COMMUTING TUPLES IN REDUCTIVE GROUPS AND
THEIR MAXIMAL COMPACT SUBGROUPS
ALEXANDRA PETTET AND JUAN SOUTO
Abstract. Let G be a connected reductive algebraic group and K €
G a maximal compact subgroup. We consider the representation spaces
HompZk , K q and HompZk , Gq with the topology induced from an embedding into K k and Gk , respectively. The goal of this paper is to prove that
HompZk , K q is a strong deformation retract of HompZk , Gq.
1. Introduction:
By a commuting k-tuple in a group G we mean a k-tuple pa1 , . . . , ak q P Gk
with ai aj aj ai for all i and j. Observe that any such commuting k-tuple
determines uniquely a homomorphism
Ñ G, pn1, . . . , nk q ÞÑ an1 . . . ank
In fact, we obtain in this way a bijection between HompZk , Gq and the set of
Zk
1
k
commuting k-tuples in G.
Suppose now that G is a Lie group and that K € G is a closed subgroup. Endowing HompZk , Gq € Gk and HompZk , K q € K k with the subspace topology,
we consider the embedding
(1.1)
HompZk , K q
ãÑ HompZ , Gq
k
induced by the inclusion of K into G. The goal of this paper is to prove that
the map (1.1) is a homotopy equivalence if G is a reductive algebraic group and
K € G is a maximal compact subgroup.
Before stating our main result precisely, recall that a weak retraction of a
topological space X to a subspace A is a continuous map
F : r0, 1s X
Ñ X, pt, xq ÞÑ Ftpxq
with F0 IdX and F1 pX q € A, and such that Ft pAq € A for all t.
Theorem 1.1. Let G be the group of complex or real points of a connected
reductive algebraic group, defined over R in the latter case, and K € G a
maximal compact subgroup. For all k there is a weak retraction of HompZk , Gq
to HompZk , K q.
The first author has been partially supported by NSF grant DMS-0856143 and NSF RTG
grant DMS-0602191. The second author has been partially supported by NSF grant DMS0706878, NSF CAREER award 0952106 and by the Alfred P. Sloan Foundation.
: Per me si va ne la cittá dolente, per me si va ne l’etterno dolore, per me si va tra la
perduta gente.
1
2
ALEXANDRA PETTET AND JUAN SOUTO
Before going further, recall that the presence of a weak retraction implies that
the inclusion of HompZk , K q into HompZk , Gq is a homotopy equivalence [15,
p.18]. Notice also that HompZk , K q and HompZk , Gq are algebraic varieties and
hence homeomorphic to CW-complexes [39]. It follows thus from Whitehead’s
theorem [15, p.346] that in fact HompZk , K q is a strong deformation retract of
HompZk , Gq.
Corollary 1.2. If G is the group of complex or real points of a connected
reductive algebraic group, defined over R in the latter case, and K € G a
maximal compact subgroup, then HompZk , K q is a strong deformation retract
of HompZk , Gq for all k.
If we replace the free abelian group Zk by the free group Fk , then the analogues of Theorem 1.1 and Corollary 1.2 follow directly from the well-known
fact that G retracts to K. It is worth noticing that such an argument cannot
be applied to prove Theorem 1.1 because in general there is no retraction of G
to K which preserves commutativity [33].
It goes without saying that for a general finitely generated group Γ, the
inclusion of HompΓ, K q into HompΓ, Gq is not a homotopy equivalence. For
example, if Γ € SL2 C is a cocompact lattice then it follows from Mostow’s
rigidity theorem that the component of HompΓ, SL2 Cq containing the tautological representation Γ Ñ SL2 C consists of conjugates of this representation
and hence is disjoint from HompΓ, SU2 q. Also, invariance of the Euler number
implies that there are components of Hompπ1 pΣq, PSL2 Rq which do not intersect Hompπ1 pΣq, SO2 q, where Σ is a closed orientable surface of genus at least
2 [13].
ã
Remark. The proof of Theorem 1.1 applies, with only minor modifications, if we
replace Zk by any other finitely generated abelian group. We wonder whether
Theorem 1.1 still holds true if Zk is replaced by a nilpotent group.
Remark. The connectedness assumption in Theorem 1.1 is probably unnecessary and we would like to stress here that we assume connectedness in the Zariski
topology, or equivalently connectedness of the Lie group of complex points. In
particular, Theorem 1.1 applies also to some disconnected Lie groups such as
for example GLn R or SOpn,1q pRq.
Consequences of Theorem 1.1: Recall that the space HompZk , Gq may be
identified with the space of flat G-connections on the k-torus modulo based
gauge transformations. As such, these and other related spaces play a rôle in,
for example, gauge theory, mathematical physics and symplectic geometry; see
Jeffrey’s nice survey describing these different points of view [16]. In fact, it
was the work of Witten [37, 38] which motivated Borel, Friedman and Morgan
[8], and Kac and Smilga [21] to initiate the systematic study of the topological
properties of the spaces HompZk , Gq. While the papers [8] and [21] concentrate on understanding the number of connected components of HompZk , Gq
for certain values of k, higher topological invariants of these spaces, such as
cohomology groups, were first investigated by Ádem and Cohen in [1]. Starting with the latter paper, there has been a wealth of results in this direction
[2, 3, 4, 5, 6, 12, 19, 35].
3
At this point it should be mentioned that all results proved in the papers just
mentioned concern, almost without exception, the space of commuting tuples
in compact groups. Combining these results with Theorem 1.1 we obtain hence
the corresponding results for reductive groups. We give just a few examples of
this strategy:
Corollary 1.3. Suppose that G is the set of complex points of a connected,
semisimple algebraic group. The space HompZk , Gq is connected if and only if
one of the following conditions is satisfied:
(1) k 1,
(2) k 2 and G is simply connected, or
(3) k ¥ 3 and G is a product of SLn C’s and Spn C’s.
Proof. Let K be a maximal compact subgroup of G, and recall that K is homotopy equivalent to G. It is due to Kac and Smilga [21] that HompZk , K q is
connected if and only if k 1, k 2 and K is simply connected, or k ¥ 3
and K is a product of SUn ’s and Spn ’s for some n ¥ 1. As K determines G
up to isomorphism (see [30, VIII]), and as SUn and Spn are maximal compact
subgroups of SLn C and Spn C, the claim follows from Theorem 1.1.
Remark. In the proof of Corollary 1.3 we have used that complex reductive
groups are determined by their maximal compact subgroups; this is not true
in the real case: On pRq is a maximal compact subgroup of both GLn R and
SOpn,1q pRq. Theorem 1.1 implies that the varieties of commuting k-tuples of
these two groups are homotopy equivalent; at first look, the authors find this
somehow surprising.
If K is a compact group, then the connected component HompZk , K q1 of
the trivial representation in HompZk , K q is much better understood than is the
whole space. For instance, its fundamental group [19] and rational cohomology
groups [5] are known.
Corollary 1.4. If G is a connected reductive group, complex or real, then
π1 pHompZk , Gq1 q π1 pGqk for all k. Here, HompZk , Gq1 is the connected
component in HompZk , Gq of the trivial representation.
Proof. Let K be a maximal compact subgroup of G. It follows from Theorem
1.1 that HompZk , Gq1 and HompZk , K q1 are homotopy equivalent and hence
have isomorphic fundamental groups. By [19] we have that π1 pHompZk , K q1 q
and π1 pK qk are isomorphic. Since K and G are also homotopy equivalent, the
result follows.
Corollary 1.5. Let G be a connected reductive group, complex or real, T € G
a maximal torus, and W NG pT q{ZG pT q the Weyl-group of G. We have
H pHompZk , Gq1 , F q H pG{T
T k , F qW
where F is any field with characteristic relatively prime to the order W .
Proof. Let K be a maximal compact subgroup of G such that TK T X K is
a maximal torus in K; notice that this implies that the inclusion TK Ñ T is a
4
ALEXANDRA PETTET AND JUAN SOUTO
homotopy equivalence. In particular, the inclusion
K {TK
TKk Ñ G{T T k
is a W -equivariant homotopy equivalence and hence we have
(1.2)
H pK {TK
TKk , F qW H pG{T T k , F qW
It is due to Baird [5, Theorem 4.3] that the left side of (1.2) is isomorphic to
H pHompZk , K q1 , F q and hence to H pHompZk , Gq1 , F q by Theorem 1.1. Remark. From the proof of Theorem 1.1 we obtain the following description of
HompZk , Gq1 : a commuting tuple pa1 , . . . , ak q belongs to HompZk , Gq1 if and
only if there is a torus T € G containing the semisimple part of the Jordan
decomposition of ai for all i.
Topological considerations aside, observe that, with notation as before, we
can also consider HompZk , Gq as an algebraic variety. In [28], Richardson proved
that HompZ2 , Gq is an irreducible variety whenever G is the group of complex
points of a connected, simply connected and semisimple algebraic group. Combining Richardson’s result with the fact that an irreducible complex variety is
connected in the Hausdorff topology, we deduce from Corollary 1.3:
Corollary 1.6. Let G be the group of complex points of a semisimple algebraic
group. The variety of commuting pairs in G is irreducible if and only if G is
simply connected.
Similarly, we obtain from Corollary 1.3 that for G simple and k ¥ 3, the variety HompZk , Gq is reducible, unless possibly for G SLn C and G Spn pCq.
This result is far from optimal: it follows easily from the work of Gerstenhaber [18] that HompZk , SLn Cq is reducible for k, n ¥ 4; see also [20] and the
references therein for more recent results.
Remark. In comparison, it is easy to see that HompZk , SUn q is an irreducible
analytic space for all k and n.
Outline of the paper: We will divide the proof of Theorem 1.1, and with it
the whole of this paper paper, into two distinct parts, as the arguments lend
themselves naturally to this partition. In the second part of the paper (Sections
7-8), we prove that there is a commutativity preserving retraction of G to Gs ,
the subset consisting of semisimple elements. It follows directly that there is a
retraction of HompZk , Gq to HompZk , Gs q, the space of commuting semisimple
k-tuples. In the first part of the paper (Sections 2-6), we prove that HompZk , K q
is a retract of HompZk , Gs q, completing the proof of Theorem 1.1.
Sections 2-6. The goal of these sections is to prove that the space HompZk , Gs q
of commuting semisimple tuples retracts to HompZk , K q.
We stratify HompZk , Gs q according to the conjugacy class of the stabilizer
under the action by conjugation G
HompZk , Gq and note that this stratification has finitely many strata; this is the content of Section 2.
To any stratum Z of HompZk , Gs q we associate an abelian subgroup AZ € G
and a surjective map
ñ
(1.3)
pG AkZ q{NGpAZ q Ñ Z
5
where NG pAZ q is the normalizer of AZ in G. The groups AZ are described in
Section 3, where we discuss the well-centered groups: these are the diagonalizable subgroups of G which are equal to the center of their centralizer. The
main result of this section is that the sets G and K of well-centered groups in G
and K are manifolds with finitely many connected components, and that there
is an injective homotopy equivalence K Ñ G. The map (1.3) and some of its
properties are discussed in Section 4. The central observation there, is that the
domain of (1.3) is a flat bundle over a connected component of the manifold G
of well-centered groups. The flatness of this bundle allows horizontal lifting of
any retraction of G to K. We prove that one can choose the retractions of G
to K in such a way that the associated horizontal lifts descend under (1.3) to a
retraction of HompZk , Gs q to HompZk , K q. These steps make up Sections 5 and
6; some of the painful details of Section 5 are suppressed until the appendix.
ã
Sections 7-8: These two sections are devoted to proving that there is a commutativity preserving retraction of G into the set Gs of semisimple elements.
The basic idea is to associate to every element of G its multiplicative Jordan
decomposition and to homotope away the unipotent part using the logarithm.
In doing so, we face the problem that the Jordan decomposition is not continuous. We bypass this difficulty by constructing a continuous approximation
of the Jordan decomposition. More concretely, letting G act on itself and on
G G by conjugation, we prove:
Theorem 1.7. Let G € SLn C be a connected algebraic group. For every ¡ 0
there is a continuous G-equivariant map
G Ñ G G,
ÞÑ pgs , gu q
satisfying the following properties for all g P G:
(1) g gs gu gu gs .
(2)
(3)
(4)
(5)
(6)
g
gs and gu are polynomials in g, i.e. they belong to Crg s € Mn C.
If h P G commutes with g P G, then h also commutes with gs and gu .
gs is semisimple and gu Id has spectral radius at most .
If K € G is a maximal compact group then gs , gu P K for all g P K.
If moreover G is defined over R, then ḡs gs and ḡu gu , where ¯ is
complex conjugation.
Remark. Notice that equivariance in Theorem 1.7 means that
phgh1qs hgs h1
phgh1qu hgu h1
”
Observe also that the closure of K G gPG gKg 1
and
for all g, h P G.
in G
contains non-trivial unipotent elements. In particular, Theorem 1.7 (5) cannot
be strengthened to get gs gs for all g P K.
We refer to the decomposition g gs gu provided by Theorem 1.7 as the approximated Jordan decomposition of g P G. It shares the commutativity properties of the standard Jordan decomposition g gs gu , and likewise it behaves
well with respect to conjugation, and has the property that both components
are polynomials in g. The distinct advantage of the approximated version is
that it is continuous. The main compromise is that we can only say that the
6
ALEXANDRA PETTET AND JUAN SOUTO
linear transformation gu Id has spectral radius bounded above by ; in the
standard decomposition, the spectral radius of gu is vanishing, i.e. gu is unipotent. Recall that the spectral radius of a matrix is the maximum of the norms
of its eigenvalues.
Concluding remark. The strategy of the proof of Theorem 1.1 can likely be
adapted to shed some light on whether or not HompΓ, Gq retracts to HompΓ, K q,
when Γ is a right angled Artin group [11]. The latter problem is particularly
interesting as right angled Artin groups in some sense interpolate between free
groups and free abelian groups; and although for these latter two classes of
groups, HompΓ, Gq retracts to HompΓ, K q, each does so for its own distinct reason. Finally, much of the content of Sections 2-6 seems well posed for the study
of other problems, such as computing cohomological invariants of HompZk , Gq.
Real vs. Complex. Most of the work in this paper is devoted to prove Theorem 1.1 for complex groups. The proof for real groups is almost identical, it
suffices to add only a few remarks here and there. When needed, we will discuss
the real case of the results in a section only after the complex case has been
completely treated.
Acknowledgements. The authors thank the universities of Chicago, Stanford
and Michigan for allowing them to teach linear algebra. We thank Alejandro
Ádem, Mladen Bestvina, Bill Fulton, José Manuel Gómez, Vincent Guirardel,
Mel Hochster, Mircea Mustata, Zinovy Reichstein, Ralf Spatzier and Enrique
Torres-Giese for very helpful and interesting conversations. This paper would
not have been possible without Michel Brion’s and Gopal Prasad’s willingness
to answer our many questions.
2. Action by conjugation
By an algebraic group G we mean the set of complex points of a linear algebraic group; we say that G is connected if it is connected in the Zariski topology.
If G is defined over R, then we denote by GpRq the set of real points. Note that
G and GpRq are both Lie groups satisfying
dimC pGq dimR pGpRqq
As Lie groups, both G and GpRq have finitely many connected components. In
fact, G is connected as a Lie group if and only if it is connected in the sense
above. Notice that the set GpRq of real points of a connected algebraic group
defined over R does not need to be a connected Lie group. We will denote by K
a maximal compact subgroup of G and by K pRq a maximal compact subgroup
of GpRq. It is well-known that maximal compact subgroups exist and that any
two are conjugate to each other.
Recall that the unipotent radical Ru pGq of an algebraic group G is the largest
normal connected unipotent subgroup thereof. If Ru pGq is trivial, then G is
said to be reductive. A real reductive group is the group of real points of a
reductive algebraic group defined over R.
Algebraic varieties will be, unless explicitly stated, complex affine varieties;
in fact, varieties will be more often than not actually embedded inside affine
7
space. In particular, variety is often just synonymous to Zariski closed subset
of Cn for some n.
Remark. Throughout most of this paper we will be interested in topological
properties of groups and spaces with respect to the analytic topology. Whenever
we work with the Zariski topology we will make this explicit by writing Zariski
closed, Zariski open, etc...
We refer to [7] and [32] for basic facts about algebraic groups and varieties.
See [9] for the relation between complex and real algebraic groups.
2.1. The variety of commuting tuples. Given a linear algebraic group G,
we identify the set HompZk , Gq of all homomorphism Zk Ñ G with the variety
of commuting k-tuples in G:
HompZk , Gq tpa1 , . . . , ak q P Gk |ai aj
aj ai @i, j u
In this way, HompZk , Gq becomes an affine complex algebraic variety. The
identification between HompZk , Gq and the variety of commuting k-tuples in G
depends on the choice of basis of Zk . It is well-known, and otherwise easy to
see, that the structure of HompZk , Gq as an algebraic variety does not depend
on the chosen embedding.
As an analytic space, the variety HompZk , Gq also has the associated Hausdorff topology. Notice that this topology agrees with the topology induced by
embedding HompZk , Gq into Gk , when considering the algebraic group G as a
complex Lie group.
Remark. We will denote elements in HompZk , Gq by ρ if we consider them as
homomorphisms and by pa1 , . . . , ak q if we consider them as tuples. With the
hope that no confusion arises, we feel free to pass from one form of notation to
the other.
The group G acts on HompZk , Gq by conjugation:
G HompZk , Gq Ñ HompZk , Gq
pg, pa1, . . . , ak qq ÞÑ pga1g1, . . . , gak g1q
In this section we study a few properties of this action. However, we first remind
the reader of a few facts on algebraic varieties and algebraic actions of algebraic
groups.
2.2. Quotients. Recall that any closed subgroup H of a Lie group G is also
a Lie group. It is well known that the coset space G{H is a manifold which
admits a unique smooth structure with respect to which the quotient map
π : G Ñ G{H
is the projection associated to a principal bundle with structure group H.
Similarly, any Zariski closed subgroup H of a (complex) algebraic group G
is also an algebraic group. In this setting, the coset space G{H has a unique
structure as a quasiprojective variety such that
the map π : G Ñ G{H is a morphism,
the Zariski topology of G{H is the quotient topology, and
8
ALEXANDRA PETTET AND JUAN SOUTO
for any Zariski open set U € G{H, the pull-back map is an isomorphism
between the algebras of functions on U and of H-invariant functions on
π 1 pU q .
While it is important to keep in mind that G{H is in general not affine, but
rather only quasiprojective, the following result due to Richardson [27] will have
the consequence that in all cases of interest in this paper, the coset space is in
fact affine:
Theorem 2.1 (Richardson [27]). Suppose G is a reductive algebraic group and
H € G is a Zariski closed subgroup. Then the following are equivalent:
The quasi-projective variety G{H is isomorphic to an affine variety.
H is reductive.
Still assuming that H is a Zariski closed subgroup of G, it is worth mentioning that the subjacent analytic space of the variety G{H is a smooth manifold which, as one would expect, agrees with the manifold structure mentioned
above.
We refer the reader to [14] and [7, 25] for basic properties of quotient spaces
in the smooth and algebraic categories.
2.3. Actions of reductive groups. Suppose that G is an algebraic group and
X an affine variety; both complex. An algebraic action G
X of G on X is a
morphism of varieties
ñ
GX
ÑX
satisfying the usual properties of an action of a group on a set. We refer to [10]
for a gentle introduction to the basic properties of algebraic actions.
Given an (algebraic) action G
X and a point x P X we denote by
OrbitG pxq and StabG pxq the orbit and the stabilizer of x, respectively. We
summarize a few properties of the orbits and stabilizers of algebraic groups:
ñ
(1) StabG pxq is Zariski closed in G; hence StabG pxq is an algebraic subgroup
of G.
(2) OrbitG pxq is a smooth quasi-projective variety.
(3) The map G{ StabG pxq Ñ OrbitG pxq given by g StabG pxq ÞÑ gx is an
isomorphism of quasi-projective varieties.
Notice that if OrbitG pxq happens to be a Zariski closed subset of the affine
variety X, then OrbitG pxq is itself an affine variety. We deduce hence from (3)
and Theorem 2.1:
ñ
Lemma 2.2. Suppose that G
X is an algebraic action of a reductive group
G on an affine variety X. If the orbit OrbitG pxq of some x P X is Zariski
closed, then StabG pxq is reductive.
It should be remarked that there are actions of reductive groups with reductive stabilizers for which the orbits are not closed; this is so because the image
of an affine variety under a morphism is in general not Zariski closed.
9
ñ
2.4. Orbit types and principal orbits. Suppose that G
X is an action of
a group on a set. We say that x, y P X have the same orbit type if the stabilizers
StabG pxq and StabG py q are conjugate in G.
Remark. Suppose for the sake of concreteness that G is an algebraic group, X
is an algebraic variety, and G
X is an algebraic action. Notice that x, y P X
have the same orbit type if and only if their orbits OrbitG pxq and OrbitG py q
are G-equivariantly isomorphic.
ñ
ñ
A topological (resp., algebraic) action G
X of a Lie group (resp., algebraic
group) G on a topological space (resp., algebraic variety) X is said to have a
principal orbit if there is a dense open (resp., Zariski dense Zariski open) set
U € X such that any two x, y P U have the same orbit type.
It is well-known that actions of compact Lie groups on compact manifolds
have principal orbits. As a general rule, reductive groups are the algebraic analogue of compact Lie groups. In this spirit, the following result due to Richardson [26] asserts that under suitable conditions, algebraic actions of reductive
groups on affine varieties have principal orbits as well:
ñ
Theorem 2.3 (Richardson [26]). Let G
X be an algebraic action of a reductive algebraic group G on an irreducible affine algebraic variety X. If there
is a smooth point x P X with StabG pxq reductive, then the action G
X has
a principal orbit.
ñ
In the setting of actions of compact Lie groups on compact manifold M ,
Mostow [23] proved that the stabilizers of points near x P M are conjugate to
subgroups of the stabilizer of x. In [26], Richardson also obtained an analogue
of this assertion for algebraic actions of reductive groups:
ñ
Theorem 2.4 (Richardson [26]). Let G
X be an algebraic action of an affine
algebraic group G on an affine algebraic variety X. Suppose that x P X is such
that its orbit OrbitG pxq is closed. Then there is a Zariski open set U € X
such that the stabilizer StabG py q of every y P U is conjugate to a subgroup of
StabG pxq.
Theorem 2.4 is also a consequence of Luna’s slice theorem [22], which asserts
namely that there is a StabG pxq-invariant affine subvariety Y € X with x P Y
such that the morphism
pY Gq{ StabGpxq Ñ X
is étale. In complex analytic terms, “étale” is synonymous with “local analytic isomorphism” [24, III.5 Corollary 2]. In particular, Theorem 2.4 can be
strengthened as follows: if xn Ñ x converges in the analytic topology, then there
are gn Ñ Id with gn StabG pxn qgn1 € StabG pxq.
2.5. Centralizers of semisimple tuples. Assume that G is a reductive algebraic group and recall that we identify elements ρ in HompZk , Gq with commuting k-tuples pa1 , . . . , ak q in G. As we mentioned earlier, the group G acts
on HompZk , Gq by conjugation. The stabilizer of a representation ρ under this
action of G is the centralizer of its image:
StabG pρq ZG pρpZk qq
10
where, for X
ALEXANDRA PETTET AND JUAN SOUTO
€ G,
ZG pX q tg
P G|gx xg @x P X u
We prove that the action G ñ HompZk , Gq has only finitely many orbit types
of semisimple tuples: A tuple pa1 , . . . , ak q P HompZk , Gq is semisimple if each
ai is.
Continuing with the same notation, it is well-known that every abelian subgroup of G generated by semisimple elements consists solely of semisimple elements [7]. In other words:
Lemma 2.5. ρ P HompZk , Gq is semisimple if and only if ρpZk q is an abelian
subgroup consisting of semisimple elements.
The key ingredient in the proof of the finiteness of orbit types of semisimple
tuples is the following result due to Richardson:
Theorem 2.6 (Richardson [29]). Let G be a reductive algebraic group and
consider the action G
HompZk , Gq by conjugation. If ρ P HompZk , Gq is
semisimple, then OrbitG pρq is Zariski closed in HompZk , Gq; hence, StabG pρq
is reductive.
ñ
We are now ready to prove that there are only finitely many orbit types of
semisimple tuples:
Proposition 2.7. Let G be a connected, reductive algebraic group. The action of G on HompZk , Gq by conjugation has only finitely many orbit types of
semisimple representations.
Proof. Let V be the set of all irreducible, G-invariant affine subvarieties V €
HompZk , Gq containing as a smooth point a semisimple tuple. Arguing by
induction on the dimension, we will show that each V P V consists of finitely
many orbit types. The base case of the induction, namely dim V 0, is clear,
for these are exactly the tuples of central elements in G.
So, suppose that we have proved the claim for all elements in V of at most
dimension d 1 and let V P V be of dimension d. By assumption, V contains as
a smooth point a semisimple tuple ρ. By Theorem 2.6, StabG pρq is reductive.
Hence, we can apply Theorem 2.3 and deduce that G
V has a principal
orbit. In other words, there is U € V Zariski open such that for any two
ρ1 , ρ2 P U the stabilizers StabG pρ1 q and StabG pρ2 q are conjugate. Hence it
suffices to show that the affine G-invariant variety V zU contains finitely many
orbit types of semisimple tuples; notice that V zU has at most dimension d 1.
Consider the Zariski closure V 1 of the set of semisimple tuples in V zU and let
V 1 V11 Y Y Vr1 be the decomposition into irreducible components. Clearly V 1
is G-invariant; by the connectedness of G, each Vi1 is also G-invariant. Moreover,
each Vi1 has at most dimension d 1 and contains a semisimple tuple as a
smooth point, as the set of singular points is a subvariety. From the induction
assumption, we deduce that each Vi1 contains only finitely many orbit types of
semisimple tuples. Since every semisimple tuple in V zU is contained in V 1 , this
proves that V also contains only finitely many orbit types of semisimple tuples.
ñ
11
At this point we know that every affine irreducible G-invariant subvariety
V € HompZk , Gq containing as a smooth point a semisimple tuple contains
only finitely many orbit types of semisimple tuples. As in the induction step,
the Zariski closure of the set of semisimple tuples in HompZk , Gq is a finite
union of such subvarieties. In particular, HompZk , Gq contains only finitely
many orbit types of semisimple tuples, as claimed.
2.6. Triangulations. In later sections we will need the existence of suitable
triangulations of the subset of HompZk , Gq consisting of semisimple representations. Before stating a precise result, we introduce some notation for the
sets of semisimple elements and of elements which can be conjugated into some
maximal compact subgroup K € G:
Gs
(2.1)
tg P G|g semisimpleu,
KG
¤
P
gKg 1
g G
We will also use the following notation:
(2.2)
HompZk , Gs q tpa1 , . . . , ak q P HompZk , Gq|ai
P Gs @iu
P K G @iu
Remark. Since every element in K is semisimple, we have K G € Gs .
(2.3)
HompZk , K G q tpa1 , . . . , ak q P HompZk , Gq|ai
We can now state the needed existence of triangulations:
Proposition 2.8. Suppose that G is a connected reductive algebraic group,
and let K € G be a maximal compact subgroup. There is a triangulation of
HompZk , Gs q such that
tρ P HompZk , Gsq|Dg P G with gHg1 € StabGpρqu
is a subcomplex for every subgroup H € G. Moreover, the subsets HompZk , K q
and HompZk , K G q are also subcomplexes.
XH
Recall that a triangulation of a topological space is a homeomorphism to
the geometric realization of a simplicial complex. Abusing notation, we will
not distinguish between simplicial complexes, their geometric realizations, and
triangulations.
Before launching into the proof of Proposition 2.8, we need a few preliminary
observations.
Lemma 2.9. The set tXH |H
€ Gu of subsets of HompZk , Gsq is finite.
Proof. By Proposition 2.7, the action G ñ HompZk , Gs q has only finitely many
orbits types. In other words, there is a finite set H1 , . . . , Hr of subgroups of
G such that for all ρ P HompZk , Gs q there are i P t1, . . . , ru and g P G with
gHi g 1 StabG pρq. It follows that XH is precisely the union of the sets XHi
for which there is g P G with gHg 1 € Hi . Clearly, only finitely many subsets
of HompZk , Gs q can be constructed as unions of subcollections of the finitely
many sets XH1 , . . . , XHr .
Lemma 2.10. XH is a closed subset of HompZk , Gs q for all H
€ G.
12
ALEXANDRA PETTET AND JUAN SOUTO
It is worth noticing that in general the set of those ρ P HompZk , Gs q such
that there is g P G with gHg 1 StabG pρq is neither open nor closed.
Proof. We prove that the complement of XH in HompZk , Gs q is open. Given
ρ R XH , recall that StabG pρq is reductive and that OrbitG pρq is Zariski closed
by Theorem 2.6. Theorem 2.4 implies hence that ρ belongs to a Zariski open
set U € HompZk , Gs q with the property that the stabilizer StabG pρ1 q of any
ρ1 P U is conjugate to a subgroup of StabG pρq. The assumption that H cannot
be conjugated into StabG pρq then implies that neither can it be conjugated into
StabG pρ1 q for ρ1 P U X HompZk , Gs q. This proves that U X XH H. Since U
is Zariski open, it is a fortiori open; the claim follows.
Lemma 2.11. The sets HompZk , K q and HompZk , K G q are closed in HompZk , Gs q.
Notice that K G is not a closed subset of G. This implies that HompZk , K G q
is not closed in HompZk , Gq.
Proof. Clearly HompZk , K q is closed because K € Gs is compact and hence
closed. Similarly, in order to prove that HompZk , K G q is closed in HompZk , Gs q,
it suffices to show that K G is closed in Gs . Choosing an embedding of G into
nC
SLn C for some suitable n, it suffices to prove that the closure of K G € SUSL
n
within the set of diagonalizable matrices consists of diagonalizable matrices
which can be conjugated into SUn . However, a diagonalizable matrix A P SLn C
can be conjugated into SUn if and only if all of its eigenvalues have unit norm;
this is a closed condition (compare with Lemma 7.2).
Proposition 2.8 will follow easily from the fact that semialgebraic sets are
triangulizable. Recall that a subset X € CN is semialgebraic if it is the finite
union of sets determined by a finite collection of real polynomial equalities and
inequalities.
Theorem 2.12. Let X be a semialgebraic set and tXi uiPI a locally finite collection of closed semialgebraic subsets of X. Then there is a simplicial complex
C and a homeomorphism t : C Ñ X such that each Xi is the image of a
subcomplex of C.
Theorem 2.12 is a version of Whitney’s stratification theorem [39, 40]; see
also [36]. We refer the interested reader to the Ph.D. thesis of Kyle Hofmann
[17] for a very pleasant discussion of these topics.
Lemma 2.13. The sets HompZk , Gs q, HompZk , K G q, HompZk , K q, and XH
are semialgebraic.
Proof. Embedding G into SLn C for some n, notice that K € G is equal to the
intersection of G with a conjugate of SUn . It follows that K is semialgebraic.
Since HompZk , K q is the intersection of the algebraic set HompZk , Gq with the
semialgebraic set K k , it is itself semialgebraic.
Let T be a maximal torus in G and consider the following algebraic subvariety
of pG Tqk :
X
tppg1, a1q, . . . , pgk , ak qq P pG Tqk |rgiaigi1, gj aj gj1s IdGu
13
Notice that X is an affine variety and hence a semialgebraic set of CN for some
suitable choice of N . We consider now the map
Ñ HompZk , Gq pgi, aiqi1,...,k ÞÑ pgiaigi1qi1,...,k
By definition, αpX q € HompZk , Gs q. On the other hand, the assumption that G
α:X
is connected implies that every semisimple element in G can be conjugated into
T. This implies that αpX q HompZk , Gs q. We have proved that HompZk , Gs q
is the image of a semialgebraic set under a polynomial map. A renowned theorem of Tarski [34] asserts that images under polynomial maps of semialgebraic
sets are semialgebraic. This proves that HompZk , Gs q is semialgebraic.
The same argument, replacing T by a maximal torus in K, shows that
HompZk , K G q is also semialgebraic.
It remains to prove that XH is semialgebraic for every subgroup H € G. To
begin with observe that
Fixs pH q tρ P HompZk , Gs q|H
€ StabGpρqu
is semialgebraic because it is the intersection of the fixpoint set of H on HompZk , Gq,
an affine variety, with the semialgebraic set Homs pZk , Gq. Consider the polynomial map
β : G Fixs pH q Ñ HompZk , Gs q
β : pg, ρq ÞÑ gρg 1
It follows directly from its definition that XH is equal to the image of β. Hence,
Tarski’s theorem shows again that XH is semialgebraic, as claimed.
Proposition 2.8 follows directly from Lemmas 2.9, 2.10, 2.11, 2.13, and Theorem 2.12.
2.7. The real case. We consider now the real case. More concretely, assume
that G € SLn C is a reductive algebraic group defined over R and denote by
GpRq tg
P G|ḡ gu
the group of real points, where ḡ is the complex conjugate of g.
The following lemma is one of the key facts needed to reduce the real case
to the complex case.
Lemma 2.14. Let G € SLn C be the group of complex points of an algebraic
group defined over R and H € G an algebraic subgroup also defined over R.
There are finitely many GpRq-conjugacy classes of subgroups H 1 € G which are
defined over R and conjugate to H by an element in G.
The reader can find a proof of Lemma 2.14 in [31, III 4.4]; for the sake of
completeness we give a simple argument.
Proof. Let N NG pH q be the normalizer of H in G and identify the smooth
variety G{N with the set of subgroups of G conjugate to H. The subgroup N
is a closed subgroup defined over R and hence G{N and the map G Ñ G{N
are also defined over R; compare with [7, Chapter II].
Suppose that H 1 P G{N and let g P G with H 1 gHg 1 . Taking into
account that H 1 ḡ H̄ ḡ 1 ḡH ḡ 1 we deduce that the set of real points
pG{N qpRq tgN P G{N |ḡN gN u
14
ALEXANDRA PETTET AND JUAN SOUTO
of G{N is exactly the set of subgroups of G defined over R and conjugate to H
by an element in G.
Notice that the left action G
G{N induces an action GpRq
pG{N qpRq;
the claim of Lemma 2.14 follows once we have proved that the latter action
has only finitely many orbits. Taking into account that pG{N qpRq is both a
submanifold of G{N and a real algebraic variety, and that as such, pG{N qpRq
has only finitely many connected components with respect to the Hausdorff
topology [39], we deduce that it suffices to prove that GpRq
pG{N qpRq has
open orbits.
Avoiding the introduction of new notation, we will just prove that the GpRqorbit of H is open; the argument is identical for any other point of pG{N qpRq.
Notice that the stabilizer of H under the GpRq action is the group N pRq N X GpRq of real points of N and that the induced map
ñ
ñ
ñ
GpRq{N pRq Ñ pG{N qpRq
(2.4)
is locally injective. Observe also that the dimension of pG{N qpRq as a manifold
is at most as large as the complex dimension of G{N :
dimR pG{N qpRq ¤ dimC G{N
On the other hand, since G and N are groups defined over R we have
dimR GpRq dimC G and dimR N pRq dimC N
and hence
dimR GpRq{N pRq dimC G{N ¥ dimR pG{N qpRq
This implies that the locally injective map (2.4) is open.
The space HompZk , GpRqq of representations of Zk in GpRq is the set of fixed
points under complex conjugation in the variety HompZk , Gq € G G; in
particular, HompZk , GpRqq is a real algebraic variety.
Notice that the action by conjugation G
HompZk , Gq induces the action by
k
conjugation of GpRq on HompZ , GpRqq. We prove that this action has finitely
many orbit types of semisimple representations:
ñ
Proposition 2.7 (Real case). Let GpRq be the set of real points of a reductive
algebraic group defined over R. The action GpRq
HompZk , GpRqq has only
finitely many orbit types of semisimple representations.
ñ
ñ
Proof. Consider the action G
HompZk , Gq by conjugation and observe that
the stabilizer StabG pρq ZG pρpZk qq of any point ρ P HompZk , GpRqq is a
closed subgroup defined over R. By Proposition 2.7, the set of all stabilizers
of semisimple points in HompZk , GpRqq falls into finitely many G-conjugacy
classes. Lemma 2.14 implies now that they also fall into finitely many GpRqconjugacy classes, as we needed to prove.
Continuing with the same notation, let GpRqs
semisimple elements and consider
Gs X GpRq be the set of real
HompZk , GpRqs q tpa1 , . . . , ak q P HompZk , Gq|ai
P GpRqs @iu
the set of real semisimple representations. As the fixed point set under complex
conjugation, HompZk , GpRqs q is closed semialgebraic subset of HompZk , Gs q.
15
This said, one can apply word-for-word the arguments used to prove Proposition
2.8 to show:
Proposition 2.8 (Real case). Suppose that GpRq is the group of real points
of a connected reductive algebraic group defined over R, and let K pRq € GpRq
be a maximal compact subgroup. There is a triangulation of HompZk , GpRqs q
such that
tρ P HompZk , GpRqsq|Dg P GpRq with gHg1 € StabGpRqpρqu
is a subcomplex for every subgroup H € GpRq. Moreover, HompZk , K pRqq and
HompZk , K pRqGpRq q are also subcomplexes.
XH
3. Well-centered groups
Throughout this section we assume that G is the group of complex points of a
reductive algebraic group and that K € G is a maximal compact subgroup; the
real case will only be discussed at the end. Recall that if H € G is a subgroup,
then ZG pH q is the centralizer of H in G. We denote by C pH q the center of H.
3.1. Diagonalizable groups. A Zariski closed subgroup A € G is diagonalizable if it is abelian and consists of semisimple elements. Notice that the Zariski
closure of every abelian subgroup consisting of semisimple elements in diagonalizable. Also, the center of a reductive group C pGq is diagonalizable. Conversely,
the centralizer ZG pAq of a diagonalizable subgroup A € G is reductive.
We refer the reader to [7, Chapter 8] for basic facts on diagonalizable groups;
we will just prove a few surely well-known facts for which we have not found a
reference.
Lemma 3.1. Suppose that G € SLn C is a reductive algebraic group and that
A € G is a diagonalizable subgroup. Then:
A contains a unique maximal compact subgroup Ā. Moreover, Ā is
Zariski dense in A.
A contains a Zariski dense subgroup generated by n elements.
Proof. Denote by A0 the connected component of the identity so that we have
an exact sequence
0 Ñ A0 Ñ A Ñ A{A0 Ñ 0
where A{A0 is a finite abelian group. This exact sequence splits [7, 8.7], and
hence A is the direct product of the torus A0 and a finite subgroup F of G
isomorphic to A{A0 . Identifying A0 with C C we obtain that
Ā S1 S1 F
is a maximal compact group which is clearly Zariski dense. Given any pa0 , f q P
n|F |
A0 F A, consider the powers pa0 , f qn|F | pa0 , 1q where |F | is the order of
the finite group F . It follows that if pa0 , f q belongs to some compact subgroup,
then a0 P S1 S1 . This proves that Ā is unique, and the first claim is
proved.
Starting now with the proof of the second claim, observe that the finite
subgroup F € G € SLn C is contained in an SLn C-conjugate of the group of
16
ALEXANDRA PETTET AND JUAN SOUTO
diagonal matrices with determinant one. It follows that F is isomorphic to a
subgroup of Z{q1 Z Z{qn1 Z with qi P Z. In particular, F can be generated
by n 1 elements. Since the torus A0 contains a Zariski dense cyclic subgroup,
it follows that A has a Zariski dense subgroup generated by n elements.
Suppose that pa1 , . . . , ak q P HompZk , K G q and recall that this just means that
each ai can be conjugated into K. Since K consists of semisimple elements,
Lemma 2.5 implies that the Zariski closure A € G of the group generated by
a1 , . . . , ak is a diagonalizable subgroup of G. Each ai belongs to some compact
subgroup of A and by Lemma 3.1, A contains a unique maximal compact subgroup Ā; hence ai P Ā for all i. Since Ā can be conjugated into K, we have
proven that the elements ai can be simultaneously conjugated into K.
Lemma 3.2. An element ρ P HompZk , Gq belongs to HompZk , K G q if and only
if there is g P G with gρpZk qg 1 € K.
3.2. Trans. Given two subgroups H, H 1 of G, we denote by
TransG pH, H 1 q tg
P G|gHg1 € H 1u
the subset of G consisting of all elements that conjugate H into H 1 . Notice that
if H is Zariski closed, then
TransG pH, H q NG pH q
is the normalizer of H in G.
Suppose that A, A1 € G are diagonalizable. It is easy to see that TransG pA, A1 q
is an algebraic variety [7, Chapter I] and hence has finitely many connected components. We show now that each is homeomorphic to the connected component
of the identity in the centralizer ZG pAq of A in G.
Lemma 3.3. Suppose that A, A1 are diagonalizable subgroups of G, and let V
be a connected component of TransG pA, A1 q. For all g0 P V , the map
Z G p Aq0
Ñ V,
h ÞÑ g0 h
is an isomorphism. Here, ZG pAq0 is the identity component of the centralizer
of A in G.
In particular, the Weyl group WA NG pAq{ZG pAq is finite for every diagonalizable subgroup A of G.
Proof. With the same notation as in the statement of the lemma, observe that
by Lemma 3.1 we have
TransG pA, A1 q TransG pĀ, Ā1 q
where Ā, Ā1 are unique maximal compact subgroups of A and A1 . Endowed
with the compact open topology, HompĀ, Ā1 q is a totally disconnected space.
Hence, the map
TranspA, A1 q Ñ HompA, A1 q, g
ÞÑ ta ÞÑ gag1u
is constant on each connected component (compare with [7, 8.10]). In other
words, it follows that
gag 1 g0 apg0 q1
17
for all a P A and all g P V . This implies in particular that g01 g
all g P V . Since on the other hand the morphism
ZG pAq0
Ñ V,
P ZGpAq0 for
h ÞÑ g0 h
is injective, the claim follows.
The following result, of capital importance in this paper, relates TransG and
TransK where K € G is a maximal compact subgroup.
€ K are subgroups of K, then the inclusion
TransK pH1 , H2 q ãÑ TransG pH1 , H2 q
Lemma 3.4. If H1 , H2
is a homotopy equivalence. In particular we have:
(1) H1 , H2 € K are conjugate in G if and only if they are in K.
(2) The inclusion NK pH1 q Ñ NG pH1 q is a homotopy equivalence.
ã
Before proving Lemma 3.4 we need to recall a few facts about the Cartan
decomposition. It is well-known that the Lie algebra g of every reductive group
G contains an AdpK q-invariant linear subspace p € g such that the map
pK
Ñ G, pX, kq ÞÑ eX k
is a diffeomorphism. In particular, we can write every element g P G uniquely
as g eXg kg , where Xg P p and kg P K; this is the Cartan decomposition of g
associated to K.
Lemma 3.5. Suppose that k P K and g eXg kg P G are such that gkg 1
Then kg kkg1 gkg 1 and Xg P FixpAdpgkg 1 qq.
Proof. We have eXg kg kkg1 eXg
P K.
gkg1 k1 P K and hence that
eXg kg kkg1
k1eX eAdpk1qpX qk1
g
g
The claim follows now from the uniqueness of the Cartan decomposition.
Proof of Lemma 3.4. By Lemma 3.5 we have for g eX k P G that gH1 g 1 H2 if and only if kH1 k 1 H2 and X P FixpAdpH2 qq. It follows that there is
a diffeomorphism
TransG pH1 , H2 q TransK pH1 , H2 q FixpAdpH2 qq
Notice that FixpAdpH2 qq is contractible because it is a linear subspace of p. It
follows that the inclusion
TransK pH1 , H2 q Ñ TransG pH1 , H2 q
is a homotopy equivalence.
18
ALEXANDRA PETTET AND JUAN SOUTO
3.3. Well-centered subgroups. We will be interested in a particular class of
diagonalizable subgroups of the reductive group G and its maximal compact
subgroup K.
A diagonalizable subgroup A € G is well-centered in G if it is the center
of its centralizer: A C pZG pAqq. Similarly, an abelian subgroup H € K is
well-centered in K if H C pZK pH qq. Recall that the Zariski closure in G of
every abelian subgroup of K is diagonalizable.
Well-centered groups will play a central rôle in this paper; we discuss here
briefly what they are in the particular case that G SLn C.
Example: well-centered groups in SLn C. It is well-known that every
diagonalizable subgroup A of SLn C is “digonalizable,” meaning that there is a
direct sum decomposition
Cn
(3.1)
V1 ` ` Vr
such that every element a P A preserves each one of the factors Vi and acts on
it by a homothety. In fact, taking the direct sum decomposition (3.1) with the
smallest possible number of factors, we have the following description of the
centralizer of A:
(3.2)
ZSLn C pAq SLn C X pGLpV1 q GLpVr qq
We might refer to the unique direct sum decomposition satisfying (3.2) as the
canonical direct sum decomposition diagonalizing A; notice that its factors are
not ordered.
The center of the centralizer of A is then the subgroup of SLn C consisting
of all elements diagonalized by the canonical direct sum decomposition (3.1)
associated to A:
(3.3) C pZSLn C pAqq tg
P SLn C|@i 1, . . . , r Dλi with gvi λivi @vi P Viu
From this discussion we deduce that
there is a natural correspondence between the sets of well-centered
groups in SLn C and of unordered direct sum decompositions of
Cn .
Returning now to the general setting, we state for further reference a few
general facts about well-centered groups:
Lemma 3.6.
(1) We have C pZG pAqq ZG pZG pAqq for any A € G abelian.
(2) Suppose that A1 and A2 are well-centered groups in G. Then A1  A2
if and only if ZG pA1 q € ZG pA2 q.
(3) Suppose that A € G is an abelian subgroup consisting of semisimple
elements. Then  C pZG pAqq is the smallest well-centered group containing A.
The proof of Lemma 3.6 is a straightforward computation in elementary
group theory; notice only that in (3), the proviso that A consist of semisimple
elements is just to ensure that  is diagonalizable. We leave the proof of Lemma
3.6 to the reader.
19
Example: well-centered groups in SLn C and polynomials. As a matter
of example we discuss the third claim of Lemma 3.6 for G SLn C and A
generated by a single semisimple element a P SLn C. Denote by (3.1) the
canonical direct sum decomposition associated to the Zariski closure of A xay
and consider
A tg
P MnC|@i 1, . . . , r Dλi with gvi λivi @vi P Viu
i.e. the linear subspace of the vector space of all n-by-n-matrices which are
diagonalized by (3.1). The space A has dimension r and the Vandermonde
determinant implies that the elements Id, a, a2 , . . . , ar1 P A are linearly independent. It follows that A can be identified with the subring Cras of Mn C
generated by a. From (3.3) we deduce hence that
C pZSLn C paqq Cras X SLn C
Taking into account that ZG paq ZSLn C paqX G for every subgroup G of SLn C
we deduce:
Lemma 3.7. Suppose that G € SLn C is an algebraic group and that a P G
is semisimple. Then C pZG paqq Cras X G is the smallest well-centered group
containing a.
After these remarks we come now to the central observation of this section:
Lemma 3.8. Suppose that H € K is well-centered in K. Then H is the unique
maximal compact subgroup in the well-centered group C pZG pH qq. In particular
H
is Zariski dense in C pZG pH qq.
K X C pZGpH qq
Proof. By Lemma 3.6, Ĥ C pZG pH qq is well-centered and in particular diagonalizable. Lemma 3.1 implies hence that Ĥ has a unique maximal compact
subgroup H̄ € Ĥ. Since H̄ is also Zariski dense, by Lemma 3.1 it will suffice
to prove that H H̄. To begin with, notice that
H
€ H̄
We have to prove the opposite inclusion. Since H̄ is compact, there is g eX k P G with g H̄g 1 € K. By Lemma 3.5 we have ghg 1 khk 1 for all
h P H. So, replacing g by k 1 g, we may assume that g P ZG pH q. Now, since
Ĥ is the center of ZG pH q we have that g Ĥg 1 Ĥ. By uniqueness of the
maximal compact subgroup H̄ of Ĥ, we have g H̄g 1 H̄. But this proves that
H̄ € K. Hence,
H̄
€ K X ZG pH q ZK pH q
Now, H̄ is central in ZG pH q and hence a fortiori in ZK pH q. This shows that
H̄
This proves the claim.
€ C pZK pH qq H
20
ALEXANDRA PETTET AND JUAN SOUTO
3.4. Manifolds of well-centered groups. Continuing with the same notation, let
G tA € G|A well-centered in Gu
K tH
€ K |H well-centered in K u
The groups G and K act, from the left, on G and K by conjugacy.
Lemma 3.9. The actions G
ñ G and K ñ K have finitely many orbits.
Proof. By Lemma 3.1, there is some fixed n such that every well-centered subgroup A of G contains a Zariski dense subgroup B generated by n elements.
Observe that A and B have the same centralizer. Since B is generated by
n elements, there is a semismple representation Zn Ñ G with image B. By
Proposition 2.7, there are only finitely many conjugacy classes of centralizers
of images of semisimple representations Zn Ñ G. This proves that there are
finitely many conjugacy classes of centralizers of well-centered groups.
Suppose now that A and A1 have conjugate centralizers. Then we have
A1
C pZGpA1qq C pgZGpAqg1q g C pZGpAqq g1 gAg1
It follows that A and A1 belong to the same orbit of G under conjugation. We
have proved that G
G has only finitely many orbits.
The claim for K
K follows from the validity of the claim for G
G and
Lemma 3.4 and Lemma 3.8.
ñ
ñ
ñ
Give G and K the topology in which each orbit has the obvious structure
as a homogenous manifold. More concretely, the connected component of G to
which some well-centered group A belongs, is diffeomorphic to G{NG pAq.
Example: manifolds of well-centered groups in SLn C. As mentioned
above, there is a natural correspondence between the sets of well-centered
groups in SLn C and of unordered direct sum decompositions of Cn . In particular, the manifolds of well-centered groups in SLn C are disjoint unions of
quotients of products of open sets in Grassmanians. Similarly, the manifolds
of well-centered groups in SUn are disjoint unions of quotients of open sets in
compact flag manifolds.
From the proof of Lemma 3.9 we deduce that two well-centered groups A, A1 P
G are in the same connected component of G if and only if they, or equivalently
their centralizers, are conjugate in G. We state this fact for later reference:
Lemma 3.10. For any two well-centered groups A, A1 P G the following are
equivalent:
A and A1 are in the same connected component of G.
A and A1 are conjugate in G.
ZGpAq and ZGpA1q are conjugate in G.
We come now to the main result of this section:
Proposition 3.11. The map ι : K
homotopy equivalence.
Ñ G, ιpH q C pZGpH qq is an injective
21
Proof. Suppose that H, H 1 P K are such that ιpH q ιpH 1 q. By Lemma 3.8,
H and H 1 are each the unique maximal compact subgroup of the well-centered
group ιpH q ιpH 1 q; in other words, H H 1 . This proves that ι is injective.
We prove next that ι induces a bijection between the connected components
of K and G. Suppose that H, H 1 P K are such that ιpH q and ιpH 1 q belong
to the same connected component of G. In other words, there is g P G with
ιpH 1 q gιpH qg 1 . We deduce again from Lemma 3.8 that H 1 gHg 1 .
Lemma 3.4 implies then that H and H 1 are conjugate within K. This shows
that H and H 1 belong to the same connected component of K. We have proven
that the map between connected components is injective. Given now A P G,
let Ā be the unique maximal compact subgroup of A, and let g P G be such
that g Āg 1 € K. Since g Āg 1 is the unique maximal compact subgroup of
the well-centered group gAg 1 , it follows from Lemma 3.8 that g Āg 1 P K and
that
ιpg Āg 1 q gAg 1
Since A and gAg 1 belong to the same connected component of G we deduce
that the map between the sets of connected components is also surjective.
It remains to prove that the restriction of ι to each connected component
K0 € K is a homotopy equivalence into the corresponding component G0 € G.
Fix H0 P K0 and let A0 ιpH0 q P G0 . By definition we have that G0 OrbitG pA0 q, that K0 OrbitK pH0 q, and that
StabK pH0 q NK pH0 q, StabG pA0 q NG pA0 q NG pH0 q
where the last equality holds by Lemma 3.1 and Lemma 3.8.
Lemma 3.4 implies that the inclusion StabK pH0 q Ñ StabG pA0 q is a homotopy
equivalence. In particular we have a diagram as follows
StabK pH0 q
/K
/ K0
ι
StabG pA0 q
/G
/ G0
where the rows are fibrations and where the first two vertical arrows are homotopy equivalences. It follows that the third vertical arrow is also a homotopy
equivalence, as claimed.
3.5. Orderings on π0 pG q and π0 pKq. We introduce a partial order on the set
π0 pG q of connected components of G as follows: if G1 and G2 are connected
components of G then we say that
G1
¥ G2
K1
¥ K2
if for some, and hence for every A1 P G1 there is A2 P G2 with A1  A2 . Recall
that Lemma 3.6 implies that this is equivalent to asking that for every A1 P G1
there is A2 P G2 with ZG pA1 q € ZG pA2 q.
On π0 pKq we consider the analogously defined partial ordering: if K1 and K2
are connected components of K, then we say that
22
ALEXANDRA PETTET AND JUAN SOUTO
if for some, and hence for every H1 P K1 , there is H2 P K2 with H1  H2 .
Lemmas 3.1, 3.6, and 3.8 imply that, as above, this is equivalent to asking that
for every H1 P K1 there is H2 P K2 with ZG pH1 q € ZG pH2 q.
The following claim follows directly from the definitions:
Lemma 3.12. The homotopy equivalence ι : K Ñ G induces an order preserving
bijection π0 pιq : π0 pKq Ñ π0 pG q.
Henceforth we will identify K and ιpKq and so feel free to write K € G.
3.6. The set RG1 ,G2 . Suppose that we have two components G1 , G2 of G with
G1 ¥ G2 and consider the set
RG1 ,G2
(3.4)
tpA1, A2q P G1 G2|A1  A2u
Our next goal is to prove that the projections of RG1 ,G2 onto G1 and G2 are
respectively a finite cover and a locally trivial fibration. Before doing so, notice
that for any pA1 , A2 q P RG1 ,G2 the groups NG pA1 q and NG pA2 q act from the
right on TransG pA2 , A1 q:
TransG pA2 , A1 q NG pA2 q Ñ TransG pA2 , A1 q
pg, hq ÞÑ gh
TransG pA2 , A1 q NG pA1 q Ñ TransG pA2 , A1 q pg, hq ÞÑ h1 g
We prove:
Lemma 3.13. Fixing pA1 , A2 q P RG1 ,G2 the following holds:
The projection p2 : RG1,G2 Ñ G2 on the second factor is a locally trivial
fiber bundle with fiber TransG pA2 , A1 q{NG pA1 q.
The projection p1 : RG1,G2 Ñ G1 on the first factor is a finite covering
with fiber TransG pA2 , A1 q{NG pA2 q.
Proof. Identifying G1 and G2 with the G-orbits of A1 and A2 , we obtain identifications
G1 G{NG pA1 q, G2 G{NG pA2 q
Consider the standard right action of NG pA1 q NG pA2 q on G G
pG Gq pNGpA1q NGpA2qq Ñ G G
ppg1, g2q, ph1, h2qq ÞÑ pg1h1, g2h2q
and notice that the set
R̃ tpg1 , g2 q P G G|g1 A1 g11
 g2A2g21u
is NG pA1 q NG pA2 q-invariant. In fact we have that
R̃{pNGpA1q NGpA2qq
The projection p̃2 : R̃ Ñ G, p̃2 pg1 , g2 q g2 has fibers
1
1
p̃
2 pg2 q tpg1 , g2 q P G G|g1 g2 P TranspA2 , A1 qu
It follows that p̃2 : R̃ Ñ G is a locally trivial fiber bundle over G, with fiber
TranspA2 , A1 q. From here we obtain that the projection
p2 : RG ,G Ñ G2
is also a locally trivial fiber bundle with fibers TranspA2 , A1 q{NG pA1 q.
RG1 ,G2
1
2
23
The same reasoning yields that the projection
p1 : RG1 ,G2
Ñ G1
is also a locally trivial fiber bundle with fibers TranspA2 , A1 q{NG pA2 q. It follows
from Lemma 3.3 that the quotient TranspA2 , A1 q{NG pA2 q is finite. As it is a
locally trivial fiber bundle with discrete fibers, p1 : RG1 ,G2 Ñ G1 is a cover, as
claimed.
Recall that we are identifying K with a subset of G. Given G1 , G2 P π0 pG q with
G1 ¥ G2 let K1 € G1 and K2 € G2 be the corresponding connected components
of K and set
(3.5)
OG1 ,G2
RG ,G X pK1 K2q tpH1, H2q P K1 K2|H1  H2u
1
2
The same argument used to prove Lemma 3.13 yields:
Lemma 3.14. Fixing pH1 , H2 q P OG1 ,G2 , the following holds:
The projection p2 : OG1,G2 Ñ K2 on the second factor is a locally trivial
fiber bundle with fiber TransK pH2 , H1 q{NK pH1 q.
The projection p1 : OG1,G2 Ñ K1 on the first factor is a finite covering
with fiber TransK pH2 , H1 q{NK pH2 q.
3.7. The real case. Suppose now that the reductive group G is defined over
R, let GpRq be the group of real points and K pRq a maximal compact subgroup
of GpRq.
We will say that A € GpRq is diagonalizable if its Zariski closure ZarpAq € G
is diagonalizable and satisfies A ZarpAqpRq. A diagonalizable subgroup A of
GpRq (resp. K pRq) is well-centered in GpRq (resp. in K pRq) if it is equal to
the center of its centralizer in GpRq (resp. in K pRq). As above, we denote by
G and K the set of all well-centered subgroups of GpRq and K pRq respectively.
We state now a few facts whose proofs we leave to the reader:
(1) If A € GpRq is an abelian subgroup consisting of semisimple elements, then
ZarpAqpRq (resp. C pZGpRq pAq) is the smallest diagonalizable subgroup of GpRq
(resp. smallest well-centered group in GpRq) containing A.
(2) If A € GpRq is diagonalizable, then the intersection of A and the unique
maximal compact subgroup of ZarpAq is the unique maximal compact subgroup
of A. Moreover, the Zariski closures of A and its unique maximal compact
subgroup agree.
(3) If A, A1 € GpRq are diagonalizable, then TransGpRq pA, A1 q is the subset of
TransG pZarpAq, ZarpA1 qq fixed by complex conjugation and hence a real algebraic variety; as such, it has only finitely many connected components.
(4) If A, A1 € GpRq are diagonalizable, V € TransGpRq pA, A1 q is a connected
component, and g0 P V , then the map ZGpRq pAq0 Ñ V, h ÞÑ g0 h is an isomorphism; here ZGpRq pAq0 is the connected component of the identity in the Lie
group ZGpRq pAq.
(5) If H1 , H2 € K pRq are subgroups, then the map TransK pRq pH1 , H2 q Ñ
TransGpRq pH1 , H2 q is a homotopy equivalence.
(6) If H P K is well-centered in K pRq, then H is the unique maximal compact
subgroup of the well-centered group C pZGpRq qpH q P G.
ã
24
ALEXANDRA PETTET AND JUAN SOUTO
(7) The actions GpRq
ñ G and K pRq ñ K have finitely many orbits.
Remarks about the proofs of (1)-(7). Claims (1),(2) and (3) are just elementary
considerations; compare with the proof of Lemma 2.14 for (3). Lemma 3.3 and
(3) imply (4). The group GpRq has a Cartan decomposition, now called polar
decomposition; this said, Lemma 3.5 holds and hence the proofs of Lemma 3.4
and Lemma 3.8 transfer word-for-word to prove (5) and (6). Claim (7) follows
from Lemma 3.9 and Lemma 2.14.
Before going any further, observe that (4) implies:
Lemma 3.3 (Real case). The Weyl group WA
for every diagonalizable subgroup A of GpRq.
NGpRqpAq{ZGpRqpAq is finite
Endow G (resp. K) with the manifold structure induced by the action of
GpRq (resp. K pRq). With facts (1)-(6) on hand, we can repeat word-for-word
the proof of Proposition 3.11 and show:
Proposition 3.11 (Real case). The map ι : K Ñ G, ιpH q C pZGpRq pH qq is
an injective homotopy equivalence.
At this point we can define orderings on π0 pG q and π0 pKq exactly as we did
in Section 3.5 in the complex case. We have then:
Lemma 3.12 (Real case). The homotopy equivalence ι : K
order preserving bijection π0 pιq : π0 pKq Ñ π0 pG q.
Ñ G induces an
As for the complex case, we identify K and ιpKq € G. For any two components
G1 , G2 of G with G1 ¥ G2 containing components K1 € G1 and K2 € G2 of
K we define the sets RG1 ,G2 and OG1 ,G2 exactly as we did in (3.4) and (3.5).
Armed with Facts (1)-(7) we can repeat word-for-word the same arguments as
in Section 3.6 to prove:
Lemma 3.13 (Real case). Fixing pA1 , A2 q P RG1 ,G2 the following holds:
The projection p2 : RG1,G2 Ñ G2 on the second factor is a locally trivial
fiber bundle with fiber TransGpRq pA2 , A1 q{NGpRq pA1 q.
The projection p1 : RG1,G2 Ñ G1 on the first factor is a finite covering
with fiber TransGpRq pA2 , A1 q{NGpRq pA2 q.
Lemma 3.14 (Real case). Fixing pH1 , H2 q P OG1 ,G2 , the following holds:
The projection p2 : OG1,G2 Ñ K2 on the second factor is a locally trivial
fiber bundle with fiber TransK pRq pH2 , H1 q{NK pRq pH1 q.
The projection p1 : OG1,G2 Ñ K1 on the first factor is a finite covering
with fiber TransK pRq pH2 , H1 q{NK pRq pH2 q.
4. The canonical bundle over G
In this section, let G be the group of either complex or real points of a
reductive algebraic group (defined over R in the latter case), let K € G be a
maximal compact subgroup, and recall that G is the manifold of all well-centered
subgroups of G. In this section we study a certain flat bundle
π:D
ÑG
25
We take a flat bundle to mean a bundle endowed with a complete flat connection; in more concrete terms, the total space D is foliated in such a way that
the restriction of π to each leaf of the foliation is a covering of G.
We also construct for all k a map
Ψ : Dpk q Ñ HompZk , Gs q
where Dpk q Ñ G is the k-fold direct product of the bundle D Ñ G with itself.
Finally, we describe how Ψ is related to the ordering of π0 pG q and the isotopies
induced by the flat connection.
4.1. Canonical bundle. Endow D tpA, aq P G G|a P Au with the subspace
topology and consider the projection
Ñ G, pA, aq ÞÑ A
onto the first factor. We claim that π : D Ñ G is a locally trivial bundle.
π:D
Fixing a connected component G0 of G set
DG0
π1pG0q tpA, aq P G0 G|a P Au
and consider the restriction of π to DG0 . In order to prove that π : D Ñ G is a
locally trivial bundle, it suffices to prove it for π : DG0 Ñ G0 .
Fixing a base point A0 P G0 recall that, by construction, G0 is equal to the
orbit of A0 under the action of G by conjugation. It follows that the map
Ñ DG , pg, aq ÞÑ pgA0g1, gag1q
is surjective. The normalizer NG pA0 q of A0 in G acts from the right on G A0
σ0 : G A0
0
as follows:
pG A0q NGpA0q Ñ G A0, ppg, aq, hq ÞÑ pg, aq h pgh, h1ahq
A simple computation shows that the fibers of the map σ0 agree with the
NG pA0 q-orbits on G A0 . It follows that σ0 descends to a homeomorphism
τ0 : pG A0 q{NG pA0 q Ñ DG0
Notice that, as G acts transitively on G0 , the base point A0 yields a canonical
identification G0 G{NG pA0 q. The projection G A0 Ñ G onto the first factor
induces a projection
π 1 : pG A0 q{NG pA0 q Ñ G{NG pA0 q G0
Moreover, the projection π 1 and the homeomorphism τ0 are such that the following diagram commutes:
pG A0q{NGSpSAq
τ0
SSS
SSS
SSS
SSS
π1
)
/D
kk G0
k
k
k
π kkk
kkk
k
k
ku kk
G{NGpA0q
In particular, in order to prove that π : DG Ñ G0 is a locally trivial bundle it
G0
0
will be sufficient to prove it for
π 1 : pG A0 q{NG pA0 q Ñ G{NG pA0 q
26
ALEXANDRA PETTET AND JUAN SOUTO
Recall that by Lemma 3.3, the group
WA0
NGpA0q{ZGpA0q
is finite. Taking into account that
pG A0q{NGpA0q pG{ZGpA0qq A0 {WA
G{NG pA0 q pG{ZG pA0 qq{WA
0
0
we obtain a commuting diagram as follows:
G{ZG pA0 q A0
G{ZG pA0 q
ñ
WA
ñ
WA
0
p A0q{NGpA0q
/ G
0
π1
{ p q
/ G NG A 0
Here the left arrow is the projection onto the first factor, and the horizontal
arrows are the orbit maps of the action of WA0 .
By construction, WA0 acts on the trivial bundle G{ZG pA0 q A0 Ñ G{ZG pA0 q
by bundle homomorphisms. Moreover, the action is free and a fortiori discrete
because WA0 is finite. It follows that the quotient π 1 : pG A0 q{NG pA0 q Ñ
G{NG pA0 q is a locally trivial bundle. We have proved that
π:D
ÑG
is a locally trivial bundle. We will refer to this bundle as the canonical bundle
over G.
Continuing with the same notation, endow now the trivial bundle pG{ZG pA0 qq
A0 Ñ G{ZG pA0 q with the flat connection induced by the product structure. In
other words, the leaves of the foliation associated to this flat connection are the
sets of the form pG{ZG pA0 qq tau. Since the group WA0 preserves the product
structure, it follows that this flat connection descends to a flat connection on
the canonical bundle. We have hence:
Proposition 4.1. The canonical bundle π : D
bundle.
Ñ
G is a flat locally trivial
4.2. Horizontal lifts. The use of having a flat connection on the bundle π :
D Ñ G is that to every curve γ : r0, 1s Ñ G and to every point a P π 1 pγ p0qq €
D we can associate the horizontal lift of γ starting at a. This is namely the
unique curve
γ̃ : r0, 1s Ñ D
with γ̃ p0q a and π pγ̃ ptqq γ ptq for all t P r0, 1s, and whose image is contained
in a single leaf of the foliation associated to the flat connection.
Our next goal is to give a concrete description of the horizontal lifts of curves
γ : r0, 1s Ñ G. To do so, let G0 be the connected component of G containing
the image of γ, fix A P G0 , and recall that the map
ÞÑ gAg1
is a fiber bundle. Hence, every curve γ : r0, 1s Ñ G admits a (far from unique)
lift r0, 1s Ñ G, t ÞÑ gt . The condition that t ÞÑ gt is a lift of γ amounts to
γ ptq gt Agt1 pgt g01 qγ p0qpgt g01 q1 for all t
G Ñ G0 , g
27
Notice that any element in π 1 pγ p0qq
γ p0q g0 Ag01 € G. The curve
P D is of the form pγ p0q, aq, where a P
γ̃ : r0, 1s Ñ D, t ÞÑ ppgt g01 qγ p0qpgt g01 q1 , pgt g01 qapgt g01 q1 q
is both parallel and a lift of γ; in other words, γ̃ is the horizontal lift of γ with
γ̃ p0q pγ p0q, aq. We have proved:
Lemma 4.2 (Horizontal lift). Let γ : r0, 1s Ñ G be a continuous curve and fix
a point A in the connected component of G containing the image of γ.
For any a P γ p0q P G, the horizontal lift γ̃ : r0, 1s Ñ D of γ starting at
γ̃ p0q pγ p0q, aq is given by
γ̃ ptq ppgt g01 qγ p0qpgt g01 q1 , pgt g01 qapgt g01 q1 q
where r0, 1s Ñ G, t ÞÑ gt is an arbitrary curve with γ ptq gt Agt1 for all t.
4.3. The evaluation map. Fix now k, recall that HompZk , Gs q is the set of
commuting semisimple k-tuples in G, and denote by
π : D pk q Ñ G
the k-th cartesian product of the bundle D
total space Dpk q is given by
Ñ G.
In more concrete terms, the
Dpk q tpA, a1 , . . . , ak q P G G G|a1 , . . . , ak
P Au
As a cartesian product of flat bundles, Dpk q is naturally endowed with a flat
connection.
Forgetting for a moment the bundle structure of Dpk q Ñ G, we consider the
following continuous map defined on the total space:
(4.1)
Ψ : Dpk q Ñ HompZk , Gs q,
pA, a1, . . . , ak q ÞÑ pa1, . . . , ak q
Here we are identifying representations of Zk with k-commuting tuples in G.
Notice that the evaluation map Ψ is well-defined because every well-centered
group A P G is, by definition, abelian and consists of semisimple elements.
Lemma 4.3. The evaluation map Ψ : Dpk q Ñ HompZk , Gs q is surjective.
Proof. Suppose that we are given ρ P HompZk , Gs q. By Lemma 2.5, the abelian
group ρpZk q consists of semisimple elements. In particular, its Zariski closure
is diagonalizable. It follows hence from Lemma 3.6 that A C pZG pρpZk qqq is
a well-centered group containing ρpZn q. Let ai ρpei q where e1 , . . . , ek is the
standard basis of Zk . Then pA, a1 , . . . , ak q P Dpk q and ρ ΨpA, a1 , . . . , ak q by
construction.
Before moving on we observe the following useful fact:
Lemma 4.4. We have ΨpgAg 1 , gαg 1 q
and g P G.
gΨpA, αqg1 for all pA, αq P Dpkq
Example: horizontal lifting and evaluation map for SLn C. In the case of
SLn C, the combination of horizontal lifting and the evaluation map just means
the following: Suppose that a matrix A P SLC is diagonalized by a direct sum
decomposition Cn V1 ` ` Vr and let λi be the homothety factor of A when
28
ALEXANDRA PETTET AND JUAN SOUTO
restricted to Vi . If we perturb each one of the spaces Vi slightly to get some
other space Vi1 , then there is a unique matrix A1 P SLn C which, restricted to
Vi1 , is a homothety with factor λi .
4.4. Compatibility of horizontal lifts. Suppose that we are given G1 , G2 P
π0 pG q with G1 ¥ G2 , and recall that this means that some, and hence every,
A2 P G2 can be conjugated into some, and hence into every, A1 P G1 . Recall
that by Lemma 3.13, the projections of the set
RG1 ,G2
tpA1, A2q P G1 G2|A1  A2u
onto G1 and G2 are respectively a finite cover and a locally trivial fibration.
Lemma 4.5 (Compatibility of horizontal lifts). Suppose that we are given a
curve
r0, 1s Ñ RG ,G , t ÞÑ pγ1ptq, γ2ptqq
and two points pγ1 p0q, α1 q P DG pk q and pγ2 p0q, α2 q P DG pk q, and consider the
1
2
1
2
horizontal lifts
γ̃1 : r0, 1s Ñ DG1 pk q, γ̃2 : r0, 1s Ñ DG2 pk q
of γ1 and γ2 , with γ̃i p0q pγi p0q, αi q for i 1, 2.
If Ψpγ1 p0q, α1 q Ψpγ2 p0q, α2 q, then Ψpγ̃1 ptqq Ψpγ̃2 ptqq for all t.
Proof. Setting pA1 , A2 q pγ1 p0q, γ2 p0qq, we obtain, as so often above, the identifications
G{NGpA2q
Consider G Ñ G1 as a fiber bundle, and let t Ñ gt be any continuous lift of the
curve t ÞÑ γ1 ptq with g1 p0q Id. Recall that by Lemma 3.13, the projection
p1 : RG ,G Ñ G1
G1
G{NGpA1q,
1
G2
2
is a finite covering. The two curves
r0, 1s Ñ RG ,G t ÞÑ pγ1ptq, γ2ptqq
r0, 1s Ñ RG ,G t ÞÑ pgtA1gt1, gtA2gt1q pγ1ptq, gtA2gt1q
are lifts of t ÞÑ γ1 ptq under the covering p1 : RG ,G Ñ G1 ; since both curves
agree for t 0, it follows that γ2 ptq gt A2 gt1 for all t. This proves that the
curve t ÞÑ gt is also a lift of the curve t ÞÑ γ2 ptq for the fiber bundle G Ñ G2 .
Lemma 4.2 shows that the horizontal lift γ̃i : r0, 1s Ñ Di of γi with γ̃i p0q pγip0q, αiq is given by
γ̃i ptq pgt γi p0qgt1 , gt αi gt1 q
for i 1, 2. Lemma 4.4 implies that
Ψpγ̃i ptqq gt Ψpγ̃i p0qqgt1
1
2
1
2
1
The claim follows.
2
29
4.5. An interesting subbundle. Continuing with the same notation, recall
that by Lemma 3.1, every well-centered group A P G contains a uniquely determined maximal compact subgroup Ā. Consider the subset
D̄
tpA, aq P G G|a P Āu € D
and notice that the uniqueness of the maximal compact subgroups implies that
gAg 1 g Āg 1 for all A P G and all g P G. It follows hence from Lemma 4.2
that for any pA, aq P D̄ and any curve γ : r0, 1s Ñ G with γ p0q A, we have
γ̃ ptq P D̄ for all t, where γ̃ is the horizontal lift of γ with γ̃ p0q pA, aq. In other
words, the subset D̄ of the total space of the bundle D Ñ G is parallel.
This implies directly that the subset
D̄pk q tpA, a1 , . . . , ak q P G G G|a1 , . . . , ak
P Āu
is a parallel subset of the bundle π : Dpk q Ñ G. Observe now that any parallel
subset of a flat bundle is a flat bundle in its own right. We have proved:
Lemma 4.6. π : D̄pk q Ñ G is a flat subbundle of π : Dpk q Ñ G.
Recall at this point that K is the manifold of well-centered subgroups of K,
and that by Proposition 3.11 the map
ι : K Ñ G, ιpH q C pZG pH qq
is an injective homotopy equivalence. Recall also the definition (2.3) of HompZk , K G q.
Lemma 4.7. With the same notation as above, we have
(1) ΨpD̄pk qq HompZk , K G q, and
(2) ΨpD̄pk q X π 1 Kq HompZk , K q.
Proof. Recall that pA, a1 , . . . , ak q P D̄pk q if and only if a1 , . . . , ak P Ā where Ā
is the unique maximal compact subgroup of A. Since Ā is compact, there is
g P G with Ā € gKg 1 . In particular, a1 , . . . , ak P K G YgPG gKg 1 ; this
proves that
ΨpA, a1 , . . . , ak q pa1 , . . . , ak q P HompZk , K G q
and hence that ΨpDpk qq € HompZk , K G q. The opposite inclusion follows from
a similar, simpler, argument (compare with Lemma 3.2); this proves (1).
Continuing with the same notation assume now that A P K € G and recall
that this means that there is a well-centered H € K with A C pZG pH qq.
Lemma 3.8 shows that H is the maximal compact subgroup of A, i.e. H Ā.
Since H € K, it follows that ΨpA, a1 , . . . , ak q pa1 , . . . , ak q P HompZk , K q for
any a1 , . . . , ak P Ā H. This proves that ΨpD̄pk q X π 1 Kq € HompZk , K q.
The opposite inclusion follows again by a similar, again simpler, argument.
This proves (2).
5. The homotopic framework
In this section we introduce some terminology and state a topological result,
Proposition 5.3, which will play a key rôle in the proof of Theorem 1.1. We
defer its proof to the Appendix.
All topological spaces in this section will be CW-complexes; or more precisely,
homeomorphic to CW-complexes.
30
ALEXANDRA PETTET AND JUAN SOUTO
5.1. Regular coverings of simplicial complexes. Suppose that X is a (finite dimensional, locally finite) simplicial complex and denote by C pX q the set
of all closed cells of X. Abusing terminology, we will not distinguish between
the abstract complex X and its geometric realization. We denote by X pdq the
d-skeleton of X, i.e. the simplicial complex consisting of all the cells of X of
dimension at most d; the set of closed d-dimensional cells we denote by Cd pX q.
Recall also that an open covering of a space is good if all the possible intersections are either empty or contractible.
Definition. An open covering U tUc ucPC pX q of a simplicial complex X with
components indexed by the set of closed cells of X is regular if the following are
satisfied:
U is a good”covering.
For all”d, cPCdpX q Ūc is a closed regular neighborhood in X zX pd1q of
X pdq z cPC pX pd1q q Uc . Here Ūc is the closure of Uc in X.
If c, c1 P C pX q are cells with Uc X Uc1 H then either c € c1 or c1 € c.
In particular, if c and c1 are distinct cells of the same dimension then
Uc X Uc1 H.
For any cell c P C pX q we have c € ”c1€c Uc1 .
The reader can find in Figure 1 an example of a regular covering of the 2dimensional simplex. Clearly, every simplicial complex has a regular covering.
Figure 1. The shaded regions are the sets Uc for the vertices.
The solid lines bound the neighborhoods Uc for the edges and
the dashed line is the boundary of Uc for the 2-dimensional cell.
Later we will need the following lemma, whose proof we leave to the reader:
Lemma 5.1 (Smashing lemma). Let X be simplicial complex and U
a regular covering. There is a continuous (non-simplicial) map
ÑX
with ΦpUc q Φpcq c for any closed cell c P C pX q.
tUcucPC pX q
Φ:X
In the particular case represented in Figure 1, the map Φ provided by Lemma
5.1 sends the shaded regions to the vertices, the regions bounded by solid lines
to the edges, and the region bounded by the dashed line onto the whole simplex.
31
5.2. Fibered relations. Recall that a fibration is a continuous surjective map
π : S Ñ S 1 with the homotopy lifting property. In this paper we are not lucky
enough to find ourselves working with fibrations. What we encounter are fibered
relations.
Definition. A subset R € S S 1 of the cartesian product S S 1 of two CWcomplexes is a fibered relation if R Ñ S 1 is a fibration.
If K € S and K 1 € S 1 are such that O R X pK K 1 q is a fibered relation
in K K 1 , then we say that pR, Oq is a fibered relation pair.
Remark. In all cases of interest in this paper, the spaces S, S 1 , K, K 1 , R, and O
are smooth manifolds, and the projections are locally trivial fiber bundles.
It is easy to construct examples of fibered relations. For instance, R S S 1
is a fibered relation. The graph tps, π psqq|s P S u € S S 1 of any fibration
π : S Ñ S 1 is also a fibered relation. Finally, if R̃ € S̃ S̃ 1 is a fibered relation
and α : S̃ Ñ S and β : S̃ 1 Ñ S 1 are coverings, then R pα β qpR̃q € S S 1 is
a fibered relation.
We will work simultaneously with various fibered relations. Suppose that we
are given the following data:
(*) Let P pP, ¥q be a partially ordered set, S tS∆ |∆ P P u and K tK∆|∆ P P u set of CW-complexes indexed by P with K∆ € S∆ for all
∆. For all ∆, ∆1 P P with ∆ ¥ ∆1 let
€ S∆ S∆1 , and
O∆,∆1 R∆,∆1 X pK∆ K∆1 q € K∆ K∆1
such that pR∆,∆1 , O∆,∆1 q is fibered relation pair. Set also
R tR∆,∆1 |∆ ¥ ∆1 , ∆, ∆1 P P u
O tO∆,∆1 |∆ ¥ ∆1 , ∆, ∆1 P P u
R∆,∆1
Assuming (*), let X be a simplicial complex and recall that the set of all
closed cells is denoted by C pX q. A P-labeling of a simplicial complex X is a
map
δ : C pX q Ñ pP, ¥q
satisfying δ pcq ¥ δ pc1 q for all faces c, c1 P C pX q with c  c1 . A P-labeled
simplicial complex is a pair pX, δ q where δ is a P-labeling of X.
Continuing with notation as in (*), suppose that pX, δ q is a P-labeled simplicial complex. To every regular cover U tUc ucPC pX q of X we associate the
bundle
§
§
S δ U
Sδpcq Uc Ñ
Uc
—
P p q
c C X
P p q
c C X
where stands for disjoint union. As a bundle S δ U has no interest: it is just
the disjoint union of trivial bundles. However, a rather non-trivial structure is
added if we take into account the intersections of the components of U and that
we have the set R of fibered relations.
32
ALEXANDRA PETTET AND JUAN SOUTO
Definition. A section of pS δ U, Rq is a collection of continuous maps
tσcucPC pX q tσc : Uc Ñ Sδpcq|c P C pX qu
with pσc pxq, σc1 pxqq P Rδpcq,δpc1 q for all c, c1 P C pX q, c  c1 , and all x P Uc X Uc1 .
Two sections tσc ucPC pX q and tσc1 ucPC pX q of pS δ U, Rq are homotopic if for
every c P C pX q there is a continuous map
r0, 1s Uc Ñ Sδpcq, pt, xq ÞÑ σct pxq
with σc0 σc and σc1 σc1 , such that tσct ucPC pX q is a section for all t. If W € X
is such that σct pxq σc pxq for all c P C pX q, all x P W X Uc , and all t, then we
say that the homotopy tσct ucPC pX q is constant on W .
Notation. In order to lighten the burden of notation we will from now on
write
S δ U pS δ U, Rq
dropping any reference to the set R of fibered relations.
5.3. Fibered relations associated to a group G. Let G be the group of
either complex or real points of a reductive algebraic group (defined over R in
the latter case) and K € G a maximal compact subgroup. As in Section 3.4
let G be the manifold of all well-centered subgroups of G and consider π0 pG q
with the partial order introduced in Section 3.5. Recall that every component
Gi contains a unique component Ki of the manifold K of well-centered groups
in K. The following is a direct consequence of Lemmas 3.13 and 3.14:
Lemma 5.2. Let G1 , G2 and K1 , K2 be connected components of the manifolds
G and K of well-centered groups in G and K, respectively; assume that G1 ¥ G2
and that Ki € Gi for i 1, 2. The pair
pRG ,G , OG ,G q € pG1 G2, K1 K2q
1
2
1
2
where RG1 ,G2 and OG1 ,G2 are as in (3.4) and (3.5), is a fibered relation pair.
To conclude the analogy with (*), set
tRG ,G |G1 ¥ G2, G1, G2 P π0pG qu
(5.2)
OG tOG ,G |G1 ¥ G2 , G1 , G2 P π0 pG qu
Given now a π0 pG q-labeled simplicial complex pX, δ q and a regular cover U of
X we consider the bundle G δ U and sections therein as defined above.
Remark. To clarify the nature of the different objects, when working with G δ U
we will denote by Gδpcq the connected component δ pcq P π0 pG q for c P C pX q.
(5.1)
RG
1
2
1
2
The following result asserts that, up to slightly reducing the domain, every
section of G δ U can be homotoped to a section of K δ U.
Proposition 5.3. Let G be the group of either complex or real points of a
reductive algebraic group, K € G a maximal compact subgroup, G and K the
manifolds of well-centered groups in G and K, and consider the sets of fibered
relations RG and OG defined in (5.1) and (5.2).
33
Let pX, δ q be a π0 pG q-labeled complex, and U tUc ucPC pX q and V tVc ucPC pX q
regular covers of X with Vc relatively compact in Uc for all c P C pX q. Finally,
let tσc ucPC pX q be a section of G δ U, and W € X with σc pwq P Kδpcq for all
c P C pX q and all w P Uc X W .
The restriction of tσc ucPC pX q to G δ V is homotopic to a section tσc1 ucPC pX q
with σc1 pVc q € Kδpcq for all c P C pX q; moreover the homotopy is constant on W .
Recall that by Proposition 3.11 the manifold G retracts to the submanifold
K. In light of this fact, the reader may be willing to believe that the proof of
Proposition 5.3 is a very tedious and painful exercise in topology; we thereby
defer it to Appendix A. Essentially the idea is to observe that the set of fibered
relation pairs pRG , OG q satisfies some conditions (transitivity, reflexivity, simultaneous lifting and relative retraction properties) which allow us to work with
sets of fibered relations very much as one would with single fibrations.
6. Proof of Theorem 1.1
In this section we prove Theorem 1.1 assuming the following result:
Proposition 6.1. Let G be the group of complex points of a connected reductive
algebraic group, K a maximal compact subgroup of G, and recall that G acts on
HompZk , Gq by conjugation. There is a continuous G-equivariant map
Φ : r0, 1s pHompZk , Gq, HompZk , K qq Ñ pHompZk , Gq, HompZk , K qq
with the following properties:
Φp0, ρq ρ and Φp1, ρq P HompZk , K Gq for all ρ P HompZk , Gq.
There is an open covering W of HompZk , K Gq such that for every W P
W one of the following non-exclusive alternatives holds:
(1) All the elements in Φpt1u W q have the same orbit type.
(2) All the elements in W have the same orbit type.
If moreover G is defined over R, then Φ is equivariant under complex conjugation.
Recall that by Lemma 3.2, HompZk , K G q is the set of representations whose
image is contained in a conjugate of K. In particular, HompZk , K G q is contained in the set HompZk , Gs q of semi-simple representations and contains
HompZk , K q. Recall also that two elements ρ, ρ1 in HompZk , Gq have the same
orbit type if their stabilizers StabG pρq and StabG pρ1 q are conjugate in G.
We will prove Proposition 6.1 in Section 8.
Theorem 1.1. Let G be the group of complex or real points of a connected
reductive algebraic group, defined over R in the latter case, and K € G a
maximal compact subgroup. For all k there is a weak retraction of HompZk , Gq
to HompZk , K q.
For the sake of concreteness, suppose that G is the group of complex points.
With the same notation as in Proposition 6.1, let
(6.1)
φ : pHompZk , K G q, HompZk , K qq Ñ pHompZk , K G q, HompZk , K qq
34
ALEXANDRA PETTET AND JUAN SOUTO
be the restriction of Φp1, q to HompZk , K G q and notice that G-equivariance
implies that
StabG pφpρqq  StabG pρq
for all ρ P HompZk , Gq.
Theorem 1.1 will follow once we show that φ is homotopic, as a map of pairs,
to a map whose image is contained in HompZk , K q. We sketch briefly how we
will show that such a homotopy exists, but before doing so refer to Section
3.4, Section 4.3 and Section 4.5 for the definitions of G, K, π : Dpk q Ñ G and
π : D̄pk q Ñ G.
The strategy: Considering X HompZk , K G q as a simplicial complex with
set of cells C pX q, we will associate to the map φ
a regular covering U tUcucPC pX q of X, and
a labeling δ : C pX q Ñ π0pG q.
In order to remind the reader of the nature of the different objects we will
denote by Gδpcq the connected component δ pcq of G; accordingly, we also set
D̄δpcq pk q π 1 pGδpcq q.
Besides the covering U and the labeling δ, we also associate to the map φ
—
—
—
a section
σc : cPC pX q Uc Ñ cPC pX q Gδpcq of G δ U, and
c
P
C
p
X
q
a map —cPC pX q σ̃c : —cPC pX q Uc Ñ —cPC pX q D̄δpcqpkq
such that the following diagram commutes:
(6.2)
—
P p; q D̄4δpcq pkq
c C X
x
44
xx
44
xx
x
44
x
x
44
π
xx
x
44
xx
x
44Ψ
x
x
— σc —
44
xx
—
44
/
G
U
δ pcq
cPC pX q c WW
44
WWWWW cPC pX q
44
WWWWW
WWWWW
44
WWWWW
4
W
W
W
WWW+ φ
/
— σ̃c
X
X
Here Ψ : D̄pk q Ñ HompZk , Gq is the evaluation map
— defined
” in Section 4.3 and
the unlabeled vertical arrow is the obvious map Uc Ñ Uc X.
These objects
in place, we apply Proposition 5.3 to obtain a homotopy of
—
the section σc to a section which takes values in K δ U. The flatness of the
bundle
π : D̄ Ñ G allows us to lift this homotopy to a homotopy of the map
—
σ̃c . Composing the elements of this last homotopy
with the evaluation map
—
Ψ, we obtain a one parameter family of maps Uc Ñ X with the property
that the image of the final map is contained in HompZk , K q. The compatibility
of horizontal lifts, Lemma 4.5, implies then that this one parameter family of
maps descends to a one parameter family of maps X Ñ X starting with φ and
ending with a map whose image is HompZk , K q; this is the desired homotopy.
Still assuming that G is the group of complex points, we now put the proof
of Theorem 1.1 into full swing:
35
The simplicial complex. Consider the triangulation T0 of HompZk , K G q provided by Proposition 2.8 and notice that by Proposition 6.1 there is a refinement
T of T0 with the property that every closed cell c P T has an open neighborhood
Wc such that:
(1) either any two ρ, ρ1 P φpWc q have the same orbit type, or
(2) any two ρ, ρ1 P Wc have the same orbit type.
Consider from now on
X pHompZk , K G q, T q
as a simplicial complex and notice that by Proposition 2.8, HompZk , K q € X
is a subcomplex.
Remark. As a triangulation, T is locally finite. In particular, we can assume
without loss of generality that Wc1 € Wc for any two closed cells c, c1 of X with
c1 € c.
The labeling. With the same notation as in Section 3, let G and K be the
manifolds of all well-centered groups in G and K and recall that by Proposition
3.11 the map
ι : K Ñ G, H ÞÑ C pZG pH qq
is an injective homotopy equivalence. Recall also that the inclusion relation
between well-centered groups yields an order on the sets π0 pG q and π0 pKq of
connected components of G and K and that the map ι induces an order preserving bijection π0 pG q Ñ π0 pKq by Lemma 3.12.
Our next goal is to define a π0 pG q-labeling on the complex X, but first we
remind the reader of a few facts:
StabGpρq ZGpρpZk qq for all ρ P HompZk , Gq.
By Lemma 3.6, the group C pZGpρpZk qqq C pStabGpρqq is well-centered,
i.e. an element of G, for any ρ P HompZk , K G q.
By Lemma 3.10, ρ, ρ1 P HompZk , K Gq have the same orbit type if and
only if C pStabG pρqq and C pStabG pρ1 qq belong to the same connected
component of G.
Notice that it follows from these observations, together with (1) and (2) above,
that for every cell c P C pX q one of the following non-mutually exclusive alternatives holds:
(1’) Either there is a component G0 of G with
C pStabG pφpρqqq, C pStabG pφpρ1 qqq P G0
for all ρ, ρ1 P Wc , or
(2’) there is a component G1 of G with
C pStabG pρqq, C pStabG pρ1 qq P G1
for all ρ, ρ1 P Wc .
Keeping the notation above, we define a function δ : C pX q Ñ π0 pG q by
δ pcq G0 if (1’) is satisfied, and
δ pcq G1 otherwise
We claim that pX, δ q is a labeled complex.
36
ALEXANDRA PETTET AND JUAN SOUTO
Lemma 6.2. The function δ : C pX q Ñ π0 pG q is a labeling, i.e. δ pcq ¥ δ pc1 q for
all c, c1 P C pX q with c  c1 .
Proof. Suppose first that (1’) holds for c. The assumption that Wc1 € Wc for
all c1 € c implies that (1’) also holds for c1 . This proves that δ pc1 q δ pcq.
Assume now that (1’) is not satisfied for c. In particular
C pStabG pρqq, C pStabG pρ1 qq P δ pcq
for all ρ, ρ1 P Wc . Notice that if (1’) is also not satisfied for c1 , then we again
have δ pc1 q δ pcq. So, suppose that (1’) is satisfied for c1 . Recall that for
ρ P Wc1 € Wc we have
StabG pρq € StabG pφpρqq
by Proposition 6.1. It follows hence from Lemma 3.6 that
C pStabG pρqq  C pStabG pφpρqqq
Since C pStabG pρqq P δ pcq and C pStabG pφpρqqq
δ pc1 q, as we needed to prove.
P δpc1q this means that δpcq ¥
From now on we consider X pX, δ q as a π0 pG q-labeled complex. We will
denote the connected component δ pcq of G by Gδpcq for c P C pX q.
The cover and the section. Fix once and for all a regular cover U tUcucPC pX q of X with Uc € Wc for all c P C pX q. Our next goal is to associate to the map φ a section tσc ucPC pX q of
G δ U
§
P p q
Gδpcq Uc
Ñ
c C X
§
P p q
Uc
c C X
For ρ P Uc we set, with the same notation as before,
σc pρq C pStabG pφpρqqq if (1’) is satisfied, and
σc pcq C pStabG pρqq otherwise
and notice that by definition σc pρq P Gδpcq . We have hence for all c P C pX q a
map
σc : Uc Ñ Gδpcq
We summarize in the following lemma a few properties of the maps σc :
Lemma 6.3. Each one of the maps σc : Uc Ñ Gδpcq is continuous. Moreover,
the collection of maps tσc ucPC pX q is a section of G δ U.
Proof. Suppose that we have a sequence pρn q in Uc converging to ρ P Uc and
suppose for the sake of concreteness that (1’) holds for c. Observe that the
continuity of φ implies that φpρn q Ñ φpρq. Noticing that
φpρq P HompZk , K G q € HompZk , Gs q
is a semisimple representation, it follows from Theorem 2.6 that the orbit of
φpρq under G is closed in the affine variety HompZk , Gq. Hence, Theorem 2.4
and its following remark imply there is a sequence pgn q in g converging to Id
such that
gn StabG pφpρn qqgn1 € StabG pφpρqq
37
On the other hand, since we are assuming that (1’) is satisfied, we know
StabG pφpρqq and Stabpφpρn qq are conjugate for all n; hence
gn StabG pφpρn qqgn1
It follows now directly from gn
StabGpφpρqq
Ñ Id and the definition of σc that
σc pρn q Ñ σc pρq
as we needed to prove. This proves that the map σc is continuous if (1’) is
satisfied for c. If (1’) is not satisfied then the argument is word-for-word the
same, once we remove the appearance of φ everywhere.
It remains to prove that the collection of maps tσc ucPC pX q is a section of
G δ U. The argument is the same as that we used in Lemma 6.2 to prove that δ is
a labeling, and we leave it to the reader to make the necessary modifications. Before moving on, notice that it follows directly from the definitions that
σc pUc X HompZk , K qq € Kδpcq
(6.3)
for all c P C pX q.
The lift. Consider as in Section 4 the k-th power
π : D pk q Ñ G
of the canonical bundle and set Dδpcq pk q
concrete terms,
π1pGδpcqq for c P C pX q.
Dδpcq pk q tpA, a1 , . . . , ak q P Gδpcq G G|ai
In more
P A@iu
Denote also by D̄δpcq pk q the intersection of Dδpcq pk q with the total space of the
flat subbundle π : D̄pk q Ñ G introduced in Section 4.5. Explicitly, D̄δpcq is the
set of those pA, a1 , . . . , ak q P Dδpcq such that the elements a1 , . . . , ak belong to
the unique maximal compact subgroup of A.
Notice now that for c P C pX q and ρ P Uc we have that
φpρq pZn q € C pStabG pφpρqqq € C pStabG pρqq
and hence that φpρq pZn q € σc pρq. Also, as φpρq P HompZk , K G q it follows that
pφpρqqpeiq is contained in the unique maximal compact subgroup of σcpρq, where
e1 , . . . , ek is the standard basis of Zk . It follows from these two considerations
that for all c P C pX q the map
Ñ D̄δpcqpkq
σ̃ pρq σc pρq, φpρq pe1 q, . . . , φpρq pek q
σ̃c : Uc
is well-defined.
It follows directly from the construction and the definitions that
pπ σ̃cqpρq σcpρq and φpρq pΨ σ̃cqpρq
for all ρ P Uc . Here, Ψ : Dpk q Ñ HompZk , Gq is the evaluation map defined in
Section 4.3; recall that ΨpD̄pk qq € HompZk , K G q by Lemma 4.7.
At this point we have constructed all the objects in the commuting diagram
(6.2) above. It remains to construct the homotopy of φ.
38
ALEXANDRA PETTET AND JUAN SOUTO
The homotopy. Proposition 5.3 asserts that, up to reducing each one of the
open sets Uc slightly, there is a continuous map
§
r0, 1s Uc Ñ
P p q
c C X
§
P p q
Gδpcq
c C X
sending pt, ρq P r0, 1s Uc to pρq such that the following properties are satisfied:
(i) for all t the collection of maps tσct ucPC pX q is a section of G δ U,
(ii) σc0 σc for all c P C pX q,
(iii) σc1 pUc q € Kδpcq for all c P C pX q.
(iv) σct pρq σc pρq for all c P C pX q and all ρ P Uc with σc pρq P Kδpcq .
σct
At this point, recall that for all c P C pX q, the bundle π : D̄δpcq
flat, and that we have the map σ̃c : Uc Ñ D̄δpcq with
π σ̃c
Ñ Gδpcq is
σc σc0
Consider for all c P C pX q the horizontal lift
r0, 1s Uc Ñ D̄δpcq, pt, ρq ÞÑ σ̃ct pρq
of the homotopy r0, 1s Uc Ñ Gδpcq , pt, ρq ÞÑ σct pρq satisfying σ̃c0 pq σ̃c pq.
As the horizontal lift of a continuous homotopy, the map pt, ρq ÞÑ σ̃ct pρq is
continuous as well.
Lemma 6.4. The map
§
P p q
r0, 1s Uc Ñ X, r0, 1s Uc Q pt, ρq ÞÑ Ψpσ̃ct pρqq
c C X
descends to a continuous map
r0, 1s X Ñ X, pt, ρq ÞÑ φtpρq
with φ0 φ. Moreover, φt pρq φpρq for all t and all ρ P HompZk , K q.
Proof. Suppose that we have two cells c, c1 with Uc X Uc1 H, and notice that,
by the definition of a regular covering, one of them is contained in the other,
say, c1 € c. This implies hence that δ pcq ¥ δ pc1 q. The condition that the maps
tσct ucPC pX q form a section of G δ U implies that for all ρ P Uc X Uc1 we have
pσct pρq, σct1 pρqq P Rδpcq,δpc1q
for all t. Recall now that we have
Φpσ̃c pρqq φpρq Φpσ̃c1 pρqq
and that σ̃ct pρq and σ̃ct1 pρq are both obtained through horizontal lifting. Lemma
4.5 asserts that we have
Φpσ̃ct pρqq Φpσ̃ct1 pρqq
for all t. We have proved that the map
§
P p q
c C X
r0, 1s Ucq Ñ X
r0, 1s Uc Q pt, ρq ÞÑ Φ̃pσct pρqq
39
descends to a map
r0, 1s X Ñ X, pt, ρq ÞÑ φtpρq
given by φt pρq Φpσ̃ct pρqq for any c P C pX q with ρ P Uc ; notice that by construction φ0 φ. The map pt, ρq ÞÑ φt pρq is continuous because its restriction
for each one of the sets r0, 1s Uc is the continuous map pt, ρq ÞÑ Φpσ̃ct pρqq.
Finally, suppose that ρ P HompZk , K q and that c P C pX q is a cell with
ρ P Uc . By (6.3) we have σc pρq P Kδpcq and hence, by (iv) above, we have that
σct pρq σc pρq for all for all t. This in turn implies that σ̃ct pρq σ̃c pρq for all t.
Hence
φt pρq Φpσ̃ct pρqq Φpσ̃c pρqq φpρq
for again all all t. This concludes the proof of Lemma 6.4.
The end. At this point, we are nearly done with the proof of Theorem 1.1.
It only remains to prove that the final map φ1 of the homotopy provided by
Lemma 6.4 takes values in HompZk , K q. In order to see that this is the case,
recall that by construction we have σc1 pUc q € Kδpcq for all c P C pX q; hence
σ̃c1 pUc q € D̄δpcq X π 1 pKδpcq q
Lemma 4.7 implies now that
φ1 pUc q Φpσ̃c1 pUc qq € HompZk , K q
The claim follows because the sets Uc cover HompZk , K G q.
We have proved that the map (6.1) is homotopic as a map of pairs to a map
which takes values in HompZk , K q. This concludes the proof of Theorem 1.1 if
G is the group of complex points.
The real case. Suppose now that G GpRq is the group of real points, and
let K pRq and K be maximal compact subgroups of GpRq and G with K pRq €
K. The map Φp1, q provided by Proposition 6.1 is equivariant under complex
conjugation. In particular, Φp1, q maps HompZk , GpRqq to HompZk , K pRqGpRq q,
and HompZk , K pRqq to itself. Consider the map
φ : HompZk , K pRqGpRq q Ñ HompZk , K pRqGpRq q
obtained by restricting Φp1, q to HompZk , K pRqGpRq q. Now we can repeat wordfor-word, referring to the appropriate results in Section 2.7 and Section 3.7, the
argument given in the complex case.
The next two sections are devoted to the proof of Proposition 6.1.
7. Approximated Jordan decomposition
The main tool needed to prove Proposition 6.1 is what we call the approximated Jordan decompositon. This construction is the goal of the present section.
40
ALEXANDRA PETTET AND JUAN SOUTO
7.1. Jordan decomposition. Recall that g P SLn C is unipotent if one of the
following three equivalent conditions are satisfied:
g has characteristic polynomial χg pT q pT 1qn.
g Id is nilpotent, meaning that pg Idqn 0.
g Id has spectral radius ρpg Idq 0 where the spectral radius of a
linear transformation is the maximum of the norms of its eigenvalues.
Given arbitrary g P SLn C, there is h P SLn C such that hgh1 is in Jordan
normal form. Denote by a the diagonal part of hgh1 , and set
gs
h1ah
and gu
ggs1
By construction, gs is semisimple and gu is unipotent. Moreover, it is wellknown that gs and gu are polynomials in g, meaning that gs , gu P Crg s € Mn C.
In particular, if h P SLn C commutes with g, then it also commutes with gs and
gu . Suppose now that G € SLn C is an algebraic group. A priori, the semisimple
part gs and the unipotent part gu of g P G only belong to the ambient group
SLn C, but in fact gs , gu P G as well [7]. To summarize, we have:
Jordan decomposition. Let G be an algebraic group. For every g P G there
are uniquely determined elements gs P G semisimple and gu P G unipotent with
g gs gu , and such that if h P G commutes with g, then it also commutes with
each of gs and gu .
It is a rather unlucky fact that the Jordan decomposition g ÞÑ pgs , gu q is not
continuous; the goal of this section is to construct a continuous approximation
of this decomposition. Letting G act on itself and on G G by conjugation, we
prove:
Theorem 1.7. Let G € SLn C be a connected algebraic group. For every ¡ 0
there is a continuous G-equivariant map
G Ñ G G,
ÞÑ pgs , gu q
satisfying the following properties for all g P G:
(1) g gs gu gu gs .
(2)
(3)
(4)
(5)
(6)
g
gs and gu are polynomials in g, i.e. they belong to Crg s € Mn C.
If h P G commutes with g P G, then h also commutes with gs and gu .
gs is semisimple and gu Id has spectral radius at most .
If K € G is a maximal compact group then gs , gu P K for all g P K.
If moreover G is defined over R, then ḡs gs and ḡu gu , where ¯ is
complex conjugation.
We refer to the decomposition
g
gs gu
provided by Theorem 1.7 as the approximated Jordan decomposition for the following reason. The elements gs and gu have the same commutativity properties
as those of the Jordan decomposition, and while gu is not unipotent, it is almost
so: recall that u P SLn C is unipotent if and only if ρpu Idq 0.
To prove Theorem 1.7 we will work in the ambient group SLn C: we construct
the desired map there and then restrict it to G. In this setting, it is easy to
41
understand the reason for the lack of continuity of the Jordan decomposition
and essentially the idea behind the construction of the proof of Theorem 1.7
is to preempt its discontinuties. The precise statement we will prove is the
following:
Proposition 7.1. For every ¡ 0 there is a continuous SLn C-equivariant map
(7.1)
SLn C Ñ SLn C SLn C,
ÞÑ pgs , gu q
g
satisfying the following properties for all g P SLn C:
(1) g gs gu gu gs .
(2) gs and gu are polynomials in g, i.e. they belong to Crg s € Mn C.
(3) gs is semisimple and gu Id has at most spectral radius .
(4) gs , gu P SUn for all g P SUn .
(5) ḡs gs and ḡu gu , where ¯ is complex conjugation.
Moreover, if G € SLn C is a connected algebraic group then the map (7.1) can
be chosen in such a way that gs , gu P G for all g P G.
Theorem 1.7 follows immediately from Proposition 7.1. Before proving the
latter, we require a few preparatory remarks.
7.2. Musings on sets of eigenvalues. We denote by Sn € SLn C the standard maximal torus of SLn C, i.e. the group of diagonal matrices with determinant 1. Recall that the Weyl-group
W pSn , SLn Cq NSLn C pSn q{ZSLn C pSn q
is the symmetric group Σn on n symbols. Here, NSLn C pSn q and ZSLn C pSn q are,
as always, the normalizer and the centralizer of Sn in SLn C.
We identify Sn with a subset of Cn :
(7.2) Sn
tpλiqi1,...,n P Cn|λ1 . . . λn 1u,
λ1
..
.
pλ1, . . . , λnq
λn
Under this identification, the action of the Weyl group Σn on Sn is just the
restriction of the standard action on Cn .
Suppose that we are given g P SLn C, and let gs be its semisimple part
with respect to the Jordan decomposition. As gs is diagonalizable, there is
h P SLn C with hgs h1 P Sn . Notice that if h1 P SLn C is another element with
h1 gs h11 P Sn , then hgs h1 and h1 gs h11 differ by an element of the Weyl group
Σn . Hence we can associate to g the uniquely determined element Eg P Sn {Σn .
Identifying Sn as above with a subset of Cn , let rλ1 , . . . , λn s P Cn {Σn be
an unordered n-tuple of complex numbers representing Eg . It follows from the
definition that Eg rλ1 , . . . , λn s is the unordered tuple of the eigenvalues of g,
counted with multiplicity.
We can also consider the unordered tuple Eg rλ1 , . . . , λn s as an atomic
measure on C with total measure n:
Eg
δλ
1
δλn
where δλi is the Dirac measure in C with atom in λi .
42
ALEXANDRA PETTET AND JUAN SOUTO
Below we will pass freely between these three incarnations of the spectral measure Eg : as an element in Sn {Σn , as an unordered tuple of n complex numbers,
and as a measure.
7.3. Continuity properties of eigenspaces. Keeping notation as before,
consider Eg as a measure and denote its support by |Eg |. Given λ P |Eg |, i.e. an
eigenvalue of g, denote by
Eλg
kerpg λ Idq
and
Ēλg kerpg λ Idqn
the λ-eigenspace and generalized λ-eigenspace of g. Observe that the Eg measure of the singleton tλu is given by Eg pλq dim Ēλg .
As C is algebraically closed, we have that Cn is the direct sum of the generalized eigenspaces Ēλg :
Cn `λP|Eg | Ēλg
We list a few well known linear algebraic facts:
(1) Suppose that g, h P SLn C commute, and let λ P |Eg | be an eigenvalue
of g. Then hEλg Eλg and hĒλg Ēλg .
(2) Suppose that h P SLn C is such that for every λ P |Eg | we have
(a) hĒλg Ēλg , and
(b) h|Ē g is a homothety.
λ
Then every g 1 P SLn C which commutes with g also commutes with h,
meaning that h P C pZSLn C pg qq.
(3) The semisimple part gs of g P SLn C is the unique linear transformation
with gs v λv for all v P Ēλg and λ P |Eg |.
It is easy to find examples showing that the eigenspaces of g, generalized or
not, do not depend continuously on g. The following well known lemmas show
the kind of continuity we do have:
Lemma 7.2. The map g
ÞÑ Eg is continuous.
Proof. The characteristic polynomial depends continuously on g; and the set
of zeroes, counted with multiplicity, of a complex polynomial depends also
continuously on the coefficients.
Lemma 7.3. Suppose that the sequence pgi q in SLn C converges to g. Suppose
also that δ ¡ 0 is such that any two distinct points in |Eg | are at distance at
least 2δ. Then for all λ P |Eg | we have
lim
i
à
Ñ8 λ1 PBδ pλq
Ēλgi1
Ēλg
Here Bδ pλq is the ball in C of radius δ centered at λ.
Sketch of proof. If λi P |Egi | is any sequence of eigenvalues of gi converging to
λ P |Eg |, then, up to passing to a subsequence, the generalized eigenspaces Ēλgii
converge to a subspace of Ēλg . It follows that
lim
i
à
Ñ8 λ1 PBδ pλq
Ēλgi1
€ Ēλg
43
The equality follows as both spaces have the same dimension by Lemma 7.2.
7.4. Smashing eigenvalues. In the construction of the approximated Jordan
decomposition we will need a technical lemma, proved next. Consider the function d : Sn Sn Ñ R
d λ1
..
, .
λ11
λn
..
.
λ1
λi
max 1
i1,...,n λ
i
1
n
The function dp, q is not a distance, but we will think anyway as such. In this
section we prove the following lemma:
Lemma 7.4. For all ¡ 0 there is a δ
P p0, q and a Σn-equivariant map
Θ : pSn , Sn X SUn q Ñ pSn , Sn X SUn q
Θ : λ pλi q ÞÑ Θpλq pΘi pλqq
with Θpq Θp¯q and such that for all λ pλ1 , . . . , λn q P Sn and all i, j P
t1, . . . , nu the following holds:
(1) dpλ,
Θpλqq
¤ , and
λ
(2) if λ 1 ¤ δ then Θi pλq Θj pλq.
Moreover, if T € Sn is a torus, then Θ can be chosen so that ΘpTq € T.
i
j
Proof. Let Σ̄n be the finite group of transformations of Sn generated by the
symmetric group Σn and by complex conjugation pλi q ÞÑ pλ̄i q. Choose a Σ̄n invariant triangulation T of Sn which contains both Sn X SUn and the given
torus T as subcomplexes, and consider X Sn {Σ̄n with the induced structure
of a simplicial complex. We may assume without loss of generality that for any
simplex c P T we have
(7.3)
max dpλ, λ1 q ¤ λ,λ1 c
P
Let now U tUc ucPC pX q be a regular covering of X and choose δ
enough so that
(7.4)
tx P X | min
dpx, y q ¤ 2δ u €
y Pc
¤
c1 c
€
P p0, 2 q small
Uc1
for any cell c P C pX q. For further use we also assume that δ 1010 .
Let now Θ1 : X Ñ X be the map provided by Lemma 5.1 and recall that
1
Θ sends every closed simplex to itself. In particular, Θ1 induces uniquely a
Σ̄n -equivariant map
Θ : Sn Ñ Sn
which sends every closed simplex of T to itself. It follows from (7.3) that Θ
satisfies (1). Notice also that as Θ maps every simplex to itself, it preserves
every subcomplex in Sn . It follows that
ΘpSn X SUn q € Sn X SUn and ΘpTq € T
44
ALEXANDRA PETTET AND JUAN SOUTO
It remains to prove that Φ satisfies (2). Given λ pλ1 , . . . , λn q P Sn , suppose
that for some i and j we have
λi
λ
j
and let r
with
1
¤ δ 1010
P C be the unique complex number in the ball of radius
r2
1
2
around 1
λλi
j
We consider now the element λ1 pλ11 , . . . , λ1n q P Sn with
λ1k λk if k i, j
λ1 λ1 rλj
i
and notice that
j
dpλ, λ1 q maxt|r 1|, |r1 1|u 2δ
Denote by c the minimal closed simplex in T containing λ1 and observe that
there is a cell c1 € c with λ P Uc1 by (7.4). Since the map Θ maps Uc1 to c1 , it
follows that Θpλq P c1 € c. Observe at this point that c is pointwise fixed by the
stabilizer of λ1 inside Σn € Σ̄n . In other words, for every λ2 pλ21 , . . . , λ2n q P c,
we have λ2i λ2j . This applies in particular to the point Θpλq; this proves that
Θ satisfies (2) and concludes the proof of Lemma 7.4.
7.5. Proof of Proposition 7.1. Suppose that G € SLn C is a connected algebraic group, and let T € G be a maximal torus, and K € G a maximal
compact subgroup, such that T X K is a maximal torus in K. Replacing G by a
conjugate of itself, we may assume that T € Sn and hence K € SUn . Suppose
from now on that
Θ : Sn Ñ Sn
is the map provided by Lemma 7.4 satisfying ΘpTq € T.
Given g P SLn C with semisimple part gs , let h P SLn C be such that hgs h1
Sn . We define
gs h1 pΘphgs h1 qqh
P
Notice that the equivariance of Θ under the action of the Weyl group of Sn
implies that gs does not depend on the choice of h. In other words, the map
SLn C Ñ SLn C, g
ÞÑ gs
is well defined. Once we have defined gs , we have no choice but to define gu as
follows:
SLn C Ñ SLn C, g ÞÑ gu g pgs q1
Notice that by construction, gs and gu behave well with respect to conjugation,
meaning that phgh1 qs hgs h1 and phgh1 qu hgu h1 for all h P SLn C. In
other words, g ÞÑ pgs , gu q is SLn C-equivariant.
Suppose now that g, h P SLn C are such that hgs h1 P Sn and notice that
ḡs
gs
and h̄ḡs h̄1
P Sn
45
Taking into account that Θ is equivariant under complex conjugation, we have
that
h̄1Θph̄ḡsh̄1qh̄ h̄1Θph̄gsh̄1qh̄ h̄1Θphgsh1qh̄ h̄1Θphgh1qh̄ h1Θphgh1qh gs
This implies that ḡu gu for any g P G.
Suppose now that we are given an element g P G. The semisimple part gs
ḡs
also belongs to G, and the assumption that G is connected implies that there
is h P G with hgs h1 P T. As ΘpTq € T it follows that
h1Θphgsh1qh
belongs to G, so that gu g pgs q1 belongs to G as well. A similar argument
shows that gs , gu P SUn for g P SUn .
It remains to prove that the maps g ÞÑ gs and g ÞÑ gu are continuous and
gs
satisfy (1), (2) and (3) in Proposition 7.1. Before doing so, we give a different
description of gs . Continuing with the same notation, identify Sn with a subset
of Cn as in (7.2), let Eg pλ1 , . . . , λn q P Cn be the tuple associated to hgs h1 P
Sn , and let pΘ1 pEg q, . . . , Θn pEg qq P Cn be the tuple associated to Θphgs h1 q.
Then gs is the unique linear map with the following property:
(7.5)
gs pv q Θi pEg q v for all v
P Ēλg
gs
i
commutes with every element h
Notice now that (7.5) implies that
that commutes with the semisimple part gs of g. In other words,
gs
P SLn C
P C pZSL Cpgsqq
n
gs
and its inverse are polynomials in gs .
It follows hence from Lemma 3.7 that
Since gs is in turn a polynomial in g we deduce that
gs , gu
gpgs q1 P Crgs
This proves (2), and (1) follows immediately.
Starting with the proof of (3), notice that by construction, gs is conjugate to
an element in Sn and hence is semisimple. To prove that gu Id has very small
spectral radius let, with the same notation as above,
pλ1, . . . , λnq P Cn
be the tuple corresponding to hgs h1 P Sn . Considering g in Jordan normal
Eg
form we see that the eigenvalues of gu are given by ΘiλpEi g q . It follows from
Lemma 7.4 that
*
"
λi
ρpgu Idq max 1 , i 1, . . . , n dpEg , ΘpEg qq Θi pEg q
This proves (3).
At this point we have proven all the claims of Proposition 7.1, except for the
continuity of the map g ÞÑ pgs , gu q pgs , g pgs q1 q. It obviously suffices to prove
that g ÞÑ gs is continuous.
46
ALEXANDRA PETTET AND JUAN SOUTO
Suppose that pgi q is a sequence in SLn C converging to some g. Let δ be
as in Lemma 7.4, and choose δ0 δ positive, such that any two atoms of the
measure Eg are at distance at least 2δ0 . Recall that, associated to g and gi , we
have direct sum decompositions of Cn into generalized eigenspaces. Given λ in
the support of Eg , we define for all i
Vλ,i
à
λ1 Bδ0 λ ,Egi λ1
P
It follows from Lemma 7.3 that
Ēλg
p q¡0
pq
Ēλgi1
ilim
V
Ñ8 λ,i
for each eigenvalue λ of g.
By construction the map pgi qs is a homothety on each one of the eigenspaces
of gi . Moreover, as δ0 is assumed to be smaller than δ, it follows from Lemma
7.4 (2) that in fact the factor of the homothety of pgi qs is equal for any two
generalized eigenspaces of gi which are contained in the same Vλ,i ; in other
words, the restriction of pgi qs to Vλ,i is a homothety. Finally, the continuity of
Θ ensures that the homothety factors of pgi qs on Vλ,i converge to the homothety
factor of gs on Eλg . It follows that the maps pgi qs converge to gs as claimed.
This concludes the proof of Proposition 7.1.
Before continuing, we wish to make a technical observation that will be important later on. Its proof, which we leave it to the reader, follows directly from
the construction of the approximated Jordan decomposition:
Proposition 7.5. Suppose that V € C is a linear subspace, and suppose g P
SLn C is such that gV V and that the restriction g |V of g to V is a homothety.
If pgj q is a sequence of semisimple elements in SLn C converging to g, with
gj V
V
for all j, then there is j0 such that for all j ¥ j0 the following holds:
pgj qsV V , and
the restriction pgj qs|V of pgj qs to V is a homothety.
8. Retraction into HompZk , K G q
In this section we prove Proposition 6.1. However, before doing so we need
to make a few observations on the exponential and the logarithm of a matrix.
8.1. The exponential and the logarithm. Recall the definition of the exponential and the logarithm as formal power series:
eX
8̧ 1
m!
m 0
X m , logpX q 8̧
m 1
p1qm 1 pX Idqm
m
If we substitute X for an arbitrary square matrix, then the exponential series
converges absolutely. The logarithmic series converges absolutely if the matrix
X Id has spectral radius ρpX Idq 1.
Lemma 8.1. Given two commuting square matrices g, h P Mn pCq such that all
the involved series converge absolutely, we have:
47
elogpgq g.
eg h eg eh.
logpghq logpgq logphq.
g commutes with both eh and logphq.
eg eḡ and logpgq logpḡq.
Aside from the simple facts listed in Lemma 8.1, we will also need the following fact, which is surely well-known to experts:
Lemma 8.2. Let G € SLn C be either an algebraic group or a compact group,
and denote its Lie algebra by g. There is P p0, 1q so that logpg q P g for all
g P G with ρpg Idq .
It is easy to construct examples showing that the constant cannot be chosen,
even after fixing the ambient group SLn C and the abstract group G, to be
independent of the embedding of G in SLn C.
Proof. It is well-known that whenever g P G is unipotent, then logpg q P g. We
consider now the semisimple case. To begin with, let T be a maximal torus
in G, and let t be its Lie algebra. Since the topology of T is the same as the
induced topology, and since log is a local inverse of
g Ñ G, X
ÞÑ eX
near Id, it follows that there is small neighborhood U
logpT X U q € t
€ G of Id with
On the other hand, T consists of semisimple commuting elements. Hence there
is a basis v1 , . . . , vn of Cn consisting of eigenvectors of t P T for all t. It follows
that any element t P T with eigenvalues very close to 1 is very close to the
identity. In other words, there is P p0, 1q so that any t P T with ρpt Idq is in U . We have proved the claim for elements in the torus T. Suppose now
that g P G is an arbitrary semisimple element. The assumption that G is
connected implies that there is some h P G with hgh1 P T. Noticing that
ρphgh1 Idq whenever ρpg Idq , we deduce that
logpg q h1 logphgh1 qh Adh1 plogphgh1 qq P Adh1 t € g
for any g P G semisimple with ρpg Idq .
To conclude the proof we consider now the case of a general element g P G
with ρpg Idq . Consider the Jordan decomposition g gs gu of g, and
notice that ρpgs Idq ρpg Idq 1, and that ρpgu Idq 0. Hence,
logpgs q and logpgu q are also absolutely converging power series. Since gs and gu
commute, and since logpgu q, logpgs q P g, we deduce from Lemma 8.1 that
logpg q logpgs q
logpgu q P g
This concludes the proof of Lemma 8.2.
From Lemma 8.2 we obtain immediately the following corollary:
Corollary 8.3. Let G € SLn C be either an algebraic group or a compact group.
There is P p0, 1q with et logpgq P G for all g P G with ρpg Idq and all t. 48
ALEXANDRA PETTET AND JUAN SOUTO
8.2. Retraction on the level of G. Recall that for n sufficiently large, there
is no retraction of SLn C to SUn preserving commutativity [33]. What we prove
now is that if G € SLn C is a connected group, then there is a weak retraction,
preserving commutativity, of G into the set Gs of semisimple elements in G.
Proposition 8.4. Suppose that G € SLn C is a connected algebraic group, and
let K € G be a maximal compact subgroup. For all ¡ 0 small enough, there
is a G-equivariant continuous map
F : r0, 1s pG, K q Ñ pG, K q,
pt, gq ÞÑ Ftpgq
with
F0 pg q g and F1 pg q gs
for all g P G, where gs is as provided by Theorem 1.7. In particular, F1 pGq € Gs
and if g, h P G commute then so do Ft pg q and Ft phq for all t.
If G is defined over R, then F is equivariant under complex conjugation.
Here, as always, G acts on itself by conjugation.
Proof. Suppose that is positive and smaller than the constant provided by
Corollary 8.3 for the groups G and K. For g P G consider the approximated
Jordan decomposition
g gs gu
provided by Theorem 1.7 and recall that ρpgu Idq . We define the map F
as follows:
F : r0, 1s G Ñ SLn C,
Ft pg q gs ep1tqlog gu
The G-equivariance and continuity of F follow directly from Theorem 1.7 and
the absolute convergence of the involved series. As gs , gu P G, we deduce from
the choice of and Corollary 8.3 that Ft pg q belongs to G for all t. The same
argument implies that whenever g P K we have Ft pg q P K for all t. Also, if G
is defined over R, then equivariance of the approximated Jordan decomposition
under complex conjugation implies that F is equivariant as well.
Notice that the G-equivariance of F implies that if g, h P G commute then
so do Ft pg q and h for all t. This implies that Ft pg q and Ft phq also commute for
all t.
8.3. Scaling. As above, let Sn be the diagonal group of SLn C, identified with
a subset of Cn as in (7.2). Consider the map
σ̄ : r0, 1s Sn
Ñ Sn, pt, pλiqq ÞÑ σ̄tppλiqq pet log |λ |λiq
i
where logpq is the standard real logarithm. Notice that σ̄ is continuous and
equivariant under the action of the Weyl group Σn , and that we have
σ̄0 pg q g and σ̄1 pg q P Sn X SUn
for all g P Sn .
Given now g P SLn C semisimple, there is h with hgh1
σ̄ under the Weyl group implies that for all t the element
σt pg q h1 σ̄t phgh1 qh
P Sn.
Invariance of
49
is independent of the choice of h. In particular, we obtain a well defined map
(8.1)
σ : r0, 1s SLsn C Ñ SLsn C,
pt, gq ÞÑ σtpgq
where SLsn C is the set of semisimple elements in SLn C. It is easy to see that
σ is continuous and that it satisfies the following properties:
”
(1) σ0 pg q g and σ1 pg q P hPSLn C h SUn h1 for all g P SLsn C,
(2) σt pg q g for all g P SUn , and
(3) σ is SLn C-equivariant; in particular, if g and h commute, then so do
σt pg q and σt phq for all t.
(4) If g P SLn R, then σt pg q P SLn R for all t.
Suppose now that G € SLn C is a connected algebraic group, and let T and
K be, respectively, a maximal torus and a maximal compact subgroup in G
such that T X K is a maximal torus in K. Up to conjugating in SLn C, we can
assume that T € S and K € SUn . The assumption that G is connected implies
that for every semisimple element g in G there is h P G with hgh1 P T. Since
the map σ̄t pq preserves T for all t, we deduce that σt pg q P G for all t and all
g P G semisimple. With notation as in (2.1), we have hence proved:
Lemma 8.5. Suppose that G is a connected linear algebraic group, and let
K € G be a maximal compact subgroup. There is a G-equivariant continuous
map
σ : r0, 1s Gs Ñ Gs , pt, g q ÞÑ σt pg q
with
(1) σ0 pg q g and σ1 pg q P K G for all g P Gs ,
(2) σt pg q g for all t and all g P K G , and
In particular, if g, h P G commute, then so do σt pg q and σt phq for all t.
If G is defined over R, then σ is equivariant under complex conjugation.
8.4. Proof of Proposition 6.1. At this point we are finally ready to prove
Proposition 6.1. Let G be the group of complex points of a connected reductive
algebraic group, K a maximal compact subgroup of G, and recall that G acts on
HompZk , Gq by conjugation. There is a continuous G-equivariant map
Φ : r0, 1s pHompZk , Gq, HompZk , K qq Ñ pHompZk , Gq, HompZk , K qq
with the following properties:
Φp0, ρq ρ and Φp1, ρq P HompZk , K Gq for all ρ P HompZk , Gq.
There is an open covering W of HompZk , K Gq such that for every W P
W one of the following non-exclusive alternatives holds:
(1) All the elements in Φpt1u W q have the same orbit type.
(2) All the elements in W have the same orbit type.
If moreover G is defined over R, then Φ is equivariant under complex conjugation.
Recall that two elements ρ, ρ1 in HompZk , Gq have the same orbit type if their
stabilizers StabG pρq and StabG pρ1 q are conjugate in G.
50
ALEXANDRA PETTET AND JUAN SOUTO
Proof. Choose ¡ 0 small enough for Proposition 8.4 to apply, and let
F : r0, 1s G Ñ G,
pt, gq ÞÑ Ftpgq
be the map provided by that proposition. Recall that F1 pg q gs , where gs P Gs
is the approximated semisimple part of g provided by Theorem 1.7. Let also
σ : r0, 1s Gs
Ñ Gs, pt, gq ÞÑ σtpgq
be the map provided by Lemma 8.5.
Considering elements in HompZk , Gq as commuting k-tuples in G, we define
Φ as follows:
1
Φpt, pa1 , . . . , ak qq pF2t pa1 q, . . . , F2t pak qq
for t ¤
2
1
Φpt, pa1 , . . . , ak qq pσ2t1 ppa1 qs q, . . . , σ2t1 ppak qs qq
for t ¥
2
By construction, the homotopy Φ is continuous, preserves HompZk , K q, and we
have Φp0, ρq ρ and Φp1, ρq P HompZk , K G q for all ρ P HompZk , Gq. Moreover,
it follows from the G-equivariance of F and σ that Φ is G-equivariant as well.
Also, if G is defined over R, then Φ is equivariant under complex conjugation
because F and σ are.
We have proved all claims of Proposition 6.1 but the existence of the covering
W. Suppose now that H € G is a subgroup, and consider the set
XH
tρ P HompZk , K Gq|Dg P G with gHg1 € StabGpρqu
It follows from Proposition 2.8 that there is a triangulation of HompZk , K G q
such that XH is a (closed) subcomplex for all H € G.
Noting that G-equivariance implies Φp1, XH q € XH , observe that the existence of the open cover W follows once we have established the following claim:
Claim. For every H
Φp1, U q € XH .
€
G there is an open neighborhood U of XH with
Proof of the Claim. Given a sequence paj1 , . . . , ajk q in HompZk , K G q converging
to pa1 , . . . , ak q P XH , notice that it follows from Theorem 2.6, Theorem 2.4
and the remark after the latter theorem, that there is a sequence pgj q in G
converging to IdG and with
StabG pgj aj1 gj1 , . . . , gj ajk gj1 q gj StabG paj1 , . . . , ajk qgj1
€ StabGpa1, . . . , ak q
for all sufficiently large j. Observe also that by the G-equivariance of Φ and
the G-invariance of XH , it suffices to prove that
Φp1, gj aj1 gj1 , . . . , gj ajk gj1 q P XH
for all j large enough. We may hence assume, without loss of generality, that
paj1, . . . , ajk q Ñ pa1, . . . , ak q
StabG paj1 , . . . , ajk q € StabG pa1 , . . . , ak q  H
for all j.
51
By assumption, the elements a1 , . . . , ak P G € SLn C commute and are diagonalizable. In particular, they are simultanously diagonalizable. Let
Cn
V1 ` ` Vr
be the coarsest direct sum decomposition of Cn such that for all i
and all s 1, . . . , r we have:
aipVsq Vs, and
the restriction ai|V
s
1, . . . , k
of ai to Vs is a homothety.
Notice that the assumption that Cn V1 ` ` Vr is the coarsest direct sum
decomposition with these properties implies that
StabG pa1 , . . . , ak q G X pGLpV1 q GLpVr qq
For sufficiently large j we have hence that the elements
aj1 , . . . , ajk
P StabGpaj1, . . . , ajk q € StabGpa1, . . . , ak q
preserve V1 , . . . , Vr . Proposition 7.5 implies that for all i and s, and for all
sufficiently large j, we have
paji qsVs Vs, and
the restriction paji qs|V of paji qs to Vs is a homothety.
This implies that ZG ppaji qs q  G X pGLpV1 q GLpVr qq for all j.
s
At this
point notice that the G-equivariance of the map σ provided by Lemma 8.5
implies that we also have
ZG pσ1 ppaji qs qq  G X pGLpV1 q GLpVr qq
for all j. This shows that
StabG pΦp1, paj1 , . . . , ajk qqq StabG pσ1 ppaj1 qs q, . . . , σ1 ppajk qs qq
Xji1ZGppaji qsq
 G X pGLpV1q GLpVr qq
StabGpa1, . . . , ak q  H
implying that Φp1, paj1 , . . . , ajk qq P XH , as we needed to prove.
Having established the claim, we have proved Proposition 6.1.
Appendix A
While reading this Appendix the reader should always keep in mind the
terminology and notation introduced in Section 5. More concretely, notation
will always be as in (*) in Section 5.
We will identify four properties of the set pR, Oq of fibered relation pairs
which ensure that if pX, δ q is a P-labeled complex and U is a regular cover of X,
then every section of S δ U is homotopic to a section of K δ U € S δ U. More
precisely, we will define transitivity, reflexivity, and the simultaneous lifting and
the relative retraction properties so that the following holds:
52
ALEXANDRA PETTET AND JUAN SOUTO
Proposition A.1. With notation as in (*), suppose that R is transitive and reflexive, that it satisfies the simultanous lifting property, and that pR, Oq satisfies
the relative retraction property. Let also pX, δ q be a P-labeled locally finite and
finite dimensional simplicial complex, and U tUc ucPC pX q and V tVc ucPC pX q
regular covers of X with Vc relatively compact in Uc for all c P C pX q. Finally,
let tσc ucPC pX q be a section of S δ U, and W € X with σc pwq P Kδpcq for all
c P C pX q and all x P Uc X W .
The restriction of tσc ucPC pX q to S δ V is homotopic to a section tσc1 ucPC pX q
with σc1 pVc q € Kδpcq for all c P C pX q; moreover the homotopy is constant on
W.
Once Proposition A.1 is proved, we will show that the sets of fibered relation
pairs pRG , OG q associated in (5.1) and (5.2) to a group G satisfies these four
properties. Proposition 5.3 follows immediately.
A.1. Transitivity and Reflexivity. The set R is transitive if
 tpx, zq P S∆ S∆2 |Dy P S∆1 with px, yq P R∆,∆1 , py, zq P R∆1,∆2 u
for all ∆, ∆1 , ∆2 P P with ∆ ¥ ∆1 ¥ ∆2 ; R is reflexive if
R∆,∆ tpx, xq P S∆ S∆ |x P S∆ u
for all ∆ P P.
R∆,∆2
Notice that transitivity and reflexivity of R imply that O is also transitive
and reflexive.
A.2. Simultanous lifting. Suppose now that pX, δ q is a P-labeled simplicial
complex, and let U pUc qcPC pX q be a regular cover. Consider the bundle
—
—
S δ U
cPC pX q Sδ pcq Uc Ñ
cPC pX q Uc . Our goal is to prove that, under
certain conditions, one can homotope sections tσc ucPC pX q of S δ U to sections
tσc1 ucPC pX q such that σc1 takes values in the subset Kδpcq € Sδpcq for all c P C pX q.
In order to do so we want to be able to construct homotopies of sections skeleton
by skeleton, i.e. starting with the maps σc : Uc Ñ Sδpcq corresponding to the
vertices of X, continuing with those corresponding to the edges, etc... We will
see shortly that the following is the property allowing us to do so.
The set R has the simultaneous lifting property if:
(SL) Suppose that ∆0 , ∆1 , . . . , ∆r P P are such that ∆i ¥ ∆0 for all i and
that W0 , . . . , Wr are contractible open sets in a CW-complex with Wi €
W0 for all i. Finally suppose that we have for each i 0, . . . , r a
continuous map σi : Wi Ñ S∆i with pσi pxq, σj pxqq P R∆i ,∆j for all
i, j P t0, . . . , ru with ∆i ¥ ∆j and all x P Wi X Wj .
For any continuous map φ0 : r0, 1s W0 Ñ S∆0 with φ0 p0, xq σ0 pxq
there are continuous maps φi : r0, 1s Wi Ñ S∆i for i 1, . . . , r with
φi p0, xq σi pxq and with
pφipt, xq, φj pt, xqq P R∆ ,∆
for all t, all i, j P t0, . . . , ru with ∆i ¥ ∆j and all x P Wi X Wj .
i
j
Moreover, if the homotopy φ0 is constant outside of some compact
set C € W0 , then φi is also constant outside of C X Wi .
53
Lemma A.2 (Lifting lemma). Suppose that R is transitive and reflexive, and
satisfies the simultaneous lifting property and let U tUc ucPC pX q be a regular
cover of a P-labeled complex pX, δ q.
Suppose moreover that tσc ucPC pX q is a section of S δ U and that for some
c0 P C pX q we have a continuous map
r0, 1s Uc Ñ Sδpc q, pt, xq Ñ σ̄ct pxq
0
0
0
σc , such that
(1) pσ̄ct pxq, σc pxqq P Rδpc q,δpcq for all c0 ‰ c and all x P Uc X Uc .
(2) σ̄ct pxq σc pxq for all x R C where C € Uc is compact.
Then there is a homotopy of sections tσct ucPC pX q of S δ U satisfying
σc0 σc for all c P C pX q,
σct σ̄ct for all t,
σct σc for all c P C pX q with c0 ˆ c and all t, and
the homotopy tσct ucPC pX q is constant on X zC.
Proof. Let c1 , . . . , cs P C pX qztc0 u be cells in X with Uc X Uc H and observe
that these are exactly the cells ci in C pX q with either c0 ˆ ci or with ci ˆ c0 .
with σ̄c00
0
0
0
0
0
0
0
0
0
0
i
We may assume that c1 , . . . , cr are those containing c0 and cr 1 , . . . , cs those
contained in c0 . In particular δ pci q ¥ δ pc0 q for all i 1, . . . , r. Applying the
simultaneous lifting property to
δpc0q, . . . , δpcr q,
the contractible open sets Uci X Uc0 for i 0, . . . , r,
the restrictions of σci to Uci X Uc0 again for i 0, . . . , r, and
the map r0, 1s Uc0 Ñ Sδpc0q, pt, xq ÞÑ σ̄ct0 pxq
we obtain for i 1, . . . , r a map
r0, 1s pUc X Uc q Ñ Sδpc q, pt, xq ÞÑ σ̄ct pxq
satisfying pσ̄ct pxq, σ̄ct pxqq P Rδpiq,δpj q for all i, j P t0, . . . , ru with ci  cj and all
x P Uc X Uc . A priori, the maps σ̄ct are only defined on Uc X Uc . However,
since σ̄ct pxq σc pxq for all x P Uc zC, we can extend σ̄ct continuously to Uc
by σ̄ct pxq σc pxq for all x P Uc zC.
Suppose now that ci ˆ c0 , i.e. i P ts 1, . . . , ru. By the assumptions in
the lemma we have that pσ̄ct pxq, σc pxqq P Rδpc q,δpc q for all x P Uc X Uc .
The transitivity of R implies thus that for all j P t1, . . . , ru we have also
pσ̄ct pxq, σc pxqq P Rδpc q,δpc q for all x P Uc X Uc . It follows that tσct ucPC pX q
0
i
i
i
i
i
0
0
0
i
0
i
i
i
0
i
i
j
j
i
j
i
j
i
i
0
j
i
0
i
j
with
σ̄ct
σct σc
σct
for c P tc0 , . . . , cr u
for c R tc0 , . . . , cr u
is a section of S δ U for all t. It has the desired properties by construction. A.3. Relative retractions. We will be interested later in pairs pS∆ , K∆ q with
the property that the inclusion K∆ Ñ S∆ is a homotopy equivalence. However,
since we are not working with single spaces but with sets of spaces, we need
54
ALEXANDRA PETTET AND JUAN SOUTO
that the homotopy equivalence between K∆ and S∆ behaves well with respect
to other pairs pS∆1 , K∆1 q.
The set pR, Oq satisfies the relative retraction property if:
(RR1) For all ∆ P P the inclusion K∆ Ñ S∆ is a homotopy equivalence.
(RR2) For all ∆, ∆1 P P with ∆ ¥ ∆1 there is a retraction of R∆,∆1 X pS∆ K∆1 qq to O∆,∆1 R∆,∆1 XpK∆ K∆1 q which preserves the fibers of the
projection R∆,∆1 X pS∆ K∆1 qq Ñ K∆1 .
We formulate the second condition in more concrete, but also more obscure
terms:
Lemma A.3 (Reformulation of (RR2)). Suppose that pR, Oq satisfies the relative retraction property. Given ∆0 , ∆1 P P with ∆0 ¥ ∆1 , an open set W in a
CW-complex, continuous maps σi : W Ñ S∆i with pσ0 pxq, σ1 pxqq P R∆0 ,∆1 for
all x P W and such that σ1 pW q € K∆1 , and finally U € W open and relatively
compact, then the following holds:
There is a continuous map φ0 : r0, 1s W Ñ S∆0 satisfying:
(1) φ0 p0, xq σ0 pxq for all x P W ,
(2) pφ0 pt, xq, σ1 pxqq P R∆0 ,∆1 for all x P W and all t,
(3) φ0 pt, xq σ0 pxq for all x outside some compact set C
x P W with σ0 pxq P K∆0 , and
(4) φ0 p1, xq P K∆0 for all x P U .
€
W and all
Before the relative retraction property can be of any use to us, we need
to extract from it something which takes into account the interplay between
multiple members of the set pR, Oq of fibered relations, and not only two of
them. This is done in the following lemma:
Lemma A.4 (Retraction lemma). Suppose that R is transitive and reflexive,
and that pR, Oq satisfies the relative retraction property. Let U tUc ucPC pX q
and V tVc ucPC pX q be regular covers of a P-labeled complex pX, δ q with Vc
relatively compact in Uc for all c P C pX q.
Suppose moreover that tσc ucPC pX q is a section of S δ U and that for some
c0 P C pX q the following holds:
σc pUc q € Kδpcq
@ c P C pX q with c ˆ c0
There is a continuous map φc : r0, 1s Uc Ñ Sδpc q satisfying:
(1) φc p0, xq σc pxq for all x P Uc ,
(2) pφc pt, xq, σc pxqq P Rδpc q,δpcq for all c0 ‰ c, x P Vc X Vc and t,
(3) φc pt, xq σc pxq for all x outside some compact set C € Uc
x P Uc with σc pxq P Kδpc q , and
(4) φc p1, xq P Kδpc q for all x P Vc .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
and all
0
0
Before proving Lemma A.4 we establish some terminology. Suppose that
φ : X Ñ Y is a map between two CW-complexes and that for some (countable)
set of indices i, j, . . . we have homotopies
φi : r0, 1s X
ÑY
55
with φi p0, xq φpxq for all x P X. We say that the homotopies φi do not
interfere with each other if for each i there is a compact set Ci € X of satisfying
the following two properties:
Ci X Cj H for all i j
and no sequence pxi q with xi
convergent subsequence.
φi pt, xq φpxq for all x R Ci and all t.
P Ci
has a
If the homotopies φi do not interfere with each other, then the map
Φ : r0, 1s X
ÑY
and Φpt, xq φpxq
”
given by Φpt, xq φi pt, xq for x P Ki
for x R i Ci is
continuous. We will refer to Φ as the combination of the homotopies φi .
Continuing with the same notation suppose that we have maps
Φ, Φ1 : r0, 1s X
ÑY
with Φp1, xq Φ1 p0, xq for all x P X. The continuous map Φ2 : r0, 1s X Ñ Y
given by Φ2 pt, xq Φp2t, xq for t ¤ 21 and Φ2 pt, xq Φ1 p2t 1, xq for t ¥ 21 is
the juxtaposition of Φ and Φ1 .
Proof of Lemma A.4. Let D dimpc0 q be the dimension of c0 and choose for
i 0, . . . , D a regular cover U i tUci ucPC pX q with U 0 U, U D V and such
that Uci 1 is relatively compact in Uci for all c P C pX q.
Suppose that c ˆ c0 is a codimension 1 face of c0 . Since the relative retraction
property is satisfied, we can apply Lemma A.3 to δ pc0 q ¥ δ pcq, the open set
Uc00 X Uc0 and the restrictions of σc0 and σc to Uc00 X Uc0 . Noticing that Uc10 X Uc1
is relatively compact in Uc00 X Uc0 we obtain hence a continuous map
φc : r0, 1s pUc00
X Uc0q Ñ Sδpc q
0
satisfying:
(1) φc p0, xq σc0 pxq for all x P Uc00 X Uc0 ,
(2) pφc pt, xq, σc pxqq P Rδpc0 q,δpcq for all x P Uc00 X Uc0 and t,
(3) φc pt, xq σc0 pxq for all x outside some compact set C
all x P Uc00 X Uc0 with σ0 pxq P Kδpc0 q , and
(4) φc p1, xq P Kδpc0 q for all x P Uc10 X Uc1 .
€ Uc0 X Uc0 and
0
Notice that defining φc pt, xq σc0 pxq for all x P Uc00 zUc0 we obtain a continuous
map φc defined on the whole of r0, 1s Uc00 . Suppose now that we are given
a further proper face c̄ of c0 with Uc00 X Uc0 X Uc̄0 H and notice that this
implies that c0  c  c̄. We obtain from (2), the assumption that tσc ucPC pX q is
a section, and the transitivity of R, that
pφcpt, xq, σc̄pxqq P Rδpc q,δpc̄q
for all x P Uc0 X Uc0 X Uc̄0 . On the other hand, if x P pUc0 zUc0 q X Uc̄0 we have by
definition that φc pt, xq σc pxq; it follows that (A.1) is actually satisfied for all
x P Uc0 X Uc̄0 and hence:
(2’) pφc pt, xq, σc̄ pxqq P Rδpc q,δpc̄q for all c0 ‰ c̄, x P Uc0 X Uc̄0 and t.
(A.1)
0
0
0
0
0
0
0
56
ALEXANDRA PETTET AND JUAN SOUTO
Suppose that we are given a second codimension one face c1 of c0 . Since
Uc0 X Uc01 H the homotopies φc and φc1 do not interfere with each other.
Combining all the homotopies corresponding to all the codimension one faces
of c0 , we have then that for d 1 there is a map
ψ d : r0, 1s Uc0
Ñ Sδpc q
0
satisfying:
(a) ψ d p0, xq σc0 pxq for all x P Uc0 Uc00 ,
(b) pψ d pt, xq, σc pxqq P Rδpc0 q,δpcq for all c0 ‰ c, x P Ucd01 X Ucd1 and t,
(c) ψ d pt, xq σc0 pxq for all x outside some compact set C € Uc0 and for all
x P Uc0 with σc0 pxq P Kδpc0 q , and
(d) ψ d p1, xq P Kδpcq for all x P Ucd0 X Ucd where c ˆ c0 has at most codimension d.
Arguing by induction, we may assume that for some d ¥ 1 we have a map
ψ d : r0, 1s Uc0 Ñ Sδpc0 q satisfying (a), (b), (c) and (d). Suppose that c € c0 is
a codimension d 1 face. Applying Lemma A.3 to δ pc0 q ¥ δ pcq, the open set
Ucd0 X Ucd and the restrictions of ψ d p1, q and σc to Ucd0 X Ucd , we obtain again a
map
φc : r0, 1s pUcd0 X Ucd q Ñ Sδpc0 q
satisfying
(1̂) φc p0, xq ψ d p1, xq for all x P Ucd0 X Ucd ,
(2̂) pφc pt, xq, σc pxqq P Rδpc0 q,δpcq for all x P Ucd0 X Ucd and t,
(3̂) φc pt, xq ψ d p1, xq for all x outside some compact set C
all x P Ucd0 X Ucd with ψ d p1, xq P Kδpc0 q , and
(4̂) φc p1, xq P Kδpc0 q for all x P Ucd0 1 X Ucd 1 .
€ Ucd X Ucd and
0
As above, we set φc pt, xq ψd p1, xq for all x P Uc0 zUcd obtaining thus a continuous extension
φc : r0, 1s Uc0 Ñ Sδpc0 q
We claim that φc satisfies the following strengthening of (2̂):
(2̂1 ) pφc pt, xq, σc̄ pxqq P Rδpc0 q,δpc̄q for all c0
‰ c̄, x P Ucd X Uc̄d and t.
0
If c̄ € c then, as above, the transitivity of the set R implies that (2̂1 ) holds
for all x P Ucd0 X Uc̄d and t. Suppose now that c ˆ c̄. By (d) we have that
ψ d p1, xq P Kδpc0 q for all x P Ucd0 X Uc̄d . It follows hence from (3̂) and (b) that
pφcpt, xq, σc̄pxqq P Rδpc0q,δpc̄q for all x P Ucd0 X Uc̄d and t. Finally observe that
Ucd X Uc̄d H if c̄ is neither contained in nor contains c. It follows that for any
such c we have φc pt, xq ψ d p1, xq for all x P Uc0 X Uc and all t. The validity of
(2̂1 ) for x P Ucd0 X Uc̄d follows hence from (b). We have proved that φc satisfies
(2̂1 ).
As above, the maps φc and φc1 do not interfere with each other for any two
distinct co-dimension d 1 faces c and c1 of c0 . The juxtaposition
ψd
1
: r0, 1s Uc0
Ñ Sδpcq
57
of ψ d and the combination of all the homotopies φc with c a co-dimension d 1
face of c0 satisfies (a), (b), (c) and (d) for d 1.
Recall that U D V; repeating this process D times we obtain hence a map
ψ
ψD : r0, 1s Uc Ñ Sδpc q
0
0
satisfying:
(a’) ψ p0, xq σc0 pxq for all x P Uc0 ,
(b’) pψ pt, xq, σc pxqq P Rδpc0 q,δpcq for all c0 ‰ c, x P Vc0 X Vc and t,
(c’) ψ pt, xq σc0 pxq for all x outside some compact set C € Uc0 and for all
x P Uc0 with σc0 pxq P Kδpc0 q , and ”
(d’) ψ p1, xq P Kδpc0 q for all x P Vc0 X p cˆc0 Vc q.
So far, we have only used the second part of the relative retraction property
(RR2). Taking now into account (d’) and that the inclusion of Kδpc0 q Ñ Sδpc0 q
is a homotopy equivalence, we can find a homotopy
ψ 1 : r0, 1s Uc0
”
Ñ Sδpc q
0
relative to Vc0 X p cc0 Vc q, to the complement of a compact subset of Uc0 ,
starting with ψ 1 p0, xq ψ p1, xq, such that ψ 1 p1, xq P Kδpc0 q for all x P Vc0 .
Juxtaposing ψ and ψ 1 we obtain finally a continuous map φc0 : r0, 1s Uc0 Ñ
Sδpc0 q satisfying conditions (1), (2), (3) and (4) in the statement of Lemma
A.4.
A.4. Proof of Proposition A.1. Starting with the proof of Proposition A.1,
let D dim X and choose a regular cover U i tUci ucPC pX q , for i 0, . . . , D
with U 0 U, U D V, such that Uci 1 is relatively compact in Uci for all
c P C pX q. We will denote by X pdq the d-skeleton of X, i.e. the simplicial
complex consisting of all the cells of dimension at most d; the set of cells of
dimension precisely d is denoted by Cd pX q.
Again we will argue by induction. Observing that X p1q H, we may
suppose that for some d ¥ 0 we have σc pUcd1 q € Kδpcq for all c P C pX q of
dimension at most d 1.
We will show that there is a homotopy of sections, constant on W , which
starts with the restriction of tσc ucPC pX q to U d and ends with a section tσc1 ucPC pX q
of S δ U d with σc1 pUcd q € Kδpcq for all c P C pX pdq q.
Let c0 P Cd pX q be a d-dimensional cell. Consider the cells c1 , . . . , cr P C pX q
which are properly contained in c0 and notice that δ pc0 q ¥ δ pci q for all i. The
retraction lemma, Lemma A.4, yields a map
σ̄c0 : r0, 1s Ucd01
Ñ Sδpc q, pt, xq ÞÑ σ̄ct pxq
0
0
σc , p q € Kδpc q, such that
pσ̄ct pxq, σc pxqq P Rδpc q,δpc q for all i ¡ 0, x P Ucd X Ucd , and t.
σ̄ct pt, xq σc pxq for all x outside some compact set C € Ucd1 and all
x P W X Ucd1 .
The lifting lemma, Lemma A.2, implies that there is a section tσc1 ucPC pX q of
S δ U d with σc1 σ̄c1 and a homotopy tσct ucPC pX q supported by K between
with
σ̄c00
0
σ̄c10
Ucd0
0
i
0
0
0
0
0
i
0
0
0
i
0
58
ALEXANDRA PETTET AND JUAN SOUTO
tσcucPC pX q and tσc1 ucPC pX q. Moreover, the homotopy t ÞÑ σct is constant unless
c0 € c. In particular, σc1 σc for all c P C pX pd1q q.
Since Ucd X Ucd1 H for every other c10 P Cd pX q, we deduce that the homotopy
just constructed for c0 and the corresponding one for c10 do not interfere with
each other. Combining the homotopies corresponding to all the d-dimensional
cells, we obtain that the restriction to U d of the section tσc u is homotopic,
through a homotopy constant on W , to a section tσc1 u of S δ U d , with σc pUcd q €
Kδpcq for all c P C pX q with dimpcq ¤ d. This concludes the proof of the
induction step and hence the proof of Proposition A.1.
A.5. Proof of Proposition 5.3. Let G be the group of complex or real points
of a connected reductive algebraic group (defined over R in the latter case),
K € G a maximal compact subgroup, and let G and K be the manifolds of all
well-centered subgroups of G and K. Consider π0 pG q with the partial ordering
introduced in Section 3.5, and RG and OG as defined in (5.1) and (5.2). By
Lemma 5.2, pRG , OG q is the set of fibered relation pairs. It follows directly
from the definitions that RG is transitive and reflexive:
Lemma A.5. The set RG of fibered relations is transitive and reflexive.
We prove that RG and pRG , OG q satisfy, respectively, the simultaneous lifting
and relative retraction properties.
Lemma A.6. The set RG satisfies the simultaneous lifting property.
Proof. Suppose that G0 , G1 , . . . , Gr P π0 pG q are such that Gi ¥ G0 for all i and
that W0 , . . . , Wr are contractible open sets in a CW-complex with Wi € W0
for all i. Finally suppose that we have for each i 0, . . . , r a continuous map
σi : Wi Ñ Gi with pσi pxq, σj pxqq P RGi ,Gj for all i, j P t0, . . . , ru with Gi ¥ Gj
and all x P Wi X Wj .
Let also φ0 : r0, 1s W0 Ñ G0 be continuous with φ0 p0, xq σ0 pxq. Choosing
a base point A0 P G0 we consider the map
p : G Ñ G0 , g
ÞÑ gA0g1
Notice that p : G Ñ G0 is a fiber bundle with fiber NG pA0 q; hence p is a
fibration. It follows from the contractibility of W0 that there is map
α : r0, 1s W0
ÑG
such that the following diagram commutes:
r0, 1s m6 G
mmm
m
m
αmmm
p
mmm
mmm
/ G0
W0
φ0
Moreover, if there is a compact set K € W0 with φ0 pt, xq φ0 p0, xq for all
x R K and all t then we may assume also that αpt, xq αp0, xq for all x R K
and all t.
We consider now the map
β : r0, 1s W0
Ñ G, β pt, xq αpt, xqαp0, xq1
59
Observe that β pt, xq Id for all x R K and all t, that β p0, xq Id for all x P W0
and that
φpt, xq αpt, xqA0 αpt, xq1
αpt, xqαp0, xq1σ0pxqαp0, xqαpt, xq1
β pt, xqσ0pxqβ pt, xq1
for all x P W0 and all t.
For i 1, . . . , r we consider the map
φi : r0, 1s Wi
Ñ Gi,
φi pt, xq β pt, xqσi pxqβ pt, xq1
and notice that t ÞÑ φi pt, xq is constant for x outside of K X Wi . Also, it follows
directly from (5.1), the definition of RGi ,Gj , that
pφipt, xq, φj pt, xqq P RG ,G
for all t, all i, j P t0, . . . , ru with Gi ¥ Gj and all x P Wi X Wj . We have proved
i
j
that RG has the simultaneous lifting property.
Lemma A.7. The set pRG , OG q satisfies the relative retraction property.
Proof. To begin with recall that for every G0 P π0 pG q the inclusion K0 Ñ G0 is a
homotopy equivalence by Proposition 3.11; in other words, the first half (RR1)
of the relative retraction property is satisfied. Fixing a base point pA0 , A1 q P
RG0 ,G1 X pK0 K1 q we obtain from Lemma 3.13 and Lemma 3.14 that the
fibrations
R X pG0 K1 q Ñ K1 and R X pK0 K1 q Ñ K1
are locally trivial fiber bundles with fibers
TransG pA1 , A0 q{NG pA1 q and TransK pA1 , A0 q{NK pA1 q
In order to prove that (RR2) is satisfied, it suffices to show that the inclusion
TransK pA1 , A0 q{NK pA1 q Ñ TransG pA1 , A0 q{NG pA1 q
is a homotopy equivalence. Consider the commuting diagram
p
q
/ TransK A1 , A2 NK A1
q
/ TransG A1 , A2 NG A1
NK p A 1 q
/ TransK A1 , A0
/ TransG A1 , A0
NG p A 1 q
p
p
p
q{
p q
q{ p q
The rows are fibrations and the first two columns are homotopy equivalences
by Lemma 3.4. It follows that the third column is also a homotopy equivalence
as we needed to show. This concludes the proof of Lemma A.7.
Proposition 5.3 follows directly from Lemmas A.5, A.6 and A.7 and Proposition A.1.
The second author wishes to quote Günter Netzer, who, to the best of his
understanding was paraphrasing Berti Vogts: Deutsche Tugenden haben wir
immer noch am besten. Gemeint waren Kampf, Kondition, nimmermüde Einsatzbereitschaft, Durchsetzungsvermögen und Siegeswille.
60
ALEXANDRA PETTET AND JUAN SOUTO
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Department of Mathematics, University of Michigan, Ann Arbor, U.S.A.
Mathematical Institute, University of Oxford, Oxford, U.K.
apettet@umich.edu
Department of Mathematics, University of Michigan, Ann Arbor, U.S.A.
jsouto@umich.edu