenceproblem
for closed connected subgroups. We show that it is sufficient to consider closed connected
subgroups with a common maximal torus. For a given connected torus with a biinvariant
Riemannian metric, we define root systems on it and get a unique maximal one among
them. We reduce the dimension datum problem (or the linear dependence problem) to
comparing certain characters (or getting linear relations among them). These characters
are associated with reduced sub-root systems of the maximal root system and a certain
finite group Γ◦. We formulate two questions in terms of the characters of reduced sub-
root systems of a given root system. By solving these two newly formulated questions, we
reach the solutions of the dimension datum problem and the linear dependence problem
for closed connected subgroups.
Chapter 4 contains two main results. One is the existence of non-trivial algebraic
vector bundles of any rank at least n on X with trivial pull-backs to Y . Another is
n n
the existence of continuous families having arbitrarily large dimension consisting of non-
isomorphic rank n algebraic vector bundles on X with trivial pull-backs to Y . These
n n
results begin to suggest how the classification of algebraic vector bundles is sensitive to
A1-homotopy equivalences between affine and non-affine varieties.
4
Zusammenfassung
Diese Arbeit besteht aus zwei Teilen. Der erste Teil behandelt Dimensionsdaten von
abgeschlossenen Untergruppen einer kompakten Lie Gruppe. Der zweite Teil untersucht
das Verhalten von algebraischen Vektorbu¨ndeln unter dem Pullbackfunktor assoziiert zur
Abbildung p : Y = SL /SL −→ X = An+1\{0}.
n n+1 n n
InKapitel1wirdeineKlassevonLieprimitivenundLiequasi-primitivenUntergruppen
von kompakten Lie Gruppen untersucht und eine Verallgemeinerung eines klassischen
Theorems von Borel und Serre bewiesen. Eines der Lemmata das hier bewiesen wird, ist
wichtig fu¨r den Beweis des Haupttheorems in Kapitel 2.
Kapitel 2 enth¨alt einen Beweis eines Rigidit¨atsresultats der oben beschriebenen Di-
mensionsdaten. Aus dieser Rigidit¨at folgt dann direkt, dass der topologische Raum der
Dimensionsdaten aller abgeschlossenen Untergruppen folgenkompakt ist. Zus¨atzlich fu¨hrt
diesesResultatzuweiterenKonsequenzenfu¨reinigeFragenderRigidit¨atundEndlichkeit.
In Kapitel 3 wird das Dimensionsdatenproblem und das Problem der linearen Ab-
h¨angigkeit dieser Dimensionsdaten fu¨r abgeschlossen Gruppen gel¨ost. Wir zeigen, dass
es ausreichend ist, abgeschlossene und zusammenh¨angende Gruppen mit einem gemein-
samen maximalen Torus zu betrachten. Gegeben sei ein zusammenh¨angender Torus mit
einerbiinvariantenRiemannschenMetrik. WirdefinierenWurzelsystemeaufdiesemTorus
und bekommen ein eindeutiges maximales Wurzelsystem. Wir werden das Dimensions-
datenproblem auf das Vergleichen von gewissen Charakteren und das Problem der lin-
earen Abh¨angigkeit dieser Dimensionsdaten auf linearen Abh¨angigkeiten zwischen diesen
Charakteren zuru¨ckfu¨hren. Diese Charaktere sind zu Unterwurzelsystemen eines gegebe-
nenWurzelsystemsundzueinergewissenendlichenGruppeassoziiert. Anschliessendwer-
den zwei Fragen bezu¨glich Charaktere von Unterwurzelsystemen formuliert. Zum Schluss
werden diese beiden Fragen beantwortet und so die urspru¨nglichen Probleme l¨osen.
Kapitel 4 enth¨alt zwei Hauptresultate. Das eine ist die Existenz von nicht-trivialen
algebraischen Vektorbu¨ndeln vom Rang mindesten n auf X mit trivialem pull-back auf
n
Y . Das andere ist die Existenz von stetigen Familien von beliebig hoher Dimension
n
von paarweise nicht-isomorphen algebraischen Vektorbu¨ndeln vom Rang n auf X , so-
n
dass alle pull-backs nach Y trivial sind. Diese Resultate suggerieren, dass die Klassi-
n
fikation von algebraischen Vektorbu¨ndeln u¨ber affinen und nicht-affinen Variet¨aten nicht
A1-Homotopieinvariant ist.
5
Contents
Acknowledgments 2
Summary 3
Zusammenfassung 4
0. Introduction 1
1. A generalization of Borel-Serre Theorem 5
1.1. Lie primitive subgroups 6
1.2. Proof of the theorem 8
1.3. Finiteness 10
2. A rigidity result for dimension data 11
2.1. Proof of the main theorem 12
2.2. Some consequences 14
3. Dimension datum problem and linear dependence problem 17
3.1. Root system in a lattice 20
3.2. The Larsen-Pink method 25
3.3. Comparison of different conjugacy conditions 32
3.4. Formulation of the problems in terms of root systems 34
3.5. Classification of sub-root systems 34
3.6. Formulas of the leading terms 38
3.7. Equalities among dimension data 39
3.8. Linear relations among dimension data 51
3.9. Comparison of Question 3.2 and Question 3.4 70
4. Algebraic vector bundles on punctured affine spaces and smooth quadrics 77
4.1. Vector bundles on affine spaces, projective spaces and punctured affine spaces 78
4.2. Unimodular and completable rows 78
4.3. Invariant e and some examples 79
4.4. Bundles with e = 1 and more examples 83
4.5. Some remarks 87
4.6. A theorem of Swan 87
References 89
1
0. Introduction
Algebraic invariant theory is an old subject in Mathematics. It originated in the 19th
centurywithimportantcontributionsmadebyCayley, Sylvester, Gordan, Kleinetal, and
it gained its modern shape since Hilbert solved Gordan’s problem by proving his famous
finitenesstheoremforinvariantrings. Inthe1960s,Mumfordinitiatedgeometricinvariant
theory, which later became a powerful tool for solving many algebraic geometric problems
by expressing the moduli of various geometric objects in terms of quotients of algebraic
reductive groups. In this thesis, we study two topics related to algebraic invariant theory
and geometric invariant theory, respectively. The first is the dimension data of closed
subgroups of compact Lie groups and the second is algebraic vector bundles on punctured
affine spaces and smooth quadrics.
Given a subgroup H of a group G, the branching rule of this pair is concerned with
the decomposition of complex linear irreducible (not necessarily finite-dimensional) rep-
resentations of G into sums of irreducible representations of H when regarded as complex
linear representations of H. To describe the branching rule is called the branching rule
problem. If we consider the multiplicities of the trivial representation of H appearing
in these restriction representations, we get the dimension datum D of H (cf. Chap-
H
ter 2). Let G = GL(n) and consider the representations {Symk(Cn)| k ≥ 0}. Then
Sym(Cn)H = ((cid:76)∞ Symk(Cn))H is the ring of H-invariant functions on the dual space
k=0
(Cn)∗. If only multiplicities are considered, algebraic invariants could be viewed as a
dimension datum. Obviously dimension datum is a part of the branching rule. On the
other hand, let G(cid:48) = H ×G and H(cid:48) = {(x,x) ∈ G(cid:48) : x ∈ H}. Then the branching rule
of H ⊂ G and the dimension datum of H(cid:48) ⊂ G(cid:48) are equivalent objects. Hence nothing is
lost if we consider dimension datum instead of branching rule.
In the converse direction, there is the induction functor taking complex linear repre-
sentations of H to complex linear representations of G. Given a G-invariant measure on
G/H, for any complex linear unitary representation ρ of H, the induced representation
can be defined as IndG(ρ) = L2(G× ρ), i.e., the complex linear space of L2-integrable
H H
sections of a vector bundle G× ρ on the homogeneous space G/H. The Frobenius reci-
H
procity shows that the restriction functor and induction functor are adjoint functors to
each other. When G and H are finite groups, the induced representation has a simple
algebraic definition. It was defined by Frobenius and used by him to study the so-called
Frobenius groups. In particular, taking the trivial representation 1 of H, the induced
H
representation IndG(1 ) equals L2(G/H). Decomposing L2(G/H) in terms of irreducible
H H
representations of G is harmonic analysis on the homogeneous space G/H.
There are three kinds of branching rules important for us. The first is to take G a
compact (connected) semisimple Lie group and H a closed subgroup, and allow π to vary
over all finite-dimensional irreducible complex linear representations of G. The second is
to take G a semisimple real Lie group and H a closed reductive subgroup, and allow π to
vary over all irreducible unitary (usually infinite-dimensional) representations of G. The
third is to take a pair H(A ) ⊂ G(A ) for linear algebraic groups H ⊂ G defined over a
F F
number field F and the algebraic groups over the ring of adeles A , and allow π to vary
F
over irreducible automorphic unitary representations of G(A ) (cf. [Bo]).
F
Several branching rules for the classical groups were determined by Weyl between suc-
cessive unitary groups in the 1940s, by Murnaghan between successive special orthogonal
groups and unitary symplectic groups in the 1940s, and by Littlewood from the unitary
groups to the symplectic groups and special orthogonal groups in the 1950s. The most
notable branching rule is probably the Littlewood-Richardson rule found in the 1930s.
It describes the tensor product decomposition of irreducible finite-dimensional complex
2
linear representations of the unitary group G = U(n). Despite a lot of important devel-
opments in this area, the study of the branching rule problem is still active today. For
example, the past 10-15 years saw the solution of describing the saturated tensor cones
(cf. [BSj], [BK]and[R])andthereexistrecentlinksofsaturatedtensorconestoQuantum
Information Theory (cf. [CDKW]). As mentioned in [Ku], there still remains the inter-
esting saturation problem (cf. [BK2]) and the need for explicit multiplicity formulas (cf.
[Lit] and [CDW]). For the branching problem of infinite-dimensional unitary representa-
tions of linear reductive Lie groups, there are the important dual pair theory (cf. [How],
[How2], [How3] and [Li]) and the admissible restriction theory (cf. [Ko] and [Ko2]). In
the harmonic analysis problem, Delorme obtains the Plancherel formula for semisimple
symmetric spaces (cf. [D]) and there has been important development of other problems
in this area (cf. [BFJS]). With regards to automorphic representations, the branching
rule is related to the periods of automorphic forms and L-functions (cf. [GP]). Here, we
just mention the important Gross-Prasad conjecture (cf. [GP]) and a recent conjectural
Plancherel formula on spherical varieties (cf. [SV]).
Given a compact Lie group G, the dimension datum D of a closed subgroup H of G
H
is the map from G(cid:98) to Z,
D : V (cid:55)→ dimVH,
H
where G(cid:98) is the set of equivalence classes of irreducible finite-dimensional complex linear
representations of G and VH is the subspace of H-invariant vectors in a complex linear
representation V of G. Dimension data arise from number theory in the determination
of monodromy groups (cf. [K]). They also appear in differential geometry: in relation
with the spectra of the Laplace operators on Riemannian homogeneous spaces (cf. [Sut]).
Moreover, Langlands has suggested to use the dimension data as a key ingredient in his
program“BeyondEndoscopy”(cf. [La],Page2,Problem(II)).Intheliterature,dimension
data have been studied in the articles [LP], [Lar] and [AYY].
Endowing the space ZGb (identified with the set of all maps from G(cid:98) to Z) with the
product topology of the discrete topologies on the factors Z, we could consider the con-
vergence of sequences of dimension data. Since 0 ≤ dimVH ≤ dimV for any complex
irreducible representation V of G and any closed subgroup H of G, each sequence of di-
mension data {D | n ≥ 1} contains a convergent subsequence as elements in ZGb. This
Hn
raises a question: is the limit still a dimension datum D and does {H | n ≥ 1} give
H n
some restriction on H? Theorem 2.1 answers this question completely. It states that the
limit must be a dimension datum D and the subgroup H is strongly controlled by a
H
subsequence {H | j ≥ 1}. To prove this theorem, we need to use Lie primitive and Lie
nj
quasi-primitive subgroups defined in Chapter 1. The main result of Chapter 1 is a gener-
alization of a theorem of Borel and Serre. In Chapter 2, we also give some consequences
of Theorem 2.1 by strengthening a theorem of Larsen and give different proofs of two
theorems proving in a joint paper [AYY].
The dimension datum problem asks: to what extent is H (up to G-conjugacy) de-
termined by its dimension data D ? In [La], Subsections 1.1 and 1.6, Langlands raised
H
anotherquestionaboutdimensiondata. Asexplainedin[AYY],Pages13-14,thisisequiv-
alent to the question: given a list of finitely many closed subgroups H ,...,H of G with
1 n
D (cid:54)= D for any i (cid:54)= j, are D ,...,D linearly independent? We call this the linear
Hi Hj H1 Hn
dependence problem.
In Chapter 3, given a compact Lie groupG we choose a bi-invariant Riemannian metric
m on it. For a closed connected torus T in G, we introduce root systems on T. Among
the root systems on T, there is a maximal one, denoted by Ψ , which contains all other
T
root systems on T. For each reduced root system Φ on T and a finite group W acting
on t = LieT satisfying some condition (cf. Definition 3.29), we define a character F .
0 Φ,W
3
In particular, this definition applies to the finite group Γ◦ := N (T)/C (T). We define
G G
another root system Ψ(cid:48) on T as the sub-root system of Ψ(cid:48) generated by the root systems
T T
of closed connected subgroups H of G with T a maximal torus of H. Moreover we show
that the Weyl group of Ψ(cid:48) is a sub-group of Γ◦. By Propositions 3.32 and 3.33, we
T
show that the dimension datum problem (or the linear dependence problem) reduces to
comparing the characters {F | Φ ⊂ Ψ } (or finding linear relations among them). We
Φ,Γ◦ T
propose Questions 3.3 and 3.4. They are concerned with the characters associated to
reduced sub-root systems of a given root system Ψ. In the next two sections we solve
these two questions completely. In Section 3.9, we give constructions showing that we get
completesolutionstothedimensiondatumproblemandtothelineardependenceproblem
once we solve these two questions.
WereducebothquestionstothecasewhereΨisanirreduciblerootsystem. Inthecase
that Ψ is a classical irreducible root system, the classification of sub-root systems of Ψ is
well-known. In[LP],LarsenandPinkdefinedanalgebraisomorphismE takingcharacters
of sub-root systems to polynomials (cf. Lemma 3.47 and Definition 3.48). In this case
we solve Questions 3.3 and 3.4 by getting all algebraic relations among polynomials in
the image of E. In the case that Ψ is an exceptional irreducible root system, Oshima
classified sub-root systems of Ψ (cf. [Os]). In [LP], the authors defined a weight 2δ(cid:48) for
Φ
each reduced sub-root system Φ of Ψ. We give the formulas of these weights in the Tables
2-9. We also define and calculate a generating function for each reduced sub-root system
of Ψ (cf. Definition 3.68). With these, we are able to find and prove all linear relations
among the characters of reduced sub-root systems of Ψ. Therefore we solve Questions 3.3
and 3.4 completely.
Geometric invariant theory is a very useful tool in the study of algebraic vector bundles
on projective smooth varieties (cf. [HL]). The past 40 years saw a rapid development
of the research of moduli spaces of stable (or semistable) vector bundles on algebraic
projective curves and algebraic projective surfaces. For a projective variety of higher
dimension, the study of algebraic vector bundles becomes much harder. There is the
interesting and difficult Hartshorne conjecture (cf. [H2]). Roughly speaking, it said:
any indecomposable algebraic vector bundle on the projective space Pn is either a line
bundle or has a relatively non-small rank. For algebraic vector bundles on smooth affine
varieties,Serre’sproblem(cf. [Se2])asksifanyalgebraicvectorbundleontheaffinespace
An(k) is a trivial bundle, where k an algebraic closed field. This was solved by Quillen
and Suslin independently and essentially simultaneously in the affirmative way (cf. [Q]
and [Su]). Another interesting question is the so called cancellation problem (cf. [Su2]).
Suslin, Murthy and Fasel-Rao-Swan (cf. [Su2], [Mu] and [FRS]) have made important
and interesting contributions on this subject. Besides affine and projective varieties,
the punctured affine space An −{0} is probably the simplest smooth algebraic variety.
Horrocks obtained interesting results about algebraic vector bundles on punctured affine
spaces (cf. [Ho]). Moreover, he invented monads, which is important for the subsequent
researches, e.g., in the description of moduli spaces of stable bundles on P2 with fixed
Chern classes (cf. [BH] and [OSS]), the splitting property of bundles on Pn (cf. [KPR]
and [Fl]) and the ADHM construction (cf. [ADHM]).
InChapter4,westudyalgebraicvectorbundlesonthepuncturedaffinespaceCn+1\{0}
and on the smooth quadric SL (C)/SL (C). For each n ≥ 1, let
n+1 n
Y = SL (C)/SL (C)
n n+1 n
and
X = Cn+1\{0}.
n
We have a map p : Y −→ X (cf. Introduction of Chapter 4), which expresses Y as
n n n
an affine vector bundle torsor over X . From the point of view of A1-homotopy theory,
n