Irreducibly represented groups

0.24 MB·English

by  Bachir Bekka

IRREDUCIBLY REPRESENTED GROUPS
8
0
0
Bachir Bekka and Pierre de la Harpe
2

n
a
J Abstract. A group is irreducibly represented if it has a faithful irreducible unitary repre-
sentation. For countable groups, a criterion for irreducible representability is given, which
4
1 generalises a result obtained for finite groups by W. Gaschu¨tz in 1954. In particular, torsion-
free groups and infinite conjugacy class groups are irreducibly represented.

] We indicate some consequencesof this for operator algebras. In particular, we charaterise
R
up to isomorphism the countable subgroups ∆ of the unitary group of a separable infinite
G dimensional Hilbert space H of which the bicommutants ∆′′ (in the sense of the theory of
h. von Neumann algebras) coincide with the algebra of all bounded linear operators on H.
t
a
m
[

2 1. Gaschu¨tz Theorem for infinite groups, and consequences
v
4
Define a group to be irreducibly represented if it has a faithful irreducible unitary repre-
1
8 sentation and irreducibly underrepresented1 if not. For example, a finite abelian group is
1
irreducibly represented if and only if it is cyclic (because finite subgroups of multiplicative
1
6 groups of fields, in particular finite subgroups of C∗, are cyclic). It is a straightforward
0 consequence of Schur’s lemma that a group of which the centre contains a non–cyclic fi-
/
h nite subgroup is irreducibly underrepresented. For finite groups, there are also standard
t
a examples of groups without centre which are irreducibly underrepresented (see Note F in
m
[Burns–11]); moreover, there exists a criterion due to Gaschu¨tz who states for finite dimen-
:
v sional representations over algebraically closed fields of characteristic zero the equivalence
i
X of Properties (i), (iv), and (v) in Theorem 2 below (see [Gasch–54], as well as [Huppe–88,
r § 42] and [P´alfy–79]).
a
The purpose of the present paper is to extend Gaschu¨tz’ result to infinite groups and
unitary representations; for the particular case of finite groups, our arguments provide a
new proof of the main result of [Gasch–54] (at least for complex representations). For a
generalisation of Gachu¨tz’ result of a rather different kind, see [Tushe–93].
Since our arguments use measure theory, it is convenient to avoid the difficulties con-
nectedwithnon–standardspaces, sothatweassumesystematicallythatthe groups involved
are countable (see also Example VII in Subsection 5.a below). Moreover and from now
on, we write “representation” for “unitary representation” and, similarly, “character” for
“unitary character”.
2000 Mathematics Subject Classification. 22D10, 20C07.
Key words and phrases. Group representations, irreducible representations, faithful representations,
infinite groups, von Neumann algebras.
The authors are grateful to the Swiss National Science Foundation for its support.
1
Onthedayofwriting,Googleshows29000000entriesforrepresented groups,2390000forunderrep-
resented groups, 641 000 for “represented groups”, 670 000 for “underrepresented groups”, and zero entry
for “irreducibly underrepresented groups”. In some sense at least, what we have to say is new.
2 BACHIR BEKKA AND PIERRE DE LA HARPE
To formulate our results, we need the following preliminaries. Let Γ be a group.
Let N be a normal subgroup of Γ. A representation σ of N is said to be Γ–faithful if
ker(σγ) = {e}, where e denotes the unit element of the group and where σγ denotes
γ∈Γ
tThe representation n 7−→ σ(γnγ−1), namely the conjugate of σ by γ. For example, if V
denotes the normal subgroup of order 4 in the symmetric group Sym(4) on four letters,
any character of V distinct from the unit character is Sym(4)–faithful (even though V
does not have any faithful character).
If {S } isa family of subsets of Γ, we denote by h{S } i the subgroup of Γ generated
i i∈I i i∈I
by S . Following [Remak–30], we define a foot of Γ to be a minimal normal subgroup
i∈I i
of ΓS, namely a normal subgroup M in Γ such that M 6= {e}, and any normal subgroup of
Γ contained in M is either M or {e}. We denote by F the set of finite feet of Γ. The
Γ
minisocle of Γ is the subgroup MS(Γ) of Γ generated by the union of its finite feet; it is
a characteristic subgroup of Γ. Let A denote the subset of F of abelian groups, and let
Γ Γ
H be the complement of A in F . We define MA(Γ) and MH(Γ) to be the subgroups
Γ Γ Γ
of Γ generated by A and H respectively; both are characteristic subgroups
A∈AΓ H∈HΓ
of Γ contained inSMS(Γ). By tShe usual convention, MS(Γ) = {e} if F is empty, and
Γ
similarly for MA(Γ) and MH(Γ).
1. Proposition. Let Γ be a group, and let the notation be as above.
(i) Each A ∈ A is isomorphic to (F )n for some prime p and some positive integer n
Γ p
(depending on A).
(ii) There exists a subset {A } of A such that MA(Γ) = A . In particular,
i i∈I Γ i∈I i
the group MA(Γ) is abelian. L
(iii)For each H ∈ H , the feetS ,...,S of H are conjugate in Γ, and simple. Moreover
Γ 1 k
H = S ⊕···⊕S .
1 k
(iv) We have MH(Γ) = H.
H∈HΓ
(v) We have MS(Γ) = MLA(Γ)⊕MH(Γ).
For some examples of minisocles, see Section 5.a. Here is our first main result.
2. Theorem. Let Γ be a countable group. Let MA(Γ) = A and
i∈I i
MS(Γ) = MA(Γ)⊕MH(Γ) be as above. The following prLoperties are equivalent:
(i) Γ is irreducibly represented;
(ii) MA(Γ) has a Γ–faithful character;
(iii) MS(Γ) has a Γ–faithful irreducible representation;
(iv) for every finite subset E of I, there exists an element x in MA (Γ) + A
E E i∈E i
such that the Γ–conjugacy class of x generates MA (Γ); L
E E
(v) for every pair of finite subsets E ⊂ I and F ⊂ H , there exists an element z
Γ E,F
in MS (Γ) + A ⊕ H such that the Γ–conjugacy class of z
E,F i∈E i H∈F E,F
generates MS (cid:0)(LΓ). (cid:1) (cid:0)L (cid:1)
E,F
In particular, a countable group Γ has a faithful irreducible representation as soon as
MA(Γ) = {e}, and a fortiori as soon as MS(Γ) = {e}.
The next corollary is a straightforward consequence of Theorem 2. Recall that a group
is icc if it is not reduced to one element and if all its conjugacy classes distinct from {e}
are infinite.
3. Corollary. For a countable group to be irreducibly represented, any of the three follow-
ing conditions is sufficient: (i) the group is torsionfree, (ii) the group is icc, (iii) the group
has a faithful primitive action on an infinite set.
IRREDUCIBLY REPRESENTED GROUPS 3
The case of icc groups is well–known, sometimes with a different proof. Indeed, a group
is icc if and only if its von Neumann algebra is a factor of type II (Lemma 5.3.4 of [RO–
1
IV]); it is then a standard fact that the reduced C∗–algebra of an icc group has a faithful
irreducible representation, so that a fortiori the group itself has a faithful irreducible
representation (see for example Proposition 21 of [Harpe–07]).
For a group Γ which has a faithful primitive action on an infinite set X (see [GelGl]),
observe that any normal subgroup of Γ not reduced to {e} is transitive on X and therefore
infinite, so that MS(Γ) = {e}.
Theorem 2 does not state anything on the dimensions of the representations which can
occur in (i). Before providing some information, let us recall that a group is virtually
abelian if it has an abelian subgroup of finite index.
4. Theorem. For a countable group Γ, the two following properties are equivalent:
(i) Γ has an infinite dimensional faithful irreducible representation;
(ii) Γ has the properties of Theorem 2 and is not virtually abelian.
In other words, the following properties are equivalent:
(iii) Γ has a faithful irreducible representation, and all its faithful irreducible represen-
tations are finite dimensional;
(iv) Γ has the properties of Theorem 2 and is virtually abelian.
Let M be a von Neumann algebra. We denote by U(M) the unitary group {X ∈ M |
X∗X = XX∗ = 1} of M and by S′′ the double commutant of a subset S of M. Recall
that M is a factor if its centre is reduced to C, a factor of type I if there exists a Hilbert
space H such that M = L(H), and a factor of type I in case H is infinite dimensional
∞
(moreover, we assume here that Hilbert spaces are separable). For factors of type I, we
write U(H) instead of U(M).
5. Corollary. Let M = L(H) be a factor of type I . For a countable group Γ, the
∞
following two properties are equivalent:
— there is a subgroup ∆ of U(H) isomorphic to Γ such that ∆′′ = M;
— Γ has the properties of Theorem 2 and is not virtually abelian.
It would be interesting to have some information of this kind for other factors. In
particular, wedo notknow anyanalogueofTheorem4forany givenfinitedimension n ≥ 2,
nor of Corollary 5 for the finite dimensional factor L(Cn). We do not know any solution
to the a priori easier problem to characterise the countable groups which have at least one
finite dimensional faithful irreducible representation.
The proof of Proposition 1 uses standard arguments (compare with Section 4.3 of
[DixMo–96]). For the convenience of the reader, we give details in Section 2. Theorem 2
is proved in Section 3. Theorem 4 and Corollary 5 are proved in Section 4. We formu-
late a few remarks in Section 5: on examples of socles and minisocles, on the comparison
between minisocles and periodic FC–kernels, on a theorem of Gelfand and Raikov, on ten-
sor products of faithful representations, and on countable groups with primitive maximal
C∗–algebras. The final Section 6 is devoted to a generalisation of Theorem 2 concerning
a countable group Γ given together with a group of automorphisms G which contains the
group of inner automorphisms.
4 BACHIR BEKKA AND PIERRE DE LA HARPE
Understanding groups of a given class includes understanding their faithful actions of
variouskinds, andthesettingoflinear(orunitary)actionsisonlyoneamongseveralothers.
For example, in the case of finite groups, the questions of classifying multiply transitive
actions and primitive actions which are faithful have been central in group theory for more
than hundred years; faithful primitive actions for infinite groups have been addressed in
[GelGl]. Faithful amenable actions are the subject of [GlaMo–07]. Our initial motivation
has been to ask some of the corresponding questions for linear actions.
We are most grateful to Yair Glasner for explaining us his work [GelGl] and for his con-
tribution to the setting out of the present work, to Yehuda Shalom for a useful observation,
and to Yves de Cornulier and John Wilson for their remarks on a preliminary version of
this paper.
2. Proof of Proposition 1
We prepare the proof of Proposition 1 by recalling two lemmas.
6. Lemma. Let Γ be a group. Let M be a minimal normal subgroup of Γ and N a normal
subgroup of Γ. Then either M ⊂ N or hM,Ni = M ⊕N.
Proof. We can assume N 6= {e}. Since M ∩ N is both in M and normal in Γ, either
M ∩N = M, and M ⊂ N, or M ∩N = {e}, and hM,Ni = M ⊕N. (cid:3)
7. Lemma. Let A be a group and let (S ) be a family of nonabelian simple groups; set
i i∈I
S = A⊕ S . Let M be a minimal normal subgroup of S.
i∈I i
Then e(cid:0)iLther M (cid:1)= S for some ℓ ∈ I, or M ⊂ A.
ℓ
Proof. Assume that M 6= S for all ℓ ∈ I. Choose i ∈ I; by Lemma 6 applied to M and
ℓ
N = S , the groups M and S commute. It follows that M is a subgroup of the centraliser
i i
of S in S, namely a subgroup of A. (cid:3)
i∈I i
L
Proof of Proposition 1. (i) Let A ∈ A . By the structure theory of finite abelian
Γ
groups, there exist a prime p and an element a ∈ A of order p. Let A∗ denote the set of
elements of order p in A. Then A∗∪{e} is a characteristic subgroup of A, and therefore a
normal subgroup of Γ. By minimality of A, we have A∗∪{e} = A, so that A is isomorphic
to (F )n for some n ≥ 1, as claimed.
p
(ii) Let L be the set of subsets {A } of A such that h{A } i = A ; we order
ℓ ℓ∈L Γ ℓ ℓ∈L ℓ∈L ℓ
L by inclusion. The crucial observation is that the ordered set L is indLuctive, so that we
can choose a maximal element, say {A } . Suppose that A is strictly contained in
i i∈I i∈I i
MA(Γ); we will arrive at a contradiction. L
Choose B ∈ A such that B is not contained in A . By Lemma 6 applied to
Γ i∈I i
M = B and N = A , we have either B ⊂ AL, which is ruled out by the choice
i∈I i i∈I i
of B, or hB,{A }Li = B ⊕ A , which iLs ruled out by the maximality of I. This
i i∈I i∈I i
is the announced contradictio(cid:0)nL. (cid:1)
(iii) Let H ∈ H . Choose a minimal normal subgroup S in H (this is possible since
Γ
H is finite). For each x ∈ Γ, the subgroup xSx−1 is minimal normal in H. Choose a set
S ,...,S of such conjugates of S in Γ which is such that hS ,...,S i = S ⊕···⊕S and
1 k 1 k 1 k
which is maximal for this property. Set N = hS ,...,S i; it is a normal subgroup of H.
1 k
Weclaimthat xSs−1 ⊂ N foreach x ∈ Γ, so that N isnormalinΓ. Indeed, by Lemma 6
applied to M = xSx−1 and N in H, either hxSx−1,S ,...,S i = xSx−1 ⊕S ⊕···⊕S ,
1 k 1 k
IRREDUCIBLY REPRESENTED GROUPS 5
but this is ruled out by the maximality of the set {S ,...,S }, or xSx−1 ⊂ N, and this
1 k
establishes the claim.
Since N is normal in Γ and N ⊂ H, we have N = H by minimality of H. Observe that,
for each i ∈ {1,...,k}, any normal subgroup of S is normal in H; it follows that S is a
i i
simple group. Finally, the set {S ,...,S } coincides with the set of all minimal normal
1 k
subgroups of H by Lemma 7.
(iv) The same argument as for (ii) shows that there exists a subset {H } of H such
k k∈K Γ
that H = MH(Γ), and Lemma 7 implies that {H } = H .
k∈K k k k∈K Γ
L
(v) Again by the same argument as for (ii), there exists a subset {M } of F such
ℓ ℓ∈L Γ
that M = MS(Γ), and Lemma 7 implies that {M } contains H . (cid:3)
ℓ∈L ℓ ℓ ℓ∈L Γ
L
3. Proof of Theorem 2
We will prove successively that
(i) =⇒ (ii) & (iii) (see Lemma 9),
(iii) =⇒ (i) (Lemma 10),
(ii) ⇐⇒ (iii) (Lemma 13),
(ii) ⇐⇒ (iv) (Lemma 14).
The equivalence (iv) ⇐⇒ (v) is straightforward, since nonabelian feet are direct products
of simple groups. Recall that we write “representation” for “unitary representation”.
Given a representation π of a countable group Γ in a Hilbert space H, there exist a
standard Borel space Ω, a bounded positive measure µ on Ω, a measurable field ω 7−→ π
ω
of irreducible representations of Γ in a measurable field ω 7−→ H of Hilbert spaces on Ω,
ω
⊕
and an isomorphism of H with H dµ(ω) which implements a unitary equivalence
Ω ω
R
⊕
π(γ) ≈ π (γ)dµ(ω)
Z ω
Ω
for all γ ∈ Γ. See [Dix–69C∗, Sections 8.5 and 18.7.6]. (Such decompositions in irre-
ducible representations carry over to continuous representations of separable locally com-
pact groups, and more generally of separable C∗–algebras. They are applications of the
reduction theory for von Neumann algebras [Dix–69vN, Chapter II]). The following lemma
is standard, but we haven’t found any appropriate reference.
8. Lemma. Let Γ be a countable group. Let Ω be a measure space with a positive mea-
sure µ. Let ω 7−→ π be a measurable field of representations of Γ in a measurable field of
ω
Hilbert spaces ω 7−→ H over Ω and let γ ∈ Γ.
ω
Then {ω ∈ Ω | π (γ) = I} is a measurable subset of Ω.
ω
Proof. Let (ξ(1),ξ(2),....) be a fundamental sequence of measurable vector fields (see [Dix-
69vN, Chapter II, Number 1.3]). For i,j ≥ 1, consider the set
Ω = {ω ∈ Ω | hπ (γ)ξ(i)(ω),ξ(j)(ω)i = hξ(i)(ω),ξ(j)(ω)i}.
i,j ω
Observe that
{ω ∈ Ω | π (γ) = I} = Ω .
ω i,j
\
i,j≥1
6 BACHIR BEKKA AND PIERRE DE LA HARPE
Therefore it suffices to show that each set Ω is measurable.
i,j
For fixed i,j ≥ 1, the functions
ω 7−→ hξ(i)(ω),ξ(j)(ω)i and ω 7−→ hπ (γ)ξ(i)(ω),ξ(j)(ω)i
ω
are measurable, by definition of a measurable vector field and of a measurable field of
representations. Hence Ω is measurable and the proof is complete. (cid:3)
i,j
Let us now recall a general fact which can be seen as a weak form of Clifford theorem for
infinite dimensional representations. (For a version of Clifford theorem concerning finite
dimensional representations but possibly infinite groups, see Theorem 2.2 in [Dixon–71].)
9. Lemma. Let Γ be a countable group, N a normal subgroup, π an irreducible represen-
tation of Γ in a Hilbert space H, and σ the restriction of π to N. Identify σ to a direct
integral of irreducible representations
⊕
σ = π| = σ dµ(ω)
N Z ω
Ω
as above.
If the representation π is faithful, then the representation σ is Γ–faithful for almost all
ω
ω ∈ Ω.
Proof. If N = {e}, there is nothing to prove. We assume from now on that N is not
reduced to one element.
Denote by {C } the family of Γ–conjugacy classes in N distinct from {e}. For each
j j∈J
j ∈ J, denote by N the subgroup of N generated by C ; observe that each N is normal
j j j
in Γ, and that the family {N } is countable (possibly finite) and nonempty. Set
j j∈J
Ω = ω ∈ Ω | N ⊂ ker σγ and Ω = Ω .
j j ω j
n (cid:16)M (cid:17)o [
γ∈Γ j∈J
e
For ω ∈ Ω, observe that σ is not Γ–faithful if and only if the kernel of σγ contains
ω γ∈Γ ω
L
one of the N ; thus Ω is the subset of Ω of the points ω such that σ is not Γ–faithful.
j ω
Each Ω is measurable in Ω as a consequence of Lemma 8; as J is countable, Ω is also
j e
measurable.
e
To end the proof, we assume that µ(Ω) > 0 and we will arrive at a contradiction.
As the family J is countable, there exists ℓ ∈ J such that µ(Ω ) > 0. Hence the unit
e ℓ
representation 1 of the group N is strongly contained in the restriction of σ to N , so
Nℓ ℓ ℓ
that the subspace of H of N –invariant vectors is not reduced to {0}. Since N is normal
ℓ ℓ
in Γ, this subspace is invariant by π(Γ); by irreducibility of π, this subspace is the whole of
H. In other words, the restriction of π to N is the unit representation. The last statement
ℓ
is a contradiction, since π is faithful. (cid:3)
Theparticular case ofLemma 9 for which N = MA(Γ)[respectively N = MS(Γ)]shows
that (i) implies (ii) [respectively (iii)] in Theorem 2. The implication (iii) =⇒ (i) follows
from the next lemma applied to N = MS(Γ) since, by definition, there does not exist any
finite foot M of Γ such that M ∩MS(Γ) = {e}.
IRREDUCIBLY REPRESENTED GROUPS 7
10. Lemma. Let Γ be a countable group, N a normal subgroup, σ an irreducible represen-
tation of N in a Hilbert space K, and π = IndΓ (σ) the corresponding induced representa-
N
⊕
tion. Let π = π dµ(ω) be a direct integral decomposition of π into irreducible represen-
Ω ω
tations. AssumRe that there does not exist any finite foot M in Γ such that M ∩N = {e}.
If the representation σ is Γ–faithful, then the representation π is faithful for almost all
ω
ω in Ω.
Proof. In the model we choose for induced representations, π acts on the Hilbert space H
of mappings f : Γ −→ K with the two following properties:
(1) f(γn) = σ(n−1)f(γ) for all γ ∈ Γ and n ∈ N,
(2) kf(γ)k2 < ∞.
X
Γ/N
(The notation of (2) indicates a summation over one representant γ ∈ Γ of each class in
Γ/N.) Then (π(x)f)(γ) = f(x−1γ) for all x,γ ∈ Γ.
Denote this time by {C } the family of conjugacy classes of Γ distinct from {e}. For
j j∈J
each j ∈ J, denote by Γ the subgroup generated by C , which is a normal subgroup of Γ
j j
not reduced to {e}; set
Ω = {ω ∈ Ω | Γ ⊂ ker(π )} and Ω = Ω .
j j ω j
[
j∈J
e
As in the proof of Lemma 9, Ω is the set of points ω such that π is not faithful, and it
ω
is measurable. To end the proof, we assume that µ(Ω) > 0, so that there exists ℓ ∈ J for
e
which µ(Ω ) > 0, and we will arrive at a contradiction.
ℓ e
Continuing as in the proof of Lemma 9, we observe that there exists a nonzero vector
⊕
f : Γ −→ K in H = H dµ(ω) which is supported in Ω (as a measurable section of the
Ω ω ℓ
field of Hilbert spacRes ω 7−→ H underlying the field of representations ω 7−→ π ), and
ω ω
which is such that π(x)f = f for all x ∈ Γ .
ℓ
Let γ ∈ Γ be such that f(γ−1) 6= 0; set ξ = f(γ−1). Using (1), we find
0 0 0
(3) ξ = f(γ−1) = f(x−1γ−1) = f γ−1(γ x−1γ−1) = σ(γ xγ−1)ξ = σγ0(x)ξ
0 0 0 0 0 0 0
(cid:0) (cid:1)
for all x ∈ Γ ∩N.
ℓ
Claim 1: Γ ∩ N = {e}. Denote by KΓℓ∩N the subspace of K of vectors invariant by
ℓ
σγ0(Γ ∩N). This is a σγ0(N)–invariant subspace of K, since Γ ∩N is a normal subgroup
ℓ ℓ
of N. Now KΓℓ∩N 6= {0} by (3) and KΓℓ∩N = K because σγ0 is irreducible. Thus Γ ∩N is
ℓ
inside the kernel of the representation σγ0 of N; as Γ ∩N is normal in Γ, the group Γ ∩N
ℓ ℓ
is also inside the kernel of the representation σγ of N for all γ ∈ Γ. As σ is Γ–faithful,
Γ ∩N = {e}, as claimed.
ℓ
Claim 2: the subgroup Γ of Γ is finite. Consider the function
ℓ
ϕ : Γ −→ R , γ 7−→ kf(γ)k.
+
8 BACHIR BEKKA AND PIERRE DE LA HARPE
We have
(4) ϕ(γ−1) 6= 0,
0
(5) ϕ is constant under right translations by elements of N,
(6) |ϕ(γ)|2 < ∞,
X
Γ/N
(7) ϕ is invariant under left translations by elements of Γ .
ℓ
Itfollowsfrom(4)to(7)thattheimageofΓ γ inΓ/N isfinite. Theimageofγ−1Γ γ = Γ
ℓ 0 0 ℓ 0 ℓ
in Γ/N is also finite, so that the index of N in Γ N is finite. Claim 2 follows since Γ N is
ℓ ℓ
isomorphic to the direct sum Γ ⊕N by Claim 1.
ℓ
Any subgroup M of Γ which is normal in Γ and minimal for this property is a finite
ℓ
foot of Γ, and M∩N = {e} by Claim1. This is in contradiction with one of the hypotheses
of the lemma. (cid:3)
The particular case N = {e} is of independent interest.
11. Proposition. Let Γ be a countable infinite group which does not contain any finite
⊕
foot, and let λ = π dµ(ω) be a direct integral decomposition of λ into irreducible
Γ Ω ω Γ
representations. TheRn π is faithful for almost all ω ∈ Ω.
ω
Next,weshowthat(ii) ⇐⇒ (iii)inTheorem2. ThiswillbeaconsequenceofLemma13,
for the proof of which we will call upon the following lemma.
For a Hilbert space H, we denote by L(H) its algebra of bounded linear operators.
12. Lemma. Let H ,H be two Hilbert spaces. Let S ∈ L(H ),S ∈ L(H ) be such that
1 2 1 1 2 2
S ⊗S ∈ L(H ⊗H ) is a non–zero multiple of the identity operator.
1 2 1 2
Then S and S are multiples of the identity.
1 2
Proof. Let λ ∈ C∗ be such that S ⊗S = λI. Let {ξ } be a Hilbert space basis of H .
1 2 i i∈I 1
Since S 6= 0, there exist η ,η ∈ H such that
2 1 2 2
hS (η ),η i =6 0.
2 1 2
For every ξ ∈ H , we have
1
h(S ⊗S )(ξ ⊗η ),ξ ⊗η i = hS (ξ),ξ ihS (η ),η i
1 2 1 i 2 1 i 2 1 2
and hence
1
hS (ξ),ξ i = h(S ⊗S )(ξ ⊗η ),ξ ⊗η i
1 i 1 2 1 i 2
hS (η ),η i
2 1 2
λ
= hξ ⊗η ,ξ ⊗η i
1 i 2
hS (η ),η i
2 1 2
λhη ,η i
1 2
= hξ,ξ i
i
hS (η ),η i
2 1 2
for all i ∈ I. It follows that
S (ξ) = hS (ξ),ξ iξ
1 1 i i
X
i∈I
λhη ,η i
1 2
= hξ,ξ iξ
i i
hS (η ),η i
2 1 2 X
i∈I
λhη ,η i
1 2
= ξ
hS (η ),η i
2 1 2
IRREDUCIBLY REPRESENTED GROUPS 9
for every ξ ∈ H , showing that S is a multiple of the identity. A similar argument applies
1 1
to S . (cid:3)
2
13. Lemma. Let Γ be a group and let N be a normal subgroup of Γ. Assume that N =
A ⊕ S, where A is an abelian normal subgroup of Γ and where S is the direct sum of a
family (S ) of finite simple nonabelian normal subgroups of S. The following properties
i i∈I
are equivalent:
(i) N has a Γ-faithful irreducible representation;
(ii) A has a Γ-faithful character.
Proof. Assume first that there exists a Γ–faithful irreducible representation π of N. Since
the factor A of N = A ⊕ S is abelian, and in particular a type I group, there exist a
character χ of A and an irreducible representation ρ of S such that π = χ⊗ρ [Dix–69C∗,
Proposition 13.1.8]. Since ker(χγ) = ker((πγ)| ) for all γ ∈ Γ, the character χ of A is
A
Γ–faithful.
Assume now that there exists a Γ–faithful character χ of A. We claim that there exists
an irreducible representation ρ of S such that, for every γ ∈ S, γ 6= e, the operator ρ(γ) is
not a multiple of the identity operator. Lemma 12 will then imply that the exterior tensor
product χ⊗ρ is a Γ–faithful representation of N = A⊕S.
For every i ∈ I, let ρ be an irreducible representation of S distinct from the unit rep-
i i
resentation, in some Hilbert space H . Choose a unit vector η ∈ H . Consider the infinite
i i i
tensor product ρ = ρ of the family (ρ ) with respect to the family (η ) . Recall
i∈I i i i∈I i i∈I
that ρ is the represeNntation of S defined on the infinite tensor product H = ⊗ (H ,η )
i∈I i i
of the family of Hilbert spaces (H ) with respect to the family (η ) by
i i∈I i i∈I
ρ (γ ) ξ ⊗ η = ρ (γ )ξ ⊗ η ,
i i∈I f i i i f i
(cid:16) (cid:17)(cid:16)(cid:0)O (cid:1) (cid:0) O (cid:1)(cid:17) (cid:0)O (cid:1) (cid:0) O (cid:1)
f∈F i∈I\F f∈F i∈I\F
for every finite subset F of I, element (γ ) ∈ S with γ = 1 whenever i ∈ I \ F, and
i i∈I i
decomposable vector (ξ ) ∈ H . The representation ρ is irreducible, since the
f f∈F f∈F f
ρ ’s are irreducible. For all this, Nsee for example [Guich–66], in particular Corollary 2.1.
i
Let us check that, for γ = (γ ) ∈ S, γ 6= e, the operator ρ(γ) is not a multiple of the
i i∈I
identity operator. Choose j ∈ I such that γ 6= e. Observe that the set
j
{δ ∈ S : ρ (δ) is a multiple of the identity operator}
j j
is an abelian normal subgroup of S and is therefore reduced to {e} since S is simple and
j j
nonabelian. The operator ρ (γ ) is therefore not a multiple of the identity operator. We
j j
can now write
H = H ⊗H′ and ρ = ρ ⊗ρ′
j j j j
where ρ′ is the tensor product of the family (ρ ) , defined on H′ = (H ,η ).
j ℓ ℓ∈I\{j} j ℓ∈I\{j} ℓ ℓ
Lemma 12 implies that ρ(γ) is not a multiple of the identity operator. N (cid:3)
It remains to show that (ii) ⇐⇒ (iv) in Theorem 2. This will be a consequence of the
following lemma.
We are most grateful to Roland L¨otscher, who pointed out a mistake at this point in a
first version of our paper; we are also grateful to Jacques Th´evenaz for a helpful discussion
on modular representations.
10 BACHIR BEKKA AND PIERRE DE LA HARPE
14. Lemma. Let Γ be a countable group; set A = MA(Γ). Let {A } be a set of finite
i i∈I
abelian feet of Γ as in Proposition 1, so that A = A . For each finite subset E of I,
i∈I i
set A = A , which is a finite abelian grouLp. Let A,A denote the dual group of
E i∈I i E
A,A respLectively. The following properties are equivalent:
E b b
(i) A has a Γ–faithful character;
(ii) there exists a character χ ∈ A such that the subgroup generated by χΓ + {χγ |
γ ∈ Γ} is dense in A;
b
(iii) for every finite subset E of I, the finite group A has a Γ–faithful character.
b E
(iv) for every finite subset E of I, there exists χ ∈ A such that A is generated by the
bE E
Γ–orbit of χ.
b b
(v) for every finite subset E of I, there exists x ∈ A such that A is generated by
E E E
the Γ–conjugacy class of x .
E
Proof. Equivalence of (i) and (ii) and equivalence of (iii) and (iv). Let N be a normal
abelian subgroup of Γ. Let χ ∈ N. Denote by H the closed subgroup of N generated by
χΓ. By Pontrjagin duality, the unitary dual of the compact abelian group N/H can be
b b
identified with the subgroup
b
H⊥ = {a ∈ N : ψ(a) = 1 for all ψ ∈ H};
observe that
H⊥ = {a ∈ N : ψ(a) = 1 for all ψ ∈ χΓ} = ker(χγ).
\
γ∈Γ
Thus χΓ is dense in N if and only if H⊥ = {e}, namely if and only if χ is Γ–faithful.
Equivalence of (ii)band (iii). It is clear that (ii) implies (iii). Let us assume that (iii)
holds; we have to check that this implies (ii). For every finite subset E of I, denote by
p : A → A the canonical projection. Consider the subset
E E
b b
X = {χ ∈ A | the Γ–orbit of p (χ) generates A }.
E E E
b b
Since the group A is finite, the subset X of A is closed. For a finite family E ,...,E of
E E 1 k
finite subsets of I, the intersection X ∩···∩X contains X . By Condition (iii),
b E1 bEk E1∪···∪Ek
X is non empty for any finite subset E of I. Since A is compact, it follows that
E
b
X 6= ∅,
E
\
E
where E runs over all finite subsets of I. Let χ ∈ X . It is easily checked that χ is
E E
Γ–faithful. T
Equivalence of (iv) and (v). Consider a finite subset E of I. Recall that each A is a
i
finite dimensional vector space over a prime field F , for a prime number p . For each
pi i
prime p, denote by V the direct sum of those A with i ∈ E which are vector spaces over
p i
F , and denote by P the set of primes p such that V 6= {0}. We have A = V .
p p E p∈P p
Since the V ’s are subgroups of A of pairwise coprime orders, every subgroup HLof A is
p E E