New York Journal of Mathematics
New York J. Math. 21 (2015) 783–800.
Induced representations arising from a
character with finite orbit in a semidirect
product
Palle Jorgensen and Feng Tian
Abstract. Making use of a unified approach to certain classes of induced representations, we establish here a number of detailed spectral
theoretic decomposition results. They apply to specific problems from
noncommutative harmonic analysis, ergodic theory, and dynamical systems. Our analysis is in the setting of semidirect products, discrete
subgroups, and solenoids. Our applications include analysis and ergodic
theory of Bratteli diagrams and their compact duals; of wavelet sets,
and wavelet representations.
Contents
1. Introduction
2. The representation IndG
B (χ)
3. Irreducibility
4. Super-representations
5. Induction and Bratteli diagrams
References
783
786
788
793
796
799
1. Introduction
The purpose of the present paper is to demonstrate how a certain induction (from representation theory) may be applied to a number of problems in
dynamics, in spectral theory and harmonic analysis; yielding answers in the
form of explicit invariants, and equivalences. In more detail, we show that a
certain family of induced representations forms a unifying framework. The
setting is unitary representations, infinite-dimensional semidirect products
and crossed products. While the details in our paper involve a variant of
Received April 19, 2015.
2010 Mathematics Subject Classification. Primary: 47L60, 46N30, 46N50, 42C15,
65R10, 05C50, 05C75, 31C20; Secondary: 46N20, 22E70, 31A15, 58J65, 81S25.
Key words and phrases. Hilbert space, unitary representation, induced representations,
Frobenius reciprocity, semidirect products, imprimitivity, Pontryagin duality, direct integral decompositions, wavelet-sets, noncommutative harmonic analysis, Halmos–Rohlin,
C ∗ -algebra, solenoid.
ISSN 1076-9803/2015
783
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PALLE JORGENSEN AND FENG TIAN
Mackey’s theory, our approach is extended, and it is constructive and algorithmic. Our starting point is a family of imprimitivity systems. By this
we mean a pair: a unitary representation, and a positive operator-valued
mapping, subject to a covariance formula for the two (called imprimitivity).
One of its uses is a characterization of those unitary representations of some
locally compact group G “which arise” as an induction from a subgroup; i.e.,
as induced from of a representation of a specific subgroup of G. (Here, the
notation “which arise” means “up to unitary equivalence.”)
Below we give a summary of notation and terminologies, and we introduce
a family of induced representations, see, e.g., [Jor88, Mac88, Ørs79].
The framework below is general; the context of locally compact (nonabelian) groups. But we shall state the preliminary results in the context of
unimodular groups, although it is easy to modify the formulas and the results given to nonunimodular groups. One only needs to incorporate suitable
factors on the respective modular functions, that of the ambient group and
that of the subgroup. Readers are referred to [Ørs79] for additional details
on this point. Another reason for our somewhat restricted setting is that our
main applications below will be to the case of induction from suitable abelian
subgroups. General terminology: In the context of locally compact abelian
groups, say B, we shall refer to Pontryagin duality (see, e.g., [Rud62]); and
so in particular, when the abelian group B is given, by “the dual” we mean
the group of all continuous characters on B, i.e., the one-dimensional unitary
representations of B. We shall further make use of Pontryagin’s theorem to
the effect that the double-dual of B is naturally isomorphic to B itself.
Our results are motivated in part by a number of noncommutative harmonic analysis issues involved in the analysis of wavelet representations,
and wavelet sets; see especially [LiPT01]; and also [LaSST06, CM11, CM13,
CMO14, MO14].
Definition 1.1. Let G be a locally compact group, B ⊂ G a closed subgroup;
both assumed unimodular. Given V ∈ Rep (B, HV ), a representation of B
in the Hilbert space HV , let
(1.1)
U = IndG
B (V )
be the induced representation. U acts on the Hilbert space HU as follows:
HU consists of all measurable functions f : G → HV s.t.
(1.2)
(1.3)
f (bg) = Vb f (g) , ∀b ∈ B, ∀g ∈ G,
Z
kf k2HU =
kf (g)k2 dµB\G (g) < ∞,
B\G
w.r.t. the measure on the homogeneous space B\G, i.e., the Hilbert space
carrying the induced representation. Set
(1.4)
(Ug f ) (x) := f (xg) ,
x, g ∈ G.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
785
Remark 1.2. If the respective groups B and G are nonunimodular, we select
respective right-invariant Haar measures db, dg; and corresponding modular
functions 4B and 4G . In this case, the modification to the above is that
Equation (1.2) will instead be
4B (b) 1/2
(1.5)
f (bg) =
Vb f (g) ; b ∈ B, g ∈ G,
4G (b)
thus a modification in the definition of H (IndG
B (V )).
We recall the following theorems of Mackey [Mac88, Ørs79].
Theorem 1.3 (Mackey). U = IndG
B (V ) is a unitary representation.
Theorem 1.4 (Imprimitivity Theorem). A representation U ∈ Rep (G, H )
is induced iff ∃ a positive operator-valued mapping
(1.6)
π : Cc (B\G) −→ B (H ) s.t.
(1.7)
Ug π (ϕ) Ug∗ = π (Rg ϕ)
∀g ∈ G, ∀ϕ ∈ Cc (B\G); where π is nondegenerate, and Rg in (1.7) denotes
the right regular action.
Proof. See [Ørs79, Jor88].
Theorem 1.5. Given U ∈ Rep (G, H ), the following are equivalent:
(1) ∃π s.t. (1.7) holds.
(2) ∃V ∈ Rep (B, HV ) s.t. U ∼
= IndG
B (V ).
∼
(3) LG (U, π) = LB (V ).
(4) If Vi ∈ Rep (B, HVi ), i = 1, 2 and Ui = IndG
B (Vi ), i = 1, 2, then
LG ((U1 , π1 ) ; (U2 , π2 )) ∼
= LB (V1 , V2 ) ;
i.e., all intertwining operators V1 → V2 lift to the pair (Ui , πi ), i =
1, 2.
∼ denotes unitary equivalence.) Specifically,
(Here, “ =”
(1.8) LG ((U1 , π1 ) , (U2 , π2 ))
n
= W : HU1 → HU2 W U1 (g) = U2 (g) W, ∀g ∈ G, and
U π1 (ϕ) = π2 (ϕ) W
o
and
(1.9)
LB (V1 , V2 ) = w : HV1 → HV2 wV1 (b) = V2 (b) w, ∀b ∈ B .
Proof. See [Ørs79, Jor88, Mac88].
We also recall the following result:
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PALLE JORGENSEN AND FENG TIAN
Lemma 1.6. Let B be a lattice in Rd , let α be an action of Z on B by
b α
automorphisms, and let (K = B,
b) represent the dual action of Z on the
compact abelian group K. Fix a d × d matrix A preserving the lattice B with
spec (A) ⊂ {λ : |λ| > 1}.
If α = αA ∈ Aut (B), then α
b ∈ Aut (K) is ergodic, i.e., if ν is the
normalized Haar measure on K and if E ⊂ K is measurable s.t. α
bE = E,
then ν (E) (1 − ν (E)) = 0.
Proof. By Halmos–Rohlin’s theorem (see [BreJ91]), we must show that if
b ∈ B and An b = b for some n ∈ N, then b = 0. This then follows from the
assumption on the spectrum of A.
2. The representation IndG
B (χ)
We now specify our notation and from this point on will restrict our study
to the following setting:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
B: a discrete abelian group (written additively);
α ∈ Aut (B);
Gα := B oα Z;
b the compact dual group;
K := B,
b dual action;
α
b ∈ Aut(B),
Lα := K oαb Z, the C ∗ -algebra crossed product [BreJ91, p.299]; also
written as C ∗ (K) oαb Z.
More generally, let K be a compact Hausdorff space, and β : K → K a
homeomorphism; then we study the C ∗ -crossed product C ∗ (K) oβ Z (see
[BreJ91]).
We define the induced representation
U χ := IndG
B (χ)
(2.1)
where χ ∈ K, i.e., a character on B, and
G := B oα Z
(2.2)
as a semi-direct product. Note that U χ in (2.1) is induced from a onedimensional representation.
Below, {δk }k∈Z denotes the canonical basis in l2 (Z), i.e.,
δk = (. . . , 0, 0, 1, 0, 0, . . . )
with “1” at the k th place.
Lemma 2.1. The representation IndG
B (χ) is unitarily equivalent to
(2.3)
U χ : G → B l2 (Z) , where
(2.4)
χ
U(j,b)
δk = χ (αk−j (b)) δk−j ,
(j, b) ∈ G.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
787
Proof. First note that both groups B and G = B oα Z are discrete and
unimodular, and the respective Haar measures are the counting measure.
Since B\G ' Z, so H χ = L2 (B\G) ' l2 (Z). Recall the multiplication rule
in G:
(k, c) (j, b) = (k + j, αk (b) + c)
(2.5)
where j, k ∈ Z, and b, c ∈ B.
The representation IndG
B (χ) acts on functions f : G → C s.t.
f ((j, b)) = χ (b) f (j, 0)
(2.6)
and
kf k2H χ =
(2.7)
X
|f (j, 0)|2 .
j∈Z
Moreover, the mapping
l2 3 ξ 7−→ W ξ ∈ H λ
(2.8)
given by
(W ξ) (j, b) = χ (b) ξj
(2.9)
is a unitary intertwining operator, i.e.,
χ
IndG
B (χ) W ξ = W U ξ
(2.10)
where
(2.11)
χ
U(j,b)
ξ
k
= χ (αk (b)) ξk+j
∀j, k ∈ Z, ∀b ∈ B, ∀ξ ∈ l2 (Z). Note that (2.11) is equivalent to (2.4).
To verify (2.10), we have
IndG
= W ξ ((k, c) (j, b))
B (χ)(j,b) W ξ (k, c)
=
(2.5)
=
(2.6)
=
(2.9)
W ξ (k + j, αk (b) + c)
χ (αk (b) + c) W ξ (k + j, 0)
χ (αk (b) + c) ξk+j
and
χ
W U(j,b)
ξ (k, c)
=
(2.9)
=
(2.11)
=
χ
χ (c) U(j,b)
ξ
k
χ (c) χ (αk (b)) ξk+j
χ (αk (b) + c) ξk+j .
In the last step we use that χ (αk (b) + c) = χ (αk (b)) χ (c), which is just the
representation property of χ : B → T = {z ∈ C | |z| = 1},
(2.12)
χ (b1 + b2 ) = χ (b1 ) χ (b2 ) ,
∀b1 , b2 ∈ B.
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PALLE JORGENSEN AND FENG TIAN
Remark 2.2. In summary, the representation IndG
B (χ) has three equivalent
forms:
(1) On l2 (Z),
(2.13)
IndG
(ξk ) ∈ l2 (Z) .
B (χ) ξ k = χ (αk (b)) ξk+j ,
(2) In the ONB {δk | k ∈ Z},
IndG
B (χ)(j,k) δk = χ (αk−j (b)) δk−j ,
(2.14)
k ∈ Z.
(3) On H χ , consisting of functions f : G → C s.t.
X
f (j, b) = χ (b) f (j, 0) , kf k2H λ =
|f (j, 0)|2 < ∞
j∈Z
where
IndG
(χ)
f
(k, c) = f ((k, c) (j, b))
B
(j,b)
= f ((k + j, αk (b) + c))
= χ (αk (b) + c) f (k + j, 0) ,
with (j, b) , (k, c) ∈ G = B oα Z, i.e., j, k ∈ Z, b, c ∈ B.
3. Irreducibility
Let G = B oα Z, and IndG
B (χ) be the induced representation. Our main
results are summarized as follows:
b has infinite order, then IndG (χ) ∈ Repirr G, l2 (Z) ,
(1) If χ ∈ K = B
B
i.e., IndG
B (χ) is irreducible. We have
(3.1)
IndG
B (χ)(j,b) = Dχ (b) Tj
where Dχ (b) is diagonal, and Tj : l2 (Z) → l2 (Z), (Tj ξ)k = ξk+j , for
all ξ ∈ l2 (Z).
(2) Suppose χ has finite order p, i.e., ∃p s.t. α
bp χ = χ, but α
bk χ 6= χ,
χ
∀1 ≤ k < p. Set Up = IndG
B (χ), then
(3.2)
Upχ (j, b) = Dχ (b) P j
where Tj δk = δk−j and P = the p × p permutation matrix. See (3.9).
Details below.
Lemma 3.1. Set (Tj ξ)k = ξk+j , for k, j ∈ Z, ξ ∈ l2 ; equivalently, Tj δk =
δk−1 . Then
(1) The following identity holds:
(3.3)
n
Tn U χ = U αb χ Tn ,
∀n ∈ Z, ∀χ ∈ K.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
789
(2) For ω = (ωi ) ∈ l∞ (Z), set
ξ ∈ l2 (Z) ; or
(π (ω) ξ)k = ωk ξk ,
π (ω) δk = ωk δk .
(3.4)
Then
χ
χ
U(j,b)
π (ω) U(j,b)
(3.5)
∗
= π (Tj ω)
∀ (j, b) ∈ G, ∀χ ∈ K.
(3) Restriction to the representation:
⊕
X
U χ =
α
bj χ .
(3.6)
B
j∈Z
Recall that G := B oα Z, and so the subgroup B corresponds to
j = 0 in {(j, b) | j ∈ Z, b ∈ B}. Note that α
bj χ is a one-dimensional
representation of B.
Proof. One checks that
χ
χ
Tn U(j,b)
ξ = U(j,b)
ξ
k
n
α
b χ
U(j,b)
Tn ξ
k
k+n
= χ (αk+n (b)) ξk+n+j , and
= (b
αn χ) (αk (b)) (Tn ξ)k+j = χ (αk+n (b)) ξk+n+j .
The other assertions are immediate.
Corollary 3.2. If α
bp χ = χ, then Tp commutes with U χ . In this case,
χ
U(j,b)
δk = α
bk−j (χ) (b) δk−j
(3.7)
induces an action on l2 (Z/pZ). If m is fixed, the same representation is
repeated, where the same action occurs in the sub-band
l2 ({δj+mp | 0 ≤ j < p}) .
(3.8)
Proof. Immediate from Lemma 3.1.
The p-dimensional representation in each band {δj+mp | 0 ≤ j < p} may
be given in matrix form:
Lemma 3.3. Let χ ∈ K, and assume the orbit of χ under the action of α
b
has finite order p, i.e., α
bp χ = χ and α
bk χ 6= χ, 1 ≤ k < p. For b ∈ B, set


χ (b)


χ (α (b))
p−1


i
=
diag
χ
α
(b)
,
Dχ (b) = 

.
0
..


χ αp−1 (b)
0
0
a p × p diagonal matrix. If χ = (χ1 , . . . , χn ), set Dχ (b) = diag (χi (b))n×n .
790
PALLE JORGENSEN AND FENG TIAN
For the permutation matrix P , we shall use its usual p × p matrix representation


0 1 0 ... 0
0 0 1 . . . 0
. .
.. 
. .

.
.
0
.
.
.
.

. . .
.

(3.9)
P =
.
 .. .. .. . . . .. 
. . .

 .. .. ..
0


0 0 0 . . . 1
1 0 0 . . . 0 p×p
We have that:
(1) The following identity holds:
(3.10)
P Dχ (b) = Dχ (α (b)) P.
(2) For all (j, b) ∈ G = B oα Z, let
Up(χ) (j, b) := Dχ (b) P j .
(3.11)
Then Upχ ∈ Rep (G, Cp ), i.e.,
Upχ (j, b) Upχ j 0 , b0 = Upχ j + j 0 , αj b0 + b
for all (j, b) , (j 0 , b0 ) in G.
Proof. It suffices to verify
(3.10), and the rest of the lemma is straightforward. Set χk := χ αk (b) , 0 ≤ k < p, and
(3.12)
Dχ (b) = diag (χ0 , χ1 , . . . , χp−1 ) .
The assertion in (3.10) follows from a direct calculation. We illustrate this
with p = 3: see Example 3.4 below.
Example 3.4. For p = 3,

0

P Dχ (b) = 0
1

χ1
Dχ (α (b)) P =  0
0
(3.10) reads:

1 0
χ0 0


0 1
0 χ1
0 0
0 0

0 0
0 1
χ2 0  0 0
0 χ0
1 0
 

0
0 χ1 0
0  =  0 0 χ2  .
χ2
χ0 0 0
 

0
0 χ1 0
1 =  0 0 χ2  .
0
χ0 0 0
Lemma 3.5. Let χ ∈ K, assume the orbit of χ under the action of α
b has
finite order p. Let Upχ be the corresponding representation. Then Upχ ∈
Repirr G, l2 (Z/pZ) , i.e., Upχ is irreducible.
Proof. Recall Upχ (j, b) = Dχ (b) P j , ∀ (j, b) ∈ G; see (3.11).
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
791
Let A : l2 (Zp ) → l2 (Zp ) be in the commutant of Upχ , so A commutes with
P . It follows that A is a Toeplitz matrix


.. ..
.
. Ap−1
A0
A2


.
.. 

. 
A0 A2 . .
 A−1
 .
.
.. 
(3.13)
A=
. 
A−1 A0 . .
 ..

 .
..
.. ..
.. 
 ..
.
.
.
. 


..
.. ..
.
.
. A0
A−p+1
relative to the ONB {δ0 , δ1 , . . . , δp−1 }, where Ai,j := hδi , Aδj il2 . Note that
Ai,j = hδi , Aδj i = hδi , AP δj+1 i = hδi , P Aδj+1 i
= hP ∗ δi , Aδj+1 i = hδi+1 , Aδj+1 i = Ai+1,j+1
where i + 1, j + 1 are the additions in Zp = Z/pZ, i.e., + mod p.
If A also commutes with D, then
(3.14)
Ak χ (b) − χ αk (b) = 0
for all k = 1, 2, . . . , p − 1, and all b ∈ B. But since χ, α
bχ, . . . , α
bp−1 χ are
distinct, Ak = 0, ∀k 6= 0, and so A = A0 Ip×p , A0 ∈ C.
Proposition 3.6. The orbit of χ ∈ K under the action of α
b has finite order
p if and only if
IndG
B (χ) =
⊕
X
(p-dimensional irreducibles) .
Proof. This follows from an application of the general theory; more specifically from an application of Mackey’s imprimitivity theorem, in the form
given to it in [Ørs79]; see also Theorem 1.4 above.
Lemma 3.7. IndG
b
B (χ) is irreducible
kiff the orbit
of χ under the action of α
has finite order p, i.e., iff the set α
b χ | k ∈ Z consists of distinct points.
Proof. We showed that if χ has finite order p, then the translation Tp operator commutes with U χ := IndG
B (χ), where Tp δk = δk−p , k ∈ Z. Hence
assume χ has infinite order.
Consider the cases U χ (j, b) of
(1) b = 0. U χ (j, 0) = Tj , j ∈ Z;
χ
(2) j = 0.
U (0, b) = Dχ (b),
δk 7−→ χ (αk (b)) δk , where χ (αk (b)) =
k
α
b χ (b), and Dχ = α
bk χ = diagonal matrix.
Thus if A ∈ B l2 (Z) is in the commutant of U χ , then
X
(Aξ)k =
ηk−j ξj , η ∈ l∞ .
j∈Z
792
PALLE JORGENSEN AND FENG TIAN
But
(χ (αs (b)) − χ (αk (b))) ηk−s = 0
(3.15)
∀b, ∀k, s, so ηt = 0 if t ∈ Z\ {0} and A = η0 I.
Note from (3.15) that if k − s 6= 0 then ∃b s.t. χ (αs (b)) − χ (αk (b)) 6= 0,
since α
bs χ 6= α
bk χ. (Compare with (3.14) in Lemma 3.5.)
Theorem 3.8. Let IndG
B (χ) be the induced representation in (2.1)–(2.2).
G
(1) IndB (χ) is irreducible iff χ has no finite periods.
(2) Suppose the orbit of χ under the action of α
b has finite order p, i.e.,
p
k
α
b χ = χ, and α
b χ 6= χ, 1 ≤ k < p. Then the commutant is as
follows:
0 ∼
(3.16)
Mp := IndG
= f (z p ) f ∈ L∞ (T)
B (χ)
where ∼
= in (3.16) denotes unitary equivalence.
b α
Theorem 3.9. Let G = B oα Z, K = B,
b ∈ Aut (K). Assume the orbit
of χ under the action of α
b has finite order p, i.e., α
bp χ = χ, α
bk χ 6= χ, for all
1 ≤ k < p. Then the representation IndG
B (χ) has abelian commutant
G
0
Mp = IndB (χ)
and Mp does not contain minimal projections.
Proof. Follows from the fact that Mp is L∞ (Lebesgue) as a von Neumann
algebra, and this implies the conclusion.
Details: Recall that l2 (Z) ' L2 (T), where
X
(3.17)
l2 3 ξ 7−→ fξ (z) =
ξj z j ∈ L2 (T)
j∈Z
T ξ 7−→ z
(3.18)
IndG
B
Note that
(3.17), where
−j
fξ (z) ,
(χ) may be realized on l2 (Z) or equivalently on L2 (T) via
kfξ k2L2 =
Z
T
On
l2 (Z),
z ∈ T.
|fξ |2 =
X
|ξk |2 .
k∈Zl
we have
IndG
B (χ)(j,b) = Dχ (b) Tj ;
See (3.1), and Lemma 3.3.
And on L2 (T), we have
b
b
IndG
B (χ)(j,b) = Dχ (b) Tj , where
b χ (b) denotes rotation on T, extended to the solenoid; and Tbj = multipliD
cation by z −j acting on L2 (T).
By (3.16), Mp is abelian and has no minimal projections. Note projections
in L∞ are given by PE = multiplication by χE (z p ), where E is measurable
in T.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
793
4. Super-representations
By “super-representation” we will refer here to a realization of noncommutative relations “inside” certain unitary representations of suitable groups
acting in enlargement Hilbert spaces; this is in the sense of “dilation” theory
[FK02], but now in a wider context than is traditional. Our present use of
“super-representations” is closer to that of [BiDP05, DJ08, DJP09, DLS11].
Definition 4.1. Let G be a locally compact group, B a given subgroup of
G, and let H0 be a Hilbert space. Let U0 : G → B (H0 ) be a positive
definite operator-valued mapping, i.e., for all finite systems {cj }nj=1 ⊂ C,
{gj }nj=1 ⊂ G, we have
XX
cj ck U0 (gj−1 gk ) ≥ 0
j
k
in the usual ordering of Hermitian operators.
If there is a Hilbert space H , an isometry V : H0 → H , and a unitary
representation U : G → (unitary operators on H ) such that
(1) U is induced from a unitary representation of B, and
(2) U0 (g) = V ∗ U (g) V , g ∈ G,
then we say that U is a super-representation.
b α ∈ Aut (B), α
Let B be discrete, abelian as before, K = B,
b ∈ Aut (K),
and G = B oα Z. Set:
• Rep (G): unitary representations of G;
• Repirr (G): irreducible representations in Rep (G).
For χ ∈ K, let O (χ) be the orbit of χ, i.e.,
j
(4.1)
O (χ) = α
b (χ) | j ∈ Z .
Given U ∈ Rep (G), let Class (U ) = the equivalence class of all unitary
representations equivalent to U ; i.e.,
(4.2) Class (U ) = V ∈ Rep (G) V ' U
= {V ∈ Rep (G) | ∃W, unitary s.t. W Vg = Ug W, g ∈ G} .
For U1 , U2 ∈ Rep (G), set
n
(4.3) L U (1) , U (2) = W : H (U1 ) → H (U2 ) | W bounded s.t.
o
W Ug(1) = Ug(2) W, g ∈ G .
Theorem 4.2. The mapping {set of all orbits O (χ)} −→ Class (Rep (G))
K 3 χ 7−→ U χ := IndG
B (χ) ∈ Rep (G)
passes to
(4.4)
O (χ) 7−→ Class (U χ ) .
794
PALLE JORGENSEN AND FENG TIAN
Proof. (Sketch) By Lemma 3.1 Equation (3.3), if χ ∈ K, j ∈ Z, then
j
Tj U χ Tj∗ = U αb χ , and so Class (U χ ) depends only on O (χ) and not on the
chosen point in O (χ).
Using (3.4)–(3.6), we can show that U χ is irreducible, but if O (χ) is finite
then we must pass to the quotient Zp := Z/pZ and realize U χ on l2 (Zp ), as
a finite-dimensional representation.
Theorem 4.3. There is a natural isomorphism:
∼ LB χ2 , U χ1 LG (U χ1 , U χ2 ) =
B
X
j
bj (χ1 ) .
b (χ1 ) = # j χ2 = α
=
LB χ2 α
j∈Z
Proof. Follows from Frobenius reciprocity.
Another application of Frobenius reciprocity:
Theorem 4.4. Let χ ∈ K, and assume the orbit of χ under the action
of α
b has finite order p, i.e., α
bp χ = χ, and α
bk χ 6= χ if 1 ≤ k < p. Let
χ
χ
2
Up ∈ Rep G, l (Zp ) . (Recall that Up is irreducible, dim Upχ = p.) Then
χ
LG Upχ , IndG
B (χ) ' LB χ, Up
B
and
dim LB χ, Upχ B = 1.
(4.5)
(χ)
Proof. Since Up (j, b) := Dχ (b) P j (see (3.11)), where P = the permutation matrix on Zp , we get
⊕
X
Upχ B =
α
bk χ;
0≤k<p
see (3.6). But notice that all the p characters
χ, α
b (χ) , . . . , α
bp−1 (χ) are
distinct, so (4.5) holds since LB χ, α
bk (χ) = 0 if k 6= 0 mod p.
We have also proved the following:
Theorem 4.5. Let χ ∈ K, and assume the orbit of χ under the action of
α
b has finite order p. Assume G is compact. Let Upχ ∈ Repirr G, l2 (Zp ) .
Then Upχ is contained in IndG
B (χ) precisely once, i.e.,
dim LG Upχ , IndG
B (χ) = 1.
But this form of Frobenius reciprocity only holds for certain groups G, e.g.,
when G is compact. Now for G = B oα Z, the formal Frobenius reciprocity
breaks down, and in fact:
Theorem 4.6. LG Upχ , IndG
B (χ) = 0 if χ ∈ K is an element of finite order
p.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
795
Proof. We sketch the details for p = 3 to simplify notation. For p = 3,




0 1 0
χ (b)
0
0

χ (α (b))
0
P = 0 0 1 , Dχ (b) =  0
1 0 0
0
0
χ (α2 (b))
and
Upχ (j, b) = Dχ (b) P j
(4.6)
while
IndG
B (χ)(j,b) ξ
k
= χ (αk (b)) ξk+j
∀ (j, b) ∈ G, ∀k ∈ Z, ∀ξ ∈ l2 (Zp).
Let W ∈ LG Upχ , IndG
B (χ) , and u0 , u1 , u2 be the canonical basis in
χ
3
H (Up ) = C , where
Upχ (j, b) uk = χ (αk+2j (b)) uk+2j
mod 3
∀ (j, b) ∈ G, k ∈ {0, 1, 2} ' Z/3Z.
P
(k)
(k)
Set W uk = ξ (k) = (ξs )s∈Z ∈ l2 (Z), where kξ (k) k2 = s∈Z |ξs |2 < ∞.
It follows that
W Upχ (j, b) uk = IndG
B (χ)(j,b) W uk
∀ (j, b) ∈ G, k ∈ {0, 1, 2}. Thus
(k+2j)3
χ (αk+2j (b)) ξs
(k)
= χ (αs (b)) ξs+j ,
∀s, j ∈ Z.
Now set j = 3t ∈ 3Z, and we get
(k)
χ (αk (b)) ξs(k) = χ (αs (b)) ξs+3t
and
(4.7)
(k) (k) ξs = ξ
s+3t ,
∀s, t ∈ Z.
(k)
Since ξ (k) ∈ l2 (Z), limt→∞ ξs+3t = 0. We conclude from (4.7) that ξ (k) = 0
in l2 (Z).
2
Remark 4.7. The decomposition of IndG
B (χ) : G −→ B l (Z) is still a bit
mysterious. Recall this representation commutes with T3 : (ξk ) 7−→ (ξk+3 );
or equivalently via f (z) 7−→ z 3 f (z).
So it is not irreducible.
χ
G
The reason for LG Up , IndB (χ) = 0, e.g., in the case of p = 3, is really
that there is no isometric version of the 3 × 3 permutation matrix P in T ,
where T δk = δk−1 .
If W : C3 −→ l2 (Z) is bounded, W P = T W , then applying the polar
decomposition
W = (W ∗ W )1/2 V
with V : C3 → l2 isometric, and V P = T V , or
P = V ∗ T V,
P j = V ∗ Tj V.
796
PALLE JORGENSEN AND FENG TIAN
So Tj has the form
j
P ∗
Tj =
.
0 ∗
Pick u ∈ C3 ; then
= u, P j u . However, hu, Tj ui → 0 by Riemann–
j ui
hu, T
Lebesgue; while u, P j u 9 0 since P 3 = I.
Definition 4.8. A group L acts on a set S if there is a mapping
L × S −→ S,
(λ, s) 7−→ λ [s] ,
λλ0 [s] = λ[λ0 [s]]
and λ−1 is the inverse of S 3 s 7−→ λ [s] ∈ S, a bijection. Often S will have
the structure of a topological space or will be equipped with a σ-algebra of
measurable sets.
Remark 4.9. While we have stressed discrete decompositions of induced
representations, the traditional literature has stressed direct integral decompositions, see, e.g., [Mac88]. A more recent use of continuous parameters in
decompositions is a construction by J. Packer et al. [LiPT01] where “wavelet
sets” arise as sets of support for direct integral measures. In more detail,
starting with a wavelet representation of a certain discrete wavelet group
(an induced representation of a semidirect product), the authors in [LiPT01]
establish a direct integral where the resulting measure is a subset of R called
“wavelet set.” These wavelet sets had been studied earlier, but independently
of representation theory. In the dyadic case, a wavelet set is a subset of R
which tiles R itself by a combination of Z-translations, and scaling by powers
of 2. In the Packer et al. case, R ,→ K, and
Z ⊕
2
E wavelet set ⇐⇒
IndG
B (χt ) dt is the wavelet representation in L (R) .
E
Theorem 4.10. The group L := K oαb Z acts on the set Rep (G) by the
following assignment:
Arrange it so that all the representations U χ := IndG
B (χ), χ ∈ K, acts on
the same l2 -space. Then Rep (G) ∼ Rep G, l2 .
5. Induction and Bratteli diagrams
A Bratteli diagram is a group G with vertices V , and edges
E ⊂ V × V \ {diagonal} .
It is assumed that
V =
∞
[
Vn ,
n=0
as a disjoint union in such a way that the edges E in G can be arranged in
a sequnce of lines Vn → Vn+1 , so no edge links pairs of vertices at the same
level Vn . With a system of inductions and restrictions one then creates these
diagrams.
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
797
Let G be created as follows: For a given Vn , let the vertices in Vn represent
irreducible representations of some group Gn , and assume Gn is a subgroup
in a bigger group Gn+1 . For the vertices in Vn+1 , i.e., in the next level in
the Bratteli diagram G, we take the irreducible representations occurring in
the decomposition of each of the restrictions:
G
(L)
IndGn+1
,
n
Gn
so the decomposition of the restriction of the induced representation; see
Figure 5.1.
Multiple lines in a Bratteli diagram count the occurrence of irreducible
representations with multiplicity; these are called multiplicity lines.
Counting multiplicity lines at each level Vn → Vn+1 we get a so called incidence matrix. For more details on the use of Bratteli diagrams in representation theory, see [BraJKR00, BraJKR01, BraJKR02, BraJO04, BeKwY14,
BeKa14, BeJ15].
Vn-1
Vn
Vn+1
Figure 5.1. Part of a Bratteli diagram.
While there is a host of examples from harmonic analysis and dynamical
systems; see the discussion above, we shall concentrate below on a certain
family of examples when the discrete group B in the construction (from
Sections 2 and 3) arises as a discrete Bratteli diagram, and, as a result, its
Pontryagin dual compact group K is a solenoid; also often called a compact
Bratteli diagram.
Example 5.2 below illustrates this in the special case of constant incidence
diagrams, but much of this discussion applies more generally.
Example 5.1. Let B = Z 12 , dyadic rationals in R. Let α2 = multiplication by 2, so that α2 ∈ Aut (B), and set G2 := B oα2 Z.
Recall the Baumslag–Solitar group G2 with two generators {u, t}, satisfying utu−1 = t2 . With the correspondence,
2 0
1 1
u ←→
, t ←→
0 1
0 1
798
PALLE JORGENSEN AND FENG TIAN
α2 acts by conjugation,
2 0
1
0 1
0
1 k
In particular, T k =
,k
0 1
We have
1
1
−1 1 2
2
0
=
.
0 1
0 1
= 1, 2, 3, . . . .
j
2 b
= Dχ (b) T j .
0 1
Moreover, 2j : j ∈ Z ' Z 12 \G2 ' Z. Set (T ξ)k := ξk+1 , ξ ∈ l2 (Z),
and let
.

..


χ (b)




χ (2b)
Dχ (b) = 
 , ∀b ∈ Z 12 .


χ (3b)


..
.
2
IndG
(χ)
Z[ 21 ]
0
0
Note that
T Dχ (b) = Dχ (α2 (b)) T = Dαb2 χ (b) T,
∀b ∈ Z
1
2
.
Example 5.2. Let A be a d × d matrix over Z, det A 6= 0; then
Zd ,→ A−1 Zd ,→ A−2 Zd ,→ · · ·
and set
BA :=
[
A−k Zd .
k≥0
Let αA = multiplication by A, so that αA ∈ Aut (BA ), and GA := BA oαA Z.
The corresponding standard
wavelet representation UA will now act on
the Hilbert space L2 Rd with d-dimensional Lebesgue measure; and UA is
a unitary representation of the discrete matrix group GA = {(j, β)}j∈Z,β∈BA ,
specified by
(5.1)
(j, β) (k, γ) = j + k, β + Aj γ , and
defined for all j, k ∈ Z, and β, γ ∈ BA . Alternatively, the group from (5.1)
may be viewed in matrix form as follows:
j
k
A β
A γ
(j, β) −→
, and (k, γ) −→
.
0 1
0 1
The wavelet representation UA of GA acting on L2 Rd is now
j
(5.2)
UA (A) (x) := (det A)− 2 f A−j (x − β) ,
defined for all f ∈ L2 Rd , ∀j ∈ Z, ∀β ∈ BA , and x ∈ Rd .
INDUCED REPRESENTATIONS ARISING FROM A CHARACTER
799
Acknowledgements. The co-authors thank the following colleagues for
helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezuglyi, Ilwoo Cho, Ka Sing Lau, Azita Mayeli, Paul Muhly, Myung-Sin Song,
Wayne Polyzou, Keri Kornelson, and members in the Math Physics seminar
at the University of Iowa.
The authors are very grateful to an anonymous referee for a list of corrections and constructive suggestions. The revised paper is now much better,
both in substance, and in its presentation.
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(Palle Jorgensen) Department of Mathematics, The University of Iowa, Iowa
City, IA 52242-1419, U.S.A.
palle-jorgensen@uiowa.edu
http://www.math.uiowa.edu/~jorgen/
(Feng Tian) Department of Mathematics, Trine University, IN 46703, U.S.A.
tianf@trine.edu
This paper is available via http://nyjm.albany.edu/j/2015/21-34.html.