Quantum Analogs of Tensor Product Representations of su(1,1)

We study representations of $U_q(su(1,1))$ 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 $U_q(su(1,1))$ 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.


Introduction
The quantum algebra U q = U q (su(1, 1)) has five classes of irreducible * -representations: the positive and negative discrete series π ± , the principal unitary series π P , the complementary series π C , and the strange series π S . 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 U q that can be considered as quantum analogs of tensor products of irreducible * -representations of su(1, 1), but the representations that we consider are not tensor products of irreducible * -representations of U q . 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 M is generated as a von Neumann algebra by the standard generators of U q and two extra generators. In this sense U q can be considered as a subalgebra of M . An irreducible discrete series representation of M restricted to U q decomposes as the sum π + ⊕ π − ⊕ π S , 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 U q -representations: π + ⊗ π + , π − ⊗ π − , π + ⊗ π − , π + ⊗ π S and π S ⊗ π − , 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 = (π − ⊗ π + ) ⊕ (π S ⊗ π S ) and T ′ = (π − ⊗ π S ) ⊕ (π S ⊗ π + ), and determine their decompositions. In a similar way the principal unitary series and complementary series representations of M restricted to U q decompose as π P ⊕ π P and π C ⊕ π C , 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 tensor products mentioned above can be described in terms of terminating basic hypergeometric 3 ϕ 2 -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 polynomials. The Clebsch-Gordan coefficients for the representations we consider in this paper turn out to be non-terminating 3 ϕ 2 -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 U q (su(1, 1)) and its irreducible * -representations. In Section 3 we consider 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 U q , but has no classical analog. In Section 4 we consider the representation (π P ⊗ π P ) ⊕ (π P ⊗ π P ). 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 − xq k ), where the empty product is equal to 1. For x = 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 The basic hypergeometric series 3 ϕ 2 is defined by The Casimir element is a central element of U q . The algebra U q is a Hopf * -algebra with comultiplication ∆ defined on the generators by and ∆ is extended to U q as an algebra homomorphism. In particular, it follows from (2.2) and (2.3) that (2.4) The * -structure on U q is defined on the generators by Note that the Casimir element is self-adjoint in U q , i.e. Ω * = Ω. There are, besides the trivial representation, five classes of irreducible * -representations of U q , see [15,Proposition 4], [2,Section 6]. The representations are given in terms of unbounded operators on ℓ 2 (N) or ℓ 2 (Z). As common domain we take the finite linear combinations of the standard orthonormal basis vectors e n . The representations are unbounded * -representations in the sense of Schmüdgen [20,Definition 8.1.9]. Below we list the actions of the generators K, K −1 , E, F on basis vectors e n . 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 π be a U q -representation acting on ℓ 2 (Z) with orthonormal basis {e n } n∈Z , and suppose that π(K)e n = q n+ε e n and π(Ω)e n = ωe n for all n ∈ Z, where ε and ω are real numbers. The commutation relations (2.1) imply that π(E)e n = c n e n+1 and π(F )e n = d n e n−1 for certain numbers c n and d n . Furthermore, the relation E * = −F implies c n = −d n+1 , so that π(F E)e n = −|c n | 2 e n . On the other hand, from (2.2) and the actions of K and Ω it follows that So c n can be determined up to a phase factor. The five classes of irreducible * -representations of U q are the following: Positive discrete series. The positive discrete series π + k are labeled by k > 0. The representation space is ℓ 2 (N) with orthonormal basis {e n } n∈N . The action is given by π + k (K)e n = q k+n e n , π + k K −1 e n = q −(k+n) e n , Negative discrete series. The negative discrete series π − k are labeled by k > 0. The representation space is ℓ 2 (N) with orthonormal basis {e n } n∈N . The action is given by π − k (K)e n = q −(k+n) e n , π − k K −1 e n = q k+n e n , Principal unitary series. The principal unitary series representations π P ρ,ε are labeled by 0 ≤ ρ < − π 2 ln q and ε ∈ [0, 1), where (ρ, ε) = (0, 1 2 ). The representation space is ℓ 2 (Z) with orthonormal basis {e n } n∈Z . The action is given by splits into a direct sum of a positive and a negative discrete series representation: π P 0, 1 . The representation space splits into two invariant subspaces: {e n | n ≥ 0} ⊕ {e n | n < 0}.
3 A quantum analog of the tensor product of negative and positive discrete series representations Let k 1 , k 2 > 0 and ε ∈ R. We consider the U q -representation For the decomposition of T into irreducible representations we need the big q-Jacobi functions [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 for y ∈ C \ (−∞, −bc] and |γ| < a. By [3, (III.9)] Φ γ 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 3 ϕ 2 -function can be transformed by [3, (III.10)] and then we have Let t > 0. We define discrete sets Γ fin and Γ inf ('fin' and 'inf' stand for 'finite' and 'infinite') by Note that the set Γ fin is empty if a, b, c > 1. Now we introduce the measure ν( · ) = ν( · ; a, b, c; t|q) by where v is the weight function on the counter-clockwise oriented unit circle T given by and w(γ) = w(γ; a, b, c; t; q) = Res The weights w in the discrete mass points can be calculated explicitly, see [13,Section 8].
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 where the quadratic norm N y is given by We define The difference equation for the big q-Jacobi functions leads to a recurrence relation for the functions φ zq k , z ∈ {−1, t}, of the form 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 a k and b k 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 ; p k (x) = p k x; a, b, c|q −1 = a −k (ab, ac; q −1 ) k (q −1 , bc; q −1 ) k 3 ϕ 2 q k , ax, a/x ab, ac ; q −1 , q −1 .
So the function φ −q k is a multiple of the continuous dual q −1 -Hahn polynomials p k . 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.
Proposition 3.1. Let p ∈ Z, σ ∈ {−, +}, and let a, b, c, t σ be defined by Then the operator Θ : extends to a unitary operator. Moreover, Θ intertwines T (Ω) with the multiplication operator 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 The intertwining property follows from comparing this with the recurrence relation (3.3) for the function φ tσq 2n (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)f σ p,n = q −2p+2k 2 −2k 1 f σ p,n .
By comparing the spectral values of T (K) and T (Ω) with (2.5)-(2.9), we obtain the following decomposition of T .
Theorem 3.1. The U q -representation T k 1 ,k 2 ,ε is unitarily equivalent to: For n, p ∈ Z and σ ∈ {−, +} the unitary intertwiner is given by 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 ℓ 2 (N) ⊗2 ⊕ ℓ 2 (Z) ⊗2 into −π/(2 ln(q)) 0 ℓ 2 (Z) dρ ⊕ N ℓ 2 (Z). In case (ii) another ℓ 2 (Z) has to be added here, and in cases (iii) and (iv) a finite number of ℓ 2 (N)-spaces has to be added. (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 U q -representation π + k 2 ⊗ π − k 1 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 q 2 -Hahn polynomials, see [11,Section 2] or [4,Theorem 2.4].
(iv) The term π − k 1 ⊗ π + k 2 ∆ occurring in the definition of T would of course also be a quantum 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 irreducible 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 selfadjoint 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 Θf σ p,n as 3 ϕ 2 -functions, and the following contiguous relations for 3 ϕ 2 -functions, see [9, (2.3), (2.4)], Aq, Bq, Cq Dq, Eq ; q, DE ABCq , We do not work out the details.

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 k 1 , k 2 > 0 and ε ∈ 1 2 Z, and define the U q -representation T ′ by ∆ is a genuine tensor product representation of U q , 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).
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 ∈ 1 2 Z (instead of ε ∈ R), because our method for constructing an intertwiner only works if basis vectors of ℓ 2 (Z) can be labeled by 2ε + m for m ∈ Z, which forces ε to be in 1 2 Z. For the diagonalization of T ′ (Ω) we need the big q-Jacobi polynomials [1], [12, § 14.5]. They are defined by P m (x; a, b, c; q) = 3 ϕ 2 q −m , abq m+1 , x aq, cq ; q, q .
We assume that m ∈ N, x ∈ {aq k+1 | k ∈ N} ∪ {cq k+1 | k ∈ N}, 0 < a, b < q −1 and c < 0, then the big q-Jacobi polynomials satisfy the orthogonality relations where the (positive) functions u(·) = u(·; a, b, c; q) and v(·) = v(·; a, b, c; q) are given by Here the Jackson q-integral is defined by We define functions r x (m), related to the big q-Jacobi polynomials, by r x (m; a, b, c; q) = |x|u(x; a, b, c; q)v(m; a, b, c; q)P m (x; a, b, c; q).
By the orthogonality relations for the big q-Jacobi polynomials we have Furthermore, from the q-difference equation for P m it follows that the functions r x (m) satisfy the following q-difference equation in x: where Now we are ready to diagonalize T ′ (Ω). For p ∈ Z and n ∈ N we define f − p,n = e n ⊗ e n+p , f + p,n = e 2ε−n+p ⊗ e n , and we write Note that p∈Z A p is a dense subspace of ℓ 2 (N) ⊗ ℓ 2 (Z) ⊕ ℓ 2 (Z) ⊗ ℓ 2 (N).
Remark 3.2. Theorem 3.2 shows that the big q-Jacobi polynomials have a natural interpretation as Clebsch-Gordan coefficients for U q -representations. In this interpretation they do not have a classical analog, since the U q -representation T ′ vanishes in the classical limit.
The set is an orthogonal basis for H. We have where the squared norm N y is given by We define ψ y (x) = ψ y (x; a, c|q) = Ψ(y, x; a, c|q) N y (a, c; q) , then {ψ z − q k } k∈Z ∪ {ψ z + q k } k∈Z is an orthonormal basis for H. Furthermore, for z ∈ {z − , z + } these functions satisfy the recurrence relation where a k = a k (a, c; z; q) We are now ready for the decomposition of T . For this we need the vector-valued big q 2 -Jacobi functions and the corresponding Hilbert space H(a, c; z − , z + |q 2 ), 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 q 2 , e.g. s is given by q|c|/|a|.
Note in particular that the continuous spectrum occurs with multiplicity 2.
The result follows from comparing T (Ω)f σ p,n with the recurrence relation (4.1).
The action of K is given by T (K)f σ p,n = q p+ε 1 +ε 2 f σ p,n , and together with Proposition 4.1 this leads to the following decomposition of T .