Symmetry, Integrability and Geometry: Methods and Applications Harmonic Analysis on Quantum Complex Hyperbolic Spaces ⋆

In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order $q$-difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.


Introduction
Consider the group SU n,m and its homogeneous space H n,m = SU n,m /S(U n,m−1 × U 1 ). The latter is called a complex hyperbolic space. The Faraut paper [5] on such pseudo-Hermitian symmetric spaces has a great impact into the theory of semisimple symmetric spaces of rank 1. Also there are numerous papers of Molchanov, van Dijk and others (see [11,12,21] and references therein) on representation theory related to these symmetric spaces and harmonic analysis on them. In particular, there is the celebrated Penrose transform which enables to relate classical bounded symmetric domains and complex hyperbolic spaces.
In this paper we develop harmonic analysis on quantum complex hyperbolic spaces. Recall that the related polynomial algebras were introduced in the early paper by Faddeev, Reshetikhin, and Takhtadjan [14]. Unfortunately, there was no further inquiry.
Nearly 10 years ago L. Vaksman and his team started the theory of quantum bounded symmetric domains. Their approach enables to formulate and solve various problems on noncommutative complex and harmonic analysis in these domains, geometric realizations of representations of quantum groups [19]. Their paper [17] establishes the missed link between quantum bounded symmetric domains and introduces this project.
The initial notions of function theory on quantum complex hyperbolic spaces H n,m were introduced in [3]. Namely, a quantum analog D(H n,m ) q,k for the algebra of U k = U s(gl n × gl m )finite smooth functions on H n,m with compact support and an U q su n,m -invariant integral dν q on it were constructed. We recall basic notations from the quantum group theory and [3] in Sections 2, 3, 4. Section 5 is devoted to the subalgebra of U q k-invariant finite functions.
In Sections 6 and 7 we introduce a quantum analog for the Laplace-Beltrami operator on H n,m . Also we consider its radial part (0) , i.e., its restriction to the space L 2 dν latter operator naturally appears to be q-difference operator and is related to a three-diagonal Jacobi matrix. Section 8 is devoted to generalized eigenfunctions of (0) and lead us towards an expected yet remarkable appearance of Al-Salam-Chihara polynomials. A spectral theorem for (0) in Section 9 is obtained as a corollary to well known results on these polynomials (see [9]).

Preliminaries on quantum group theory
Everywhere in the sequel we suppose q ∈ (0, 1). All algebras are associative and unital.
The Hopf algebra U q sl N is given by its generators K i , K −1 i , E i , F i , i = 1, 2, . . . , N − 1, and the relations: The comultiplication ∆, the antipode S, and the counit ε are defined on the generators by see [7,Chapter 4]. We need also the Hopf algebra C[SL N ] q of matrix elements of finite dimensional weight U q sl N -modules. Recall that C[SL N ] q can be defined by the generators t ij , i, j = 1, . . . , N (the matrix elements of the vector representation in a weight basis) and the relations together with one more relation where det q t is a q-determinant of the matrix t = (t ij ) i,j=1,...,N : with l(s) = card{(i, j)|i < j & s(i) > s(j)}. The algebra C[SL N ] q is endowed with the standard structure of U op q sl N ⊗ U q sl N -module algebra (here 'op' reflects the fact that we should change the multiplication by the opposite one).
Let also U q su n,m , m + n = N , denotes the Hopf * -algebra (U q sl N , * ) given by with j = 1, . . . , N − 1 [14,15]. Recall the notion of an algebra of 'regular functions on the quantum principal homogeneous space' X constructed in [15]. Put Pol( X) q def = (C[SL N ] q , * ), where the involution * is defined by Here det q is the quantum determinant [4], and the matrix T ij is derived from the matrix t = (t kl ) by discarding its i's row and j's column. In [15] it is proved that Pol( X) q is a U q su n,m -module * -algebra.
Let m, n ∈ N, m ≥ 2, and N def = n + m. Recall that the classical complex hyperbolic space H n,m can be obtained by projectivization of the domain Now we pass from the classical case q = 1 to the quantum case 0 < q < 1. Let us consider the well known [14] q-analog of the polynomial algebras. Let Pol( H n,m ) q be the unital * -algebra with the generators t 1 , t 2 , . . . , t N and the commutation relations as follows: Obviously, Moreover, c is not a zero divisor in Pol( H n,m ) q . This allows one to embed the * -algebra Pol( H n,m ) q into its localization Pol( H n,m ) q,c with respect to the multiplicative system c N . The * -algebra Pol( H n,m ) q,c admits the following bigrading: deg t j = (1, 0), deg t * j = (0, 1), j = 1, 2, . . . , N.
Introduce the notation This * -algebra Pol(H n,m ) q will be called the algebra of regular functions on the quantum hyperbolic space.
We are going to endow the * -algebra Pol(H n,m ) q with a structure of U q su n,m -module algebra [4]. For this purpose, we embed it into the U q su n,m -module * -algebra Pol( X) q .
By a q-analog of the Laplace expansion of det q t along the first row [8, Section 9.2] and (1), one can obtain from det q t = 1 that Thus the map J : t j → t 1j , j = 1, 2, . . . , N , admits a unique extension to a homomorphism of * -algebras J : Pol( H n,m ) q,c → Pol( X) q . Its image will be denoted by Pol( H n,m ) q . It is easy to verify that the * -algebra Pol(H n,m ) q is embedded this way into Pol( H n,m ) q and its image is just the subalgebra in Pol( H n,m ) q generated by t 1j t * 1k , j, k = 1, 2, . . . , N (recall that c goes to det q t = 1). In what follows we will identify Pol(H n,m ) q with its image under the map J.
Consider the subalgebra U q s(gl 1 × gl N −1 ) generated by K ±1 i , i = 1, . . . , N − 1, E j , F j , j = 2, . . . , N − 1. By obvious reasons, where L is the left action of U op q sl N in Pol( X) q . Let I ϕ , ϕ ∈ R/2πZ, be the * -automorphism of the * -algebra Pol( H n,m ) q defined on the generators {t j } j=1,...,N by (2) We use the notation t j instead of t 1j for the generators of Pol( H n,m ) q . Then one more description of Pol(H n,m ) q is as follows: At the end of this section we list explicit formulas for the action of U q su n,m on Pol( H n,m ): 4 Algebras of generalized and f inite functions on the quantum H n,m Let us construct a faithful * -representation T of Pol(H n,m ) q in a pre-Hilbert space H (our method is well known; see, for example, [15]). The space H is a linear span of its orthonormal basis {e(i 1 , i 2 , . . . , The * -representation T is a restriction to Pol(H n,m ) q of the * -representation of Pol( H n,m ) q defined by for n < j < N , and, finally, Define the elements {x j } j=1,...,N as follows: Obviously, The vectors e(i 1 , . . . , i N −1 ) are joint eigenvectors of the operators T (x j ), j = 1, 2, . . . , N : The joint spectrum of the pairwise commuting operators T (x j ), j = 1, 2, . . . , N , is The next proposition was proved in [3].
Let us now introduce the notion of generalized functions on the quantum complex hyperbolic space H n,m . Evidently, using the commutation relations, one can decompose every polynomial f ∈ Pol( H n,m ) q as follows: Due to (3), the latter can be reduced to the decomposition where f IJ (x 2 , . . . , x N ) are polynomials. One can equip Pol( H n,m ) q with the weakest topology such that the functionals are continuous. The completion of Pol( H n,m ) q w.r.t. this topology will be considered as the space of generalized functions on the quantum H n,m and denoted by D( H n,m ) ′ q . Naturally, one can extend T to a representation of D( H n,m ) ′ q by continuity. Now (6) allows one to identify D( H n,m ) ′ q with the space of formal series The topology on this space of formal series is the topology of pointwise convergence of the functions f IJ . Denote by f 0 the following function (Recall that spec x n+1 = q −2Z + .) Thus f 0 is a q-analog of the characteristic function of the submanifold Introduce now a * -algebra Fun( H n,m ) q ⊂ D( H n,m ) ′ q generated by Pol( H n,m ) q and f 0 . Easy computations from (4) show that f 0 satisfies the following relations: The relation I ϕ f 0 = f 0 allows one to extend the * -automorphism I ϕ (2) of the algebra Pol( H n,m ) q to the * -automorphism of Fun( H n,m ) q . Let Let D(H n,m ) q,k be the two-sided ideal of Fun(H n,m ) q generated by f 0 . We call this ideal the algebra of finite functions on the quantum hyperbolic space. It is a quantum analog for the algebra of U k = U s(gl n × gl m )-finite smooth functions on H n,m with compact support. Remark 1. Let us explain the adjective 'finite'. If f is a finite function, T (f ) is an operator with only a finite number of nonzero entries. However, we do not consider all possible finite functions (and, therefore, all operators with finite number of nonzero entries) but only U q k-finite ones, cf. [5].
It was proved in [3] that This computation, together with (7), allows one to consider the element Thus a multiple application of (8) leads to the following claim: D(H n,m ) q,k contains all finite functions of x n+1 (i.e., such functions f that f (q −n ) = 0 for all but finitely many n ∈ N).
The U q su n,m -module algebra structure is established on D(H n,m ) q,k by the following: Now we present an explicit formula for a positive invariant integral on the space of finite functions D(H n,m ) q,k and thereby establish its existence.
Let ν q : D(H n,m ) q,k → C be a linear functional defined by where Q : H → H is the linear operator given on the basis elements e(i 1 , . . . , i N −1 ) by ). The functional ν q is well defined, positive, and U q su n,m -invariant.
One has to normalize this integral in a some way. In [3] we put Hn,m f 0 dν q = 1, so the constant in the previous theorem equals In this section we restrict ourselves to subalgebras of U q k-invariant elements. It is well known that H n,m is a pseudo-Hermitian symmetric space of rank 1 [12]. The following proposition is a natural quantum analog for this fact.
n ] ⊗ U q sl n ⊗ U q sl m , the proof of this proposition follows from the next statement on U q sl n × U q sl m -isotypic components in D(H n,m ) q,k . Proposition 3. U q sl n × U q sl m -isotypic components of D(H n,m ) q,k correspond to the modules Remark 3. The sign ⊠ reflects the fact that the multipliers are modules of different algebras U q sl n and U q sl m .
Proof . Let us describe U q sl n ×U q sl m -highest weight vectors in D(H n,m ) q,k . One can decompose every finite function f ∈ D(H n,m ) q,k in the following way (just by applying the commutation relations, cf. (5)). Let L (n) (λ) be the finite dimensional U q sl nmodule with highest weight λ and ̟ j are fundamental weights of sl n . By standard arguments, l.s. t l 1 1 t l 2 2 · · · t ln n | l 1 + · · · + l n = a = L (n) (a̟ 1 ), l.s. t * l 1 1 t * l 2 2 · · · t * ln n | l 1 + · · · + l n = a = L (n) (a̟ n−1 ), l.s. t l n+1 n+1 t l n+2 n+2 · · · t l N N | l n+1 + · · · + l N = a = L (m) (a̟ 1 ), l.s. t * l n+1 n+1 t * l n+2 n+2 · · · t * l N N | l n+1 + · · · + l N = a = L (m) (a̟ m−1 ). Thus we have an epimorphism n+1 t * jn n · · · t * j 1 1 . Recall that, as in the classical case, see [13], in the category of U q sl n -modules. By explicit calculations one can show that the U q sl n -highest weight vector in L (n) ((a − i)̟ 1 + (b − i)̟ n−1 ) ⊂ L (n) (a̟ 1 ) ⊗ L (n) (b̟ n−1 ) has the form Now a routine application of the commutation relations (similar to the ones in Remark 2) allows one to reduce every highest weight vector to a linear span of the vectors t a 1 t * b N ϕ(x n+1 )t c n+1 t * d n , where ϕ(x n+1 ) is a finite function. Now let us obtain an explicit form of the restriction of ν q to the space D(H n,m ) Uqk q,k of U q kinvariant elements of D(H n,m ) q,k .

Proposition 4.
For any function f (x n+1 ) ∈ D(H n,m ) q,k one has Hn,m and const 2 = 1

Proof . By explicit calculations,
Hn,m Let us verify that for every j, k ∈ Z + Denote the l.h.s. of the previous equation by Ψ(j, k). One can verify the recurrence relation for the q-Pascal triangle Ψ(j, k) = q −2k Ψ(j − 1, k) + Ψ(j, k − 1) and the boundary values explicitly. Thus Ψ(j, k) are the corresponding entries of the q-Pascal triangle. Now we can complete our calculations and obtain that

Now one can complete D(H n,m )
Uqk q,k with respect to the norm ||f The resulting Hilbert space will be denoted by L 2 dν In this section we introduce holomorphic and antiholomorphic covariant first order differential calculi over Pol( H n,m ) q .
First of all, recall a general definition of a covariant first order differential calculus following [8].
Let F be a unital algebra. A first order differential calculus over F is a pair (M, d) where M is an F -bimodule and d : F → M is a linear map such that 2. M is a linear span of the vectors f 1 · df 2 · f 3 , where f 1 , f 2 , f 3 ∈ F . Now suppose that A is a Hopf algebra and F is an A-module algebra. A first order differential calculus (M, d) over F is called covariant if the following conditions hold: 2. d is a morphism of A-modules.

M is an
and ω is an antiautomorphism of U q sl N defined on the generators by Let Ω (1,0) ( H n,m ) q ⊂ Pol( X) q be the Pol( H n,m ) q -submodule generated by ∂t i , i = 1, . . . , N .
Lemma 1. Ω (1,0) ( H n,m ) q is a U q sl N -covariant first order differential calculus over Pol( H n,m ) q .
Proof . One has to verify the Leibniz rule which immediately follows from the formulas for the comultiplication in U q sl N . Since left and right action of U q sl N in C[SL N ] q commute, ∂ is a morphism of U q sl N -modules, so the calculus is covariant. Now we define∂ : Pol( H n,m ) q → Pol( X) q by the rule∂f = (∂f * ) * . Let Ω (0,1) ( H n,m ) q ⊂ Pol( X) q be the Pol( H n,m ) q -submodule generated by∂t i , i = 1, . . . , N . Lemma 2. Ω (0,1) ( H n,m ) q is a U q sl N -covariant first order differential calculus over Pol( H n,m ) q .
This lemma can be proved similarly to the previous one.
Remark 4. The introduced first order differential calculi can be obtained in another way. One should start from the canonical Wess-Zumino calculi on a quantum complex space introduced in [14] and then turn to a localization of the corresponding algebras of functions. Now we introduce a Hermitian pairing Ω (0,1) ( H n,m ) q × Ω (0,1) ( H n,m ) q → Pol(H n,m ) q . Let P : Pol( X) q → Pol(H n,m ) q be the projection parallel to a sum of other U q s(gl 1 × gl N −1 )isotypic components of L. Now we define By obvious commutativity of the left and right actions of U q sl N we have Proof . Since U q s(gl 1 × gl N −1 ) = C[K 1 , K −1 1 ] ⊗ U q sl N −1 , one can decompose the projection P as follows: P = P 0 P 1 , where P 1 is a projection to the subspace of U q sl N −1 -invariant elements (w.r.t. the L-action) and P 0 is a projection to the subspace of elements that are preserved by the L(K 1 )-action.
Let u 1 , . . . , u k be the standard basis in the U q sl k -module L(̟ 1 ), and v 1 , . . . , v k the dual basis in the U q sl k -module L(̟ k−1 ), where ̟ 1 and ̟ k−1 are the fundamental weights. A standard argument on finite dimensional U q sl k -modules allows one to prove that is a projection to the subspace of U q sl k -invariant elements parallel to other isotypic components. By obvious reasons, for j = 1, . . . , N − 1 the maps admit extensions to morphisms of U q sl N −1 -modules (w.r.t. the L-action). By definition (1), t * ij = (−q) j−i det q T ij for i ≤ m and j ≥ n + 1. Thus one can apply the map (10) to compute for l ≥ n + 1 Since C[SL N ] q is a Hopf algebra, one has N j=1 t lj S(t j k) = δ lk and S(t jk ) = (−q) j−k det q T kj . Thus The other cases can be verified in a similar way.
Proof . An easy application of the previous lemma allows one to compute that Thus for j, k ≥ n + 1 one has By easy computations, q −2(n+1) x n+1 = N j=n+1 q −2j t * 1j t 1j , which enables to prove the claim.
Let us fix notation for q-difference operators: Lemma 4.
Proof . By explicit calculations in C[SL N ] q we obtain for i ≥ n + 1 Let us verify the second identity. One has∂x n+1 = N j=n+1 t 1j t * 2j , so and the claim follows from the previous proposition.
Using the above formal arguments, we can extend∂ to D(H n,m ) q,k .

The Laplace-Beltrami operator and its radial part
In this section we introduce a U q sl N -invariant operator in the space of finite functions D(H n,m ) q,k . This operator will be considered as a quantum analog for the invariant Laplace-Beltrami operator on the complex hyperbolic space H n,m . Also we compute an explicit formula for the restriction (0) of to the space D(H n,m ) Uqk q,k , the so called radial part of . The restriction appears to be a q-difference operator in variable x = x n+1 . Now we define by the formula Proof . The self-adjointness follows from the definition. The U q sl N -invariance of the form (·, ·) and the linear functional Hn,m ·dν q implies the U q sl N -invariance of .
Let us find the restriction (0) of the operator on the subspace based on the equation for every f with finite support. Using the q-analog of the partial integration formulas one can prove that operators B − and −q 2 B + are formally dual. Exactly, for every functions u(x), v(x) with finite support on q 2Z holds where const(q, n, N ) = q 2n 1 − q 2 1 − q 2(N −1) , and ρ(x) is defined in (9). Lemma 4 allows us to extend the Hermitian U q sl N -invariant pairing to first order differential forms with coefficients in D(H n,m ) q,k .
Proof . Consider the operator It differs from (0) by a compact operator, so it is sufficient to prove that is bounded. The boundness of the latter operator can be proved by the direct evaluation.
By the previous proposition, one can extend (0) from D(H n,m ) Uqk q,k to a bounded self-adjoint operator in L 2 dν (0) q . The latter extension will also be denoted by (0) .

Generalized eigenfunctions of (0)
and Al-Salam-Chihara polynomials In this section we obtain the initial results on the bounded self-adjoint operator (0) , namely we obtain its formal eigenfunctions and eigenvalues explicitly. Note that explicit computations of the asymptotics of these eigenfunctions, as in the classical case, allow us to consider a quantum analog of the Harish-Chandra c-function (see Appendix A). By the direct computation we obtain the following lemma.
in D(H n,m ) ′ q is a generalized eigenfunction for (0) : Theorem 3. The bounded self-adjoint linear operator (0) is unitary equivalent to the operator of multiplication by independent variable in the Hilbert space L 2 (dσ). The unitary equivalence is given by the operator Proof . Let us consider the finite functions on These functions form an orthogonal system in D(H n,m ) Uqk q,k and the completion of its linear span is L 2 dν (0) q . By standard arguments [1], the bounded self-adjoint linear operator (0) is unitary equivalent to the multiplication operator f (λ) → λf (λ) in the Hilbert space L 2 (dµ(λ)) of square integrable functions with respect to a certain measure dµ(λ) with compact support in R. Let us find explicitly the corresponding measure and the operator of unitary equivalence U . One can fix the unitary equivalence operator by the condition U f 0 = 1.
On the other hand, where z = 1 2 q 2l+N −1 + q −(2l+N −1) . Hence for every function f (x) on q −2Z + with finite support one has Now the claim of the theorem follows from the orthogonality relations for the Al-Salam-Chihara polynomials (11).