Tempered irreducible representations of
the free group from vector-valued
multiplicative functions
work in progress with Tim Steger
1
Γ = the (nonabelian) free group on a fixed set
A+ of r generators.
A = A+ ∪ A−
will denote the set of generators and their inverses.
Every x ∈ Γ has a unique expression as a reduced word:
x = a1 . . . an
where ai ∈ A and ai ai+1 6= e
For this x, the length of x, denoted |x|, is n,
the number of letters in its reduced word expression. Set |e| = 0.
2
The Cayley Graph T of Γ with respect to the
set A is a tree, defined as usual:
• V = the set of vertices coincides with Γ.
• E = the set of undirected edges consists
of pairs (x, xa) with x ∈ Γ, a ∈ A.
To every reduced word x = a1 . . . an corresponds
a unique geodesic in T :
{e, a1, a1a2, . . . a1a2 . . . an} “starting” at e and
“ending” at x.
3
Let π be a (continuous) unitary irreducible representation of Γ.
π is tempered if it is weakly contained in the
regular representation of Γ.
[U. Haagerup, 1979]
π is tempered iff ∀ε > 0 and ∀v ∈ Hπ
φπ,v (ε) =
X
e−ε|x||hπ(x)v, vi|2 < +∞
x∈Γ
1
φπ,v (ǫ) = O( 3 )
ε
Fix v ∈ Hπ with kvk = 1. We are interested in
the growth of φπ (ε).
4
Assume that π = πs belongs to the principal
isotropic/anisotropic series
[A. Figà-Talamanca, A. M. Picardello, 1982]
[A. Figà-Talamanca, T. Steger, 1994]
• φπs (ε) ≃ 1ε if s is not an endpoint
• φπs (ε) ≃ 13 if s is an endpoint
ε
Assume that π = πh is constructed from a
multiplicative function as described in
[K-Steger, 1996]:
1
φπh (ε) ≃ 2
ε
5
Boundary Realizations
The Boundary Ω:
all infinite reduced words ω = a1 . . . anan+1 . . .
“limit points” for the sequences ωn = a1 . . . an
Ω can be given the structure of a compact,
totally disconnected, metrizable space; Ω is
homeomorphic to the Cantor set. Fix any
x ∈ Γ let:
Ω(x) =
(
ω ∈ Ω : ω “starts” with x
ω = xωk ωk+1 . . . as reduced 0 word
The Γ action on T extends to an action on Ω.
6
Scalar-valued multiplicative functions
For every a ∈ A choose a nonzero complex
constant h(a). Define
h(a1a2 . . . an) = h(a1)h(a2) . . . h(an )
for x = a1a2 . . . an
and h(e) = 1
From the multiplicative function h we construct
• a probability measure ν on Ω,
• a cocycle P (x, ω),
• a unitary representation πh acting on L2(Ω, dν).
7
Let
P (a, ω) =

 h(a−1 )−1
 h(a)
if ω ∈ Ω(a)
if ω ∈
/ Ω(a)
One has P (a, aω)P (a−1 , ω) = 1.
Extend P by means of the cocycle identity
P (xy, ω) = P (x, ω)P (y, x−1ω).
We want πh acting on L2(Ω, dν) according to
(πh(x)f )(ω) = P (x, ω)f (x−1ω)
We need a suitable measure ν which makes
the action unitary.
νa =
(1 − |h(a)|2)|h(a−1)|
2
2
(1 − |h(a)h(a−1 )| )
We must require
νa > 0,
−1
|h(a)h(a
)| 6= 1,
X
νa = 1
a∈A
8
Isotropic series:
1
1
• h(a) = q − 2 +is or h(a) = q − 2 −is
(where ♯A = q + 1 = 2r )
1 for all a.
• νa = q+1
Anisotropic series: There exists a self adjoint
(µ(x) = µ(x−1)) measure µ supported on A
such that h is a Green function for µ:
h ∗ µ = w(λ)δe
λ=
X
µ(a)h(a−1 ) + w(λ)
a∈A
∗ (Γ).
for some λ in the spectrum of µ in Creg
The representations of More: No such a
measure exists!
9
C(Ω)= the C ∗ algebra of continuous complex
valued functions on Ω. C(Ω) acts on L2(Ω, dν):
π(F )f (ω) = F (ω)f (ω)
for F ∈ C(Ω) and f ∈ L2(Ω, dν).
π(x)π(F )π(x−1) = π(λ(x)F )
where (λ(x)F )(ω) = F (x−1ω).
A representation π ′ of the crossed product C ∗′ , π ′ ) such that
algebra Γ ⋉ C(Ω) is a pair (πΓ
Ω
′ is a unitary representation of Γ on H ′ .
• πΓ
′ is a ∗-representation of C(Ω) on H ′
• πΩ
′ (x)π ′ (F )π ′ (x−1) = π ′ (λ(x)F )
• πΓ
Ω
Γ
Ω
Any of the above defined representations extends to a representation of Γ ⋉ C(Ω).
10
Boundary Realizations.
Given an irreducible representation (π, H) of Γ,
we say that (π ′, H ′) is a boundary realization if
• (π ′, H ′) is an irreducible representation of
Γ ⋉ C(Ω).
• There is an isometry U : H → H ′ which is
also a Γ-map: U π(x) = π ′(x)U .
Fact: π is tempered if and only if it admits
a boundary realization, equivalently iff π extends to a representation π ′ of Γ ⋉ C(Ω).
11
In passing from π to π ′ many things may happen:
• The representation space Hπ must be “enlarged” to H ′ (U is not surjective).
• The representation space Hπ needn’t to be
enlarged (for example Hπ = L2(Ω, dν)).
• The same representation of Γ admits more
than one extention to a representation of
Γ ⋉ C(Ω).
• When U is not surjective it may happen
that π ′|Γ is NOT irreducible.
12
We say that a boundary realization is perfect
if U is surjective.
Phenomenoum: In all the known cases π admits no more than two inequivalent perfect
boundary realizations.
When φπ (ε) ≃ 1ε there are only two possible
cases:
• π admits exactly TWO inequivalent (as representations of Γ ⋉ C(Ω)) perfect boundary
realizations.
• π admits only one boundary realization (π ′, H ′)
which is not perfect. In this case π ′|Γ =
π ⊕ πc is the direct sum of two inequivalent irreducible Γ representations. Moreover (π ′, H ′) is the only boundary realization also for πc.
If φπ (ε) ≃ ǫ12 or φπ (ε) ≃ ǫ13 there is only ONE
boundary realization which is also perfect.
13
Uniform construction for “many” tempered representations.
For each a ∈ A choose a finite dimentional
complex vector space Va.
For each a, b ∈ A with ab 6= e choose a linear
map Hba : Va → Vb.
Define first H∞ : H will be the completion of
H∞.
H∞ = the space of generalized multiplicative
functions.
14
A generalized multiplicative function is a
`
function f : Γ → a∈A Va such that there exists
N = N (f ) so that, for |x| ≥ N
f (xa) ∈ Va
if |xa| = |x| + 1
f (xab) = Hbaf (xa)
if |xab| = |x| + 2
(1)
We declare that f ≃ g if f (x) = g(x) for all but
finitely many elements of Γ.
In short, an element of H∞ is a vector-valued
multiplicative function:
f (abca) = HacHcb Hbava
if f is multiplicative for |x| ≥ 1.
15
The Γ action is obvious: left translation.
Problem: want π unitary, tempered and irreducible.
About irreducibility: assume that there exists
a system of linear subspaces Wa ⊂ Va such that
HbaWa ⊂ Wb
in this case π will be certainly reducible:
f : Γ → a Wa will be an invariant subspace
(closed under any “reasonable” norm).
`
A system is irreducible if the ONLY subspaces
satisfying the above condition are either 0 or
Wa = Va for all a.
16
Inner product on H∞ : A reasonable norm
will come from a tuple (Ba )a∈A where each Ba
is a norm (positive form) on Va. If f = fa we
want to set
kfak2 = Ba (va, va)
Since fa = b ,ab6=e fab, in order to be consistent we must require:
P
Ba (va, va) =
X
Bb(Hbava, Hbava)
b ,ab6=e
The existence is due to the irreducibility condition and to a generalized version of PerronFrobenius theorem.
17
Let P = (Pa )a∈A be a tuple of positive (semidefinite) sesquilinear forms on (Va × Va). Define
(T P )a (wa, wa) =
X
Pb(Hbawa, Hbawa) .
b; ba6=e
for wa ∈ Va
Let Sa be the real vector space of symmetric
sesquilinear forms on Va × Va. Let Ka ⊂ Sa
consist of positive (semidefinite) forms. Then
Ka is a solid cone for every a and the same is
true for the direct sum K = ⊕a∈AKa.
18
Theorem.[Vandergraft 1968] Assume that the
system (Hba, Va) is irreducible. Then the spectral radius ρ of T is an eigenvalue and there
exists a (unique) eigenvector corresponding to
ρ which lies in K.
We shall now assume that an irreducible system is normalized so that ρ = 1. In this case
there exists a tuple of strictly positive forms
(Ba ) so that
(T B)a (wa, wa) = Ba(wa, wa)
=
X
Bb(Hbawa, Hbawa) .
b; ba6=e
In other words, if we let
kfak2 = Ba (va, va)
we get a norm in H∞.
19
Irreducibility and Inequivalence
When (Hba, Va, Ba ) and (Ĥba, V̂a, B̂a) give rise
to irreducible/inequivalent representations of
Γ ⋉ C(Ω) ?
Γ?
The case Γ ⋉ C(Ω):
Theorem.[K-Steger] An irreducible system gives
raise to an irreducible representation of Γ ⋉
C(Ω). Two irreducible systems (Hba, Va, Ba )
and (Ĥba, V̂a, B̂a) give raise to the same representation of Γ ⋉ C(Ω) iff there exists a tuple
of maps Ja : Va → V̂a such that
ĤbaJa = JbHba.
In this case we also have Ja∗Ja = λI for a suitable λ and all a.
20
The case of Γ.
Given two different irreducible systems (Hba, Va, Ba)
and (Ĥba, V̂a, B̂a ) we say that they are equivalent if there exists a tuple of maps Ja : Va → V̂a
such that
ĤbaJa = JbHba.
Theorem.[K-Steger]
Assume that we started from an irreducible
system (Hba, Va, Ba ). Denote by π|Γ the Γ representation arising from this system. Assume
that
φπ (ε) =
1
|hπ(x)v, vi|2e−ε|x| ≃ 2 .
ε
x∈Γ
X
Then π|Γ is irreducible and no other system
(Ĥba, V̂a, B̂a ) gives rise to π|Γ.
21
The case
φπ (ε) =
X
1
2
−ε|x|
|hπ(x)v, vi| e
≃ .
x∈Γ
ε
Start with a system (Hba, Va, Ba).
Let
Ĥba = Ha∗−1b−1
V̂a = Va∗−1
B̂a = Ba−1
Theorem[K-Steger] Assume that (Hba, Va, Ba )
and (Ĥba, V̂a, B̂a) are inequivalent systems. Then
the two Γ representations π|Γ and π̂|Γ are equivalent and irreducible.
Theorem[K-Steger] Assume that (Hba, Va, Ba )
and (Ĥba, V̂a, B̂a) are equivalent systems. Then
the Γ representation π|Γ splits into the sum of
two inequivalent and irreducible Γ representations.
22
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—, 1983, Harmonic Analysis on Free Groups,
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K.,T. Steger, 1996, More Irreducible Boundary
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23
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