Induced Representations Arising from a Character with Finite Orbit in a Semidirect Product

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 783 G 2. The representation IndB (χ) 786 3. Irreducibility 788 4. Super-representations 793 5. Induction and Bratteli diagrams 796 References 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 783 784 PALLE JORGENSEN AND FENG TIAN

Mackey’s theory, our approach is extended, and it is constructive and algo- rithmic. 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 (non- abelian) 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 G (1.1) U = IndB (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) f (bg) = Vbf (g) , ∀b ∈ B, ∀g ∈ G, Z (1.3) kfk2 = kf (g)k2 dµ (g) < ∞, HU B\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) (Ugf)(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 1/2 4B (b) (1.5) f (bg) = Vbf (g); b ∈ B, g ∈ G, 4G (b) G thus a modification in the definition of H (IndB (V )). We recall the following theorems of Mackey [Mac88, Ørs79].

G Theorem 1.3 (Mackey). U = IndB (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) Ugπ (ϕ) Ug = π (Rgϕ)

∀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. ∼ G (2) ∃V ∈ Rep (B, HV ) s.t. U = IndB (V ). ∼ (3) LG (U, π) = LB (V ). G (4) If Vi ∈ Rep (B, HVi ), i = 1, 2 and Ui = IndB (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. (Here, “=∼” denotes unitary equivalence.) Speciﬁcally,

(1.8) LG ((U1, π1) , (U2, π2)) n = W : HU1 → HU2 WU1 (g) = U2 (g) W, ∀g ∈ G, and o Uπ1 (ϕ) = π2 (ϕ) W 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: 786 PALLE JORGENSEN AND FENG TIAN

d Lemma 1.6. Let B be a lattice in R , let α be an action of Z on B by automorphisms, and let (K = B,b αb) 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 αb ∈ Aut (K) is ergodic, i.e., if ν is the normalized Haar measure on K and if E ⊂ K is measurable s.t. αEb = E, then ν (E) (1 − ν (E)) = 0. Proof. By Halmos–Rohlin’s theorem (see [BreJ91]), we must show that if n b ∈ B and A b = b for some n ∈ N, then b = 0. This then follows from the assumption on the spectrum of A.

G 2. The representation IndB (χ) We now specify our notation and from this point on will restrict our study to the following setting: (i) B: a discrete abelian group (written additively); (ii) α ∈ Aut (B); (iii) Gα := B oα Z; (iv) K := Bb, the compact dual group; (v) αb ∈ Aut(Bb), dual action; (vi) L := K , the C∗-algebra crossed product [BreJ91, p.299]; also α oαb Z 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 χ G (2.1) U := IndB (χ) where χ ∈ K, i.e., a character on B, and

(2.2) G := B oα Z as a semi-direct product. Note that U χ in (2.1) is induced from a one- dimensional representation. Below, {δ } denotes the canonical basis in l2 ( ), i.e., k k∈Z Z

δk = (..., 0, 0, 1, 0, 0,... ) with “1” at the kth place.

G Lemma 2.1. The representation IndB (χ) is unitarily equivalent to χ 2 (2.3) U : G → B l (Z) , where

χ (2.4) 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. χ 2 2 Since B\G ' Z, so H = L (B\G) ' l (Z). Recall the multiplication rule in G:

(2.5) (k, c)(j, b) = (k + j, αk (b) + c) where j, k ∈ Z, and b, c ∈ B. G The representation IndB (χ) acts on functions f : G → C s.t. (2.6) f ((j, b)) = χ (b) f (j, 0) and 2 X 2 (2.7) kfkH χ = |f (j, 0)| . j∈Z Moreover, the mapping (2.8) l2 3 ξ 7−→ W ξ ∈ H λ given by

(2.9) (W ξ)(j, b) = χ (b) ξj is a unitary intertwining operator, i.e., G χ (2.10) IndB (χ) W ξ = WU ξ where χ (2.11) U ξ = χ (αk (b)) ξk+j (j,b) k 2 ∀j, k ∈ Z, ∀b ∈ B, ∀ξ ∈ l (Z). Note that (2.11) is equivalent to (2.4). To verify (2.10), we have

G IndB (χ)(j,b) W ξ (k, c) = W ξ ((k, c)(j, b))

= W ξ (k + j, αk (b) + c) (2.5)

= χ (αk (b) + c) W ξ (k + j, 0) (2.6)

= χ (αk (b) + c) ξk+j (2.9) and WU χ ξ (k, c) = χ (c) U χ ξ (j,b) (2.9) (j,b) k

= χ (c) χ (αk (b)) ξk+j (2.11)

= 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. 788 PALLE JORGENSEN AND FENG TIAN

G Remark 2.2. In summary, the representation IndB (χ) has three equivalent forms: 2 (1) On l (Z), G 2 (2.13) IndB (χ) ξ k = χ (αk (b)) ξk+j, (ξk) ∈ l (Z) .

(2) In the ONB {δk | k ∈ Z}, G (2.14) IndB (χ)(j,k) δk = χ (αk−j (b)) δk−j, k ∈ Z. χ (3) On H , consisting of functions f : G → C s.t. 2 X 2 f (j, b) = χ (b) f (j, 0) , kfkH λ = |f (j, 0)| < ∞ j∈Z where G IndB (χ)(j,b) f (k, c) = f ((k, c)(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 G Let G = B oα Z, and IndB (χ) be the induced representation. Our main results are summarized as follows: G 2 (1) If χ ∈ K = Bb has inﬁnite order, then IndB (χ) ∈ Repirr G, l (Z) , G i.e., IndB (χ) is irreducible. We have G (3.1) IndB (χ)(j,b) = Dχ (b) Tj 2 2 where Dχ (b) is diagonal, and Tj : l (Z) → l (Z), (Tjξ)k = ξk+j, for 2 all ξ ∈ l (Z). p k (2) Suppose χ has ﬁnite order p, i.e., ∃p s.t. αb χ = χ, but αb χ 6= χ, χ G ∀1 ≤ k < p. Set Up = IndB (χ), then χ j (3.2) Up (j, b) = Dχ (b) P

where Tjδk = δk−j and P = the p × p permutation matrix. See (3.9). Details below.

2 Lemma 3.1. Set (Tjξ)k = ξk+j, for k, j ∈ Z, ξ ∈ l ; equivalently, Tjδk = δk−1. Then (1) The following identity holds: χ αnχ (3.3) TnU = U b Tn, ∀n ∈ Z, ∀χ ∈ K. INDUCED REPRESENTATIONS ARISING FROM A CHARACTER 789

∞ (2) For ω = (ωi) ∈ l (Z), set 2 (π (ω) ξ)k = ωkξk, ξ ∈ l (Z); or (3.4) π (ω) δk = ωkδk. Then χ χ ∗ (3.5) U(j,b)π (ω) U(j,b) = π (Tjω) ∀ (j, b) ∈ G, ∀χ ∈ K. (3) Restriction to the representation:

⊕ X (3.6) U χ = αjχ . B b j∈Z

Recall that G := B oα Z, and so the subgroup B corresponds to j j = 0 in {(j, b) | j ∈ Z, b ∈ B}. Note that αb χ is a one-dimensional representation of B. Proof. One checks that χ χ TnU ξ = U ξ = χ (αk+n (b)) ξk+n+j, and (j,b) k (j,b) k+n

n αb χ n U Tnξ = (α χ)(αk (b)) (Tnξ)k+j = χ (αk+n (b)) ξk+n+j. (j,b) k b The other assertions are immediate. p χ Corollary 3.2. If αb χ = χ, then Tp commutes with U . In this case, χ k−j (3.7) U(j,b)δk = αb (χ)(b) δk−j 2 induces an action on l (Z/pZ). If m is ﬁxed, the same representation is repeated, where the same action occurs in the sub-band 2 (3.8) l ({δj+mp | 0 ≤ j < p}) .

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 αb p k has ﬁnite order p, i.e., αb χ = χ and αb χ 6= χ, 1 ≤ k < p. For b ∈ B, set χ (b)   χ (α (b)) 0  p−1 D (b) =   = diag χ αi (b) , χ  ..  0  .  0 χ αp−1 (b) 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) PDχ (b) = Dχ (α (b)) P.

(2) For all (j, b) ∈ G = B oα Z, let (χ) j (3.11) Up (j, b) := Dχ (b) P . χ p Then Up ∈ Rep (G, C ), i.e., χ χ 0 0 χ 0 0 Up (j, b) Up j , b = Up j + j , αj b + b for all (j, b) , (j0, b0) in G. Proof. It suﬃces to verify (3.10), and the rest of the lemma is straightfor- k ward. Set χk := χ α (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,(3.10) reads:       0 1 0 χ0 0 0 0 χ1 0 PDχ (b) = 0 0 1  0 χ1 0  =  0 0 χ2 . 1 0 0 0 0 χ2 χ0 0 0       χ1 0 0 0 1 0 0 χ1 0 Dχ (α (b)) P =  0 χ2 0  0 0 1 =  0 0 χ2 . 0 0 χ0 1 0 0 χ0 0 0 Lemma 3.5. Let χ ∈ K, assume the orbit of χ under the action of α has χ b χ ﬁnite order p. Let Up be the corresponding representation. Then Up ∈ 2 χ Repirr G, l (Z/pZ) , i.e., Up is irreducible. χ j Proof. Recall Up (j, b) = Dχ (b) P , ∀ (j, b) ∈ G; see (3.11). INDUCED REPRESENTATIONS ARISING FROM A CHARACTER 791

2 2 χ Let A : l (Zp) → l (Zp) be in the commutant of Up , so A commutes with P . It follows that A is a Toeplitz matrix  .. ..  A0 A2 . . Ap−1  . .   .. ..   A−1 A0 A2   . . .  (3.13) A =  ......   A−1 A0   . . . . .   ......    ...... A−p+1 . . . A0 relative to the ONB {δ0, δ1, . . . , δp−1}, where Ai,j := hδi, Aδjil2 . Note that

Ai,j = hδi, Aδji = hδi, AP δj+1i = hδi, P Aδj+1i ∗ = hP δi, Aδj+1i = hδi+1, Aδj+1i = 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

k (3.14) Ak χ (b) − χ α (b) = 0

p−1 for all k = 1, 2, . . . , p − 1, and all b ∈ B. But since χ, αχ,b . . . , αb χ are distinct, Ak = 0, ∀k 6= 0, and so A = A0Ip×p, A0 ∈ C.

Proposition 3.6. The orbit of χ ∈ K under the action of αb has finite order p if and only if ⊕ G X IndB (χ) = (p-dimensional irreducibles) . Proof. This follows from an application of the general theory; more specif- ically from an application of Mackey’s imprimitivity theorem, in the form given to it in [Ørs79]; see also Theorem 1.4 above. G Lemma 3.7. IndB (χ) is irreducible iff the orbit of χ under the action of αb k has finite order p, i.e., iff the set αb χ | k ∈ Z consists of distinct points.

Proof. We showed that if χ has ﬁnite order p, then the translation Tp op- χ G erator commutes with U := IndB (χ), where Tpδk = δk−p, k ∈ Z. Hence assume χ has inﬁnite order. Consider the cases U χ (j, b) of χ (1) b = 0. U (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 k αb χ (b), and Dχ = αb χ = diagonal matrix. 2 χ Thus if A ∈ B l (Z) is in the commutant of U , then X ∞ (Aξ)k = ηk−jξj, η ∈ l . j∈Z 792 PALLE JORGENSEN AND FENG TIAN

But

(3.15) (χ (αs (b)) − χ (αk (b))) ηk−s = 0

∀b, ∀k, s, so ηt = 0 if t ∈ Z\{0} and A = η0I. Note from (3.15) that if k − s 6= 0 then ∃b s.t. χ (αs (b)) − χ (αk (b)) 6= 0, s k since αb χ 6= αb χ. (Compare with (3.14) in Lemma 3.5.) G Theorem 3.8. Let IndB (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 αb has finite order p, i.e., p k αb χ = χ, and αb χ 6= χ, 1 ≤ k < p. Then the commutant is as follows: G 0 ∼ p ∞ (3.16) Mp := IndB (χ) = f (z ) f ∈ L (T) where =∼ in (3.16) denotes unitary equivalence.

Theorem 3.9. Let G = B oα Z, K = Bb, αb ∈ Aut (K). Assume the orbit of χ under the action of α has ﬁnite order p, i.e., αpχ = χ, αkχ 6= χ, for all b G b b 1 ≤ k < p. Then the representation IndB (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. 2 2 Details: Recall that l (Z) ' L (T), where 2 X j 2 (3.17) l 3 ξ 7−→ fξ (z) = ξjz ∈ L (T) j∈Z −j (3.18) T ξ 7−→ z fξ (z) , z ∈ T. G 2 2 Note that IndB (χ) may be realized on l (Z) or equivalently on L (T) via (3.17), where Z 2 2 X 2 kfξkL2 = |fξ| = |ξk| . T k∈Zl 2 On l (Z), we have G IndB (χ)(j,b) = Dχ (b) Tj; See (3.1), and Lemma 3.3. 2 And on L (T), we have G IndB (χ)(j,b) = Dbχ (b) Tbj, where

Dbχ (b) denotes rotation on T, extended to the solenoid; and Tbj = multipli- −j 2 cation by z acting on L (T). By (3.16), Mp is abelian and has no minimal projections. Note projections ∞ p in L are given by PE = multiplication by χE (z ), 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 n definite operator-valued mapping, i.e., for all finite systems {cj}j=1 ⊂ C, n {gj}j=1 ⊂ G, we have X X −1 cjckU0(gj 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.

Let B be discrete, abelian as before, K = Bb, α ∈ Aut (B), αb ∈ 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) = αb (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. WVg = UgW, g ∈ G} .

For U1,U2 ∈ Rep (G), set

(1) (2) n (4.3) L U ,U = W : H (U1) → H (U2) | W bounded s.t.

(1) (2) o WUg = Ug W, g ∈ G . Theorem 4.2. The mapping {set of all orbits O (χ)} −→ Class (Rep (G)) χ G K 3 χ 7−→ U := IndB (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 χ ∗ αj χ χ TjU Tj = U b , and so Class (U ) 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 χ 2 then we must pass to the quotient Zp := Z/pZ and realize U on l (Zp), as a ﬁnite-dimensional representation. Theorem 4.3. There is a natural isomorphism:

χ1 χ2 ∼ χ1 LG (U ,U ) = LB χ2,U B X j j = LB χ2 αb (χ1) = # j χ2 = αb (χ1) . 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 p k of αb has ﬁnite order p, i.e., αb χ = χ, and αb χ 6= χ if 1 ≤ k < p. Let χ 2 χ χ Up ∈ Rep G, l (Zp) . (Recall that Up is irreducible, dim Up = p.) Then χ G χ LG Up , IndB (χ) ' LB χ, Up B and χ (4.5) dim LB χ, Up B = 1.

(χ) j Proof. Since Up (j, b) := Dχ (b) P (see (3.11)), where P = the permutation matrix on Zp, we get ⊕ X U χ = αkχ; p B b 0≤k

Proof. We sketch the details for p = 3 to simplify notation. For p = 3, 0 1 0 χ (b) 0 0  P = 0 0 1 ,Dχ (b) =  0 χ (α (b)) 0  1 0 0 0 0 χ (α2 (b)) and χ j (4.6) Up (j, b) = Dχ (b) P while G IndB (χ)(j,b) ξ = χ (αk (b)) ξk+j k 2 ∀ (j, b) ∈ G, ∀k ∈ Z, ∀ξ ∈ l (Zp). χ G Let W ∈ LG Up , IndB (χ) , and u0, u1, u2 be the canonical basis in χ 3 H (Up ) = C , where χ Up (j, b) uk = χ (αk+2j (b)) uk+2j mod 3 ∀ (j, b) ∈ G, k ∈ {0, 1, 2}' Z/3Z. Set W u = ξ(k) = (ξ(k)) ∈ l2 ( ), where kξ(k)k2 = P |ξ(k)|2 < ∞. k s s∈Z Z s∈Z s It follows that χ G WUp (j, b) uk = IndB (χ)(j,b) W uk ∀ (j, b) ∈ G, k ∈ {0, 1, 2}. Thus

(k+2j)3 (k) χ (αk+2j (b)) ξs = χ (αs (b)) ξs+j, ∀s, j ∈ Z. Now set j = 3t ∈ 3Z, and we get (k) (k) χ (αk (b)) ξs = χ (αs (b)) ξs+3t and (k) (k) (4.7) ξs = ξs+3t , ∀s, t ∈ Z. (k) 2 (k) (k) Since ξ ∈ l (Z), limt→∞ ξs+3t = 0. We conclude from (4.7) that ξ = 0 2 in l (Z). G 2 Remark 4.7. The decomposition of IndB (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−→ z3f (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. 3 2 If W : C −→ l (Z) is bounded, WP = TW , then applying the polar decomposition W = (W ∗W )1/2 V 3 2 with V : C → l isometric, and VP = TV , or ∗ j ∗ P = V TV,P = V TjV. 796 PALLE JORGENSEN AND FENG TIAN

So Tj has the form P j ∗ T = . j 0 ∗ 3 j Pick u ∈ C ; then hu, Tjui = u, P u . However, hu, Tjui → 0 by Riemann– j 3 Lebesgue; while u, P u 9 0 since P = 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 ⊕ G 2 E wavelet set ⇐⇒ IndB (χt) dt is the wavelet representation in L (R) . E Theorem 4.10. The group L := K acts on the set Rep (G) by the oαb Z following assignment: χ G Arrange it so that all the representations U := IndB (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:

IndGn+1 (L) , Gn 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 Sections2 and3) 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.

1 Example 5.1. Let B = Z 2 , 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}, satisfy- ing 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 1 2−1 0 1 2 = . 0 1 0 1 0 1 0 1 1 k In particular, T k = , k = 1, 2, 3,... . 0 1 We have j G2 2 b j Ind 1 (χ) = Dχ (b) T . Z[ 2 ] 0 1 j 1 2 Moreover, 2 : j ∈ Z ' Z 2 \G2 ' Z. Set (T ξ)k := ξk+1, ξ ∈ l (Z), and let  .  ..    χ (b) 0 D (b) =  χ (2b)  , ∀b ∈ 1 . χ   Z 2  χ (3b)    . 0 .. Note that TD (b) = D (α (b)) T = D (b) T, ∀b ∈ 1 . χ χ 2 αb2χ Z 2 Example 5.2. Let A be a d × d matrix over Z, det A 6= 0; then d −1 d −2 d Z ,→ A Z ,→ A Z ,→ · · · and set [ −k d BA := A Z . 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 2 d the Hilbert space L R with d-dimensional Lebesgue measure; and UA is a unitary representation of the discrete matrix group G = {(j, β)} , A j∈Z,β∈BA speciﬁed by (5.1) (j, β)(k, γ) = j + k, β + Ajγ , 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: Aj β Ak γ (j, β) −→ , and (k, γ) −→ . 0 1 0 1

2 d The wavelet representation UA of GA acting on L R is now

− j −j (5.2) UA (A)(x) := (det A) 2 f A (x − β) , 2 d d deﬁned for all f ∈ L R , ∀j ∈ Z, ∀β ∈ BA, and x ∈ R . INDUCED REPRESENTATIONS ARISING FROM A CHARACTER 799

Acknowledgements. The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezug- lyi, 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 correc- tions 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. [email protected] http://www.math.uiowa.edu/~jorgen/

(Feng Tian) Department of Mathematics, Trine University, IN 46703, U.S.A. [email protected] This paper is available via http://nyjm.albany.edu/j/2015/21-34.html.