Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 077, 17 pages
Quantum Analogs
of Tensor Product Representations of su(1, 1) ?
Wolter GROENEVELT
Delft Institute of Applied Mathematics, Technische Universiteit Delft,
PO Box 5031, 2600 GA Delft, the Netherlands
E-mail: w.g.m.groenevelt@tudelft.nl
URL: http://fa.its.tudelft.nl/~groenevelt/
Received April 28, 2011, in final form August 04, 2011; Published online August 09, 2011
http://dx.doi.org/10.3842/SIGMA.2011.077
Abstract. We study representations of Uq(su(1, 1)) that can be considered as quantum
analogs of tensor products of irreducible ∗-representations of the Lie algebra su(1, 1). We
determine the decomposition of these representations into irreducible ∗-representations of
Uq(su(1, 1)) by diagonalizing the action of the Casimir operator on suitable subspaces of the
representation spaces. This leads to an interpretation of the big q-Jacobi polynomials and
big q-Jacobi functions as quantum analogs of Clebsch–Gordan coefficients.
Key words: tensor product representations; Clebsch–Gordan coefficients; big q-Jacobi func-
tions
2010 Mathematics Subject Classification: 20G42; 33D80
1 Introduction
The quantum algebra Uq = Uq(su(1, 1)) has five classes of irreducible ∗-representations: the posi-
tive and negative discrete series pi±, the principal unitary series piP, the complementary series piC,
and the strange series piS. The first four classes of representations can be considered “classical”
in the sense that they are natural quantum analogs of the four irreducible ∗-representations
of the Lie algebra su(1, 1). In the classical limit q ↑ 1 these representations all tend to their
classical counterparts. The fifth class has no classical analog, hence the name “strange series”.
This class of representations disappears in the classical limit.
In this paper we study representations of Uq that can be considered as quantum analogs of
tensor products of irreducible ∗-representations of su(1, 1), but the representations that we con-
sider are not tensor products of irreducible ∗-representations of Uq. The motivation for studying
such representations comes from corepresentation theory of the locally compact quantum group
analog M of the normalizer of SU(1, 1) in SL(2,C). The dual quantum group M̂ is generated
as a von Neumann algebra by the standard generators of Uq and two extra generators. In this
sense Uq can be considered as a subalgebra of M̂ . An irreducible discrete series representation
of M̂ restricted to Uq decomposes as the sum pi+ ⊕ pi− ⊕ piS, with appropriate representation
labels, see [8, Section 5]. A tensor product of such representations consists of a sum of nine
simple tensor products. For five of these simple tensor products it is known how to decompose
them into irreducible Uq-representations: pi+⊗pi+, pi−⊗pi−, pi+⊗pi−, pi+⊗piS and piS⊗pi−, see
e.g. [10, Section 4], [11, Section 2], [5, Section 8]. In this paper we consider the remaining terms
as two “indivisible” representations, T = (pi−⊗pi+)⊕ (piS⊗piS) and T ′ = (pi−⊗piS)⊕ (piS⊗pi+),
?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe-
cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSF.html
2 W. Groenevelt
and determine their decompositions. In a similar way the principal unitary series and com-
plementary series representations of M̂ restricted to Uq decompose as piP ⊕ piP and piC ⊕ piC,
respectively. Taking tensor products of these we end up again with “indivisible” sums of simple
tensor products that can be considered as natural analogs of the tensor product of two principal
unitary series or complementary series of su(1, 1). It is remarkable that the decomposition of
the representations we consider here were already announced in [14, Section 12].
The Clebsch–Gordan coefficients with respect to standard bases for the five simple ten-
sor products mentioned above can be described in terms of terminating basic hypergeometric
3ϕ2-series. The orthogonality relations of the Clebsch–Gordan coefficients correspond to the
orthogonality relations of the (dual) q-Hahn polynomials and the continuous dual q-Hahn poly-
nomials. The Clebsch–Gordan coefficients for the representations we consider in this paper
turn out to be non-terminating 3ϕ2-series, and consequently the corresponding orthogonality
relations are (in general) not related to orthogonal polynomials, but to non-polynomial unitary
transform pairs.
Let us now briefly describe the contents of this paper. In Section 2 we recall the definition
of the quantum algebra Uq(su(1, 1)) and its irreducible ∗-representations. In Section 3 we con-
sider the decomposition of T . Our choice of representation labels is slightly more general than
allowed in the context of the locally compact quantum group M . The representation T can
be considered as a quantum analog of the tensor product of a negative and a positive discrete
series representation of su(1, 1). We diagonalize the action of the Casimir operator, and this
naturally leads to the interpretation of big q-Jacobi functions as quantum analogs of Clebsch–
Gordan coefficients. We also consider the representation T ′, which completes T to a genuine
tensor product representation of Uq, but has no classical analog. In Section 4 we consider the
representation (piP⊗ piP)⊕ (piP⊗ piP). The diagonalization of the Casimir operator leads in this
case to vector-valued big q-Jacobi functions as Clebsch–Gordan coefficients. Finally in Section 5
we give the decompositions of several other quantum analogs of tensor product representations.
Notations. We use N = {0, 1, 2, . . .} and q is a fixed number in (0, 1). We use standard
notations for q-shifted factorials, theta functions and basic hypergeometric series from the book
of Gasper and Rahman [3]. For x ∈ C and n ∈ N ∪ {∞} the q-shifted factorial is defined
by (x; q)n =
n−1∏
k=0
(1 − xqk), where the empty product is equal to 1. For x 6= 0 the normalized
Jacobi θ-function is defined by θ(x; q) = (x; q)∞(q/x; q)∞. For products of q-shifted factorial
and products of θ-functions we use the notations
(x1, x2, . . . , xm; q)n =
m∏
k=1
(xk; q)n, θ(x1, x2, . . . , xm; q) =
m∏
k=1
θ(xk; q)
and
(
x±1; q
)
n = (x, 1/x; q)n, θ
(
x±1; q
)
= θ(x, 1/x; q).
The basic hypergeometric series 3ϕ2 is defined by
3ϕ2
(
a, b, c
d, e
; q, z
)
=
∞∑
k=0
(a, b, c; q)k
(q, d, e; q)k
zk.
2 The quantum algebra Uq(su(1, 1))
The quantized universal enveloping algebra Uq = Uq
(
su(1, 1)
)
is the unital, associative, complex
algebra generated by K, K−1, E, and F , subject to the relations
KK−1 = 1 = K−1K, KE = qEK, KF = q−1FK,
Quantum Analogs of Tensor Product Representations of su(1, 1) 3
EF − FE =
K2 −K−2
q − q−1
. (2.1)
The Casimir element
Ω =
q−1K2 + qK−2 − 2
(q−1 − q)2
+ EF =
q−1K−2 + qK2 − 2
(q−1 − q)2
+ FE (2.2)
is a central element of Uq. The algebra Uq is a Hopf ∗-algebra with comultiplication ∆ defined
on the generators by
∆(K) = K ⊗K, ∆(E) = K ⊗ E + E ⊗K−1,
∆(K−1) = K−1 ⊗K−1, ∆(F ) = K ⊗ F + F ⊗K−1, (2.3)
and ∆ is extended to Uq as an algebra homomorphism. In particular, it follows from (2.2)
and (2.3) that
∆(Ω) =
1
(q−1 − q)2
[
q−1
(
K2 ⊗K2
)
+ q
(
K−2 ⊗K−2
)
− 2(1⊗ 1)
]
+K2 ⊗ EF +KF ⊗ EK−1 + EK ⊗K−1F + EF ⊗K−2. (2.4)
The ∗-structure on Uq is defined on the generators by
K∗ = K, E∗ = −F, F ∗ = −E, (K−1)∗ = K−1.
Note that the Casimir element is self-adjoint in Uq, i.e. Ω∗ = Ω.
There are, besides the trivial representation, five classes of irreducible ∗-representations of Uq,
see [15, Proposition 4], [2, Section 6]. The representations are given in terms of unbounded
operators on `2(N) or `2(Z). As common domain we take the finite linear combinations of the
standard orthonormal basis vectors en. The representations are unbounded ∗-representations in
the sense of Schmu¨dgen [20, Definition 8.1.9]. Below we list the actions of the generators K,
K−1, E, F on basis vectors en. The Casimir element Ω plays an important role in this paper,
therefore we also list the action of Ω.
The representation listed below are completely characterized, up to unitary equivalence, by
the actions of K and Ω. Let us briefly describe how these actions determine the actions of E
and F up to a phase factor. Let pi be a Uq-representation acting on `2(Z) with orthonormal
basis {en}n∈Z, and suppose that pi(K)en = qn+εen and pi(Ω)en = ωen for all n ∈ Z, where ε
and ω are real numbers. The commutation relations (2.1) imply that pi(E)en = cnen+1 and
pi(F )en = dnen−1 for certain numbers cn and dn. Furthermore, the relation E∗ = −F implies
cn = −dn+1, so that pi(FE)en = −|cn|2en. On the other hand, from (2.2) and the actions of K
and Ω it follows that
pi(FE)en =
(
ω −
q2n+2ε+1 + q−2n−2ε−1 − 2
(q−1 − q)2
)
en.
So cn can be determined up to a phase factor.
The five classes of irreducible ∗-representations of Uq are the following:
Positive discrete series. The positive discrete series pi+k are labeled by k > 0. The
representation space is `2(N) with orthonormal basis {en}n∈N. The action is given by
pi+k (K)en = q
k+nen, pi
+
k
(
K−1
)
en = q
−(k+n)en,
(
q−1 − q
)
pi+k (E)en = q
− 12−k−n
√
(1− q2n+2)(1− q4k+2n)en+1,
(
q−1 − q
)
pi+k (F )en = −q
1
2−k−n
√
(1− q2n)(1− q4k+2n−2)en−1,
4 W. Groenevelt
(
q−1 − q
)2
pi+k (Ω)en =
(
q2k−1 + q1−2k − 2
)
en. (2.5)
Negative discrete series. The negative discrete series pi−k are labeled by k > 0. The
representation space is `2(N) with orthonormal basis {en}n∈N. The action is given by
pi−k (K)en = q
−(k+n)en, pi
−
k
(
K−1
)
en = q
k+nen,
(
q−1 − q
)
pi−k (E)en = −q
1
2−k−n
√
(1− q2n)(1− q4k+2n−2)en−1,
(
q−1 − q
)
pi−k (F )en = q
− 12−k−n
√
(1− q2n+2)(1− q4k+2n)en+1,
(
q−1 − q
)2
pi−k (Ω)en =
(
q2k−1 + q1−2k − 2
)
en. (2.6)
Principal unitary series. The principal unitary series representations piPρ,ε are labeled by
0 ≤ ρ < − pi2 ln q and ε ∈ [0, 1), where (ρ, ε) 6= (0,
1
2). The representation space is `
2(Z) with
orthonormal basis {en}n∈Z. The action is given by
piPρ,ε(K)en = q
n+εen, pi
P
ρ,ε
(
K−1
)
en = q
−(n+ε)en,
(
q−1 − q
)
piPρ,ε(E)en = q
− 12−n−ε
√
(1− q2n+2ε+2iρ+1)(1− q2n+2ε−2iρ+1)en+1,
(
q−1 − q
)
piPρ,ε(F )en = −q
1
2−n−ε
√
(1− q2n+2ε+2iρ−1)(1− q2n+2ε−2iρ−1)en−1,
(
q−1 − q
)2
piPρ,ε(Ω)en =
(
q2iρ + q−2iρ − 2
)
en. (2.7)
For (ρ, ε) = (0, 12) the representation pi
P
0, 12
splits into a direct sum of a positive and a negative
discrete series representation: piP
0, 12
= pi+1
2
⊕pi−1
2
. The representation space splits into two invariant
subspaces: {en |n ≥ 0} ⊕ {en |n < 0}.
Complementary series. The complementary series representations piCλ,ε are labeled by λ
and ε, where ε ∈ [0, 12) and λ ∈ (−
1
2 ,−ε), or ε ∈ (
1
2 , 1) and λ ∈ (−
1
2 , ε− 1). The representation
space is `2(Z) with orthonormal basis {en}n∈Z. The action of the generators is the same as for
the principal unitary series, with iρ replaced by λ+ 12 :
piCλ,ε(K)en = q
n+ε en, pi
C
λ,ε
(
K−1
)
en = q
−(n+ε)en,
(
q−1 − q
)
piCρ,ε(E)en = q
− 12−n−ε
√
(1− q2n+2ε+2λ+2)(1− q2n+2ε−2λ)en+1,
(
q−1 − q
)
piCλ,ε(F )en = −q
1
2−n−ε
√
(1− q2n+2ε+2λ)(1− q2n+2ε−2λ−2)en−1,
(
q−1 − q
)2
piCλ,ε(Ω)en =
(
q2λ+1 + q−1−2λ − 2
)
en. (2.8)
Strange series. The fifth class consists of the strange series representations piSσ,ε, labeled
by σ > 0 and ε ∈ [0, 1), with representation space `2(Z). The actions of the generators are the
same as in principal unitary series with iρ replaced by − ipi2 ln q + σ:
piSσ,ε(K)en = q
n+εen, pi
S
σ,ε
(
K−1
)
en = q
−(n+ε)en,
(
q−1 − q
)
piSσ,ε(E)en = q
− 12−n−ε
√
(1 + q2n+2ε+2σ+1)(1 + q2n+2ε−2σ+1)en+1,
(
q−1 − q
)
piSσ,ε(F )en = −q
1
2−n−ε
√
(1 + q2n+2ε+2σ−1)(1 + q2n+2ε−2σ−1)en−1,
(
q−1 − q
)2
piSσ,ε(Ω)en = −
(
q2σ + q−2σ + 2
)
en. (2.9)
The representations piPρ,ε, pi
C
λ,ε, pi
S
σ,ε with ε ∈ R \ [0, 1) are equivalent to the representations
piPρ,ε+m, pi
C
λ,ε+m, pi
S
σ,ε+m respectively, where m ∈ Z is such that ε + m ∈ [0, 1). Therefore
Quantum Analogs of Tensor Product Representations of su(1, 1) 5
we will use these representations with any ε ∈ R. Furthermore, the principal unitary series
representation piPρ,ε is equivalent to pi
P
−ρ,ε and to pi
P
ρ′,ε with ρ
′ = ρ + mpiln q , m ∈ Z, and the
complementary series representation piCλ,ε is equivalent to pi
C
−λ−1,ε.
It will be convenient to use the convention e−n = 0 for n = 1, 2, . . ., if en is a basis vector
of `2(N). Furthermore, we sometimes denote a basis vector of the representation space of the
positive discrete series by e+n , and similarly for basis vectors of other representation spaces.
3 A quantum analog of the tensor product of negative
and positive discrete series representations
Let k1, k2 > 0 and ε ∈ R. We consider the Uq-representation
T = Tk1,k2,ε =
((
pi−k1 ⊗ pi
+
k2
)
⊕
(
piSk1− 12 ,−ε−k1
⊗ piSk2− 12 ,ε+k2
))
∆. (3.1)
For the decomposition of T into irreducible representations we need the big q-Jacobi func-
tions [13]. Let a, b, c be parameters satisfying the conditions a, b, c > 0 and ab, ac, bc > 1. The
big q-Jacobi functions are defined by
Φγ(y; a, b, c|q) =
(γ/a, 1/bc,−y/abcγ; q)∞
(1/ab, 1/ac,−y/bc; q)∞
3ϕ2
(
1/bγ, 1/cγ,−y/bc
1/bc,−y/abcγ
; q, γ/a
)
,
for y ∈ C \ (−∞,−bc] and |γ| < a. By [3, (III.9)] Φγ is symmetric in γ, γ−1, so for |γ| ≥ a we
define the big q-Jacobi function by the same formula with γ replaced by γ−1. If |y| < bc, the
3ϕ2-function can be transformed by [3, (III.10)] and then we have
Φγ(y; a, b, c|q) = 3ϕ2
(
γ/a, 1/aγ,−1/y
1/ab, 1/ac
; q,−y/bc
)
. (3.2)
Let t > 0. We define discrete sets Γfin and Γinf (‘fin’ and ‘inf’ stand for ‘finite’ and ‘infinite’)
by
Γfin =
{
qk
e
∣
∣
∣ e ∈ {a, b, c}, k ∈ N such that
qk
e
> 1
}
,
Γinf =
{
−
abcqk
t
∣
∣
∣ k ∈ Z such that
abcqk
t
> 1
}
,
Note that the set Γfin is empty if a, b, c > 1. Now we introduce the measure ν( · ) = ν( · ; a, b, c; t|q)
by
∫
f(γ)dν(γ) =
1
4pii
∫
T
f(γ)v(γ)
dγ
γ
+
∑
γ∈Γfin∪Γinf
f(γ)w(γ),
where v is the weight function on the counter-clockwise oriented unit circle T given by
v(γ) = v(γ; a, b, c; t|q) =
θ(−t; q)
θ(−t/ab,−t/ac,−t/bc; q)(q, 1/ab, 1/ac, 1/bc; q)∞
×
(γ±2; q)∞
(γ±1/a, γ±1/b, γ±1/c; q)∞ θ(−tγ±1/abc; q)
,
and
w(γ) = w(γ; a, b, c; t; q) = Res
γ′=γ
v(γ′)
γ′
, γ ∈ Γfin ∪ Γinf .
The weights w in the discrete mass points can be calculated explicitly, see [13, Section 8].
6 W. Groenevelt
Let H = H(a, b, c; t|q) be the Hilbert space consisting of functions that satisfy f(γ) = f(γ−1)
(ν-a.e.) with inner product
〈f, g〉 =
∫
f(γ)g(γ) dν(γ).
The set of functions {γ 7→ Φγ(x; a, b, c) |x ∈ {−qk | k ∈ N} ∪ {tqk | k ∈ Z}} is an orthogonal
basis for H, i.e.
〈Φ·(y),Φ·(y
′)〉 = Nyδyy′ , y, y
′ ∈
{
−qk | k ∈ N
}
∪
{
tqk | k ∈ Z
}
,
where the quadratic norm Ny is given by
Ny = Ny(a, b, c; q) =



(q, 1/bc; q)k
(1/ab, 1/ac; q)k
q−k, y = −qk, k ∈ N,
(q, 1/bc,−tqk/ab,−tqk/ac; q)∞
(1/ab, 1/ac,−tqk/bc,−tqk+1; q)∞
q−k
t , y = tq
k, k ∈ Z.
We define
φy(x) = φy(x; a, b, c|q) =
Φx(y; a, b, c|q)
√
Ny(a, b, c; q)
,
then {φ−qk}k∈N ∪ {φtqk}k∈Z is an orthonormal basis for H. For ease of notations, it is useful to
define φ−qk = 0 for k = −1,−2, . . ..
The difference equation for the big q-Jacobi functions leads to a recurrence relation for the
functions φzqk , z ∈ {−1, t}, of the form
(
x+ x−1
)
φzqk(x) = akφzqk+1(x) + bkφzqk(x) + ak−1φzqk−1(x), (3.3)
where
ak = ak(a, b, c; z|q) =
abcq−
1
2−2k
z2
√
(
1 + zqk+1
)
(
1 +
zqk
ab
)(
1 +
zqk
ac
)(
1 +
zqk
bc
)
,
bk = bk(a, b, c; z|q) = a+ a
−1 −
abcq−2k
z2
(
1 +
zqk
ab
)(
1 +
zqk
ac
)
−
abcq1−2k
z2
(
1 + zqk
)
(
1 +
zqk−1
bc
)
.
Here we assume k ∈ N if z = −1 and k ∈ Z if z = t, and we use a−1(a, b, c;−1|q) = 0. Note that
ak and bk are both symmetric in a, b, c.
Let us remark that for z = −1 the recurrence relation corresponds to the three term recurrence
relation for the continuous dual q−1-Hahn polynomials, i.e. Askey–Wilson polynomials with one
parameter equal to zero and base q−1;
pk(x) = pk
(
x; a, b, c|q−1
)
= a−k
√
(ab, ac; q−1)k
(q−1, bc; q−1)k
3ϕ2
(
qk, ax, a/x
ab, ac
; q−1, q−1
)
.
So the function φ−qk is a multiple of the continuous dual q
−1-Hahn polynomials pk. The moment
problem corresponding to the continuous dual q−1-Hahn polynomials is indeterminate, so the
measure ν defined above is a (non-extremal) solution for this moment problem.
Quantum Analogs of Tensor Product Representations of su(1, 1) 7
We now turn to the decomposition of T . First we diagonalize T (Ω) on a suitable subspace
of `2(N)⊗2 ⊕ `2(Z)⊗2. We define subspaces Ap, p ∈ Z, by
Ap = span{f
−
p,n | n ∈ N} ⊕ span{f
+
p,n | n ∈ Z},
where
f−p,n =
{
en ⊗ en−p, p ≤ 0,
en+p ⊗ en, p ≥ 0,
f+p,n =
{
(−1)ne−n ⊗ ep+n, p ≤ 0,
(−1)n+pe−n−p ⊗ en, p ≥ 0.
It is useful to observe that f−p,n = 0 for n = −1,−2, . . .. Furthermore, note that Ap ∼= `
2(N) ⊕
`2(Z) and A =
⊕
p∈ZAp is a dense subspace of `
2(N)⊗2 ⊕ `2(Z)⊗2. From (2.4) it follows that
each subspace Ap is invariant under T (Ω).
Proposition 3.1. Let p ∈ Z, σ ∈ {−,+}, and let a, b, c, tσ be defined by
a =
{
q2k2−2k1−2p−1, p ≥ 0,
q2k1−2k2+2p−1, p ≤ 0,
b = q1−2k1−2k2 , c =
{
q2k1−2k2−1, p ≥ 0,
q2k2−2k1−1, p ≤ 0,
tσ =
{
−1, σ = −,
q2ε, σ = +.
(3.4)
Then the operator Θ : Ap → H(a, b, c; t+|q2) defined by
fσp,n 7→
(
x 7→ φtσq2n
(
x; a, b, c|q2
))
, n ∈ Z,
extends to a unitary operator. Moreover, Θ intertwines T (Ω) with the multiplication operator
(q−1 − q)−2Mx+x−1−2.
Proof. First of all, Θ extends to a unitary operator since it maps one orthonormal basis onto
another.
To check the intertwining property, we consider the explicit action of Ω. First assume p ≥ 0.
From (2.4), (2.5), (2.6) and (2.9) it follows that the action of the Casimir operator is given by
(
q−1 − q
)2
[T (Ω) + 2]fσp,n = a
σ
nf
σ
p,n+1 + b
σ
nf
σ
p,n + a
σ
n−1f
σ
p,n−1,
where
aσn = t
−2
σ q
−2−2k2−2k1−2p−4n
×
√(
1 + tσq4k1+2p+2n
)(
1 + tσq2p+2n+2
)(
1 + tσq4k2+2n
)(
1 + tσq2n+2
)
,
bσn = q
−2p−2k1+2k2−1 + q2p+2k1−2k2+1 − t−2σ q
1−2k1−2k2−2p−4n
(
1 + tσq
2n+4k2−2
)(
1 + tσq
2n)
+ t−2σ q
−1−2k1−2k2−2p−4n
(
1 + tσq
4k1+2p+2n
)(
1 + tσq
2p+2n+2).
The intertwining property follows from comparing this with the recurrence relation (3.3) for the
function φtσq2n(x; a, b, c|q
2). For p < 0 the proof runs along the same lines. 
By Proposition 3.1 Θ intertwines T (Ω) with a multiplication operator on H. This allows
us to read off the spectrum of T (Ω) from the support of the measure ν. Since the irreducible
∗-representations are characterized by the actions of Ω and K, we now only need to consider
the action of K to find the decomposition of T into irreducible representations;
T (K)fσp,n = q
−2p+2k2−2k1fσp,n.
By comparing the spectral values of T (K) and T (Ω) with (2.5)–(2.9), we obtain the following
decomposition of T .
8 W. Groenevelt
Theorem 3.1. The Uq-representation Tk1,k2,ε is unitarily equivalent to:
(i)
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ , |k1 − k2| ≤
1
2
, k1 + k2 ≥
1
2
,
(ii)
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕ pi
C
−k1−k2,ε′ , k1 + k2 <
1
2
,
(iii)
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈N
k+j >
1
2
pi+
k+j
, k1 − k2 +
1
2
< 0,
(iv)
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈N
k−j >
1
2
pi−
k−j
, k2 − k1 +
1
2
< 0,
where ε′ = k2−k1, σj = k1 +k2 + ε+ 12 + j, k
+
j = k2−k1− j and k
−
j = k1−k2− j. For n, p ∈ Z
and σ ∈ {−,+} the unitary intertwiner is given by
(i) fσp,n 7→ Λf
σ
p,n,
(ii) fσp,n 7→ Λf
σ
p,n + (Θf
σ
p,n)
(
q2k1+2k2−1
)√
w
(
q2k1+2k2−1; a, b, c; t+; q2
)
eC−p,
(iii) fσp,n 7→ Λf
σ
p,n +



∑
j∈N
k+j+p>
1
2
(Θfσp,n)
(
q2k
+
j+p−1
)
√
w
(
q2k
+
j+p−1; a, b, c; t+; q2
)
e+j , p ≥ 0,
∑
j∈N
k+j >
1
2
(Θfσp,n)
(
q2k
+
j −1
)
√
w
(
q2k
+
j −1; a, b, c; t+; q2
)
e+j−p, p ≤ 0,
(iv) fσp,n 7→ Λf
σ
p,n +



∑
j∈N
k−j >
1
2
(Θfσp,n)
(
q2k
−
j −1
)
√
w
(
q2k
−
j −1; a, b, c; t+; q2
)
e−j+p, p ≥ 0,
∑
j∈N
k−j−p>
1
2
(Θfσp,n)
(
q2k
−
j−p−1
)
√
w
(
q2k
−
j −1; a, b, c; t+; q2
)
e−j , p ≤ 0,
where
Λfσp,n =
∫ − pi2 ln q
0
(
Θfσp,n
)(
q2iρ
)√
v
(
q2iρ; a, b, c; t+|q2
)
eP−p dρ
+
∑
j∈Z
σj>0
(
Θfσp,n
)(
−q2σj
)√
w
(
−q2σj ; a, b, c; t+; q2
)
eS−p,
Θ is as in Proposition 3.1 and a, b, c, t+ are given by (3.4).
Note that in case (i) the intertwiner maps from `2(N)⊗2⊕ `2(Z)⊗2 into
∫ −pi/(2 ln(q))
0 `
2(Z) dρ⊕
⊕
N `
2(Z). In case (ii) another `2(Z) has to be added here, and in cases (iii) and (iv) a finite
number of `2(N)-spaces has to be added.
Quantum Analogs of Tensor Product Representations of su(1, 1) 9
Remark 3.1. (i) Letting q ↑ 1 we (formally) obtain the decomposition of the tensor product of
a negative and a positive discrete series representation of su(1, 1), see e.g. [19, Theorem 7.3] or
[7, Theorem 2.2].
(ii) Theorem 3.1 shows that the big q-Jacobi functions have an interpretation as quantum
analogs of Clebsch–Gordan coefficients. In the classical case the Clebsch–Gordan coefficients
for the tensor product of negative and positive discrete series are essentially continuous dual
Hahn polynomials, see [17, Section 4] or [7, Theorem 2.2]. So in the context of Clebsch–Gordan
coefficients the big q-Jacobi transform pair should be considered as a q-analog of the transform
pair corresponding to continuous dual Hahn polynomials.
(iii) The Uq-representation pi
+
k2
⊗ pi−k1 can be decomposed similar as in Theorem 3.1, but the
infinite sum of strange series representations does not occur in this situation. The Clebsch–
Gordan coefficients in this case are essentially continuous dual q2-Hahn polynomials, see [11,
Section 2] or [4, Theorem 2.4].
(iv) The term
(
pi−k1 ⊗ pi
+
k2
)
∆ occurring in the definition of T would of course also be a quan-
tum analogue of the tensor product of a negative and positive discrete series representation
of su(1, 1), but we consider it unlikely that this representation can be decomposed into irre-
ducible representations. Indeed, in this case the action of the Casimir operator corresponds to
the Jacobi operator for the continuous dual q−2-Hahn polynomials, see also [5, Remark 8.1]. The
corresponding moment problem is indeterminate (so the Casimir operator is not essentially self-
adjoint in this case!) and no explicit N -extremal solutions are known. Even if such a measure
was known, it would have discrete support, implying that the decomposition would be a direct
sum of irreducible representations, and not a direct integral as in the classical case.
Let us denote the intertwiner from Theorem 3.1 by I. The actions I ◦T (X)◦ I−1, X = E,F ,
are given by the appropriate actions of E and F in (2.5)–(2.9), up to a phase factor. It is possible
to determine the actions explicitly using the explicit expressions for the weight functions v and w,
the explicit expressions for Θfσp,n as 3ϕ2-functions, and the following contiguous relations for
3ϕ2-functions, see [9, (2.3), (2.4)],
3ϕ2
(
Aq,B,C
D,E
; q,
DE
ABCq
)
− 3ϕ2
(
A,B,C
D,E
; q,
DE
ABC
)
=
DE(1−B)(1− C)
ABCq(1−D)(1− E) 3
ϕ2
(
Aq,Bq,Cq
Dq,Eq
; q,
DE
ABCq
)
,
(
1−
D
A
)(
1−
E
A
)
3ϕ2
(
A/q,B,C
D,E
; q,
DEq
ABC
)
−
(
1−
q
A
)(
1−
DE
ABC
)
3ϕ2
(
A,B,C
D,E
; q,
DE
ABC
)
=
q
A
(
1−
D
q
)(
1−
E
q
)
3ϕ2
(
A/q,B/q, C/q
D/q,E/q
; q,
DEq
ABC
)
.
We do not work out the details.
3.1 Completing T to a genuine tensor product representation
In this subsection we define a representation T ′ that completes T defined by (3.1) to a genuine
tensor product. Let k1, k2 > 0 and ε ∈ 12Z, and define the Uq-representation T
′ by
T ′ = T ′k1,k2,ε =
((
pi−k1 ⊗ pi
S
k2− 12 ,ε+k2
)
⊕
(
piSk1− 12 ,−ε−k1
⊗ pi+k2
))
∆.
Now the sum T ⊕T ′ =
((
pi−k1⊕pi
S
k1− 12 ,−ε−k1
)
⊗
(
pi+k2⊕pi
S
k2− 12 ,ε+k2
))
∆ is a genuine tensor product
representation of Uq, which can also be considered as a quantum analog of the tensor product
of a negative and a positive discrete series representation of su(1, 1).
10 W. Groenevelt
The decomposition of T ′ into irreducible ∗-representations is established in the same way as
the decomposition of T , therefore we omit most of the details. We remark that we need here the
condition ε ∈ 12Z (instead of ε ∈ R), because our method for constructing an intertwiner only
works if basis vectors of `2(Z) can be labeled by 2ε+m for m ∈ Z, which forces ε to be in 12Z.
For the diagonalization of T ′(Ω) we need the big q-Jacobi polynomials [1], [12, § 14.5]. They
are defined by
Pm(x; a, b, c; q) = 3ϕ2
(
q−m, abqm+1, x
aq, cq
; q, q
)
.
We assume that m ∈ N, x ∈ {aqk+1 | k ∈ N} ∪ {cqk+1 | k ∈ N}, 0 < a, b < q−1 and c < 0, then
the big q-Jacobi polynomials satisfy the orthogonality relations
∫ aq
cq
Pm1(x)Pm2(x)u(x)dqx =
δm1m2
v(m1)
,
∞∑
m=0
Pm(x1)Pm(x2)v(m) =
δx1x1
|x1|u(x1)
,
where the (positive) functions u(·) = u(·; a, b, c; q) and v(·) = v(·; a, b, c; q) are given by
u(x; a, b, c; q) =
(x/a, x/c; q)∞
(x, bx/c; q)∞
,
v(m; a, b, c; q) =
(1− abq2m+1)
aq(1− abq)
(aq, bq, cq, abq/c; q)∞
(q, abq2; q)∞θ(c/a)
×
(aq, abq, cq; q)m
(q, bq, abq/c; q)m
(
−acq2
)−m
q−
1
2m(m−1).
Here the Jackson q-integral is defined by
∫ aq
cq
f(x)dqx = (1− q)
∞∑
k=0
f
(
aqk+1
)
aqk+1 − (1− q)
∞∑
k=0
f
(
cqk+1
)
cqk+1.
We define functions rx(m), related to the big q-Jacobi polynomials, by
rx(m; a, b, c; q) =
√
|x|u(x; a, b, c; q)v(m; a, b, c; q)Pm(x; a, b, c; q).
By the orthogonality relations for the big q-Jacobi polynomials we have
∞∑
n=0
rx1(m)rx2(m) = δx1x2 ,
∞∑
k=0
raqk(m1)raqk(m2) +
∞∑
k=0
rcqk(m1)rcqk(m2) = δm1m2 .
Furthermore, from the q-difference equation for Pm it follows that the functions rx(m) satisfy
the following q-difference equation in x:
1
√
abq
(
1− q−m
)(
1− abqm+1
)
rx(m) = A(x)rqx(m) +B(x)rx(m) +A(x/q)rx/q(m),
where
A(x) = −
c
√
a
x2
√
b
√
(1− x)(1− x/a)(1− x/c)(1− bx/c),
B(x) =
c
√
aq
x2
√
b
(
(1− x)(1− bx/c) + q(1− x/aq)(1− x/cq)
)
=
c
√
aq
x2
√
b
(
(1− x)(1− x/a) + q(1− bx/cq)(1− x/cq)
)
Quantum Analogs of Tensor Product Representations of su(1, 1) 11
+
√
abq +
1
√
abq
− c
√
q
ab
−
1
c
√
ab
q
.
Now we are ready to diagonalize T ′(Ω). For p ∈ Z and n ∈ N we define
f−p,n = en ⊗ en+p, f
+
p,n = e2ε−n+p ⊗ en,
and we write
Ap = span{f
−
p,n | n ∈ N} ⊕ span{f
+
p,n | n ∈ N}.
Note that
⊕
p∈ZAp is a dense subspace of `
2(N)⊗ `2(Z)⊕ `2(Z)⊗ `2(N).
Proposition 3.2. Let p ∈ Z and define Θ : Ap → `2(N) by
f−p,n 7→
(
m 7→ rq2n+4k1
(
m; q4k1−2, q4k2−2,−q4k1−2p−2ε−2; q2
))
,
f+p,n 7→
(
m 7→ (−1)nr−q2n+4k1−2p−2ε
(
m; q4k1−2, q4k2−2,−q4k1−2p−2ε−2; q2
))
,
then Θ extends to a unitary operator. Furthermore, Θ intertwines T ′(Ω) with the multiplication
operator −(q−1 − q)−2Mq2k1+2k2−1+2m+q−(2k1+2k2−1+2m)+2.
Theorem 3.2. The Uq-representation T ′k1,k2,ε is unitarily equivalent to
∞⊕
m=0
piSk1+k2− 12+m,ε−k1+k2
.
The unitary intertwiner `2(N)⊗ `2(Z)⊕ `2(Z)⊗ `2(N)→
⊕∞
m=0 `
2(Z) is given by
fσp,n 7→
∞∑
m=0
(
Θfσp,n
)
(m)eSp , p ∈ Z, n ∈ N, σ ∈ {−,+}.
Remark 3.2. Theorem 3.2 shows that the big q-Jacobi polynomials have a natural interpreta-
tion as Clebsch–Gordan coefficients for Uq-representations. In this interpretation they do not
have a classical analog, since the Uq-representation T ′ vanishes in the classical limit.
4 A quantum analog of the tensor product
of two principal unitary series representations
Let ρ1, ρ2 ∈ (0,− pi2 ln q ) and ε1, ε2, ε ∈ R. In this section we consider the representation
T = Tρ1,ρ2,ε1,ε2,ε =
(
(piPρ1,ε1+ε ⊗ pi
P
ρ2,ε2−ε)⊕ (pi
P
ρ′1,ε1−ε
⊗ piPρ′2,ε2+ε)
)
∆,
where ρ′j = ρj −
pi
2 ln q , j = 1, 2. The representation space is `
2(Z)⊗2 ⊕ `2(Z)⊗2. Observe
that for ε1, ε2, ε ∈ 12Z the representation Tρ1,ρ′2,ε1,ε2,ε ⊕ Tρ1,ρ2,ε1,ε2,ε is a genuine tensor product
representation, using the equivalence piPρ,ε ∼= pi
P
ρ,ε+1.
For the decomposition of T into irreducible representations we need the vector-valued big
q-Jacobi functions [6]. Let a, c ∈ C, z+ > 0 and z− < 0. We set b = a, d = c and s =
√
q| ca |. For
a function f depending on the parameter a and b, f = f( · ; a, b), we write
f † = f †( · ; a, b) = f( · ; b, a).
12 W. Groenevelt
We define a discrete set Γ depending on the parameters a, c, z+, z− by
Γ = Γfins ∪ Γ
fin
q/s ∪ Γ
inf ,
Γfinα =
{
1
αqk
∣
∣
∣ k ∈ N, αqk > 1
}
,
Γinf =
{
z−z+q
k− 12 |ac|
∣
∣
∣ k ∈ Z, −z−z+qk−
1
2 |ac| < 1
}
.
Let v be the matrix-valued function given by
γ 7→ v(γ) =
(
v2(γ) v
†
1(γ)
v1(γ) v2(γ)
)
,
v1(γ) =
(cq/a, dq/a; q)2∞θ(bz+, bz−; q)
(1− q)abz2−z
2
+θ(z−/z+, z+/z−, a/b, b/a; q)
×
(γ±2; q)∞
(sγ±1, cqγ±1/as, dqγ±1/as; q)∞θ(sγ±1, absz−z+γ±1; q)
×
(
z−θ(az+, cz+, dz+, bz−, asz−γ
±1; q)− z+θ(az−, cz−, dz−, bz+, asz+γ
±1; q)
)
,
v2(γ) =
(cq/a, dq/a, cq/b, dq/b; q)∞θ(az+, az−, bz+, bz−, cdz−z+; q)
abz2−z+(1− q)θ(z+/z−, a/b, b/a; q)
×
(γ±2; q)∞
(sγ±1; q)∞θ(sγ±1, absz−z+γ±1; q)
.
For γ ∈ T we have v†1(γ) = v1(γ). In this case the matrix v(γ) is positive definite and we can
write
v(γ) = S(γ)TS(γ),
where ST is the conjugate transpose of S, and the matrix-valued function S is given by
S(γ) = S(γ; a, c|q) =





√
v2(γ) + |v1(γ)|
2
v1(γ)
|v1(γ)|
√
v2(γ) + |v1(γ)|
2
−
√
v2(γ)− |v1(γ)|
2
v1(γ)
|v1(γ)|
√
v2(γ)− |v1(γ)|
2





.
We also define
v(γ) = v(γ; a, c; z−, z+|q) = −
θ(cz−, dz−, cz+, dz+; q)
z+(1− q)θ(z−/z+)
×
(γ2, qγ2; q)∞
(cqγ/as, dqγ/as, cqγ/bs, dqγ/bs, sγ, qγ/s; q)∞θ(absz−z+/qγ; q)
.
With this function we define a positive weight function w on Γ by
w(γ) = w(γ; a, c; z−, z+|q) =
1
b(γ)
Res
γ′=γ
v(γ′)
γ′
, γ ∈ Γ,
where
b(γ) =



(
z+
z−
)k+1 θ(cz−, dz−; q)
θ(cz+, dz+; q)
, γ = z−z+qk−
1
2 |ac| ∈ Γinf ,
(
z−
z+
)k
, γ = s−1q−k ∈ Γfins ,
(
z−
z+
)k θ(az+, bz+, cz−, dz−; q)
θ(az−, bz−, cz+, dz+; q)
, γ = sq−1−k ∈ Γfinq/s.
Quantum Analogs of Tensor Product Representations of su(1, 1) 13
Explicit expressions for w can be obtained by evaluating the residues, see [6, Section 4]. We
denote by F (T∪Γ) the vector space of functions that are C2-valued on T and C-valued on Γ. We
define H = H(a, c; z−, z+|q) to be the Hilbert space consisting of functions in F (T∪Γ) satisfying
f(γ) = f(γ−1) almost everywhere on T, that have finite norm with respect to the inner product
〈f, g〉 =
1
4pii
∫
T
g(γ)Tv(γ)f(γ)
dγ
γ
+
∑
γ∈Γ
f(γ)g(γ)w(γ).
The vector-valued big q-Jacobi functions are defined by
Ψ(y, γ) = Ψ(y, γ; a, c|q) =



(
ϕγ(y)
ϕ†γ(y)
)
, γ ∈ T,
d(γ)ϕγ(y) + d†(γ)ϕ
†
γ(y), γ ∈ Γinf ∪ Γfinq/s,
c(γ)ϕγ(y), γ ∈ Γfins .
Here ϕγ is the reparametrized big q-Jacobi function given by
ϕγ(y) = ϕγ(y; a, c|q) = Φγ(−1/by; 1/s, as/cq, as/dq|q),
so explicitly ϕγ is given by, see (3.2),
ϕγ(y) = 3ϕ2
(
q/ay, sγ, s/γ
cq/a, dq/a
; q, by
)
, |y| <
1
|b|
,
and furthermore
d(γ) =
(cq/a, dq/a; q)∞θ(bz+)
θ(a/b, cz+, dz+)
(cqγ/sb, dqγ/sb; q)∞θ(asz+/qγ)
(qγ2, s/γ; q)∞
,
c(γ) =
(cq/a, dq/a, 1/γ2; q)∞θ(bz)
(s/γ, cq/asγ, dq/asγ; q)∞θ(bszγ)
.
For γ ∈ Γfins ∪ Γ
fin
q/s the function Ψ( · ; γ) is actually a multiple of a big q-Jacobi polynomial, see
[6, Lemma 3.9].
The set
{
γ 7→ Ψ(y, γ) | y ∈
{
z−q
k | k ∈ Z
}
∪
{
z+q
k | k ∈ Z
}}
is an orthogonal basis for H. We have
〈Ψ(y, · ),Ψ(y′, · )〉 = Nyδyy′ ,
where the squared norm Ny is given by
Ny = Ny(a, c; q) =
1
|y|
∣
∣
∣
∣
(cy; q)∞
(ay; q)∞
∣
∣
∣
∣
2
.
We define
ψy(x) = ψy(x; a, c|q) =
Ψ(y, x; a, c|q)
√
Ny(a, c; q)
,
then {ψz−qk}k∈Z ∪ {ψz+qk}k∈Z is an orthonormal basis for H. Furthermore, for z ∈ {z−, z+}
these functions satisfy the recurrence relation
(
x+ x−1
)
ψzqk(x) = akψzqk+1(x) + bkψzqk(x) + ak−1ψzqk−1(x), (4.1)
14 W. Groenevelt
where
ak = ak(a, c; z; q) =
∣
∣
∣
∣
(
1−
q−k
az
)(
1−
q−k
cz
)∣
∣
∣
∣ ,
bk = bk(a, c; z; q) = s
−1 + s− s−1
∣
∣
∣
∣1−
q1−k
az
∣
∣
∣
∣
2
− s
∣
∣
∣
∣1−
q−k
cz
∣
∣
∣
∣
2
.
We are now ready for the decomposition of T . For this we need the vector-valued big q2-
Jacobi functions and the corresponding Hilbert space H(a, c; z−, z+|q2), for certain values of the
parameters a, c, z−, z+. Note that in this case the q in all the formulas above must be replaced
by q2, e.g. s is given by q|c|/|a|.
As before, we first diagonalize T (Ω) on a suitable subspace. For p ∈ Z we define the subspace
Ap = span{f
−
p,n | n ∈ Z} ⊕ span{f
+
p,n | n ∈ Z},
where
fσp,n = (−1)
ne−n ⊗ ep+n, n, p ∈ Z,
and we assume that f+p,n is an element of the representation space of pi
P
ρ1,ε1+ε⊗pi
P
ρ2,ε2−ε, and f
−
p,n
an element of the representation space of piPρ′1,ε1−ε
⊗ piPρ′2,ε2+ε
. Note that Ap ∼= `2(Z)⊕ `2(Z).
Proposition 4.1. Let a, c, z−, z+ be given by
a = q2iρ2+2ε2+2p+1, c = q2iρ1−2ε1+1, zσ = σq
−2σε, σ ∈ {−,+}. (4.2)
Then the operator Θ : Ap → H(a, c; z+, z−|q2) defined by
fσp,n 7→
(
x 7→ ψzσq2n
(
x; a, c|q2
))
, σ ∈ {−,+},
extends to a unitary operator. Furthermore, Θ intertwines T (Ω) with (q − q−1)−2Mx+x−1−2.
Note in particular that the continuous spectrum occurs with multiplicity 2.
Proof. From (2.4) and (2.7) we obtain
(
q − q−1
)2
[T (Ω) + 2]fσp,n = a
σ
nf
σ
p,n+1 + b
σ
nf
σ
p,n + a
σ
n−1f
σ
p,n−1,
where
aσn =
∣
∣(1− σq−2n+2ε1+2σε+2iρ1−1)(1− σq−2n−2p−2ε2+2σε+2iρ2−1)
∣
∣ ,
bσn = q
1−2ε1−2ε2−2p + q−1+2ε1+2ε2+2p − q1−2ε1−2ε2−2p
∣
∣1− σq−2n+2ε1+2σε+2iρ1−1
∣
∣2
− q−1+2ε1+2ε2+2p
∣
∣1− σq−2n−2p+2σε+2iρ2+1
∣
∣2 .
The result follows from comparing T (Ω)fσp,n with the recurrence relation (4.1). 
The action of K is given by
T (K)fσp,n = q
p+ε1+ε2fσp,n,
and together with Proposition 4.1 this leads to the following decomposition of T .
Quantum Analogs of Tensor Product Representations of su(1, 1) 15
Theorem 4.1. The Uq-representation Tρ1,ρ2,ε1,ε2,ε is unitarily equivalent to
2
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈Z
k+j >
1
2
pi+
k+j
⊕
⊕
j∈Z
k−j >
1
2
pi−
k−j
,
where σj = ε2 − ε1 + j − 12 , ε
′ = ε1 + ε2, k
+
j = j + ε1 + ε2, k
−
j = j − ε1 − ε2. The unitary
intertwiner is given by
fσp,n 7→
∫ − pi2 ln q
0
(
ePp 0
0 ePp
)
S
(
q2iρ; a, c|q2
)(
Θfσp,n
)(
q2iρ
)
dρ
+
∑
j∈Z
σj>0
(
Θfσp,n
)(
−q2σj
)√
w
(
−q2σj ; a, c; q2
)
eSp
+
∑
j∈N
k+p−j>
1
2
(
Θfσp,n
)(
q2k
+
p−j−1
)
√
w
(
q2k
+
p−j−1; a, c; q2
)
e+j
+
∑
j∈N
k−−p−j>
1
2
(
Θfσp,n
)(
q2k
−
−p−j−1
)
√
w
(
q2k
−
−p−j−1; a, c; q2
)
e−j
where a, c, z−, z+ are given by (4.2).
Remark 4.1. (i) In this classical limit q ↑ 1 the infinite sum of strange series vanish, and
we (formally) recover the decomposition of the tensor product of two principal unitary series
of su(1, 1), see [18, Theorem II] and [19, Theorem 4.6].
(ii) Theorem 4.1 shows that the vector-valued big q-Jacobi functions have an interpretation as
quantum analogs of Clebsch–Gordan coefficients for Uq. In this interpretation the vector-valued
big q-Jacobi function transform pair should be considered as a quantum analog of Neretin’s [16]
integral transform pair that has two 3F2-functions as kernels.
(iii) Note that the label ε does not appear in the decomposition of Theorem 4.1.
5 More quantum analogs of tensor product representations
Using the (vector-valued) big q-Jacobi functions we can decompose several other quantum
analogs of tensor product representations. We list a few decompositions here. The proofs
are similar to the proofs of Theorems 3.1 and 4.1.
• A quantum analog of the tensor product of two complementary series: for λ1, λ2 ∈ (−12 , 0),
ε1, ε2, ε ∈ R,
((
piCλ1,ε1+ε ⊗ pi
C
λ2,ε2−ε
)
⊕
(
piSλ1+ 12 ,ε1−ε
⊗ piSλ2+ 12 ,ε2+ε
))
∆
∼= 2
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈Z
k+j >
1
2
pi+
k+j
⊕
⊕
j∈Z
k−j >
1
2
pi−
k−j
⊕ piCλ1+λ2,ε′ ,
under the condition λ1 + λ2 < −12 , where σj = ε2 − ε1 + j −
1
2 , ε
′ = ε1 + ε2, k
+
j = j + ε1 + ε2,
k−j = j − ε1 − ε2. If λ1 + λ2 ≥ −
1
2 the complementary series pi
C
λ1+λ2,ε′
does not occur in the
decomposition.
16 W. Groenevelt
• A quantum analog of the tensor product of a principal unitary series and a complementary
series: for ρ ∈ (0,− pi2 ln q ), λ ∈ (−
1
2 , 0), ε1, ε2, ε ∈ R,
((
piPρ,ε1+ε ⊗ pi
C
λ,ε2−ε
)
⊕
(
piPρ′,ε1−ε ⊗ pi
S
λ+ 12 ,ε2+ε
))
∆
∼= 2
− pi2 ln q∫ ⊕
0
piPρ′′,ε′dρ
′′ ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈Z
k+j >
1
2
pi+
k+j
⊕
⊕
j∈Z
k−j >
1
2
pi−
k−j
,
where ρ′ = ρ− pi2 ln q , σj = ε2 − ε1 + j −
1
2 , ε
′ = ε1 + ε2, k
+
j = j + ε1 + ε2, k
−
j = j − ε1 − ε2.
• A quantum analog of the tensor product of a principal unitary series and a positive discrete
series: for ρ ∈ (0,− pi2 ln q ), k > 0, ε1, ε ∈ R,
((
piPρ,ε1 ⊗ pi
+
k
)
⊕
(
piPρ′,ε1−ε ⊗ pi
S
k− 12 ,k+ε
))
∆ ∼=
− pi2 ln q∫ ⊕
0
piPρ′′,ε′dρ
′′ ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈Z
kj>
1
2
pi+kj
where ρ′ = ρ− pi2 ln q , σj = k − ε1 + ε+ j +
1
2 , ε
′ = ε1 + k, kj = k + ε1 + j.
• A quantum analog of the tensor product of a complementary series and a positive discrete
series: for λ ∈ (−12 , 0), k > 0, ε1, ε ∈ R,
((
piCλ,ε1 ⊗ pi
+
k
)
⊕
(
piSλ+ 12 ,ε1−ε
⊗ piSk− 12 ,k+ε
))
∆
∼=
− pi2 ln q∫ ⊕
0
piPρ,ε′dρ ⊕
⊕
j∈Z
σj>0
piSσj ,ε′ ⊕
⊕
j∈Z
kj>
1
2
pi+kj ⊕ pi
C
−λ−k,ε′
under the condition λ+ k < −12 , where σj = k − ε1 + ε+ j +
1
2 , ε
′ = ε1 + k, kj = k + ε1 + j. If
λ+ k ≥ −12 the complementary series pi
C
−λ−k,ε′ does not occur in the decomposition.
Let us remark that for the first two cases we need the vector-valued big q-Jacobi functions
and corresponding orthogonality relations as described in Section 4, but a 6= b and c 6= d. In
this case one extra discrete mass point can appear, and this occurs if qd/as < 1, see [6]. In
the third case we need the big q-Jacobi functions and corresponding orthogonality relations as
described in Section 3, but with parameters a, b, c satisfying c = b, a > 0 and ab, ac, bc > 1.
In this case the only discrete mass points are of the form aqk < 1, with k ∈ N. Orthogonality
relations for the big q-Jacobi functions with these conditions on a, b, c can be obtained by very
minor adjustments of the proof given in [13].
References
[1] Andrews G.E., Askey R., Classical orthogonal polynomials, in Orthogonal Polynomials and Applications
(Bar-le-Duc, 1984), Lecture Notes in Math., Vol. 1171, Springer, Berlin, 1985, 36–62.
[2] Burban I.M., Klimyk A.U., Representations of the quantum algebra Uq(su1,1), J. Phys. A: Math. Gen. 26
(1993), 2139–2151.
[3] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Appli-
cations, Vol. 96, Cambridge University Press, Cambridge, 2004.
[4] Groenevelt W., Bilinear summation formulas from quantum algebra representations, Ramanujan J. 8 (2004),
383–416, math.QA/0201272.
Quantum Analogs of Tensor Product Representations of su(1, 1) 17
[5] Groenevelt W., Wilson function transforms related to Racah coefficients, Acta Appl. Math. 91 (2006),
133–191, math.CA/0501511.
[6] Groenevelt W., The vector-valued big q-Jacobi transform, Constr. Approx. 29 (2009), 85–127,
math.CA/0612643.
[7] Groenevelt W., Koelink E., Meixner functions and polynomials related to Lie algebra representations,
J. Phys. A: Math. Gen. 35 (2002), 65–85, math.CA/0109201.
[8] Groenevelt W., Koelink E., Kustermans J., The dual quantum group for the quantum group analog of the
normalizer of SU(1, 1) in SL(2,C), Int. Math. Res. Not. 2010 (2010), no. 7, 1167–1314, arXiv:0905.2830.
[9] Gupta D.P., Ismail M.E.H., Masson D.R., Contiguous relations, basic hypergeometric functions, and ortho-
gonal polynomials. III. Associated continuous dual q-Hahn polynomials, J. Comput. Appl. Math. 68 (1996),
115–149, math.CA/9411226.
[10] Kalnins E.G., Manocha H.L., Miller W., Models of q-algebra representations: tensor products of special
unitary and oscillator algebras, J. Math. Phys. 33 (1992), 2365–2383.
[11] Kalnins E.G., Miller W., A note on tensor products of q-algebra representations and orthogonal polynomials,
J. Comput. Appl. Math. 8 (1996), 197–207.
[12] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues,
Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
[13] Koelink E., Stokman J.V., The big q-Jacobi function transform, Constr. Approx. 19 (2003), 191–235,
math.CA/9904111.
[14] Korogodsky L.I., Vaksman L.L., Quantum G-spaces and Heisenberg algebra, in Quantum Groups
(Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 56–66.
[15] Masuda T., Mimachi K., Nakagami Y., Noumi M., Saburi Y., Ueno K., Unitary representations of the
quantum group SUq(1, 1). II. Matrix elements of unitary representations and the basic hypergeometric
functions, Lett. Math. Phys. 19 (1990), 195–204.
[16] Neretin Yu.A., Some continuous analogues of the expansion in Jacobi polynomials, and vector-valued or-
thogonal bases, Funct. Anal. Appl. 39 (2005), 106–119, math.CA/0309445.
[17] Ørsted B., Zhang G., L2-versions of the Howe correspondence. I, Math. Scand. 80 (1997), 125–160.
[18] Puka´nszky L., On the Kronecker products of irreducible representations of the 2×2 real unimodular group. I,
Trans. Amer. Math. Soc. 100 (1961), 116–152.
[19] Repka J., Tensor products of unitary representations of SL2(R), Amer. J. Math. 100 (1978), 747–774.
[20] Schmu¨dgen K., Unbounded operator algebras and representation theory, Operator Theory: Advances and
Applications, Vol. 37, Birkha¨user Verlag, Basel, 1990.