Equivariant Fredholm Modules for the Full

Quantum Flag Manifold of SU_{q}(3)

Christian Voigt^{1} and Robert Yuncken^{2}

Received: January 9, 2015

Communicated by Joachim Cuntz

Abstract. We introduce C^{∗}-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct SL_{q}(3, ℂ)-equivariant Fredholm modules for the full quantum flag manifold X_{q} = SU_{q}(3)/T of SU_{q}(3), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold X_{q} satisfies Poincaré duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to SU_{q}(3).

2010 Mathematics Subject Classification: Primary 20G42; Secondary 46L80, 19K35. Keywords and Phrases: Noncommutative geometry, quantum groups, quantum flag manifolds, Poincaré duality, Bernstein-Gelfand-Gelfand complex, Kasparov theory, Baum-Connes Conjecture.

Abstract

We introduce C^{∗}-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct SL_{q}(3, ℂ)-equivariant Fredholm modules for the full quantum flag manifold X_{q} = SU_{q}(3)/T of SU_{q}(3), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold X_{q} satisfies Poincaré duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to SU_{q}(3).

1C. Voigt was supported by the Engineering and Physical Sciences Research Council Grant EP/L013916/1 and the Polish National Science Centre grant no. 2012/06/M/ST1/00169.

2R. Yuncken was partially supported by the project SINGSTAR of the Agence Nationale de la Recherche, ANR-14-CE25-0012-01.

Documenta Mathematica 20 (2015) 433–490

1 Introduction

In noncommutative differential geometry [Con94], the notion of a smooth manifold is extended beyond its classical scope by adopting a spectral point of view. This is centred around the idea of constructing Dirac-type operators associated with possibly noncommutative algebras, capturing the underlying Riemannian structure of geometric objects for which ordinary differential geometry breaks down. The key concept in this theory, introduced by Connes, is the notion of a spectral triple [Con96]. Quantum groups provide provide a large class of examples of noncommutative spaces, and they have been studied extensively within the framework of noncommutative differential geometry. Among the many contributions in this direction let us only mention a few. Chakraborty and Pal [CP03] defined an equivariant spectral triple on SU_{q}(2), which was studied in detail by Connes [Con04]. Later Dabrowski, Landi, Sitarz, van Suijlekom and Várilly [DLS^{+}05], [vSDL^{+}05] defined and studied a deformation of the classical Dirac operator on SU(2), thus obtaining a different spectral triple on SU_{q}(2). The techniques used in these papers rely on explicit estimates involving Clebsch-Gordan coefficients. In a different direction, Neshveyev and Tuset exhibited a general mechanism for transporting the Dirac operator on an arbitrary compact simple Lie group to its quantum deformation, based on Drinfeld twists and properties of the Drinfeld associator [NT10]. The resulting spectral triples inherit various desirable properties from their classical counterparts, although unfortunately they are difficult to study directly since this requires a certain amount of control of the twisting procedure. This article is concerned with the quantized full flag manifolds associated to the q-deformations of compact semisimple Lie groups, and in particular the flag manifold of SU_{q}(3), the simplest example in rank greater than one. In the rank-one case, that is for SU_{q}(2), the flag manifold SU_{q}(2)/T is known as the standard Podleś sphere, and Dirac operators on it have been defined and studied by several authors [Owc01, DS03, SW04]. A version of the local index formula for the Podleś sphere is exhibited in [NT05, Wag09, RS], although slight modifications must be made to Connes’ original formalism. The higher rank situation has proven to be considerably more difficult. Krähmer [Krä04] gave an algebraic construction of Dirac operators on quantized irreducible flag manifolds in higher rank. These retain a certain rank-one character in their geometry. In particular, the construction in [Krä04] does not cover the case of full flag manifolds. On the other hand, the Dirac operator defined by Neshveyev and Tuset can be used to write down spectral triples for arbitrary full quantum flag manifolds. However, the most direct way to do so, which was indicated already in [NT10], does not suffice to describe the equivariant K-homology group of the quantum flag manifold using Poincaré duality. More precisely, one only obtains certain multiples of the class of the Dirac operator in this way. In this paper, we describe a construction of a Dirac-type class in equivariant K-homology for the full flag manifold X_{q} = SU_{q}(3)/T of SU_{q}(3) as a bounded Fredholm module. This does not give the full “noncommutative Riemannian” structure on X_{q} that a Connes-type spectral triple would give. In fact, a key philosophical point behind our construction is that the natural geometric structure on quantized flag manifolds in higher rank is not Riemannian but parabolic, in the sense of [ČS00]. Correspondingly, the construction of our Dirac-type class is based not upon the Dirac or Dolbeault operator but upon the Bernstein-Gelfand-Gelfand (BGG) complex, see [BGG75, BE89, ČSS01]. The quantum version of the BGG complex for SL_{q}(n, ℂ), in its algebraic form, first appeared in [Ros91]; see also [HK07a, HK07b]. It has not been much studied from an analytical point of view so far. In fact, developing a complete unbounded noncommutative version of parabolic geometries seems to be difficult. For instance, the BGG complex is neither elliptic nor order 1, although it does exhibit a kind of subellipticity. In the present work, we convert the BGG complex into a bounded K-homology cycle. Such a construction was achieved for a classical flag manifold in [Yun10, Yun11a]. A major goal of the present work is to demonstrate that the necessary analysis can also be carried out for a quantized flag manifold. In particular, our K-homology class is equivariant not only with respect to SU_{q}(3), but with respect to the complex quantum group SL_{q}(3, ℂ) = D(SU_{q}(3)), the Drinfeld double of SU_{q}(3). Drinfeld doubles play an important role in the definition of equivariant Poincaré duality [NV10] and the proof of the Baum-Connes conjecture for the dual of SU_{q}(2), see [Voi11]. It is worth pointing out that the verification of SL_{q}(3, ℂ)-equivariance of our cycle is somewhat simpler than in the classical situation. We also remark that in the construction of our K-homology class we use some properties of principal series representations of SL_{q}(3, ℂ) which will be discussed in a separate paper [VY]. Our main result can be formulated as follows.

Theorem 1.1. The BGG complex for the full flag manifold X_{q} = SU_{q}(3)/T of SU_{q}(3) can be normalized to give a bounded equivariant K-homology cycle in the Kasparov group KK^{SL_{q}(3,ℂ)}(C(X_{q}), ℂ). The equivariant index of this element with respect to SU_{q}(3) is the class of the trivial representation in KK^{SU_{q}(3)}(ℂ, ℂ) = R(SU_{q}(3)).

We refer to Theorem 10.6 for the precise statement of this result. The main idea behind our construction can be sketched as follows. Firstly, corresponding to each of the two simple roots of SU_{q}(3) there is a fibration of the quantized flag manifold whose fibres are Podleś spheres. These fibrations carry families of Dirac-type operators analogous to the operators constructed by Dabrowski-Sitarz. As is common in Kasparov’s KK-theory, we replace these longitudinal operators by their bounded transforms. We then use a variant of the Kasparov product, inspired by the BGG complex, to assemble them into a single SL_{q}(3, ℂ)-equivariant K-homology cycle for X_{q}. At present, there is only one ingredient which prevents us from carrying out our construction for the full flag manifold of SU_{q}(n) for any n ≥ 2, namely the operator partition of unity in Lemma 10.5. We develop all the harmonic analysis in the generality of SU_{q}(n).

Using Theorem 1.1 we derive two consequences regarding equivariant KKtheory. Firstly, we conclude that the quantum flag manifold X_{q} satisfies equivariant Poincaré duality in KK-theory in the sense of [NV10].

Corollary 1.2. The flag manifold X_{q} is SU_{q}(3)-equivariantly Poincaré dual to itself. That is, there is a natural isomorphism

KK_{∗}^{D(SU_{q}(3))}(C(X_{q}) ⊠ A, B) ≅ KK_{∗}^{D(SU_{q}(3))}(A, C(X_{q}) ⊠ B)

for all D(SU_{q}(3))-C^{∗}-algebras A and B, where ⊠ denotes the braided tensor product with respect to SU_{q}(3).

For the definition and properties of braided tensor products we refer to [NV10]. We note that it is crucial here that the class obtained in Theorem 1.1 is equivariant with respect to D(SU_{q}(3)) and not just SU_{q}(3). Let us also remark that we have chosen to write D(SU_{q}(3)) instead of SL_{q}(3, ℂ) in Corollary 1.2 in order to make notation consistent with [NV10]. Secondly, we discuss an analogue of the Baum-Connes conjecture for the discrete quantum group dual to SU_{q}(3). In [MN06], Meyer and Nest have developed an approach to the Baum-Connes conjecture [BCH94] which allows one to construct assembly maps in rather general circumstances, and which applies in particular to duals of q-deformations. As already mentioned above, the simplest case of SU_{q}(2) was studied in [Voi11], and here we show how to go one step further as follows.

Corollary 1.3. The Baum-Connes conjecture with trivial coefficients ℂ holds for the discrete quantum group dual to SU_{q}(3).

This result is significantly weaker than the analogous statement for the dual of SU_{q}(2) in [Voi11]. However, let us point out that one cannot hope to carry over the arguments used in [Voi11] to the higher rank situation. Indeed, according to work of Arano [Ara], the Drinfeld double of SU_{q}(3) has property (T). This forbids the existence of continuous homotopies along the complementary series representations to the trivial representation in the unitary dual. Such homotopies are at the heart of the arguments in [Voi11]. In other words, the problem is similar to well-known obstacles to proving the Baum-Connes conjecture with coefficients for the classical groups SL(n, ℂ) in higher rank.

Let us now explain how the paper is organized. In Section 2 we collect some preliminaries on quantum groups and fix our notation. Sections 3 and 4 contain the definition and basic properties of certain ideals of C^{∗}-algebras associated to the canonical fibrations of a quantum flag manifold. These C^{∗}-ideals are defined in terms of the harmonic analysis of the block diagonal quantum subgroups of SU_{q}(n), and are the basis of all the analysis that follows.

In Section 5 we formulate the main technical results about these ideals. These results are parallel to classical facts from the calculus of longitudinally elliptic pseudodifferential operators. The proofs are deferred to subsequent sections, which may be skipped on a first reading. Specifically, Section 6 collects some facts about Gelfand-Tsetlin bases, and in particular the effect of reversing the order of roots used in their definition. Section 7 introduces the notion of essentially orthotypical quantum subgroups, in analogy with the considerations in [Yun10]. In Section 8 the analytic properties of longitudinal pseudodifferentialtype operators are established. Section 9 contains some definitions and facts related to complex quantum groups and their representations, and it is checked that our constructions are compatible with the natural action of SL_{q}(n, ℂ). In section 10 we describe the analytical quantum BGG complex for the flag manifold of SU_{q}(3), and we prove our main theorem. The final section 11 contains the corollaries stated above. That is, we show that X_{q} is equivariantly Poincaré dual to itself, and we verify the Baum-Connes conjecture with trivial coefficients for the dual of SU_{q}(3).

Let us conclude with some remarks on notation. The dual of a vector space V is denoted V^{∗}. We write 𝕃(H, H′) for the space of bounded operators between Hilbert spaces H and H′, and 𝕂(H, H′) denotes the space of compact operators. When H = H′ we abbreviate these as 𝕃(H) and 𝕂(H). Depending on the context, the symbol ⊗ denotes either an algebraic tensor product, the tensor product of Hilbert spaces or the minimal tensor product of C^{∗}-algebras. All Hilbert spaces in this paper are separable.

It is a pleasure to thank Uli Krähmer for inspiring discussions on quantized flag manifolds.

2 Preliminaries

In this section we discuss some preliminaries on quantum groups in general and q-deformations in particular. For more details and background we refer to the literature [CP95], [KS97], [Maj05].

2.1 Some notation

Let K = SU(n) or U(n) with n ≥ 2. We write T for the standard maximal torus of K, that is, the diagonal subgroup, and 𝔱 for its Lie algebra. We write 𝔰𝔩_{n} for 𝔰𝔩(n, ℂ) and 𝔤𝔩_{n} for 𝔤𝔩(n, ℂ). In either case we denote by 𝔥 = 𝔱_{ℂ} the Cartan subalgebra. We write P for the set of weights of K, viewed as a lattice in 𝔥^{∗}. If V is a K-representation and μ ∈ P, the subspace of vectors of weight μ in V will be denoted V_{μ}. It will be convenient to identify the weight lattice of U(n) with ℤ^{n}, where an element μ = (μ_{1}, …, μ_{n}) ∈ ℤ^{n} corresponds to the weight μ ∈ 𝔥^{∗} given by

μ(diag(t_{1}, …, t_{n})) = μ_{1}t_{1} + ⋯ + μ_{n}t_{n}.

The corresponding character of T will be denoted by e^{μ} ∈ C(T). We equip 𝔥^{∗} with the bilinear form which extends the standard pairing on P ≅ ℤ^{n}:

( (μ_{1}, …, μ_{n}), (μ′_{1}, …, μ′_{n}) ) = ∑_{i} μ_{i}μ′_{i}.

For SU(n), the weight lattice identifies with the quotient ℤ^{n}/ℤ(1, …, 1), and the bilinear form on 𝔥^{∗} is obtained from that above by identifying 𝔥^{∗} with the orthogonal complement of ℂ(1, …, 1) in ℂ^{n}. We write Δ for the set of roots of SU(n) or U(n); they are the same in both cases. We fix the set of simple roots Σ = {α_{1}, …, α_{n−1}} where α_{i} : diag(t_{1}, …, t_{n}) ↦ t_{i} − t_{i+1}.

2.2 Quantized universal enveloping algebras

We shall use the quantized universal enveloping algebras which are denoted Ŭ_{q}(𝔤𝔩_{n}) and Ŭ_{q}(𝔰𝔩_{n}) in [KS97] (pages 212 and 164, respectively), since these are the versions used in the literature on Gelfand-Tsetlin theory. We briefly recall their definitions. Fix q ∈ (0, 1). For any a ∈ ℂ we write [a]_{q} = \frac{q^{a}−q^{−a}}{q−q^{−1}} , and for a ∈ ℕ,

[a]_{q}! = ∏_{k=1}^{a}[k]_{q},  \begin{matrix}a \\ m \end{matrix} _{q} = \frac{[a]_{q}!}{[a − m]_{q}! [m]_{q}!}.

Often, we shall drop the subscript q in the notation. The Hopf ∗-algebra Ŭ_{q}(𝔤𝔩_{n})is generated by elements E_{i}, F_{i} (i = 1, …, n − 1) and G_{j}, G_{j}^{−1} (j = 1, …, n) with the relations

G_{j}G_{k} = G_{k}G_{j},  G_{j}^{−1}G_{j} = 1 = G_{j}G_{j}^{−1}

G_{j}E_{i}G_{j}^{−1} = \{ \begin{matrix}q^{\frac{1}{2}} E_{i}, & j = i, \\ q^{− \frac{1}{2}} E_{i}, & j = i + 1, \\ E_{i}, & otherwise, \end{matrix}  G_{j}F_{i}G_{j}^{−1} = \{ \begin{matrix}q^{− \frac{1}{2}} F_{i}, & j = i, \\ q^{\frac{1}{2}} F_{i}, & j = i + 1, \\ F_{i}, & otherwise, \end{matrix}

[E_{i}, F_{j}] = δ_{ij} \frac{G_{i}^{2}G_{i+1}^{−2} − G_{i}^{−2}G_{i+1}^{2}}{q − q^{−1}} ,

E_{i}^{2}E_{i±1} − [2]_{q}E_{i}E_{i±1}E_{i} + E_{i±1}E_{i}^{2} = 0 = F_{i}^{2}F_{i±1} − [2]_{q}F_{i}F_{i±1}F_{i} + F_{i±1}F_{i}^{2}

[E_{i}, E_{j}] = 0 = [F_{i}, F_{j}], |i − j| ≥ 2.

The formulas for the coproduct Δ̂ : Ŭ_{q}(𝔤𝔩_{n}) → Ŭ_{q}(𝔤𝔩_{n}) ⊗ Ŭ_{q}(𝔤𝔩_{n}) are

Δ̂(E_{i}) = E_{i} ⊗ G_{i}G_{i+1}^{−1} + G_{i}^{−1}G_{i+1} ⊗ E_{i},
Δ̂(F_{i}) = F_{i} ⊗ G_{i}G_{i+1}^{−1} + G_{i}^{−1}G_{i+1} ⊗ F_{i},
Δ̂(G_{i}) = G_{i} ⊗ G_{i},

the counit ϵ̂: Ŭ_{q}(𝔤𝔩_{n}) → ℂ is given by

ε̂(E_{i}) = 0,  ε̂(F_{i}) = 0,  ε̂(G_{i}) = 1,

and the antipode is determined by

Ŝ(E_{i}) = −qE_{i},  Ŝ(F_{i}) = −q^{−1}F_{i},  Ŝ(G_{i}) = G_{i}^{−1}.

Finally, the ∗-structure is given by

E_{i}^{∗} = F_{i}, G_{i}^{∗} = G_{i}.

Throughout, we will use the Sweedler notation Δ̂(X) = X_{(1)} ⊗ X_{(2)} for the coproduct. We note that with this definition of Ŭ_{q}(𝔤𝔩_{n}), weight spaces are defined by saying that G_{i} acts on vectors of weight μ = (μ_{1}, …, μ_{n}) by multiplication by q^{\frac{1}{2} μ_{i}}. The Hopf ∗-algebra Ŭ_{q}(𝔰𝔩_{n}) is the Hopf ∗-subalgebra of Ŭ_{q}(𝔤𝔩_{n}) generated by the elements E_{i}, F_{i}, K_{i} = G_{i}G_{i+1}^{−1} and K_{i}^{−1}, for i = 1, …, n − 1. The element K_{i} acts on vectors of weight μ ∈ P by multiplication by q^{\frac{1}{2} (α_{i},μ)}.

2.3 Quantized algebras of functions

Fix K_{q} = SU_{q}(n) for n ≥ 2. The quantized algebra of functions O(K_{q}) is the space of matrix coefficients of finite-dimensional type 1 representations of Ŭ_{q}(𝔰𝔩_{n}); see [KS97] for more details. If σ is a type 1 representation of Ŭ_{q}(𝔰𝔩_{n}) and ξ ∈ V^{σ}, ξ^{∗} ∈ V^{σ∗}, we denote the associated matrix coefficient by the bra-ket notation

⟨ξ^{∗}| · |ξ⟩ : X ↦ (ξ^{∗}, σ(X)ξ), for X ∈ Ŭ_{q}(𝔤).

We shall use the ∗-Hopf algebra structure on O(K_{q}) which makes the evaluation map Ŭ_{q}(𝔰𝔩_{n}) × O(SU_{q}) → ℂ into a skew-pairing of ∗-Hopf algebras, or equivalently, a Hopf pairing of Ŭ_{q}(𝔰𝔩_{n})^{cop} and O(K_{q}). Thus, for all X, Y ∈ Ŭ_{q}(𝔰𝔩_{n}), f, g ∈ O(K_{q}),

(XY, f) = (X, f_{(1)})(Y, f_{(2)}),
(X, fg) = (X_{(1)}, g)(X_{(2)}, f)
(Ŝ(X), f) = (X, S^{−1}(f)),

where we use Sweedler notation Δ(f) = f_{(1)} ⊗ f_{(2)} for f ∈ O(K_{q}). In terms of matrix coefficients the multiplication is given by

⟨ξ^{∗}| · |ξ⟩ ⟨η^{∗}| · |η⟩ = ⟨η^{∗} ⊗ ξ^{∗}| · |η ⊗ ξ⟩.  (2.1)

where ξ ∈ V^{σ}, ξ^{∗} ∈ V^{σ∗}, η ∈ V^{τ}, η^{∗} ∈ V^{τ∗} for type 1 representations σ, τ. The comultiplication of O(K_{q}) defines left and right corepresentations of O(K_{q}) on itself. They will play very different roles in what follows: the left regular corepresentation will be used to define representations of K_{q}, while the right regular representation will be used to define K_{q}-invariant differential operators and carry out their harmonic analysis. The right regular corepresentation of O(K_{q}) gives rise to a left action of Ŭ_{q}(𝔰𝔩_{n}) according to the formula

X ⇀ f = f_{(1)}(X, f_{(2)}),  for X ∈ Ŭ_{q}(𝔤𝔩_{n}), f ∈ O(U_{q}(n)).  (2.2)

We shall usually write this simply as Xf. The Hilbert space L^{2}(K_{q}) is the completion of O(K_{q}) with respect to the inner product

⟨f, g⟩ = ϕ(f^{∗}g),

where ϕ is the Haar state of O(K_{q}). The left and right multiplication action of f ∈ O(K_{q}) on L^{2}(K_{q}) will be denoted by M_{l}(f) and M_{r}(f), respectively. The left multiplication action defines a ∗- homomorphism O(K_{q}) → 𝕃(L^{2}(K_{q})). By definition, the C^{∗}-completion C(K_{q}) of O(K_{q}) is the norm closure of the image of O(K_{q}) under this representation. In this way one obtains the compact quantum group structure of K_{q}. The algebra O(U_{q}(n)) is defined analogously, as matrix coefficients of type 1 representations of Ŭ_{q}(𝔤𝔩_{n}). All the above constructions carry over to U_{q}(n).

2.4 Representations and duality

Let K_{q} = SU_{q}(n). By definition, a unitary representation of K_{q} on a Hilbert space H is a unitary element U ∈ M(C(K_{q}) ⊗ 𝕂(H)) such that (Δ ⊗ Id)(U) = U_{13}U_{23}. Here we are using leg numbering notation. We shall often designate unitary K_{q}-representations simply by the Hilbert spaces underlying them. If H, H′ are unitary representations of K_{q} we write Hom_{K_{q}}(H, H′) for the space of intertwiners, that is, for the set of all bounded linear maps T : H → H′ satisfying (Id ⊗T)U = U′(Id ⊗T). A unitary representation H of K_{q} is irreducible if and only if Hom_{K_{q}}(H, H) = ℂ. All irreducible unitary representations of K_{q} are finite dimensional, and we write Irr(K_{q}) for the set of their equivalence classes. In the context of harmonic analysis, elements of Irr(K_{q}) will be referred to as K_{q}-types. We shall usually blur the distinction between a specific irreducible representation and its class in Irr(K_{q}). Unless otherwise stated, the Hilbert space underlying a K_{q}-representation σ ∈ Irr(K_{q}) will be denoted V^{σ}. We use 𝟙_{K_{q}} to denote the trivial representation of K_{q}. For σ ∈ Irr(K_{q}), we denote by σ^{c} the (unitary) conjugate representation. If a K_{q}-representation π contains σ as an irreducible subrepresentation, we write σ ≤ π. We define C_{c}(K̂_{q}) as the algebraic direct sum

C_{c}(K̂_{q}) = ⨁_{σ∈Irr(K_{q})}𝕃(V^{σ}).

Its enveloping C^{∗}-algebra is denoted C_{0}(K̂_{q}), this identifies with the C^{∗}-algebra of functions on the dual discrete quantum group K̂_{q}. We will also work with the algebraic direct product

C(K̂_{q}) = ∏_{σ∈Irr(K_{q})}𝕃(V^{σ}),

which can be identified with the algebraic dual space O(K_{q})^{∗} of O(K_{q}). In particular, the quantized universal enveloping algebra Ŭ_{q}(𝔰𝔩_{n}) is naturally a ∗- subalgebra of C(K̂_{q}), and we will routinely use the same notation for elements of Ŭ_{q}(𝔰𝔩_{n}) and their images in C(K̂_{q}). In our context, the main reason to consider the algebra C(K̂_{q}) is that it contains some elements outside Ŭ_{q}(𝔰𝔩_{n}) which we shall need. In particular, the universal enveloping algebra U(𝔥) of the Cartan subalgebra 𝔥 of 𝔰𝔩_{n} embeds into C(K_{q}) if we identify X ∈ 𝔥 with the operator which acts as μ(X) on the weight space (V^{σ})_{μ} for each σ ∈ Irr(K_{q}), μ ∈ P.

2.5 Quantum subgroups

Let K_{q} = SU_{q}(n). Given a set I ⊆ Σ of simple roots, we let 𝔥^{I⊥} denote the subspace of 𝔥 annihilated by the α_{i} ∈ I, and let 𝔥^{I} be its orthocomplement with respect to the invariant bilinear form. We let 𝔤^{I} denote the following block-diagonal Lie subalgebra of 𝔰𝔩_{n}:

𝔤^{I} = 𝔥 ⊕ ⨁_{α∈Δ∩ℤI}𝔤_{α}.

This subalgebra admits the decomposition 𝔤^{I} = 𝔰^{I} ⊕ 𝔥^{I⊥} where 𝔰^{I} = 𝔥^{I} ⊕ ⨁ _{α∈Δ∩ℤI} 𝔤_{α} is semisimple and 𝔥^{I⊥} is central. The subalgebra 𝔨^{I} = 𝔤^{I} ∩ 𝔰𝔲_{n} is the Lie algebra of a block-diagonal subgroup K^{I} ⊆ SU(n). The analogous families of closed quantum subgroups of SU_{q}(n) are defined as follows. Here, we use the notation ⟨x_{j}⟩ to denote the σ(C(K̂_{q}), O(K_{q}))-closed subalgebra of C(K̂_{q}) generated by a collection of elements x_{j} ∈ C(K̂_{q}). For each I ⊆ Σ, we define

\begin{matrix}C(K̂_{q}^{I}) = ⟨X ∈ U(𝔥), E_{i}, F_{i} (i ∈ I)⟩, & C(Ŝ_{q}^{I}) = ⟨X ∈ U(𝔥^{I}), E_{i}, F_{i} (i ∈ I)⟩, \\ C(T̂^{I}) = ⟨X ∈ U(𝔥^{I})⟩, & C(T̂^{I⊥}) = ⟨X ∈ U(𝔥^{I⊥})⟩. \end{matrix}

We then define O(K_{q}^{I}), O(S_{q}^{I}), O(T^{I}) and O(T^{I⊥}) to be the images of O(K_{q}) under the induced surjection of C(K̂_{q})^{∗} onto C(K̂_{q}^{I})^{∗}, C(Ŝ_{q}^{I})^{∗}, C(T̂^{I})^{∗} and C(T̂^{I⊥})^{∗}, respectively. They are Hopf *-algebras under the induced operations. In particular, O(K_{q}^{∅}) is isomorphic to O(T). We write π_{T} for the projection homomorphism O(K_{q}) ↠ O(T), and for its extension to the C^{∗}-algebras. At the other extreme, we have O(K_{q}^{Σ}) = O(K_{q}).

The quantum subgroups corresponding to the singleton subsets I = {α_{i}} with i = 1, …, n − 1 will play a particularly important role. In this case, we will write K_{q}^{i}, S_{q}^{i}, T^{i}, T^{i⊥} for the above quantum groups. Note that S_{q}^{i} ≅ SU_{q}(2). We will also write Ŭ_{q}(𝔰_{q}^{i}) for the Hopf subalgebra of Ŭ_{q}(𝔰𝔩_{n}) generated by E_{i}, F_{i}, K_{i} and K_{i}^{−1}.

2.6 The quantized flag manifold

Here, we summarize the basic definitions and properties of quantum flag manifolds. For more details see [CP95], [SD99], [HK04], [Sto03]. The full flag manifold of K_{q} = SU_{q}(n) is the quantum space X_{q} = K_{q}/T, defined via its algebra of functions as follows. The algebra O(K_{q}) is a right O(T)-comodule algebra by restriction of the canonical right coaction of O(K_{q}) along the projection homomorphism π_{T} : O(K_{q}) → O(T). By definition, the algebra O(X_{q}) is the ∗-subalgebra of O(T)-coinvariant elements, that is,

O(X_{q}) = {f ∈ O(K_{q}) | (Id ⊗π_{T})Δ(f) = f ⊗ 1}
= {f ∈ O(K_{q}) | K_{i}f = f for all i = 1, …, n − 1}.

More generally, for any μ = (m_{1}, …, m_{n}) ∈ P we define the section space of the induced line bundle E_{μ} over X_{q} by

O(E_{μ}) = {f ∈ O(K_{q}) | (Id ⊗π_{T})Δ(f) = f ⊗ e^{μ}}
= \{f ∈ O(K_{q}) | K_{i}f = q^{\frac{1}{2} (m_{i}−m_{i+1})}f for all i = 1, …, n − 1\}.

In other words, O(E_{μ}) is the μ-weight space of the right regular action of T. Similarly, L^{2}(E_{μ}) and C(E_{μ}) are the right μ-weight spaces of L^{2}(U_{q}(n)) and C(U_{q}(n)), respectively. They are the closures of O(E_{μ}) in L^{2}(U_{q}(n)) and C(U_{q}(n)), respectively. We will abbreviate direct sums of the form O(E_{μ}) ⊕ O(E_{ν}) as O(E_{μ} ⊕ E_{ν}), and use analogous notation for their completions. Multiplication in O(K_{q}) restricts to a map O(E_{μ}) ⊗ O(E_{ν}) → O(E_{μ+ν}) for any μ, ν ∈ P. In particular, each O(E_{μ}) is a bimodule over O(E_{0}) = O(X_{q}). These modules are projective as either left or right O(X_{q})-modules since O(K_{q}/T) ⊂ O(K_{q}) is a faithfully flat Hopf-Galois extension [MS99]. Later on we will need an analogue of trivializing partitions of unity for the line bundles E_{μ}. These are described in the following lemma, which is an immediate consequence of Hopf-Galois theory, see [Sch04].

Lemma 2.1. For any μ ∈ P, there exists a finite collection of sections f_{1}, …, f_{k} ∈ O(E_{μ}) and g_{1}, …, g_{k} ∈ O(E_{−μ}) such that ∑ _{j=1}^{k} f_{j}g_{j} = 1 ∈ O(X_{q}).

We will be interested in operators arising from the action of Ŭ_{q}(𝔰𝔩_{n}) on the above line bundles. Let X ∈ Ŭ_{q}(𝔤) be of weight ν = (k_{1}, …, k_{n}) for the left adjoint action, i.e., K_{i}XK_{i}^{−1} = q^{\frac{1}{2} (k_{i}−k_{i+1})}X for all i. Then the right regular action X : O(K_{q}) → O(K_{q}) given by Xf = X ⇀ f restricts to a map X : O(E_{μ}) → O(E_{μ+ν}) for every μ ∈ P. In this way, X defines an unbounded operator from L^{2}(E_{μ}) to L^{2}(E_{μ+ν}) with dense domain O(E_{μ}), such that X^{∗}X is essentially self-adjoint. It should be thought of as a K_{q}-invariant differential operator.

3 Isotypical decompositions and associated C^{∗}-categories

In this section and the next, we introduce the fundamental analytical structures which will be used throughout the remainder of the paper. The idea is to describe the behaviour of certain linear operators with respect to a decomposition into isotypical subspaces.

3.1 Isotypical decompositions

Let K_{q} be a compact quantum group. Any unitary representation π of K_{q} on a Hilbert space H can be decomposed into a direct sum of its isotypical components,

H = ⨁_{σ∈Irr(K_{q})} H_{σ},

where H_{σ} ≅ Hom_{K_{q}}(V^{σ}, H) ⊗ V^{σ}. We denote by p_{σ} the orthogonal projection onto H_{σ}. More generally, for any set S ⊆ Irr(K_{q}), we write p_{S} = ∑ _{σ∈S} p_{σ}, so that p_{S} is the orthogonal projection onto H_{S} = ⨁ _{σ∈S} H_{σ}. An important observation for what follows is that certain sufficiently nice subspaces of H, such as weight spaces, still admit a K_{q}-isotypical decomposition even though they may not be K_{q}-subrepresentations. This is the point of the following definition.

Definition 3.1. A K_{q}-harmonic space is a Hilbert space of the form H = P H, where H is a unitary K_{q}-representation space and P : H → H is an orthogonal projection which commutes with every isotypical projection p_{σ} for σ ∈ Irr(K_{q}). In this case each p_{σ} restricts to a projection on H, and we call H_{σ} = p_{σ}P H the σ-isotypical subspace of H.

For us, the key example of a K_{q}-harmonic space will be the L^{2}-section space of a homogeneous line bundle over the quantized flag manifold. The corresponding Hilbert space is not a subrepresentation of the right regular representation of SU_{q}(n), but it is a SU_{q}(n)-harmonic space with respect to the right regular representation, see Example 4.2 below.

3.2 Harmonically finite and harmonically proper operators

Let H, H′ be K_{q}-harmonic spaces and let T ∈ 𝕃(H, H′) be a bounded linear operator between them. We denote by T_{στ} = p_{σ}T p_{τ} for σ, τ ∈ Irr(K_{q}) the matrix components of T with respect to the K_{q}-isotypical decompositions.

Definition 3.2. Let H, H′ be K_{q}-harmonic spaces. With the notation above, we say that a) T is K_{q}-harmonically finite if T_{στ} = 0 for all but finitely many pairs σ, τ ∈

Irr(K_{q});

b) T is K_{q}-harmonically proper if the matrix of T is row- and column-finite, that is, if for each fixed σ we have T_{στ} = 0 for all but finitely many τ ∈ Irr(K_{q}) and T_{τσ} = 0 for all but finitely many τ ∈ Irr(K_{q}).

Definition 3.3. Let H, H′ be K_{q}-harmonic spaces.

a) We define 𝕂_{K_{q}}(H, H′) to be the norm-closure of the set of K_{q}-harmonically finite operators in 𝕃(H, H′)

b) We define 𝔸_{K_{q}}(H, H′) to be the norm-closure of the set of K_{q}-harmonically proper operators in 𝕃(H, H′).

If H = H′ we will simply write 𝕂_{K_{q}}(H) and 𝔸_{K_{q}}(H), respectively.

These definitions can be thought of as defining the Hom-sets of C^{∗}-categories 𝕂_{K_{q}} and 𝔸_{K_{q}} whose objects are K_{q}-harmonic spaces. This observation will serve us as a notational convenience, since it allows us to write statements such as T ∈ 𝕂_{K_{q}} if the domain and target spaces of the operator T are understood.

Remark 3.4. The above definitions can be reinterpreted in the language of coarse geometry. A K_{q}-harmonic space H is a geometric | Irr(K_{q})|-Hilbert space, which is merely to say that it admits a representation of C_{0}(Irr(K_{q})) = Z(C_{0}(K̂_{q})). The algebra 𝔸_{K_{q}}(H) is basically the Roe algebra with respect to the indiscrete coarse structure on the discrete space Irr(K_{q}), see [Roe03] for more information. The fact that Roe algebras over dual spaces enter into our K-homology construction is no surprise: see the discussions in [Roe97], [Luu05], [Yun11b].

3.3 Alternative characterizations

We write S ⊂⊂ Irr(K_{q}) if S is a finite set of K_{q}-types. Recall that we write p_{S} = ∑ _{σ∈S} p_{σ}. It is convenient to regard (p_{S})_{S⊂⊂Irr(K_{q})} as a net of projections, where the indexing set is ordered by inclusion of subsets. The following two lemmas are exact analogues of Lemmas 3.4 and 3.5 of [Yun10], with essentially the same proofs.

Lemma 3.5. Let T ∈ 𝕃(H, H′), where H, H′ are K_{q}-harmonic spaces. The following conditions are equivalent:

a) T ∈ 𝕂_{K_{q}}(H, H′),

b) lim_{S⊂⊂Irr(K_{q})}(1 − p_{S})T = 0 = lim_{S⊂⊂Irr(K_{q})} T (1 − p_{S}) in the norm topology.

c) lim_{S⊂⊂Irr(K_{q})} p_{S}T p_{S} = T in the norm topology.

Lemma 3.6. Let A ∈ 𝕃(H, H′), where H, H′ are K_{q}-harmonic spaces. The following conditions are equivalent: a) A ∈ 𝔸_{K_{q}}(H, H′),

b) For any finite set S′ ⊂⊂ Irr(K_{q}),

lim_{S⊂⊂Irr(K_{q})}(1 − p_{S})Ap_{S′} = 0 = lim_{S⊂⊂Irr(K_{q})} p_{S′}A(1 − p_{S})

in the norm topology,

c) For any finite set S ⊂⊂ Irr(K_{q}), Ap_{S} ∈ 𝕂_{K_{q}}(H, H′) and p_{S}A ∈ 𝕂_{K_{q}}(H, H′).

d) A is a two-sided multiplier of 𝕂_{K_{q}}, that is, for any K_{q}-harmonic space H′′, T A ∈ 𝕂_{K_{q}}(H, H′′) for all T ∈ 𝕂_{K_{q}}(H′, H′′) and AT ∈ 𝕂_{K_{q}}(H′′, H′) for all T ∈ 𝕂_{K_{q}}(H′′, H).

3.4 Basic properties

If the K_{q}-isotypical components of a K_{q}-harmonic space H are all finite dimensional, we shall say that H has finite K_{q}-multiplicities. In this case, the family (p_{σ})_{σ∈Irr(K_{q})} is a complete system of mutually orthogonal finite-rank projections on H, so the following result follows from Lemma 3.5.

Lemma 3.7. If either H or H′ has finite K_{q}-multiplicities then 𝕂_{K_{q}}(H, H′) = 𝕂(H, H′), the set of compact operators from H to H′.

If H is a K_{q}-representation and K′_{q} is a closed quantum subgroup of K_{q}, then H is a K′_{q}-representation by restriction. We thus have projections p_{S′} on H for every S′ ⊂ Irr(K′_{q}). The following result is a straightforward consequence of considering successive isotypical decompositions.

Lemma 3.8. Let K′_{q} ⊆ K_{q} be a closed quantum subgroup. For any S ⊆ Irr(K_{q}), S′ ⊆ Irr(K′_{q}), the projections p_{S} and p_{S′} commute. In particular, if H is a unitary K_{q}-representation space and τ ∈ Irr(K′_{q}) then p_{τ} H is a K_{q}-harmonic space.

Lemma 3.9. Let K′_{q} ⊆ K_{q} be a closed quantum subgroup. Suppose that H_{1} and H_{2} are simultaneously K_{q}-harmonic and K′_{q}-harmonic spaces, in the sense that H_{i} = P_{i}H_{i} for i = 1, 2 where H_{i} is a unitary K_{q}-representation, and P_{i} : H_{i} → H_{i} is an orthogonal projection which commutes with both the K_{q}- and the K′_{q}-isotypical projections. Then 𝕂_{K_{q}}(H_{1}, H_{2}) ⊆ 𝕂_{K_{q}}′(H_{1}, H_{2}).

Proof. Let T ∈ 𝕃(H_{1}, H_{2}) be K_{q}-harmonically finite, so p_{S}T p_{S} = T for some finite set S ⊂⊂ Irr(K_{q}). Only finitely many K′_{q}-types occur in each σ ∈ S, so T is also K′_{q}-harmonically finite. The claim follows.

3.5 Commuting generating quantum subgroups

Definition 3.10. Let K_{1,q}, K_{2,q} be closed quantum subgroups of a compact quantum group K_{q}, defined by projections π_{i} : O(K_{q}) ↠ O(K_{i,q}) for i = 1, 2. We shall say that K_{1,q} and K_{2,q} are commuting and generating if (π_{1} ⊗ π_{2})Δ = (π_{1} ⊗π_{2})Δ^{cop} holds, and this map is an injection of O(K) into O(K_{1})⊗O(K_{2}).

The subgroups K_{i,q} give rise to injections π_{i}^{∗} : C(K̂_{1,q}) → C(K̂_{q}). One can check that K_{1,q} and K_{2,q} are commuting and generating if and only if π_{1}^{∗}(C(K̂_{1,q})) and π_{2}^{∗}(C(K̂_{2,q})) commute and generate a subalgebra of C(K̂_{q}) which is separating for O(K_{q}). The latter condition is often easier to check. Consider the direct product K_{1,q} ×K_{2,q} defined by the tensor product O(K_{1,q} × K_{2,q}) = O(K_{1,q}) ⊗ O(K_{2,q}). Note that Irr(K_{1,q} × K_{2,q}) = Irr(K_{1,q}) × Irr(K_{2,q}), where a pair (σ_{1}, σ_{2}) ∈ Irr(K_{1,q}) × Irr(K_{2,q}) is identified with the obvious corepresentation σ_{1} × σ_{2} of O(K_{1,q}) ⊗ O(K_{2,q}) on 𝕂(V^{σ_{1}} ⊗ V^{σ_{2}}). Thanks to the embedding (π_{1} ⊗π_{2})Δ : O(K_{q}) → O(K_{1,q})⊗O(K_{2,q}), any corepresentation σ of K_{q} defines a corepresentation σ̃ of K_{1,q} × K_{2,q}. If σ is irreducible, an application of Schur’s Lemma shows that σ̃ = σ_{1} × σ_{2} for some σ_{i} ∈ Irr(K_{i,q}), and moreover σ is uniquely determined by (σ_{1}, σ_{2}). We therefore have an injection Irr(K_{q}) ↪ Irr(K_{1,q}) × Irr(K_{2,q}).

Lemma 3.11. Let K_{1,q}, K_{2,q} be commuting and generating closed quantum subgroups of a compact quantum group K_{q}. Then for any K_{q}-representations H, H′ we have

𝕂_{K_{q}}(H, H′) = 𝕂_{K_{1,q}}(H, H′) ∩ 𝕂_{K_{2,q}}(H, H′),  (3.1)

and

𝔸_{K_{q}}(H, H′) ⊇ 𝔸_{K_{1,q}}(H, H′) ∩ 𝔸_{K_{2,q}}(H, H′).  (3.2)

Proof. For σ ∈ Irr(K_{q}), let σ̃ = σ_{1} × σ_{2} be the associated representation of K_{1,q} × K_{2,q}. The isotypical projection for σ is given by p_{σ} = p_{σ_{1}}p_{σ_{2}}. It follows that an operator T : H → H′ is K_{q}-harmonically finite if and only if it is both K_{1,q}- and K_{2,q}-harmonically finite. This proves Equation (3.1). Equation (3.2) follows from the characterization of 𝔸_{K_{q}} as multipliers of 𝕂_{K_{q}}, as in Lemma 3.6.

3.6 Harmonic properties of tensor products

If H_{1} = P_{1}H_{1} and H_{2} = P_{2}H_{2} are K_{q}-harmonic spaces, following the notation of Definition 3.1, then the tensor product H_{1} ⊗ H_{2} = (P_{1} ⊗ P_{2})H_{1} ⊗ H_{2} is naturally a K_{q}-harmonic space with respect to the tensor product representation of K_{q} on H_{1} ⊗ H_{2}.

Lemma 3.12. Let K_{q} be a compact quantum group. Then 𝕂_{K_{q}} ⊗ 𝕂_{K_{q}} ⊆ 𝕂_{K_{q}}, in the sense that for any K_{q}-harmonic spaces H_{1}, H_{2}, H′_{1}, H′_{2} and any T_{1} ∈ 𝕂_{K_{q}}(H_{1}, H′_{1}) and T_{2} ∈ 𝕂_{K_{q}}(H_{2}, H′_{2}) we have T_{1}⊗T_{2} ∈ 𝕂_{K_{q}}(H_{1}⊗H_{2}, H′_{1}⊗H′_{2}). Similarly, 𝔸_{K_{q}} ⊗ 𝕂_{K_{q}} ⊆ 𝔸_{K_{q}} and 𝕂_{K_{q}} ⊗ 𝔸_{K_{q}} ⊆ 𝔸_{K_{q}}.

Proof. Suppose T_{1} and T_{2} are K_{q}-harmonically finite, so that for i = 1, 2 there are finite sets S_{1}, S_{2} ⊂ Irr(K_{q}) such that p_{S_{i}}T_{i}p_{S_{i}} = T_{i}. If S denotes the set of all irreducible K_{q}-types which occur in some σ_{1} ⊗ σ_{2} with σ_{i} ∈ S_{i}, then T_{1} ⊗ T_{2} = p_{S}(T_{1} ⊗ T_{2})p_{S}. From this we deduce 𝕂_{K_{q}} ⊗ 𝕂_{K_{q}} ⊆ 𝕂_{K_{q}}. Now suppose A is K_{q}-harmonically proper and T is K_{q}-harmonically finite. Fix S ⊂ Irr(K_{q}) a finite set of K_{q}-types such that p_{S}T p_{S} = T. Take σ ∈ Irr(K_{q}) arbitrary. Then

(A ⊗ T)p_{σ} = (A ⊗ T)(1 ⊗ p_{S})p_{σ}.

Let τ ∈ S. For any τ′ ∈ Irr(K_{q}), we have σ ≤ τ′ ⊗ τ if and only if τ′ ≤ σ ⊗ τ^{c}. This implies that there are only finitely many τ′ ∈ Irr(K_{q}) for which (p_{τ′} ⊗ p_{S})p_{σ} ≠ 0. Letting S′ ⊂⊂ Irr(K_{q}) denote the set of such τ′, we have

(A ⊗ T)p_{σ} = (A ⊗ T)(p_{S′} ⊗ p_{S})p_{σ},

From the K_{q}-harmonic properness of A and T we can deduce that (A ⊗ T)p_{σ} ∈ 𝕂_{K_{q}}. A similar argument shows p_{σ}(A ⊗ T) ∈ 𝕂_{K_{q}} for all σ ∈ Irr(K_{q}), whence Lemma 3.6 shows that A ⊗ T ∈ 𝔸_{K_{q}}. Clearly, a similar argument works for T ⊗ A.

Recall that 𝟙_{K_{q}} denotes the trivial representation of K_{q}. Later we shall make much use of the following trick, which allows us to replace an arbitrary isotypical projection by the trivial one.

Lemma 3.13. Let σ ∈ Irr(K_{q}) and let V be any finite dimensional representation of K_{q} which contains σ as a subrepresentation. There exist intertwiners ι : ℂ → V^{c} ⊗V and ῑ: V^{c} ⊗V → ℂ such that on any unitary K_{q}-representation H, the isotypical projection p_{σ} factorizes as

\xymatrix{p_{σ} : H \ar[r]^{Id_{H} ⊗ι} & H ⊗ V^{c} ⊗ V \ar[r]^{p_{𝟙_{Kq}} ⊗Id_{V}} & H ⊗ V^{c} ⊗ V \ar[r]^{Id_{H} ⊗ῑ} & H.}

Proof. If V = V^{σ}, this follows from standard facts about the contragredient representation. If V^{σ} is merely a subrepresentation of V, then we can use the inclusion map V^{σ} → V and the projection V → V^{σ}, as well as the corresponding maps for the contragredient representation, to reduce to the previous situation.

4 The lattice of C^{∗}-ideals

We now specialize to the quantum group K_{q} = SU_{q}(n), although we note that the constructions and results of this section translate naturally to more general q-deformed compact semisimple Lie groups. We recall the family of quantum subgroups K_{q}^{I} ⊆ K_{q} defined in Section 2.5.

Definition 4.1. A fully K_{q}-harmonic space is a Hilbert space of the form H = P H, where H is a unitary representation of K_{q} and P is an orthogonal projection which commutes with all isotypical projections of each K_{q}^{I}, I ⊆ Σ.

Thus, a fully K_{q}-harmonic space is simultaneously a K_{q}^{I}-harmonic space for every I ⊆ Σ. Between fully K_{q}-harmonic spaces H and H′, we have the spaces 𝕂_{K_{q}^{I}} (H, H′) and 𝔸_{K_{q}^{I}} (H, H′) for every I ⊆ Σ. To avoid unwieldy subscripts, we shall write 𝕂_{I} and 𝔸_{I} for 𝕂_{K_{q}^{I}} and 𝔸_{K_{q}^{I}} in the sequel. When I = {i} is a singleton, we shall write 𝕂_{i} and 𝔸_{i}. The only examples of fully K_{q}-harmonic spaces we shall actually need are the following.

Example 4.2. a) Any unitary representation of K_{q} is a fully K_{q}-harmonic space.

b) By Lemma 3.8, any weight space of a K_{q}-representation is a fully K_{q}- harmonic space.

c) In particular, the L^{2}-section space L^{2}(E_{μ}) of a homogeneous line bundle over the quantized flag manifold of K_{q} is a fully K_{q}-harmonic space. Note that the harmonic structure here comes from the right regular corepresentation.

Lemma 3.9 shows that we have a whole lattice of C^{∗}-categories (𝕂_{I})_{I⊆Σ} for the fully K_{q}-harmonic spaces. Note that the ordering is reversed: 𝕂_{I_{1}} ⊆ 𝕂_{I_{2}} if I_{1} ⊇ I_{2}. We point out, however, that this is typically not a lattice of C^{∗}-ideals, that is, given sets I_{1} ⊃ I_{2} of simple roots and a fully K_{q}-harmonic space H we typically do not have 𝕂_{I_{1}}(H) ◁ 𝕂_{I_{2}}(H). To obtain a lattice of ideals, we must reduce 𝕂_{I} slightly, by restricting the class of operators we are working with.

Definition 4.3. For fully K_{q}-harmonic spaces H, H′, we define

𝔸(H, H′) = ⋂_{I⊆Σ} 𝔸_{I}(H, H′).

We also define

𝕁_{I}(H, H′) = 𝕂_{I}(H, H′) ∩ 𝔸(H, H′)

for each I ⊆ Σ.

In other words, 𝔸 is the simultaneous multiplier category of all the C^{∗}- categories 𝕂_{I}. Again, we view the spaces defined in definition 4.3 as the morphism sets of C^{∗}-categories 𝔸 and 𝕁_{I} whose objects are fully K_{q}-harmonic spaces. It is immediate from Lemma 3.9 that the 𝕁_{I} form a lattice of ideals, as we record in the following lemma.

Lemma 4.4. If I_{1} ⊇ I_{2} then 𝕁_{I_{1}} ◁ 𝕁_{I_{2}}.

In particular, we have 𝕁_{I_{1}∪I_{2}} ⊆ 𝕁_{I_{1}} ∩ 𝕁_{I_{2}} for any I_{1}, I_{2} ⊆ Σ. In fact, it will be shown later that 𝕁_{I_{1}∪I_{2}} = 𝕁_{I_{1}} ∩ 𝕁_{I_{2}}, see Theorem 5.1 and its proof in Section 7.3.

Lemma 4.5. Let H, H′ be fully K_{q}-harmonic spaces. Then 𝕁_{Σ}(H, H′) = 𝕂_{Σ}(H, H′). In particular, if either H or H′ has finite K_{q}-multiplicities then 𝕁_{Σ}(H, H′) = 𝕂(H, H′), the set of compact operators from H to H′.

Proof. By Lemma 3.9, we have 𝕂_{Σ} ⊆ ⋂ _{I⊆Σ} 𝕂_{I} ⊆ ⋂ _{I⊆Σ} 𝔸_{I} = 𝔸. This proves the first statement. The second follows using Lemma 3.7.

The spaces of interest to us will have finite K_{q}-multiplicities, however they will usually not have finite K_{q}^{I}-multiplicities for I ≠ Σ.

5 Longitudinal pseudodifferential operators: statement of results

In this section, we give statements of the necessary results concerning the lattice of C^{∗}-categories (𝕁_{I})_{I⊆Σ} and the “pseudodifferential” operators ph(E_{i}) and ph(F_{i}). All of these results will be discussed at the generality of SU_{q}(n). The proofs of the theorems below will be deferred until Sections 7.3 and 8. The reader willing to accept their veracity may safely skip forward to Section 9 after this section.

We begin with general results on the lattice of ideals (𝕁_{I})_{I⊆Σ}.

Theorem 5.1. Let K_{q} = SU_{q}(n) for n ≥ 2 and let H, H′ be fully K_{q}-harmonic spaces.

a) 𝔸(H, H′) = ⋂ _{i∈Σ} 𝔸_{i}(H, H′).

b) For any I ⊆ Σ and any σ ∈ Irr(K_{q}^{I}), p_{σ} ∈ 𝕁_{I}(H).

c) For any I, I′ ⊆ Σ, 𝕁_{I}(H, H′) ∩ 𝕁_{I′}(H, H′) = 𝕁_{I∪I′}(H, H′).

d) If either H or H′ has finite K_{q}-multiplicities then 𝕁_{Σ}(H, H′) = 𝕂(H, H′), the compact operators from H to H′, and hence ⋂ _{i∈Σ} 𝕁_{i}(H, H′) = 𝕂(H, H′).

Next we consider longitudinal pseudodifferential operators along various fibrations of the quantum flag manifold. The guiding philosophy is that for μ, ν ∈ P, 𝔸_{I}(L^{2}(E_{μ}), L^{2}(E_{ν})) should be thought of as containing the order zero longitudinal pseudodifferential operators along the leaves of the fibration K_{q}/T ↠ K_{q}/K_{q}^{I}, while 𝕂_{I}(L^{2}(E_{μ}), L^{2}(E_{ν})) should be thought of as the ideal of negative order longitudinal pseudodifferential operators. For instance, in the case q = 1 the space 𝔸_{I}(L^{2}(E_{μ}), L^{2}(E_{ν})) contains all order zero longitudinal pseudodifferential operators along the fibration, although it also contains many other operators, such as translations by the group action. Nevertheless, the reader should keep the analogy in mind when interpreting the next theorem. Let μ ∈ P and i ∈ Σ. We introduce the unbounded operator

D_{i} = \begin{matrix}0 & F_{i} \\ E_{i} & 0 \end{matrix} on L^{2}(E_{μ} ⊕ E_{μ+α_{i}})

which is defined by the right regular representation on the dense domain O(E_{μ}⊕ E_{μ+α_{i}}). It is to be thought of as a longitudinal differential operator along the leaves of the fibration X_{q} ↠ K_{q}/K_{i,q}. Notice that these operators are essentially families of Dirac operators of the type considered by Dabrowski-Sitarz in [DS03], over the base space K_{q}/K_{i,q}. We denote the operator phase of D_{i} by

ph(D_{i}) = \begin{matrix}0 & ph(F_{i}) \\ ph(E_{i}) & 0 \end{matrix} ,

where again ph(E_{i}) : L^{2}(E_{μ}) → L^{2}(E_{μ+α_{i}}) and ph(F_{i}) : L^{2}(E_{μ}) → L^{2}(E_{μ+α_{i}}) are acting by the right regular representation. These operators should be thought of as longitudinal pseudodifferentials operator of order 0. Recall that if f ∈ C(E_{ν}) for some ν ∈ P, then the left and right multiplication actions M_{l}(f), M_{r}(f) define operators in 𝕃(L^{2}(E_{μ}), L^{2}(E_{μ+ν})).

Theorem 5.2. Let K_{q} = SU_{q}(n) for n ≥ 2. Let μ ∈ P, i ∈ Σ and f ∈ O(E_{ν}) for some ν ∈ P. Then the following hold.

a) M_{l}(f) and M_{r}(f) are in 𝔸(L^{2}(E_{μ}), L^{2}(E_{μ+ν})).

b) ph(D_{i}) ∈ 𝔸(L^{2}(E_{μ} ⊕ E_{μ+α_{i}}))

c) For any ψ ∈ C_{0}(ℝ), we have ψ(D_{i}) ∈ 𝕁_{i}(L^{2}(E_{μ} ⊕ E_{μ+α_{i}})), or equivalently, D_{i} has resolvent in 𝕁_{i}(L^{2}(E_{μ} ⊕ E_{μ+α_{i}})).

d) The following diagram

\xymatrix{L^{2}(E_{μ} ⊕ E_{μ+α_{i}}) \ar[r]^{ph(D_{i})} \ar[d]_{M_{l}(f)} & L^{2}(E_{μ} ⊕ E_{μ+α_{i}}) \ar[d]^{M_{l}(f)} \\ L^{2}(E_{μ+ν} ⊕ E_{μ+ν+α_{i}}) \ar[r]_{ph(D_{i})} & L^{2}(E_{μ+ν} ⊕ E_{μ+ν+α_{i}})}

commutes up to an element of 𝕁_{i}(L^{2}(E_{μ} ⊕ E_{μ+α_{i}}), L^{2}(E_{μ+ν} ⊕ E_{μ+ν+α_{i}})).

Remark 5.3. By slight abuse of notation, we will usually abbreviate part d) of Theorem 5.2 by writing [ph(D_{i}), M_{l}(f)] ∈ 𝕁_{i}(L^{2}(E_{μ} ⊕ E_{μ+α_{i}}), L^{2}(E_{μ+ν} ⊕ E_{μ+ν+α_{i}})) in the sequel. Notice that above statements about ph(D_{i}) can be restated as results about ph(E_{i}) and ph(F_{i}). In particular, part d) is equivalent to the commutativity of the diagrams

\xymatrix{L^{2}(E_{μ}) \ar[r]^{ph(E_{i})} \ar[d]_{M_{l}(f)} & L^{2}(E_{μ+α_{i}}) \ar[d]^{M_{l}(f)} & L^{2}(E_{μ+α_{i}}) \ar[r]^{ph(F_{i})} \ar[d]_{M_{l}(f)} & L^{2}(E_{μ}) \ar[d]^{M_{l}(f)} \\ L^{2}(E_{μ+ν}) \ar[r]_{ph(E_{i})} & L^{2}(E_{μ+ν+α_{i}}) & L^{2}(E_{μ+ν+α_{i}}) \ar[r]_{ph(F_{i})} & L^{2}(E_{μ+ν})}

modulo 𝕁_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+ν+α_{i}})) and 𝕁_{i}(L^{2}(E_{μ+α_{i}}), L^{2}(E_{μ+ν})), respectively.

6 Comparisons of Gelfand-Tsetlin bases

This section and the next provide the technical results from harmonic analysis which will be used to prove Theorems 5.1 and 5.2. It will be convenient to work with the quantum group U_{q}(n) rather than SU_{q}(n).

6.1 Quantum subgroups of U_{q}(n)

We first introduce notation for the block diagonal quantum subgroups of U_{q}(n). These definitions follow the notation and conventions of Sections 2.2 and 2.5. We define C( ; U_{q}(n)) = O(U_{q}(n))^{∗}. For I ⊆ Σ, we define the σ(C( ; U_{q}(n)), O(U_{q}(n)))-closed subalgebra

C( ; U_{q}^{I}(n)) = ⟨E_{i}, F_{i} (i ∈ I), G_{j} (j = 1, …, n)⟩.

and denote the associated closed quantum subgroup of U_{q}(n) by U_{q}^{I}(n). In the particular cases I = {1, …, k − 1}, we will decompose U_{q}^{I}(n)

as follows. Let C( ; U_{q}^{↑}(k)) = ⟨E_{i}, F_{i} (i = 1, …, k − 1), G_{j} (j = 1, …, k)⟩

and C(Ẑ_{k}^{↑}) = ⟨G_{j}(j = k + 1, …, n)⟩, and let U_{q}^{↑}(k) and Z_{k}^{↑} be the dual closed quantum subgroups of U_{q}(n). Then U_{q}^{{1,...,k−1}}(n) = U_{q}^{↑}(k) × Z_{k}^{↑}. The superscript ↑ refers to the fact that U_{q}^{↑}(k) ≅ U_{q}(k) is embedded in the “upper-left corner” of U_{q}(n). We likewise decompose U_{q}^{{n−k+1,...,n−1}}(n) = U_{q}^{↓}(k) × Z_{k}^{↓} where the two components are dual to

C( ; U_{q}^{↓}(k)) = ⟨E_{i}, F_{i} (i = n − k + 1, …, n − 1), G_{j} (j = n − k + 1, …, n)⟩ and

C(Ẑ_{k}^{↓}) = ⟨G_{j} (j = 1, …, n − k)⟩, respectively.

6.2 Upper and lower Gelfand-Tsetlin bases

Consider the nested family of quantum groups

T = U_{q}^{∅}(n) ⊂ U_{q}^{{1}}(n) ⊂ U_{q}^{{1,2}}(n) ⊂ ⋯ ⊂ U_{q}^{Σ}(n) = U_{q}(n).

The isotypical projections of these quantum subgroups are mutually commuting. Gelfand-Tsetlin theory is based upon the observation that the simultaneous isotypical decomposition for all of these subgroups yields components of dimension one, and thus provides a basis which is well-adapted for all of them. We shall refer to this as the upper Gelfand-Tsetlin basis. We recall the main facts about the Gelfand-Tsetlin basis here, and refer to [KS97, §7.3] for the details. The highest weights of type 1 representations of Ŭ_{q}(𝔤𝔩_{n}) are given by those μ = (μ_{1}, …, μ_{n}) ∈ ℤ^{n} with μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{n}. We denote the irreducible representation with highest weight μ by σ^{μ}. The Gelfand-Tsetlin basis for V^{σ^{μ}} is indexed by tableaux of integers of the form

(M) =  \begin{matrix}m_{n,1} & m_{n,2} & ⋯ & m_{n,n−1} & m_{nn} \\ m_{n−1,1} & ⋯ & m_{n−1,n−1} \\ ... & ... \\ m_{21} & m_{22} \\ m_{11} \end{matrix}  ,

where the top row is equal to μ and the lower rows satisfy the interlacing conditions m_{i+1,j} ≤ m_{ij} ≤ m_{i+1,j+1} for all i, j. The corresponding basis element, which will be denoted |(M)^{↑}⟩, is determined up to phase by the fact that for each k = 1, …, n, the vector |(M)^{↑}⟩ belongs to a U_{q}^{↑}(k)-subrepresentation with highest weight (m_{k1}, …, m_{kk}). Moreover, |(M)^{↑}⟩ is a weight vector with weight

(s_{1} − s_{0}, s_{2} − s_{1}, …, s_{n} − s_{n−1}),  (6.1)

where s_{i} = ∑ _{j=1}^{i} m_{ij} is the sum of the ith row and s_{0} = 0 by convention. There is an alternative basis of V^{σ^{μ}} adapted to the lower-right inclusions

T = U_{q}^{∅}(n) ⊂ U_{q}^{{n−1}}(n) ⊂ U_{q}^{{n−2,n−1}}(n) ⊂ ⋯ ⊂ U_{q}^{Σ}(n) = U_{q}(n).

This basis is most easily introduced by invoking the Hopf ∗-automorphism Ψ of Ŭ_{q}(𝔤𝔩_{n}) defined by:

Ψ(G_{j}) = G_{n+1−j}^{−1},  Ψ(E_{i}) = E_{n−i},  Ψ(F_{i}) = F_{n−i}.  (6.2)

Note that a highest weight vector for σ^{μ} is also a highest weight vector for the irreducible representation σ^{μ} ∘ Ψ, but with weight μ′ = (−μ_{n}, …, −μ_{1}). By Schur’s Lemma, there is a unitary ψ_{μ} : V^{σ^{μ′}} → V^{σ^{μ}} (unique up to scalar multiple) which intertwines σ^{μ′} ∘Ψ and σ^{μ}. We define the lower Gelfand-Tsetlin basis vectors by |(M)^{↓}⟩ = ψ_{μ}|(M)^{↑}⟩, where (M) is a Gelfand-Tsetlin tableau for the representation σ^{μ′}.

6.3 Class 1 representations

Often, we will only be interested in the irreducible U_{q}(n)-representations which contain a trivial U_{q}^{{1,...,n−2}}(n)-subrepresentation. These are a special case of the class 1 representations (see [KS97, §7.3.4]). A Gelfand-Tsetlin vector is contained in a trivial subrepresentation of U_{q}^{↑}(n − 1) if and only if it is of the form

ξ_{m} = | \begin{matrix}m & & 0 & ⋯ & 0 & & −m′ \\ & 0 & & 0 ⋯ 0 & & 0 \\ & & ... & & ... \\ 0 & 0 \\ & & & 0 \end{matrix} ↑

for some m, m′ ∈ ℕ. To be contained in a trivial U_{q}^{{1,...,n−2}}(n)-representation, it must additionally be of weight 0, which is to say m = m′. Thus, the representations of interest are precisely those with highest weight of the form μ = (m, 0, …, 0, −m). Note that in this case, σ^{μ} ≅ σ^{μ} ∘ Ψ, so that the upper and lower Gelfand-Tsetlin bases are indexed by the same set of tableaux. We state the Gelfand-Tsetlin formulae for such representations; compare [KS97, §7.3.4]. The generic basis vector is

|(M)^{↑}⟩ = |  \begin{matrix}m_{n} & 0 & ⋯ & 0 & −m′_{n} \\ m_{n−1} & 0 ⋯ 0 & m′_{n−1} \\ ... & ... \\ m_{2} & m′_{2} \\ & m_{1} \end{matrix} ↑  ,

where we are putting m_{n} = m′_{n} = m for ease of notation. We write (M ± δ_{ij}) to denote the Gelfand-Tsetlin tableau obtained from (M) by adding ±1 to the (i, j)-entry. The action of the generators of Ŭ_{q}(𝔤𝔩_{n}) is given by

E_{k−1}|(M)^{↑}⟩

= ( \frac{[m_{k}−m_{k−1}][m_{k−1}−m′_{k}+k−1][m_{k−1}−m_{k−2}+1][m_{k−1}−m′_{k−2}+k−2]}{[m_{k−1}−m′_{k−1}+k−1][m_{k−1}−m′_{k−1}+k−2]}) ^{\frac{1}{2}} |(M +δ_{k−1,1})^{↑}⟩

+ ( \frac{[m_{k}−m′_{k−1}+k−2][m′_{k−1}−m′_{k}+1][m_{k−2}−m′_{k−1}+k−3][m′_{k−2}−m′_{k−1}]}{[m_{k−1}−m′_{k−1}+k−2][m_{k−1}−m′_{k−1}+k−3]}) ^{\frac{1}{2}} |(M +δ_{k−1,k−1})^{↑}⟩, (6.3)

F_{k−1}|(M)^{↑}⟩

= ( \frac{[m_{k}−m_{k−1}+1][m_{k−1}−m′_{k}+k−2][m_{k−1}−m_{k−2}][m_{k−1}−m′_{k−2}+k−3]}{[m_{k−1}−m′_{k−1}+k−2][m_{k−1}−m′_{k−1}+k−3]}) ^{\frac{1}{2}} |(M −δ_{k−1,1})^{↑}⟩

+ ( \frac{[m_{k}−m′_{k−1}+k−1][m′_{k−1}−m′_{k}][m_{k−2}−m′_{k−1}+k−2][m′_{k−2}−m′_{k−1}+1]}{[m_{k−1}−m′_{k−1}+k−1][m_{k−1}−m′_{k−1}+k−2]}) ^{\frac{1}{2}} |(M −δ_{k−1,k−1})^{↑}⟩, (6.4)

G_{i}|(M)^{↑}⟩ = q^{\frac{1}{2} (s_{i}−s_{i−1})}|(M)^{↑}⟩,  (6.5)

where, as before, s_{i} is the sum of the ith row of (M) and s_{0} = 0.

6.4 Change of basis formula

We now describe certain cases of the change of basis transformation between the upper and lower Gelfand-Tsetlin bases introduced in Section 6.2. We shall concentrate entirely on the family of representations of highest weight μ = (m, 0, 0, …, 0, −m) for m ∈ ℕ. In either choice of Gelfand-Tsetlin basis, the zero-weight subspace of σ^{μ} is spanned by the vectors |(M_{m})^{↑}⟩ or |(M_{m})^{↓}⟩ with tableaux

M_{m} = \begin{matrix}m & 0 & ⋯ & 0 & −m \\ m_{n−1} & 0 ⋯ 0 & −m_{n−1} \\ ... & ... \\ m_{2} & −m_{2} \\ & 0 \end{matrix} .

Here we use m to denote the increasing n-tuple m = (m_{1} = 0, m_{2}, …, m_{n} = m). Our first goal is to compute the coefficients of the U_{q}^{↓}(n − 1)-invariant vector |(M_{(m,0,...,0)})^{↓}⟩ with respect to the upper Gelfand-Tsetlin basis. We write

|(M_{(m,0,...,0)})^{↓}⟩ = ∑_{m} a_{m}|(M_{m})^{↑}⟩.  (6.6)

Let us apply E_{k} to this. The Gelfand-Tsetlin formula (6.3) shows that the coefficient of |(M_{m} + δ_{k,1})^{↑}⟩ in E_{k}|(M_{(m,0,...,0)})^{↓}⟩ is

\frac{[m_{k+1}−m_{k}][m_{k}+m_{k+1}+k][m_{k}−m_{k−1}+1][m_{k}+m_{k−1}+k−1]}{[2m_{k}+k][2m_{k}+k−1]} ^{\frac{1}{2}} a_{m}

+ \frac{[m_{k+1}+m_{k}+k][−m_{k}+m_{k+1}][m_{k−1}+m_{k}+k−1][−m_{k−1}+m_{k}+1]}{[2m_{k}+k+1][2m_{k}+k]} ^{\frac{1}{2}} a_{m+δ_{k}}, (6.7)

where m + δ_{k} denotes the n-tuple obtained by adding 1 to the kth entry of m. Since E_{k}|(M_{(m,0,...,0)})^{↓}⟩ = 0 for all 2 ≤ k ≤ n − 1, we obtain the recurrence relation

a_{m+δ_{k}} = −\frac{[2m_{k} + k + 1]^{\frac{1}{2}}}{[2m_{k} + k − 1]^{\frac{1}{2}}} a_{m}  (6.8)

for all m and all 2 ≤ k ≤ n − 1. This multi-parameter recurrence relation has the solution

a_{m} = (−1)^{|m|}A ∏_{k=2}^{n−1}[2m_{k} + k − 1]^{\frac{1}{2}} ,  (6.9)

where |m| = m_{1} + ⋯ + m_{n} and A ∈ ℂ is some overall constant. This constant is determined up to a phase by the fact that |(M_{(m,0,...,0)})^{↓}⟩ has norm one. We will assume a choice of phases for the Gelfand-Tsetlin bases such that ⟨(M_{(m,0,...,0)})^{↓}|(M_{(m,0,...,0)})^{↑}⟩ is positive. Then A is positive. From (6.9), we calculate

1 = ∑_{m} |a_{m}|^{2} = A^{2} ∑_{m} ∏_{k=2}^{n−1}[2m_{k} + k − 1],

where the sum is over all n-tuples m with 0 = m_{1} ≤ m_{2} ≤ ⋯ ≤ m_{n} = m. An inductive argument shows that

∑_{m} ∏_{k=2}^{n−1}[2m_{k} + k − 1] = [n − 2]! \begin{matrix}m_{n} + n − 2 \\ n − 2 \end{matrix} ^{2} ,

and one obtains that A = [n − 2]!^{− \frac{1}{2}} \begin{matrix}m + n − 2 \\ n − 2 \end{matrix} ^{−1}. In summary, we have

proved the following formula.

Proposition 6.1. In the irreducible representation of U_{q}(n) with highest weight (m, 0, …, 0, −m),

|(M_{(m,0,...,0)})^{↓}⟩ = ∑_{m} \frac{(−1)^{|m|} ∏ _{k=2}^{n−1}[2m_{k} + k − 1]^{\frac{1}{2}}}{[n − 2]!^{\frac{1}{2}} \begin{matrix}m + n − 2 \\ n − 2 \end{matrix}} |(M_{m})^{↑}⟩,

where the sum is over all n-tuples m with 0 = m_{1} ≤ m_{2} ≤ ⋯ ≤ m_{n} = m.

6.5 Change of basis formula for U_{q}(3)

In the case of U_{q}(3), the above calculation gives the following change-of-basis coefficients for the trivial U_{q}^{{2}}(3)-type:

\begin{matrix}m & & 0 & −m \\ & 0 & & 0 \\ & & 0 \end{matrix} ^{↓}| \begin{matrix}m & & 0 & −m \\ & j & & −j \\ & & 0 \end{matrix} ^{↑} = (−1)^{j+m} \frac{[2j + 1]^{\frac{1}{2}}}{[m + 1]} . (6.10)

The complete change-of-basis coefficients between the two Gelfand-Tsetlin bases of any U_{q}(3)-representation were computed in [MSK95]. They are given by q-Racah coefficients. We will only need the following special cases.

Proposition 6.2. In the representation of U_{q}(3) with highest weight (m, 0, −m), consider the vectors

|x_{j}⟩ = [2j + 1]^{− \frac{1}{2}} | \begin{matrix}m & & 0 & −m \\ & j & & −j \\ & & 0 \end{matrix} ^{↑} ,

|y_{k}⟩ = [2k + 1]^{− \frac{1}{2}} | \begin{matrix}m & & 0 & −m \\ & k & & −k \\ & & 0 \end{matrix} ^{↓} .

Then

⟨y_{k}|x_{j}⟩ = \frac{(−1)^{j+k+m}}{[m + 1]}_{4}\overline{ϕ}_{3}  \begin{matrix}q^{−2k}, q^{2(k+1)}, q^{−2j}, q^{2(j+1)} \\ q^{−2m}, q^{2(m+2)}, q^{2} \end{matrix} | q^{2}; q^{2} , (6.11)

where _{4}\overline{ϕ}_{3} denotes the q-hypergeometric function.

In the q-Racah notation of [KLS10, §14.2], this translates as

⟨y_{k}|x_{j}⟩ = (−1)^{j+k+m}[m + 1]^{−1}R_{k}(μ(j); q^{2(m+1)}, q^{−2(m+1)}, 1, 1|q^{2}),

though we shall not actually use this. It is rather cumbersome to reconcile the notation and terminology of [MSK95] with ours. For this reason, we outline a short proof of Proposition 6.2 in Appendix A.1.

6.6 Action of the phase of E_{1} on the lower Gelfand-Tsetlin basis

The final task of this section is to compute the action of ph(E_{1}) with respect to the lower Gelfand-Tsetlin basis. Obtaining an explicit formula is difficult. Instead, we compute the action asymptotically as the highest weight μ goes to infinity, which is all that will be necessary for our purposes. We shall make use of the following basic estimate for products of values near 1, whose proof we leave to the reader.

Lemma 6.3. Fix q ∈ (0, 1) and N ∈ ℕ. There is a constant c_{N} with the following property: For any real numbers 0 ≤ d_{1}, …, d_{N} ≤ q and −1 ≤ r_{1}, …, r_{N} ≤ 1,

|1 − ∏_{i=1}^{N}(1 − d_{i})^{r_{i}} | ≤ c_{N} ∑_{i=1}^{N} |r_{i}|d_{i}.

Next we prove an estimate on the change of basis coefficients from Proposition 6.2 which will allow us to reduce the q-hypergeometric function from _{4}\overline{ϕ}_{3} to 2\overline{ϕ}_{1}.

Lemma 6.4. Fix k ∈ ℕ. There is a constant C(k) such that for all j, m ∈ ℕ

|(−1)^{j+k+m} \frac{[2j+1]}{[j]^{\frac{1}{2}} [j+1]^{\frac{1}{2}}} ⟨y_{k}|x_{j}⟩ − q^{m−j+ \frac{1}{2}} (q − q^{−1})p_{k}(q^{2(m−j)}|q^{2})|

≤ C(k)q^{m+j}, (6.12)

where |x_{j}⟩ and |y_{k}⟩ are as in Proposition 6.2, and p_{k}( · |q^{2}) is the little q^{2}- Legendre polynomial:

p_{k}(x|q^{2}) = _{2}\overline{ϕ}_{1} \begin{matrix}q^{−2k}, q^{2(k+1)} \\ q^{2} \end{matrix} | q^{2}; q^{2}x .

Proof. Proposition 6.2 says

(−1)^{j+k+m} \frac{[2j + 1]}{[j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}}} ⟨y_{k}|x_{j}⟩

= ∑_{l=0}^{k} \frac{1}{[m + 1]} \frac{[2j + 1]}{[j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}}} \frac{(q^{−2k}, q^{2(k+1)}, q^{−2j}, q^{2(j+1)}; q^{2})_{l}}{(q^{−2m}, q^{2(m+2)}, q^{2}, q^{2}; q^{2})_{l}} q^{2l}. (6.13)

On the other hand

q^{m−j+ \frac{1}{2}} (q − q^{−1})p_{k}(q^{2(m−j)}|q^{2})

= ∑_{l=0}^{k}(q − q^{−1})\frac{(q^{−2k}, q^{2(k+1)}; q^{2})_{l}}{(q^{2}, q^{2}; q^{2})_{l}} q^{(2l+1)(m−j)+(2l+ \frac{1}{2} )}. (6.14)

Denote by A_{l} and B_{l} the lth summand of (6.13) and (6.14), respectively. Note that |B_{l}| ≤ C_{1}(k)q^{m−j} for some constant C_{1}(k).

If j < l, then the Pochammer symbol in the numerator of |A_{l}| is zero, so

|A_{l} − B_{l}| = |B_{l}| ≤ C_{1}(k)q^{m−j} ≤ C_{1}(k)q^{−2k}q^{m+j}.  (6.15)

If j ≥ l, we have

|A_{l} − B_{l}|

= |B_{l}| |1 − \frac{q^{−(2l+1)(m−j)− \frac{1}{2}}}{(q − q^{−1})} \frac{1}{[m + 1]} \frac{[2j + 1]}{[j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}}} \frac{(q^{−2j}, q^{2(j+1)}; q^{2})_{l}}{(q^{−2m}, q^{2(m+2)}; q^{2})_{l}} |

= |B_{l}| |1 − \frac{1}{(1 − q^{2(m+1)})} \frac{(1 − q^{2(2j+1)})}{(1 − q^{2j})^{\frac{1}{2}} (1 − q^{2(j+1)})^{\frac{1}{2}}}

×\frac{∏ _{i=1}^{l}(1 − q^{2j−2i+2}) ∏ _{i=1}^{l}(1 − q^{2j+2i})}{∏ _{i=1}^{l}(1 − q^{2m−2i+2}) ∏ _{i=1}^{l}(1 − q^{2m+2i+2})}| .

In the latter expression, all the exponents of q are positive and bounded below by 2(j − l). The estimate of Lemma 6.3 yields

|A_{l} − B_{l}| ≤ C_{1}(k)q^{m−j}(4l + 3)q^{2(j−l)}
≤ C_{1}(k)q^{m−j}(4k + 3)q^{2(j−k)}
≤ C_{1}(k)(4k + 3)q^{−2k}q^{m+j}.  (6.16)

Taken together, the estimates (6.15) and (6.16) yield a constant C_{2}(k) such that |A_{l} − B_{l}| ≤ C_{2}(k)q^{m+j} for all l, j, m. The left hand side of (6.12) is then bounded by

∑_{l=0}^{k} |A_{l} − B_{l}| ≤ kC_{2}(k)q^{m+j}.

This yields the claim.

Finally, we describe the coefficients of ph(E_{1}) | \begin{matrix}m & & 0 & −m \\ & 0 & & 0 \\ & & 0 \end{matrix} ^{↓} asymptot-

ically as m → ∞.

Proposition 6.5. For any k ∈ ℕ,

lim_{m→∞} \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓}| ph(E_{1}) | \begin{matrix}m & & 0 & −m \\ & 0 & & 0 \\ & & 0 \end{matrix} ^{↓}

= (−1)^{k} \frac{[k]}{[2k]^{\frac{1}{2}}} \frac{2k − 1}{2} ^{−1} − \frac{2k + 1}{2} ^{−1} . (6.17)

Proof. From Equation (6.10) we have

| \begin{matrix}m & & 0 & −m \\ & 0 & & 0 \\ & & 0 \end{matrix} ^{↓} = ∑_{j=0}^{m}(−1)^{j+m} \frac{[2j + 1]}{[m + 1]}|x_{j}⟩,

where the |x_{j}⟩ are as defined in Proposition 6.2. Now |x_{j}⟩ belongs to a U_{q}^{↑}(2)-subrepresentation of highest weight (j, −j), and it has weight 0. By the standard formulae for Ŭ_{q}(𝔰𝔩_{2})-representations, |x_{j}⟩ is an eigenvector of |E_{1}| = (F_{1}E_{1})^{\frac{1}{2}} with eigenvalue [j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}} . Thus,

ph(E_{1}) | \begin{matrix}m & & 0 & −m \\ & 0 & & −0 \\ & & 0 \end{matrix} ^{↓} = E_{1} · ( ∑_{j=1}^{m} \frac{(−1)^{j+m}}{[m + 1]} \frac{[2j + 1]}{[j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}}} |x_{j}⟩) .

Hence the inner product in the statement of the proposition is equal to

∑_{j=1}^{m} \frac{(−1)^{j+m}}{[m + 1]} \frac{[2j + 1]}{[j]^{\frac{1}{2}} [j + 1]^{\frac{1}{2}}}  \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓}| E_{1}| x_{j} .  (6.18)

Now E_{1}^{∗} = F_{1} acts on the lower Gelfand-Tsetlin basis by Formula (6.4) for Ψ(F_{1}) = F_{2}:

E_{1}^{∗} | \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓} = \frac{[m−k+1]^{\frac{1}{2}} [m+k+1]^{\frac{1}{2}} [k]}{[2k−1]^{\frac{1}{2}} [2k]^{\frac{1}{2}}} | \begin{matrix}m & 0 & −m \\ k−1 & −k+1 \\ 0 \end{matrix} ^{↓}

+ \frac{[m−k+1]^{\frac{1}{2}} [m+k+1]^{\frac{1}{2}} [k]}{[2k+1]^{\frac{1}{2}} [2k]^{\frac{1}{2}}} | \begin{matrix}m & & 0 & −m \\ & k & & −k \\ & & 0 \end{matrix} ^{↓}

= \frac{[m−k+1]^{\frac{1}{2}} [m+k+1]^{\frac{1}{2}} [k]}{[2k]^{\frac{1}{2}}} (|y_{k−1}⟩ + |y_{k}⟩) .

Putting this into (6.18) yields

\begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓}| ph(E_{1}) | \begin{matrix}m & & 0 & −m \\ & 0 & & −0 \\ & & 0 \end{matrix} ^{↓}

= ^{\frac{[m−k+1]^{\frac{1}{2}} [m+k+1]^{\frac{1}{2}}}{[m+1]}} \frac{[k]}{[2k]^{\frac{1}{2}}} ( ∑_{j=1}^{m}(−1)^{j+m} \frac{[2j+1]}{[j]^{\frac{1}{2}} [j+1]^{\frac{1}{2}}} (⟨y_{k−1}|x_{j}⟩ + ⟨y_{k}|x_{j}⟩)) . (6.19)

Documenta Mathematica 20 (2015) 433–490

Now we let m → ∞. Since lim_{m→∞} ^{\frac{[m−k+1]^{\frac{1}{2}} [m+k+1]^{\frac{1}{2}}}{[m+1]}} = 1, it only remains to estimate the sum in (6.19). Lemma 6.4 gives us the estimate

∑_{j=1}^{m}(−1)^{j+m} \frac{[2j+1]}{[j]^{\frac{1}{2}} [j+1]^{\frac{1}{2}}} ⟨y_{k}|x_{j}⟩

= ∑_{j=1}^{m} (−1)^{k}q^{m−j+ \frac{1}{2}} (q − q^{−1}) p_{k}(q^{2(m−j)}|q^{2}) + R_{m}, (6.20)

where

|R_{m}| ≤ ∑_{j=1}^{m} C(k)q^{m+j} ⟶ 0 as m → ∞.

Also,

∑_{j=1}^{m}(−1)^{k}q^{m−j+ \frac{1}{2}} (q − q^{−1})p_{k}(q^{2(m−j)}|q^{2})

= (−1)^{k+1}q^{− \frac{1}{2}} (1 − q^{2}) ∑_{i=0}^{m−1} q^{i}p_{k}(q^{2i}|q^{2})

which is a partial sum for the q-integral

(−1)^{k+1}q^{− \frac{1}{2}} ∫_{0}^{1} x^{− \frac{1}{2}} p_{k}(x|q^{2}) d_{q^{2}}x = (−1)^{k+1} [k + \frac{1}{2}]_{q}^{−1} .

The calculation of this q-integral is explained in Appendix A.2. Putting all this into (6.19), we arrive at

lim_{m→∞} \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓}| ph(E_{1}) | \begin{matrix}m & & 0 & −m \\ & 0 & & −0 \\ & & 0 \end{matrix} ^{↓}

= (−1)^{k} \frac{[k]}{[2k]^{\frac{1}{2}}} [k − \frac{1}{2}]_{q}^{−1} − [k + \frac{1}{2}]_{q}^{−1} .

This finishes the proof.

7 Essential orthotypicality

The notion of essential orthotypicality was introduced in [Yun10] as a tool for studying harmonic analysis on manifolds with multiple fibrations. In brief, two closed subgroups of a compact group are essentially orthotypical if their isotypical subspaces are approximately mutually orthogonal. In [Yun13] it is shown that in a compact Lie group two connected subgroups are essentially orthotypical if and only if they generate the entire group. Since we do not have an analogous characterization in the quantum case, we shall prove essential orthotypicality for quantum subgroups of SU_{q}(n) by direct calculation.

7.1 Definitions and basic properties

Definition 7.1. Two closed quantum subgroups K_{1,q}, K_{2,q} of a compact quantum group K_{q} are essentially orthotypical if for any τ_{1} ∈ Irr(K_{1,q}), τ_{2} ∈ Irr(K_{2,q}) and any ϵ > 0 there are only finitely many σ ∈ Irr(K_{q}) for which

sup{|⟨p_{τ_{1}}ξ, p_{τ_{2}}η⟩| | ξ, η ∈ V^{σ}, ∥ξ∥ = ∥η∥ = 1} ≥ ϵ.

Lemma 7.2. Let K_{1,q}, K_{2,q} be closed quantum subgroups of a compact quantum group K_{q}. The following conditions are equivalent.

a) K_{1,q} and K_{2,q} are essentially orthotypical.

b) For any τ_{1} ∈ Irr(K_{1,q}) and any ϵ > 0, there are only finitely many irreducible unitary K_{q}-representations σ ∈ Irr(K_{q}) for which

sup{|⟨p_{τ_{1}}ξ, p_{𝟙_{K_{2,q}}} η⟩| | ξ, η ∈ V^{σ}, ∥ξ∥ = ∥η∥ = 1} ≥ ϵ,

where 𝟙_{K_{2,q}} denotes the trivial representation of K_{2,q}.

c) For any finite sets S_{1} ⊂⊂ Irr(K_{1,q}) and S_{2} ⊂⊂ Irr(K_{2,q}), p_{S_{2}}p_{S_{1}} ∈ 𝕂_{K_{q}}(H) on any unitary K_{q}-representation space H.

Proof. This is essentially Lemma 5.1 from [Yun10]. Here, we will only prove the implication b) ⇒ c). The other implications can easily be adapted from the proof in [Yun10]. Let τ_{1} ∈ Irr(K_{1,q}). Fix ϵ > 0 and let S ⊂⊂ Irr(K_{q}) be the finite set of K_{q}-types which satisfy the condition in b). Then (1 − p_{S})p_{𝟙_{K_{2,q}}} p_{τ_{1}} = p_{𝟙_{K_{2,q}}} p_{τ_{1}}(1 − p_{S}) has norm at most ϵ. By Lemma 3.5, we therefore have p_{𝟙_{K_{2,q}}} p_{τ_{1}} ∈ 𝕂_{K_{q}}(H). From this, we obtain p_{𝟙_{K_{2,q}}} p_{S_{1}} ∈ 𝕂_{K_{q}}(H) for any finite set S_{1} ⊂⊂ Irr(K_{1,q}). Now let τ_{2} ∈ Irr(K_{2,q}) be arbitrary. Choose a finite-dimensional K_{q}- representation V in which τ_{2} occurs as a K_{2,q}-type. By Lemma 3.13 there are linear maps ι : ℂ → V^{c} ⊗ V and ῑ: V^{c} ⊗ V → ℂ so that p_{τ_{2}} factorizes as

\xymatrix{p_{τ_{2}} : H \ar[r]^{Id_{H} ⊗ι} & H ⊗ V^{c} ⊗ V \ar[r]^{p_{𝟙_{K_{2,q}}} ⊗Id_{V}} & H ⊗ V^{c} ⊗ V \ar[r]^{Id_{H} ⊗ῑ} & H.}

Let S_{1} be the finite set of all K_{1,q}-types occurring in p_{τ_{1}}(H) ⊗ V^{c}. We get the factorization

\xymatrix{p_{τ_{2}}p_{τ_{1}} : H \ar[r]^{p_{τ_{1}} ⊗ι} & H ⊗ V^{c} ⊗ V \ar[r]^{p_{𝟙_{K_{2,q}}} p_{S_{1}} ⊗Id_{V}} & H ⊗ V^{c} ⊗ V \ar[r]^{Id_{H} ⊗ῑ} & H.}

But p_{𝟙_{K_{2,q}}} p_{S_{1}} ∈ 𝕂_{K_{q}}(H ⊗ V^{c}) by the preceding paragraph, and since V is finite-dimensional, we obtain p_{𝟙_{K_{2,q}}} p_{S_{1}} ⊗ Id_{V} ∈ 𝕂_{K_{q}}(H ⊗ V^{c} ⊗ V). We also have p_{τ_{1}} ⊗ ι ∈ 𝔸_{K_{q}}(H, H ⊗ V^{c} ⊗ V) and Id_{H} ⊗ῑ ∈ 𝔸_{K_{q}}(H ⊗ V^{c} ⊗ V, H), since they preserve K_{q}-types. We deduce that p_{τ_{2}}p_{τ_{1}} ∈ 𝕂_{K_{q}}(H, H).

Corollary 7.3. Let K_{1,q} and K_{2,q} be essentially orthotypical quantum subgroups of K_{q}. Suppose H, H′, H′′ are unitary K_{q}-representations and that H′ has finite K_{q}-multiplicities. Then 𝕂_{K_{2,q}}(H′, H′′)𝕂_{K_{1,q}}(H, H′) ⊆ 𝕂(H, H′′) ⊆ 𝕂_{K_{q}}(H, H′′).

Proof. Suppose that A ∈ 𝕂_{K_{1,q}}(H, H′) is K_{1,q}-harmonically finite and that B ∈ 𝕂_{K_{2,q}}(H′, H′′) is K_{2,q}-harmonically finite. Then there are finite sets S_{1} ⊂ ⊂ Irr(K_{1,q}), S_{2} ⊂⊂ Irr(K_{2,q}) such that A = p_{S_{1}}A and B = Bp_{S_{2}}. By essential orthotypicality and Lemma 3.7 we have p_{S_{2}}p_{S_{1}} ∈ 𝕂_{K_{q}}(H′) = 𝕂(H′). Thus BA = Bp_{S_{2}}p_{S_{1}}A is compact. The result follows.

Remark 7.4. It is important that H′ has finite K_{q}-multiplicities in the above statement. Corollary 7.3 can fail when H′ has infinite K_{q}-multiplicities.

Lemma 7.5. Let K_{1,q}, K′_{1,q}, K_{2,q}, K′_{2,q} be closed quantum subgroups of K_{q}, with K_{1,q} ⊆ K′_{1,q} and K_{2,q} ⊆ K′_{2,q}. If K_{1,q} and K_{2,q} are essentially orthotypical then K′_{1,q} and K′_{2,q} are essentially orthotypical.

Proof. For i = 1, 2, let τ_{i} be an irreducible K′_{i,q}-type, and let S_{i} ⊆ Irr(K_{i,q}) be the finite collection of K_{i,q}-types which occur non-trivially in τ_{i}. Then on any K_{q}-representation H, we have

p_{τ_{1}}p_{τ_{2}} = p_{τ_{1}}p_{S_{1}}p_{S_{2}}p_{τ_{2}}.

The product p_{S_{1}}p_{S_{2}} belongs to 𝕂_{K_{q}}(H) by Lemma 7.2, and the other projections belong to 𝔸_{K_{q}}(H) since they commute with K_{q}-isotypical projections. Hence the claim follows from Lemma 7.2.

7.2 Essential orthotypicality of subgroups of U_{q}(n)

We now specialize to U_{q}(n). Recall that we defined block-diagonal quantum subgroups U_{q}^{I}(n) for any I ⊆ Σ in section 6.1.

Lemma 7.6. The quantum subgroups U_{q}^{{1,...,n−2}}(n) and U_{q}^{{2,...,n−1}}(n) are essentially orthotypical in U_{q}(n).

Proof. By Lemma 7.5, it suffices to prove that the quantum subgroups U_{q}^{↑}(n−1) and U_{q}^{{2,...,n−1}}(n) are essentially orthotypical. To show this, we will verify condition b) of Lemma 7.2. Fix τ_{1} an irreducible representation of U_{q}^{↑}(n − 1) and ϵ > 0. Let σ be an irreducible U_{q}(n)-representation. Arguing as in Section 6.3, we observe that the trivial U_{q}^{{2,...,n−1}}(n)-type does not occur in σ unless σ has highest weight of the form μ = (m, 0, …, 0, −m) for some m ∈ ℕ, in which case the trivial U_{q}^{{2,...,n−1}}(n)-isotypical subspace is spanned by the lower Gelfand-Tsetlin vector

|(M_{(m,0,...,0)})^{↓}⟩ = |  \begin{matrix}m & & 0 & ⋯ & 0 & & −m \\ & 0 & & 0 ⋯ 0 & & 0 \\ & & ... & & ... \\ 0 & 0 \\ & & & 0 \end{matrix} ↓  .

By proposition 6.1,

|(M_{(m,0,...,0)})^{↓}⟩ = ∑_{{0≤m_{2}≤··· \atop ···≤m_{n−1}≤mn=m}} \frac{(−1)^{|m|} ∏ _{k=2}^{n−1}[2m_{k} + k − 1]^{\frac{1}{2}}}{[n − 2]!^{\frac{1}{2}} \begin{matrix}m + n − 2 \\ n − 2 \end{matrix}}

× |  \begin{matrix}m & 0 & ⋯ & 0 & −m \\ & m_{n−1} & 0 ⋯ 0 & −m_{n−1} \\ & ... & ... \\ m_{2} & −m_{2} \\ & & 0 \end{matrix} ↑  . (7.1)

We see that the τ_{1}-isotypical subspace of σ will be orthogonal to |(M_{(m,0,...,0)})^{↓}⟩ unless τ_{1} has highest weight of the form (m_{n−1}, 0, …, 0, −m_{n−1}) for some m_{n−1} ∈ ℕ. So, let τ_{1} be the U_{q}^{↑}(n − 1)-type with highest weight (m_{n−1}, 0, …, 0, −m_{n−1}). Regardless of m, the sum in (7.1) contains at most a fixed finite number of vectors of U_{q}^{↑}(n − 1)-type τ_{1}, since they all have to verify m_{2} ≤ ⋯ ≤ m_{n−2} ≤ m_{n−1}. Moreover, the coefficient of each of these terms is bounded in absolute value by

\frac{[2m_{n−1} + (n − 1) − 1]^{\frac{1}{2} (n−2)}}{[n − 2]!^{\frac{1}{2}} \begin{matrix}m + n − 2 \\ n − 2 \end{matrix}},

which tends to zero as m → ∞. It follows that there are only finitely many m ∈ ℕ for which there exists a unit vector ξ of Ŭ_{q}(𝔤𝔩_{n−1}^{↑})-type τ_{1} such that |⟨ξ|(M_{m,0,...,0})^{↓}⟩| > ϵ. This completes the proof.

Proposition 7.7. Let n ≥ 2 and let I_{1}, I_{2} be sets of simple roots of U_{q}(n). If I_{1} ∪ I_{2} = Σ then U_{q}^{I_{1}}(n) and U_{q}^{I_{2}}(n) are essentially orthotypical in U_{q}(n).

Proof. The result is trivial for n = 2. Suppose now that n > 2 and that the result has been proven for U_{q}(n − 1). We claim first that U_{q}^{J_{1}}(n) and U_{q}^{J_{2}}(n) are essentially orthotypical in U_{q}^{{1,...,n−2}}(n) whenever J_{1} ∪ J_{2} = {1, …, n − 2}. Recall that U_{q}^{{1,...,n−2}}(n) = U_{q}^{↑}(n − 1) × Z_{n−1}^{↑}. Moreover, we have U_{q}^{J}(n) = U_{q}^{J}(n − 1) × Z_{n−1}^{↑} for any J ⊆ {1, …, n−2}. Note that U_{q}^{J}(n−1) and Z_{n−1}^{↑} are commuting and generating quantum subgroups of U_{q}^{J}(n). Using Lemma 3.11, the inductive hypothesis implies that the quantum subgroups U_{q}^{J_{1}}(n), U_{q}^{J_{2}}(n) ⊆ U_{q}^{{1,...,n−2}}(n) satisfy condition c) of Lemma 7.2 whenever J_{1} ∪ J_{2} = {1, …, n − 2}. This proves the claim. Suppose now that I_{1} ∪ I_{2} = {1, …, n − 1}. Assume without loss of generality that n − 1 ∈ I_{1}. Let τ_{i} ∈ Irr(U_{q}^{I_{i}}(n)) for i = 1, 2. Moreover, let S_{i} denote the set of U_{q}^{I_{i}\{n−1}}(n)-types that occur in τ_{i}, so that we have p_{τ_{i}} = p_{τ_{i}}p_{S_{i}} on any U_{q}(n)-representation H. By the claim above, p_{S_{1}}p_{S_{2}} is in 𝕂_{U_{q}^{{1,...,n−2}}(n)}(H),

so for any ϵ > 0 there exists a finite set F_{1} ⊂⊂ Irr(U_{q}^{{1,...,n−2}}(n)) such that ∥(1 − p_{F_{1}})p_{S_{1}}p_{S_{2}}∥ < ϵ. Note that p_{F_{1}} commutes with both p_{S_{1}} and p_{S_{2}}. We therefore obtain

∥p_{τ_{1}}p_{τ_{2}} − p_{τ_{1}}p_{F_{1}}p_{τ_{2}}∥ = ∥p_{τ_{1}}p_{S_{1}}(1 − p_{F_{1}})p_{S_{2}}p_{τ_{2}}∥ < ϵ.  (7.2)

Now we repeat this trick, this time removing the first simple root instead of the last. Let T_{1} denote the finite collection of U_{q}^{I_{1}\{1}}(n)-types which occur in τ_{1}, and let T_{2} denote the finite collection of U_{q}^{{2,...,n−2}}(n)-types which occur in any of the U_{q}^{{1,...,n−2}}(n)-types in F_{1}. Since we assumed that n − 1 ∈ I_{1}, we have (I_{1} \ {1}) ∪ {2, …, n − 2} = {2, …, n − 1}. Another application of the above claim implies that p_{T_{1}}p_{T_{2}} ∈ 𝕂_{U_{q}^{{2,...,n−1}}(n)}(H). Thus, there is a

finite subset F_{2} ⊂⊂ Irr(U_{q}^{{2,...,n−1}}(n)) such that ∥(1 − p_{F_{2}})p_{T_{1}}p_{T_{2}}∥ < ϵ, and we obtain

∥p_{τ_{1}}p_{F_{1}}p_{τ_{2}} − p_{τ_{1}}p_{F_{2}}p_{F_{1}}p_{τ_{2}}∥ = ∥p_{τ_{1}}p_{T_{1}}(1 − p_{F_{2}})p_{T_{2}}p_{F_{1}}p_{τ_{2}}∥ < ϵ.  (7.3)

Combining Equations (7.2) and (7.3) gives

∥p_{τ_{1}}p_{τ_{2}} − p_{τ_{1}}p_{F_{2}}p_{F_{1}}p_{τ_{2}}∥ < 2ϵ.

By Lemma 7.6, p_{F_{2}}p_{F_{1}} ∈ 𝕂_{U_{q}(n)}(H), so p_{τ_{1}}p_{F_{2}}p_{F_{1}}p_{τ_{2}} ∈ 𝕂_{U_{q}(n)}(H). Since ϵ was arbitrary, p_{τ_{1}}p_{τ_{2}} ∈ 𝕂_{U_{q}(n)}(H). This completes the proof.

Lemma 7.8. For any I_{1}, I_{2} ⊆ Σ, U_{q}^{I_{1}}(n) and U_{q}^{I_{2}}(n) are essentially orthotypical as quantum subgroups of U_{q}^{I_{1}∪I_{2}}(n).

Proof. Write I = I_{1} ∪ I_{2}. The quantum group U_{q}^{I}(n) has a block diagonal decomposition, which we shall write as U_{q}^{I}(n) = ∏ _{k} U_{q}(n_{k}). Let Σ_{k} ⊆ Σ be the set simple roots of the block U_{q}(n_{k}), and put I_{i,k} = I_{i} ∩ Σ_{k} for i = 1, 2. We obtain decompositions U_{q}^{I_{i}}(n) = ∏ _{k} U_{q}^{I_{i,k}}(n_{k}). For each k, I_{1,k} ∪ I_{2,k} = Σ_{k}, so Proposition 7.7 says that the subgroups U_{q}^{I_{1,k}}(n_{k}) and U_{q}^{I_{2,k}}(n_{k}) are essentially orthotypical in U_{q}(n_{k}). A repeated application of Lemma 3.11 completes the proof.

We finish this section with the analogous result for quantum subgroups of K_{q} = SU_{q}(n).

Proposition 7.9. For any I_{1}, I_{2} ⊆ Σ, K_{q}^{I_{1}} and K_{q}^{I_{2}} are essentially orthotypical as quantum subgroups of K_{q}^{I_{1}∪I_{2}}.

Proof. Let T be the diagonal maximal torus of SU(n) and let Z be the centre of U(n). Both T and Z can be naturally identified with subgroups of the maximal torus of U_{q}(n). Fix τ_{1} ∈ Irr(K_{q}^{I_{1}}) and let 𝟙_{2} denote the trivial representation of K_{q}^{I_{2}}. Suppose σ ∈ Irr(K_{q}^{I_{1}∪I_{2}}) contains both of these as subrepresentations. Then in particular, T acts trivially on the trivial K_{q}^{I_{2}}-isotypical subspace, and by Schur’s Lemma T ∩Z acts trivially on all of V^{σ}. By comparison with the representation theory of the classical groups, we therefore know that σ and τ_{1} extend uniquely to representations σ̃ ∈ Irr(U_{q}^{I_{1}∪I_{2}}(n)) and τ̃_{1} ∈ Irr(U_{q}^{I_{1}}(n)), respectively, in which Z acts trivially. Denote by ˜𝟙_{2} the trivial representation of U_{q}^{I_{2}}(n). By Lemma 7.8, for any ϵ > 0, there are only finitely many σ̃ ∈ Irr(U_{q}^{I_{1}∪I_{2}}(n)) for which

sup{|⟨pτ̃_{1}ξ, p˜𝟙_{2} η⟩| | ξ, η ∈ V σ̃, ∥ξ∥ = ∥η∥ = 1} ≥ ϵ.

The result therefore follows from Lemma 7.2.

Corollary 7.10. Let I ⊆ Σ and let H be a unitary representation of K_{q}. For any τ ∈ Irr(K_{q}^{I}) the isotypical projection p_{τ} belongs to 𝔸(H).

Proof. Let I′ ⊆ Σ. For any τ′ ∈ Irr(K_{q}^{I′}), Proposition 7.9 implies that p_{τ} p_{τ′} and p_{τ′}p_{τ} are in 𝕂_{I∪I′}(H) ⊆ 𝕂_{I′}(H). From Lemma 3.6 we deduce that p_{τ} ∈ 𝔸_{I′}(H). Since I′ was arbitrary p_{τ} ∈ 𝔸(H).

7.3 Application to the lattice of ideals

Essential orthotypicality is the crucial property for proving Theorem 5.1.

Lemma 7.11. Let K_{q} = SU_{q}(n) and let I_{1}, I_{2} ⊆ Σ. Then

a) 𝕂_{I_{1}} ∩ 𝕂_{I_{2}} = 𝕂_{I_{1}∪I_{2}},

b) 𝔸_{I_{1}} ∩ 𝔸_{I_{2}} ⊆ 𝔸_{I_{1}∪I_{2}}.

Proof. From Lemma 3.9 we have 𝕂_{I_{1}∪I_{2}} ⊆ 𝕂_{I_{1}} ∩𝕂_{I_{2}}. For the reverse inclusion, suppose T ∈ 𝕂_{I_{1}}(H, H′) ∩ 𝕂_{I_{2}}(H, H′) for some fully K_{q}-harmonic spaces H and H′. Thus for any ϵ > 0, there are finite sets S_{i} ⊂⊂ Irr(K_{q}^{I_{i}}) such that ∥T − p_{S_{i}}T p_{S_{i}}∥ < ϵ for i = 1, 2, and we obtain ∥T − p_{S_{1}}p_{S_{2}}T p_{S_{2}}p_{S_{1}}∥ < 2ϵ. By Proposition 7.9, there is a finite subset F ⊆ Irr(K_{q}^{I_{1}∪I_{2}}) such that ∥p_{S_{1}}p_{S_{2}} − p_{F} p_{S_{1}}p_{S_{2}}p_{F} ∥ < ϵ/∥T ∥, from which

∥T − p_{F} p_{S_{1}}p_{S_{2}}p_{F} T p_{F} p_{S_{2}}p_{S_{1}}p_{F} ∥ < 4ϵ.

This proves the first statement. The second claim follows by using the characterization of 𝔸_{I} as multipliers of 𝕂_{I} in Lemma 3.6.

Now we are ready to assemble the above results in order to prove Theorem 5.1. Indeed, parts a) and c) of the theorem now follow as a corollary of Lemma 7.11, and part d) is contained in Lemma 4.5. To prove part b), note that if σ ∈ Irr(K_{q}^{I}) for some I ⊆ Σ then p_{σ} is in 𝕂_{I}(H) for any fully K_{q}-harmonic space H, so the result follows from Corollary 7.10. This completes the proof of Theorem 5.1.

8 Longitudinal pseudodifferential operators

In this section we prove Theorem 5.2.

8.1 Multiplication operators

We shall begin with Theorem 5.2 a), which is a consequence of the next proposition. Let us recall once again that we are equipping L^{2}(K_{q}), and its weight spaces L^{2}(E_{μ}) for μ ∈ P, with the structure of a fully K_{q}-harmonic space coming from the right regular representation. Thus, if τ ∈ Irr(K_{q}^{I}) for some I ⊆ Σ and g = ⟨η^{∗}| · |η⟩ is a matrix coefficient, then p_{τ} g = ⟨η^{∗}| · |p_{τ} η⟩.

Proposition 8.1. For any f ∈ O(K_{q}), the left and right multiplication operators M_{l}(f) and M_{r}(f) belong to 𝔸(L^{2}(K_{q})).

Proof. Fix I ⊆ Σ. We may assume that f = ⟨ξ^{∗}| · |ξ⟩ is a matrix coefficient of an irreducible K_{q}-representation. Moreover we may assume that ξ belongs to a K_{q}^{I} subrepresentation, say of type σ. Let τ ∈ Irr(K_{q}^{I}). From the formula (2.1) for the product of matrix coefficients, one sees that M_{l}(f)p_{τ} = p_{S}M_{l}(f)p_{τ} where S is the finite set of K_{q}^{I}-types which occur in τ ⊗ σ. This means M_{l}(f)p_{τ} ∈ 𝕂_{i}(L^{2}(K_{q})). We therefore obtain M_{l}(O(K_{q}))p_{τ} ⊆ 𝕂_{i}(L^{2}(K_{q})). Taking adjoints shows that p_{τ} M_{l}(O(K_{q})) ⊆ 𝕂_{i}(L^{2}(K_{q})). By Lemma 3.6, this implies M_{l}(O(K_{q})) ∈ 𝔸_{I}(L^{2}(K_{q})). Since I was arbitrary this yields the claim for left multiplication operators. The proof for right multiplication operators is similar.

8.2 Basic properties of the phase of E_{i}, F_{i}

Before specializing to the section spaces of bundles over the quantum flag manifold, we will first consider the abstract properties of the operators

D_{i} = \begin{matrix}0 & F_{i} \\ E_{i} & 0 \end{matrix} on H_{μ} ⊕ H_{μ+α_{i}}

for any K_{q}-representation H and any μ ∈ P. This operator D_{i} is essentially self-adjoint with domain the linear span of the K_{q}-isotypical components.

Lemma 8.2. With the notation above, ψ(D_{i}) ∈ 𝕂_{i}(H_{μ} ⊕ H_{μ+α_{i}}) for any function ψ ∈ C_{0}(ℝ), and ϕ(D_{i}) ∈ 𝔸_{i}(H_{μ} ⊕ H_{μ+α_{i}}) for any bounded function ϕ ∈ C_{b}(ℝ).

Proof. Recall from Section 2.5 the subgroup S_{q}^{i} ≅ SU_{q}(2) which is associated to the simple root α_{i}. The standard formulae for irreducible Ŭ_{q}(𝔰𝔩(2))- representations show that the S_{q}^{i}-isotypical subspaces of H_{μ} ⊕ H_{μ+α_{i}} are precisely the eigenspaces for D_{i}^{2}, and that the spectrum of D_{i}^{2} is discrete. It follows that if the function ψ has compact support, then ψ(D_{i}) annihilates all but finitely many S_{q}^{i}-isotypical subspaces. On a weight space, the S_{q}^{i}-isotypical and the K_{q}^{i}-isotypical decompositions are identical, so ψ(D_{i}) ∈ 𝕂_{i}(H_{μ} ⊕ H_{μ+α_{i}}). By density, ψ(D_{i}) ∈ 𝕂_{i}(H_{μ} ⊕ H_{μ+α_{i}}) for any ψ ∈ C_{0}(ℝ). If ϕ is a bounded function then ϕ(D_{i}) is bounded, and it preserves K_{q}^{i}-types. This proves the second statement.

Remark 8.3. It follows from this proof that if ψ ∈ C_{c}(ℝ), then ψ(D_{i}^{2}) is a finite linear combination of isotypical projections for K_{i,q}. By Corollary 7.10, these isotypical projections are in 𝔸(H). We can therefore deduce that ψ(D_{i}) ∈ 𝕁_{i}(H) for any even function ψ ∈ C_{0}(ℝ). Unfortunately, showing that ψ(D_{i}) ∈ 𝕁_{i}(H) for an odd ψ ∈ C_{0}(ℝ), as required for Theorem 5.2 c), is more difficult. Now let V be a finite dimensional unitary representation of K_{q}. Then D_{i} acts on H ⊗ V, and in particular on (H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}, as

Δ̂(D_{i}) = \begin{matrix}0 & F_{i} ⊗ K_{i} \\ E_{i} ⊗ K_{i} & 0 \end{matrix} + \begin{matrix}0 & K_{i}^{−1} ⊗ F_{i} \\ K_{i}^{−1} ⊗ E_{i} & 0 \end{matrix} .

We will abbreviate this expression as D_{i} ⊗ K_{i} + K_{i}^{−1} ⊗ D_{i}.

Lemma 8.4. Let H, V be unitary K_{q}-representations with V finite dimensional and let μ ∈ P. As operators on (H ⊗V)_{μ} ⊕(H ⊗V)_{μ+α_{i}}, we have ph(Δ̂(D_{i})) ≡ ph(D_{i}) ⊗ Id_{V} modulo 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}).

Proof. Since K_{i} is strictly positive, we have ph(D_{i} ⊗ K_{i}) = ph(D_{i}) ⊗ Id_{V}. Let us set A = Δ̂(D_{i}), B = D_{i} ⊗ K_{i}. Then A − B = K_{i}^{−1} ⊗ D_{i}, which is bounded on (H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}} since V is finite dimensional, and so A − B ∈ 𝔸_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}). Let ϕ ∈ C_{b}(ℝ) be the function ϕ(x) = x(1 + x^{2})^{− \frac{1}{2}} . Lemma 8.2 implies that ph(A) ≡ ϕ(A) modulo 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}). We claim that also ph(B) ≡ ϕ(B) modulo 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}). To see this, note that (H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}} = ⨁ _{ν}(H_{μ−ν} ⊕ H_{μ+α_{i}−ν}) ⊗ V_{ν}, where the sum is over all weights of V. This decomposition is invariant for B, and on each summand B acts as q^{\frac{1}{2} (α_{i},ν)}D_{i} ⊗ Id_{V_{ν}}. Lemma 8.2 implies that ψ(B) ∈ 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}) for any ψ ∈ C_{0}(ℝ), and the claim follows. Therefore, it suffices to prove that ϕ(A)−ϕ(B) ∈ 𝕂_{i}((H ⊗V)_{μ} ⊕(H ⊗V)_{μ+α_{i}}). Now,

ϕ(A) − ϕ(B) = (A − B)(1 + A^{2})^{− \frac{1}{2}} + B((1 + A^{2})^{− \frac{1}{2}} − (1 + B^{2})^{− \frac{1}{2}} ).

The first term (A − B)(1 + A^{2})^{− \frac{1}{2}} is contained in 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}) by Lemma 8.2. For the second term we use the integral formula

(1 + x^{2})^{− \frac{1}{2}} = \frac{1}{π} ∫_{0}^{∞} t^{− \frac{1}{2}} (1 + x^{2} + t)^{−1} dt,

which gives

B((1 + A^{2})^{− \frac{1}{2}} − (1 + B^{2})^{− \frac{1}{2}} )

= \frac{1}{π} B ∫_{0}^{∞} t^{− \frac{1}{2}} ((1 + A^{2} + t)^{−1} − (1 + B^{2} + t)^{−1}) dt

= \frac{1}{π} B ∫_{0}^{∞} t^{− \frac{1}{2}} (1 + B^{2} + t)^{−1}(B^{2} − A^{2})(1 + A^{2} + t)^{−1} dt

= \frac{1}{π} B ∫_{0}^{∞} t^{− \frac{1}{2}} (1 + B^{2} + t)^{−1}B(B − A)(1 + A^{2} + t)^{−1} dt

+ \frac{1}{π} B ∫_{0}^{∞} t^{− \frac{1}{2}} (1 + B^{2} + t)^{−1}(B − A)A(1 + A^{2} + t)^{−1} dt. (8.1)

By Lemma 8.2, we have B(1+B^{2}+t)^{−1}B ∈ 𝔸_{i} with norm at most 1, B−A ∈ 𝔸_{i}, and (1 + A^{2} + t)^{−1} ∈ 𝕂_{i} with norm at most (1 + t)^{−1}, so the first integral on the right hand side of equation (8.1) converges in norm in 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}). For the second integral, we can write

B(1 + B^{2} + t)^{−1}(B − A)A(1 + A^{2} + t)^{−1}

= B(1 + B^{2} + t)^{− \frac{1}{2}} (1 + B^{2} + t)^{− \frac{1}{2}} (B − A)A(1 + A^{2} + t)^{− \frac{1}{2}} (1 + A^{2} + t)^{− \frac{1}{2}} ,

where B(1 + B^{2} + t)^{− \frac{1}{2}} and A(1 + A^{2} + t)^{− \frac{1}{2}} are in 𝔸_{i} with norm at most 1, B −A ∈ 𝔸_{i}, and (1+B^{2} +t)^{− \frac{1}{2}} and (1+A^{2} +t)^{− \frac{1}{2}} are in 𝕂_{i} with norm at most (1 + t)^{− \frac{1}{2}} , so we again have norm convergence in 𝕂_{i}((H ⊗ V)_{μ} ⊕ (H ⊗ V)_{μ+α_{i}}). This completes the proof.

Considering the matrix entries of the operators in Lemma 8.4 gives the following result for ph(E_{i}) and ph(F_{i}).

Corollary 8.5. Let H and V be unitary K_{q}-representations, with V finite dimensional. For any weight μ ∈ P the operators (ph(Δ̂(E_{i})) − ph(E_{i}) ⊗ Id_{V})p_{μ} and p_{μ}(ph(Δ̂(E_{i})) − ph(E_{i}) ⊗ Id_{V}) belong to 𝕂_{i}(H ⊗ V). Likewise, (ph(Δ̂(F_{i})) − ph(F_{i}) ⊗ Id_{V})p_{μ} and p_{μ}(ph(Δ̂(F_{i})) − ph(F_{i}) ⊗ Id_{V}) belong to 𝕂_{i}(H ⊗ V).

8.3 The phase of the longitudinal Dirac operators

In this section we prove Theorem 5.2 b) and c). It is easy to see that ph(E_{i}) and ph(F_{i}) are multipliers of 𝕂_{i}, but Theorem 5.2 b) claims a more subtle fact, namely that ph(E_{i}) and ph(F_{i}) are multipliers of 𝕂_{j} for every j ∈ Σ. We will prove this fact in a series of Lemmas, beginning with the case of SU_{q}(3). We use the notation for subgroups of U_{q}(3) which was introduced in Section 6.1.

Lemma 8.6. Let 𝟙_{2} be the trivial representation of U_{q}^{{2}}(3). On any unitary U_{q}(3)-representation H, the operators ph(E_{1})p_{𝟙_{2}} and ph(F_{1})p_{𝟙_{2}} belong to 𝕂_{U_{q}^{{2}}(3)}(H).

Proof. We will show that for any ϵ > 0 there is a finite collection S ⊂⊂ Irr(U_{q}^{{2}}(3)) such that on every irreducible U_{q}(3)-representation V^{σ} the estimate

∥(1 − p_{S}) ph(E_{1})p_{𝟙_{2}} ∥ < ϵ  (8.2)

holds. Since S does not depend on σ in this statement, the lemma will follow by decomposing H into irreducibles for U_{q}(3).

As before, we write σ^{μ} for the irreducible U_{q}(3)-representation with highest weight μ. It follows from Section 6.3 that the operator p_{𝟙_{2}} is zero on V^{σ^{μ}} unless μ = (m, 0, −m) for some m ∈ ℕ, in which case p_{𝟙_{2}} V^{σ} is spanned by the lower Gelfand-Tsetlin vector

|(M_{(m,0,0)})^{↓}⟩ = | \begin{matrix}m & & 0 & −m \\ & 0 & & 0 \\ & & 0 \end{matrix} ^{↓} .

Note that |(M_{(m,0,0)})^{↓}⟩ has weight 0, so that ph(E_{1})|(M_{(m,0,0)})^{↓}⟩ is contained in the weight space (V^{σ^{μ}})_{α_{1}}, which is spanned by the vectors

| \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓}  (8.3)

for k = 1, …, m.

Let us denote by τ_{k} the U_{q}^{{2}}(3)-type of the vector (8.3), and let S_{l} = {τ_{1}, …, τ_{l}}. On V^{(m,0,−m)}, the operator p_{S_{l}} ph(E_{1})p_{𝟙_{2}} satisfies

∥p_{S_{l}} ph(E_{1})p_{𝟙_{2}} ∥^{2} = ∑_{k=1}^{l} \begin{matrix}m & 0 & −m \\ k & −k+1 \\ & 0 \end{matrix} ^{↓} | ph(E_{1})| (M_{(m,0,0)})^{↓} ^{2} .

Using Proposition 6.5, we have that

lim_{m→∞} ∥p_{S_{l}} ph(E_{1})p_{𝟙_{2}} ∥^{2}

= ∑_{k=1}^{l} \frac{[k]^{2}}{[2k]} k − \frac{1}{2} ^{−1} − k + \frac{1}{2} ^{−1} ^{2}

= ∑_{k=1}^{l} \frac{1}{[2k] [k − \frac{1}{2}]^{2} [k + \frac{1}{2}]^{2}} [k] k + \frac{1}{2} − [k] k − \frac{1}{2} ^{2}

= ∑_{k=1}^{l} \frac{1}{[2k] [k − \frac{1}{2}]^{2} [k + \frac{1}{2}]^{2}} \frac{1}{2} [2k] ^{2}

= ∑_{k=1}^{l} \frac{1}{2} ^{2} \frac{1}{[k − \frac{1}{2}]^{2}} − \frac{1}{[k + \frac{1}{2}]^{2}}

= 1 − \frac{[\frac{1}{2}]^{2}}{[l + \frac{1}{2}]^{2}} .

Let l be sufficiently large that \frac{[ \frac{1}{2} ]^{2}}{[l+ \frac{1}{2} ]^{2}} < \frac{1}{2}ϵ. Then

lim_{m→∞} ∥p_{S_{l}} ph(E_{1})p_{𝟙_{2}} ∥^{2} > 1 − \frac{1}{2}ϵ.

This implies that for all m greater than some m_{0} we have

∥p_{S_{l}} ph(E_{1})p_{𝟙_{2}} ∥^{2} > 1 − ϵ

on the representation V^{σ^{(m,0,−m)}}. Therefore we obtain

∥(1 − p_{S_{l}}) ph(E_{1})p_{𝟙_{2}} ∥^{2} = ∥ ph(E_{1})p_{𝟙_{2}} ∥^{2} − ∥p_{S_{l}} ph(E_{1})p_{𝟙_{2}} ∥^{2} < ϵ

for all m > m_{0}. Let S ⊂⊂ Irr(U_{q}^{{2}}(3)) be the finite set containing S_{l} as well as the finite collection of U_{q}^{{2}}(3)-types which appear in any of the representations of highest weight (m, 0, −m) for m = 0, …, m_{0}. By construction, we have:

• (1 − p_{S}) ph(E_{1})p_{𝟙_{2}} = 0 on every V^{σ^{μ}} with μ not of the form (m, 0, −m);

• (1 − p_{S}) ph(E_{1})p_{𝟙_{2}} = 0 on every V^{σ^{(m,0,−m)}} with m ≤ m_{0};

• ∥(1 − p_{S}) ph(E_{1})p_{𝟙_{2}} ∥ < \sqrt{ϵ} on every V^{σ^{(m,0,−m)}} with m > m_{0}.

We conclude that on any unitary U_{q}(3)-representation H, the operator ph(E_{1})p_{𝟙_{2}} is approximated to within \sqrt{ϵ} by p_{S} ph(E_{1})p_{𝟙_{2}} . This proves that ph(E_{1})p_{𝟙_{2}} ∈ 𝕂_{U_{q}^{{2}}(3)}(H). The proof that ph(F_{1})p_{𝟙_{2}} ∈ 𝕂_{U_{q}^{{2}}(3)}(H) is similar.

Corollary 8.7. With K_{q} = SU_{q}(3), let 𝟙_{2} denote the trivial corepresentation of K_{q}^{{2}}. On any unitary K_{q}-representation H, the operators ph(E_{1})p_{𝟙_{2}} and ph(F_{1})p_{𝟙_{2}} belong to 𝕂_{2}(H).

Proof. We reuse the notation from the proof of Proposition 7.9. Recall that T ∩ Z acts trivially on any irreducible K_{q}-representation which contains the trivial K_{q}^{{2}}-type. Putting H′ = p_{𝟙_{T ∩Z}} H, we have p_{𝟙_{2}} = 0 on H′^{⊥}, so it suffices to prove the result with H′ in place of H. The K_{q}-representation on H′ extends to a U_{q}(3)-representation in which Z acts trivially. With this extension, 𝕂_{Z}(H′) = 𝕃(H′). One can check that Z and K_{q}^{{2}} are commuting and generating quantum subgroups of U_{q}^{{2}}(3), so the result follows from Lemma 8.6 and Lemma 3.11.

Lemma 8.8. With K_{q} = SU_{q}(3), let H be a unitary K_{q}-representation. Then we have ph(E_{i}) ∈ 𝔸(H) and ph(F_{i}) ∈ 𝔸(H) for i = 1, 2.

Proof. Let us first assume that H has finite K_{q}-multiplicities. In order to prove ph(E_{1}) ∈ 𝔸(H), we only need to show ph(E_{1}) ∈ 𝔸_{2}(H) since ph(E_{1}) ∈ 𝔸_{1}(H) is clear. Let τ ∈ Irr(K_{q}^{{2}}). Choose a finite dimensional K_{q}-representation V which contains τ as a K_{q}^{{2}}-type, and use Lemma 3.13 to factorize ph(E_{1})p_{τ} on H as

ph(E_{1})p_{τ} = ph(E_{1})(Id_{H} ⊗ῑ)(p_{𝟙_{2}} ⊗ Id_{V})(Id_{H} ⊗ι)
= (Id_{H} ⊗ῑ)(ph(E_{1}) ⊗ Id_{V^{c}} ⊗ Id_{V})(p_{𝟙_{2}} ⊗ Id_{V})(Id_{H} ⊗ι).  (8.4)

Write

(ph(E_{1}) ⊗ Id_{V^{c}})p_{𝟙_{2}} = ph(Δ̂(E_{1}))p_{𝟙_{2}} + (ph(E_{1}) ⊗ Id_{V^{c}} − ph(Δ̂(E_{1})))p_{𝟙_{2}} .

We have ph(Δ̂(E_{1}))p_{𝟙_{2}} ∈ 𝕂_{2}(H ⊗ V) by Corollary 8.7. We also have that (ph(E_{1}) ⊗ Id_{V^{c}} − ph(Δ̂(E_{1})))p_{𝟙_{2}} ∈ 𝕂_{2}(H ⊗ V) 𝕂_{1}(H ⊗ V) by Corollary 8.5, and since H ⊗ V has finite K_{q}-multiplicities, Lemma 3.7 shows that this is in 𝕂(H ⊗ V) ⊆ 𝕂_{2}(H ⊗ V). We conclude that ph(E_{1})p_{τ} ∈ 𝕂_{2}(H). One can similarly show that ph(F_{1})p_{τ} ∈ 𝕂_{2}(H). Moreover, by taking adjoints, we obtain p_{τ} ph(E_{1}), p_{τ} ph(F_{1}) ∈ 𝕂_{2}(H). Using Lemma 3.6, we conclude that ph(E_{1}) and ph(F_{1}) are in 𝔸_{2}(H). Suppose now that H does not necessarily have finite K_{q}-multiplicities. We can embed H into the universal K_{q}-representation H_{0} = L^{2}(K_{q}) ⊗ ℓ^{2}(ℕ), where ℓ^{2}(ℕ) is equipped with the trivial K_{q}-representation. Since ℓ^{2}(ℕ) contains only the trivial K_{q}^{{2}}-type we have Id_{ℓ^{2}(ℕ)} ∈ 𝕂_{2}(ℓ^{2}(ℕ)). Now ph(E_{1}) acts on H_{0} as ph(E_{1}) ⊗ Id_{ℓ^{2}(ℕ)}. This operator belongs to 𝔸_{2}(L^{2}(K_{q})) ⊗ 𝕂_{2}(ℓ^{2}(ℕ)), and hence to 𝔸_{2}(H_{0}) by Lemma 3.12. It follows that the restriction of ph(E_{1}) to H belongs to 𝔸_{2}(H). A similar argument shows that ph(F_{1}) ∈ 𝔸_{2}(H). To show that ph(E_{2}), ph(F_{2}) ∈ 𝔸_{1}(H) it suffices to use the automorphism Ψ of Equation (6.2) to interchange the simple roots. This completes the proof.

Proposition 8.9. Let K_{q} = SU_{q}(n) and let H be any unitary K_{q}- representation. For each i = 1, …, n − 1, the operators ph(E_{i}) : H → H and ph(F_{i}) : H → H belong to 𝔸(H).

Proof. We need to prove ph(E_{i}) ∈ 𝔸_{j}(H) for every j ∈ Σ. Note that S_{q}^{j} and T^{j⊥} are commuting and generating quantum subgroups of K_{q}^{j}. If v ∈ H is of T^{j⊥}-type λ, then ph(E_{i})v is of T^{j⊥}-type λ + α_{i}|_{T^{j⊥}}, where by abuse of notation we are identifying α_{i} ∈ P with its exponential in T̂. It follows that ph(E_{i}) ∈ 𝔸_{T^{j⊥}}(H). By Lemma 3.11 it remains only to prove ph(E_{i}) ∈ 𝔸_{S_{q}^{j}}(H). If i = j, this is immediate. If |i − j| = 1, then E_{i}, F_{i}, E_{j}, F_{j} belong to a subalgebra of Ŭ_{q}(𝔰𝔩_{n}) isomorphic to Ŭ_{q}(𝔰𝔩_{3}), and the result follows from Lemma 8.8. Finally, if |i − j| > 1, then ph(E_{i}) commutes with Ŭ_{q}(𝔰^{j}) so it preserves S_{q}^{j}-types and the result follows. By taking adjoints, we also obtain ph(F_{i}) ∈ 𝔸(H).

We can now prove parts b) and c) of Theorem 5.2. Consider D_{i} = \begin{matrix}0 & F_{i} \\ E_{i} & 0 \end{matrix}

acting on L^{2}(E_{μ} ⊕ E_{μ+α_{i}}) for some μ ∈ P. Theorem 5.2 b) follows directly from Proposition 8.9. In order to prove part c) let ϕ ∈ C_{b}(ℝ) be a continuous odd function such that ϕ(D_{i}) = ph(D_{i}). We know from Remark 8.3 that (1+D_{i}^{2})^{−1} ∈ 𝕁_{i}(L^{2}(E_{μ}⊕E_{μ+α_{i}})). Since ϕ(D_{i}) ∈ 𝔸(L^{2}(E_{μ}⊕E_{μ+α_{i}})) we also have ϕ(D_{i})(1 + D_{i}^{2})^{−1} ∈ 𝕁_{i}(L^{2}(E_{μ} ⊕ E_{μ+α_{i}})). By the Stone-Weierstrass Theorem the functions x ↦ (1 + x^{2})^{−1} and x ↦ ϕ(x)(1 + x^{2})^{−1} generate a dense subalgebra of C_{0}(ℝ), so Theorem 5.2 c) follows.

8.4 Commutator of functions with the phase of a longitudinal Dirac operator

In this subsection we prove Theorem 5.2 d). For any λ ∈ 𝔥^{∗} one may define an element K_{λ} in C(K̂_{q}) by declaring that K_{λ} acts on the weight ν subspace of any irreducible K_{q}-representation by multiplication by q^{\frac{1}{2} (λ,μ)}. If λ = α_{i} is a simple root, then K_{α_{i}} is the generator K_{i} of Ŭ_{q}(𝔰𝔩_{n}). The element K_{2ρ} ∈ C(K̂_{q}), where ρ is the half-sum of all positive roots, shows up in the Schur orthogonality relations. Specifically, the L^{2}-norms of the matrix coefficients of an irreducible unitary representation σ of K_{q} satisfy

∥⟨ξ^{∗}| · |ξ⟩∥ = \frac{1}{dim_{q}(σ)^{\frac{1}{2}}} ∥K_{2ρ} · ξ^{∗}∥∥ξ∥  (8.5)

for ξ ∈ V^{σ} and ξ^{∗} ∈ V^{σ∗}, where dim_{q} denotes the quantum dimension. We remark that the Hilbert space structure on V^{σ∗} is induced from the canonical isometric isomorphism of V^{σ∗} with the conjugate Hilbert space of V^{σ}. Moreover K_{2ρ} ∈ C(K̂_{q}) acts by the transpose action on V^{σ∗}. Let us derive an estimate on slightly more general matrix coefficients.

Lemma 8.10. Fix σ ∈ Irr(K_{q}). For any τ ∈ Irr(K_{q}) and ζ ∈ V^{τ} ⊗ V^{σ}, ζ^{∗} ∈ V^{τ∗} ⊗ V^{σ∗}, we have

∥⟨ζ^{∗}| · |ζ⟩∥ ≤ \frac{dim_{q}(σ)^{\frac{1}{2}}}{dim_{q}(τ)^{\frac{1}{2}}} ∥K_{2ρ} · ζ^{∗}∥ ∥ζ∥.

Here all norms are Hilbert space norms.

Proof. Take an orthogonal decomposition τ ⊗ σ = ⨁ _{j} τ_{j} where the τ_{j} are irreducible K_{q}-subrepresentations of τ ⊗ σ. Correspondingly, we decompose ζ = ∑ _{j} ζ_{j} and ζ^{∗} = ∑ _{j} ζ_{j}^{∗} where ζ_{j} ∈ V^{τ_{j}}, ζ_{j}^{∗} ∈ V^{τ_{j}∗}. Since τ ≤ τ_{j} ⊗ σ^{c} we have dim_{q}(τ) ≤ dim_{q}(τ_{j}) dim_{q}(σ). We obtain

∥⟨ζ^{∗}| · |ζ⟩∥^{2} = ∑_{j} \frac{1}{dim_{q}(τ_{j})}∥K_{2ρ} · ζ_{j}^{∗}∥^{2}∥ζ_{j}∥^{2} ≤ ∑_{j} \frac{dim_{q}(σ)}{dim_{q}(τ)}∥K_{2ρ} · ζ_{j}^{∗}∥^{2}∥ζ_{j}∥^{2}.

The result follows.

Let μ, ν ∈ P, i ∈ Σ and f ∈ C(E_{ν}). We will use the bracket [ph(E_{i}), M_{l}(f)] to denote the operator ph(E_{i})M_{l}(f) − M_{l}(f) ph(E_{i}) : L^{2}(E_{μ}) → L^{2}(E_{μ+ν+α_{i}}) of Theorem 5.2 d). From the other parts of Theorem 5.2, we know that this operator belongs to 𝔸(L^{2}(E_{μ}), L^{2}(E_{μ+ν+α_{i}})), so Theorem 5.2 d) is a consequence of the following lemma.

Lemma 8.11. Fix μ, ν ∈ P, i ∈ Σ and f ∈ C(E_{ν}). Then [ph(E_{i}), M_{l}(f)] ∈ 𝕂_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+ν+α_{i}})). Similarly, [ph(F_{i}), M_{l}(f)] ∈ 𝕂_{i}(L^{2}(E_{μ+α_{i}}), L^{2}(E_{μ+ν})).

Proof. Again, we write σ^{λ} for the irreducible K_{q}-representation with highest weight λ ∈ P^{+}. Let Wts(σ^{λ}) ⊂ P denote the set of weights occurring in σ^{λ}. We will assume that f = ⟨ξ^{∗}| · |ξ⟩ is a matrix coefficient of σ^{λ} ∈ Irr(K_{q}), and that ξ has weight ν and K_{q}^{i}-type β for some β ∈ Irr(K_{q}^{i}). Such f span a dense subspace of C(E_{ν}). Let ϵ > 0. By Corollary 8.5 we can find a finite set S ⊂⊂ Irr(K_{q}^{i}) such that for any unitary K_{q}-representation H we have the following estimates, where all operators are acting on H ⊗ V^{σ^{λ}}:

∥(ph(Δ̂E_{i}) − ph(E_{i}) ⊗ Id_{V^{σ^{λ}}})p_{μ+ν}(Id −p_{S})∥
< ϵ/(|Wts(σ^{λ})| dim_{q}(σ^{λ})^{\frac{1}{2}} ∥K_{2ρ} · ξ^{∗}∥∥ξ∥),
∥(Id −p_{S})(ph(Δ̂E_{i}) − ph(E_{i}) ⊗ Id_{V^{σ^{λ}}})p_{μ+ν}∥
< ϵ/(|Wts(σ^{λ})| dim_{q}(σ^{λ})^{\frac{1}{2}} ∥K_{2ρ} · ξ^{∗}∥∥ξ∥).

Let S′ be the set of all γ′ ∈ Irr(K_{q}^{i}) such that γ′ ⊗ β contains some K_{q}^{i}-type γ belonging to S. This is a finite set, since any such γ′ is a subrepresentation of γ ⊗ β^{c} with γ ∈ S.

Let g ∈ L^{2}(E_{μ}) be a matrix coefficient g = ⟨η^{∗}| · |η⟩ of an irreducible K_{q}- representation σ^{κ}. Using Lemma 8.10 we obtain

∥[ph(E_{i}), M_{l}(f)](Id −p_{S′})g∥

= ∥⟨η^{∗} ⊗ ξ^{∗}| · |(ph(Δ̂E_{i}) − ph(E_{i}) ⊗ Id)(((Id −p_{S′})η) ⊗ ξ)⟩∥

= ∥⟨η^{∗} ⊗ ξ^{∗}| · |(ph(Δ̂E_{i}) − ph(E_{i}) ⊗ Id)p_{μ+ν}(Id −p_{S})((Id −p_{S′})η) ⊗ ξ⟩∥

≤ \frac{dim_{q}(σ^{λ})^{\frac{1}{2}}}{dim_{q}(σ^{κ})^{\frac{1}{2}}} ∥K_{2ρ} · η^{∗}∥∥K_{2ρ} · ξ^{∗}∥

× ∥(ph(Δ̂E_{i}) − ph(E_{i}) ⊗ Id)p_{μ+ν}(Id −p_{S})∥∥η∥∥ξ∥

< \frac{ϵ}{|Wts(σ^{λ})| dim_{q}(σ^{κ})^{\frac{1}{2}}} ∥K_{2ρ} · η^{∗}∥∥η∥

= \frac{ϵ}{|Wts(σ^{λ})|}∥g∥.  (8.6)

A similar calculation shows that

∥(Id −p_{S})[ph(E_{i}), M_{l}(f)]g∥ < \frac{ϵ}{|Wts(σ^{λ})|}∥g∥.  (8.7)

If g ∈ L^{2}(E_{μ}) is instead a sum of irreducible matrix coefficients, then the inequalities (8.6) and (8.7) do not necessarily hold. To resolve this, we will take advantage of the fact that the operator [ph(E_{i}), M_{l}(f)](Id −p_{S′}) is banddiagonal with respect to K_{q}-types in the following sense. The operators ph(E_{i}) and (Id −p_{S}) commute with the K_{q}-isotypical projections, while M_{l}(f) satisfies p_{σ^{κ′}} M_{l}(f)p_{σ^{κ}} = 0 unless σ^{κ′} occurs as an irreducible subrepresentation of σ^{κ} ⊗ σ^{λ}. We note that if σ^{κ′} ≤ σ^{κ} ⊗ σ^{λ} then κ′ = κ + ω for some weight ω of σ^{λ}. Therefore we have a decomposition

[ph(E_{i}),M_{l}(f)](Id −p_{S′})
= ∑_{ω∈Wts(σ^{λ})} ( ∑_{κ∈P^{+}} p_{σ^{κ+ω}}[ph(E_{i}), M_{l}(f)](Id −p_{S′})p_{σ^{κ}}) ,  (8.8)

where we take the convention that p_{σ^{κ+ω}} = 0 if κ + ω is not dominant. By the calculation (8.6) for irreducible matrix coefficients, p_{σ^{κ+ω}}[ph(E_{i}), M_{l}(f)](Id −p_{S′})p_{σ^{κ}} is norm bounded by ϵ/|Wts(σ^{λ})| for each κ, ω. Since the projections p_{σ^{κ}} are mutually orthogonal, the sum in parentheses in (8.8) is bounded by ϵ/|Wts(σ^{λ})| for each fixed ω. We conclude that ∥[ph(E_{i}), M_{l}(f)](Id −p_{S′})∥ < ϵ. Similarly, one obtains ∥(Id −p_{S})[ph(E_{i}), M_{l}(f)]∥ < ϵ. This completes the proof that [ph(E_{i}), M_{l}(f)] ∈ 𝕂_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+ν+α_{i}})) for all f ∈ C(E_{μ}). Finally, one can obtain [ph(F_{i}), M_{l}(f)] ∈ 𝕂_{i}(L^{2}(E_{μ+α_{i}}), L^{2}(E_{μ+ν})) by taking adjoints.

9 The action of SL_{q}(n, ℂ)

In this section we recall the definition of the quantum group SL_{q}(n, ℂ) and its principal series representations, and prove some estimates that will be used later.

9.1 The complex quantum group SL_{q}(n, ℂ)

The quantized complex semisimple Lie group G_{q} = SL_{q}(n, ℂ) is defined as the quantum double of K_{q} = SU_{q}(n). More precisely, the C^{∗}-algebra of functions on G_{q} is given by

C_{0}(G_{q}) = C(K_{q}) ⊗ C_{0}(K̂_{q})

with the comultiplication

Δ_{G_{q}} = (Id ⊗σ ⊗ Id)(Id ⊗ ad(W) ⊗ Id)(Δ ⊗ Δ̂),

where ad(W) is conjugation with the multiplicative unitary W ∈ M(C(K_{q}) ⊗ C^{∗}(K_{q})) of K_{q} and σ denotes the flip map. In the special case n = 2 this quantum group has been studied in detail by Podleś and Woronowicz [PW90]. The unitary representations of G_{q} are in one-to-one correspondence with unitary Yetter-Drinfeld modules for K_{q}, compare [NV10]. Passing to the subspace of K_{q}-finite vectors, one can study such representations algebraically, namely in terms of Yetter-Drinfeld modules over the Hopf algebra O(K_{q}). We recall that a Yetter-Drinfeld module over O(K_{q}) is a vector space V equipped with both a left action and a left coaction of O(K_{q}) in the purely algebraic sense, satisfying the compatibility condition

f_{(1)}ξ_{(−1)}S(f_{(3)}) ⊗ f_{(2)} · ξ_{(0)} = (f · ξ)_{(−1)} ⊗ (f · ξ)_{(0)}

for f ∈ O(K_{q}) and ξ ∈ V. Here we use the Sweedler notation ξ ↦ ξ_{(−1)} ⊗ ξ_{(0)} for the coaction V → O(K_{q})⊗V, and we write f ·v for the action of f ∈ O(K_{q}) on v ∈ V.

9.2 Principal series representations of SL_{q}(n, ℂ)

As mentioned before, For λ ∈ 𝔥^{∗} one may define an element K_{λ} of C(K̂_{q}) by declaring that K_{λ} acts on the weight ν subspace of any irreducible K_{q}- representation by multiplication by q^{\frac{1}{2} (λ,ν)}. We note that K_{λ} = K_{λ′} if λ ≡ λ′ modulo 2iℏ^{−1}Q, where ℏ = \frac{log(q)}{2π} and Q is the root lattice. We write 𝔥_{q}^{∗} = 𝔥^{∗}/2iℏ^{−1}Q and i𝔱_{q}^{∗} = i𝔱^{∗}/2iℏ^{−1}Q. Our conventions here are adjusted to the quantized enveloping algebra Ŭ_{q}(𝔰𝔩_{n}); recall that in the notation of [KS97, §6.1.2] the element K_{i} in U_{q}(𝔰𝔩_{n}) corresponds to K_{i}^{2} in Ŭ_{q}(𝔰𝔩_{n}). The principal series Yetter-Drinfeld modules are parametrized by pairs (μ, λ) ∈ P × 𝔥_{q}^{∗}. We will denote the principal series Yetter-Drinfeld module with parameter (μ, λ) by O(E_{μ,λ}). As a K_{q}-representation it is just O(E_{μ}) with the left regular coaction. The action of O(K_{q}) is given by

π_{μ,λ}(a)g = a_{(1)} g (K_{λ+ρ}^{2} ⇀ S(a_{(2)}))  (9.1)

for a ∈ O(K_{q}) and g ∈ O(E_{μ}). For λ ∈ i𝔱_{q}^{∗} the representations π_{μ,λ} are ∗-representations with respect to the standard inner product on O(E_{μ}), and the resulting unitary representations of G_{q} on the completion L^{2}(E_{μ}) of O(E_{μ}) are called unitary principal series representations. We will write L^{2}(E_{μ,λ}) = L^{2}(E_{μ}) if we want to emphasize the corresponding Yetter-Drinfeld structure. We will only need the unitary principal series representations with parameters (μ, 0), and we shall make use of the following properties.

Theorem 9.1. a) The representations L^{2}(E_{μ,0}) are irreducible.

b) The representations L^{2}(E_{μ,0}) and L^{2}(E_{ν,0}) are equivalent if and only if μ and ν belong to the same orbit of the Weyl group action on P. In particular, if ν = w_{i}μ where w_{i} is the reflection associated to a simple root α_{i}, then the representations are intertwined by the operator

ph(E_{i})^{n} : L^{2}(E_{μ,0}) ⟶^{≅} L^{2}(E_{w_{i}μ,0}),  if w_{i}μ − μ = nα_{i} with n > 0,
ph(F_{i})^{n} : L^{2}(E_{μ,0}) ⟶^{≅} L^{2}(E_{w_{i}μ,0}),  if w_{i}μ − μ = nα_{i} with n < 0.

The above facts are at least partially “known to experts,” although they do not appear in this form in the literature. They are quantum analogues of results about the principal series representations of classical complex semisimple Lie groups as described, for instance, in [Duf75, Ch. III & IV]; see also [Yun11a, Section 4.2] for a formulation along the present lines. For SL_{q}(2, ℂ) the above results are essentially contained in Theorems 5.2 and 6.3 of [PW00]. The results for SL_{q}(n, ℂ) can be deduced from this case. We refer to [VY] for a detailed exposition.

9.3 Almost SL_{q}(n, ℂ)-equivariance of the phases of E_{i} and F_{i}

A straightforward computation shows that the multiplication operators on L^{2}(E_{μ}) satisfy the following covariance property with respect to principal series representations.

Lemma 9.2. Let μ, ν ∈ P and f ∈ O(E_{ν}), so that M_{l}(f) defines an operator from L^{2}(E_{μ}) to L^{2}(E_{μ+ν}). Then for any a ∈ O(K_{q}),

π_{μ+ν,0}(a) M_{l}(f) = M_{l}(a_{(1)}fS(a_{(2)})) π_{μ,0}(a_{(3)}).

The next result will be used in the proof of the equivariance properties of our K-homology cycle.

Theorem 9.3. Let K_{q} = SU_{q}(n) for n ≥ 2. Moreover let μ ∈ P and i ∈ Σ. a) For any a ∈ O(K_{q}), we have π_{μ,0}(a) ∈ 𝔸(L^{2}(E_{μ})).

b) The operators ph(E_{i}) : L^{2}(E_{μ,0}) → L^{2}(E_{μ+α_{i},0}) and ph(F_{i}) : L^{2}(E_{μ+α_{i},0}) → L^{2}(E_{μ,0}) are K_{q}-equivariant. Moreover, they are SL_{q}(n, ℂ)-equivariant modulo 𝕁_{i}, in the sense that for any a ∈ O(K_{q}),

π_{μ+α_{i},0}(a) ph(E_{i}) − ph(E_{i})π_{μ,0}(a) ∈ 𝕁_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+α_{i}})),  (9.2)
π_{μ,0}(a) ph(F_{i}) − ph(F_{i})π_{μ+α_{i},0}(a) ∈ 𝕁_{i}(L^{2}(E_{μ+α_{i}}), L^{2}(E_{μ})).  (9.3)

Proof. The Yetter-Drinfeld action of a on L^{2}(E_{μ,0}) can be written as π_{μ,0}(a) = M_{l}(a_{(1)}) M_{r}(K_{ρ} ⇀ S(a_{(2)})), so a) follows from Proposition 8.1 The K_{q}-equivariance of ph(E_{i}) and ph(F_{i}) is immediate from the fact that they are defined via the right regular representation. Let i ∈ Σ. We have w_{i}ρ = ρ − α_{i}, so according to Theorem 9.1 the operator ph(E_{i}) : L^{2}(E_{ρ−α_{i},0}) → L^{2}(E_{ρ,0}) is an intertwiner. Thus, the differences in (9.2) and (9.3) are zero when μ = ρ−α_{i}. For general μ, we use Lemma 2.1 to obtain f_{1}, …, f_{k} ∈ O(E_{μ+α_{i}−ρ}) and g_{1}, …, g_{k} ∈ O(E_{ρ−α_{i}−μ}) such that ∑ _{j} g_{j}f_{j} = 1. We can then use Theorem 5.2 and Lemma 9.2 to compute

π_{μ+α_{i},0}(a) ph(E_{i})

= ∑_{j} π_{μ+α_{i},0}(a) ph(E_{i})M_{l}(f_{j})M_{l}(g_{j})

≡ ∑_{j} π_{μ+α_{i},0}(a)M_{l}(f_{j}) ph(E_{i})M_{l}(g_{j}) (mod 𝕁_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+α_{i}})))

= ∑_{j} M_{l}(a_{(1)}f_{j}S(a_{(2)}))π_{ρ,0}(a_{(3)}) ph(E_{i})M_{l}(g_{j}),

noting that all operators involved belong to 𝔸. A similar computation yields

∑_{j} ph(E_{i})π_{μ,0}(a) ≡ ∑_{j} M_{l}(a_{(1)}f_{j}S(a_{(2)})) ph(E_{i})π_{ρ−α_{i},0}(a_{(3)})M_{l}(g_{j})

(mod 𝕁_{i}(L^{2}(E_{μ}), L^{2}(E_{μ+α_{i}}))).

Thus Equation (9.2) is reduced to the case μ = ρ − α_{i} which we have just proved. Equation (9.3) follows by taking adjoints.

10 BGG elements in K-homology

In [Yun11a] it was shown how an equivariant Fredholm module can be constructed from the geometric BGG complex for the full flag manifold of SU(3). Given the results of the previous sections, that construction can now be applied also to the quantized flag manifold of SU_{q}(3). The construction carries over almost word for word, so we shall merely give an outline of the steps involved here.

10.1 The normalized BGG complex

The reader can consult [BGG75] or [BE89] for the general combinatorial structure underlying the BGG complex of a complex semisimple Lie group. Since we only need a bounded version of the BGG complex for SL_{q}(3, ℂ), we will proceed in an ad hoc manner.

Lemma 10.1. The following is a commuting diagram of intertwining operators between SL_{q}(3, ℂ) principal series representations:

\xymatrix{ & L^{2}(E_{−α_{2},0}) \ar[ddr]^{ph(E_{2})^{2}} & L^{2}(E_{α_{1},0}) \ar@{-->}[dr]^{ph(E_{2})} \\ L^{2}(E_{−ρ,0}) \ar@{-->}[ur]^{ph(E_{1})} \ar@{-->}[dr]_{ph(E_{2})} &  &  & L^{2}(E_{ρ,0}) \\  & L^{2}(E_{−α_{1},0}) \ar[uur]_{ph(E_{1})^{2}} & L^{2}(E_{α_{2},0}) \ar@{-->}[ur]_{ph(E_{1})}}  (10.1)

Proof. That these operators are intertwiners results from Theorem 9.1. By Schur’s Lemma, the diagram commutes up to a scalar. By checking on the minimal K_{q}-type, one can verify that the diagram commutes on the nose.

To define the normalized BGG complex, we displace all the weights in the above construction by ρ = α_{1} + α_{2}.

Lemma 10.2. The following diagram commutes modulo 𝕁_{1} + 𝕁_{2}:

\xymatrix{ & L^{2}(E_{α_{1},0}) \ar[ddr]^{ph(E_{2})^{2}} & L^{2}(E_{2α_{1}+α_{2},0}) \ar@{-->}[dr]^{ph(E_{2})} \\ L^{2}(E_{0,0}) \ar@{-->}[ur]^{ph(E_{1})} \ar@{-->}[dr]_{ph(E_{2})} &  &  & L^{2}(E_{2ρ,0}) \\  & L^{2}(E_{α_{2},0}) \ar[uur]_{ph(E_{1})^{2}} & L^{2}(E_{α_{1}+2α_{2},0}) \ar@{-->}[ur]_{ph(E_{1})}}

Proof. According to Lemma 2.1 we find f_{1}, …, f_{k} ∈ C(E_{ρ}), g_{1}, …, g_{k} ∈ C(E_{−ρ}) such that ∑ _{j} f_{j}g_{j} = 1. Consider the composition ph(E_{1}) ph(E_{2})^{2} ph(E_{1}) : L^{2}(E_{0,0}) → L^{2}(E_{2ρ,0}). By Theorem 5.2,

ph(E_{1}) ph(E_{2})^{2} ph(E_{1}) = ∑_{i} M_{l}(f_{i})M_{l}(g_{i}) ph(E_{1}) ph(E_{2})^{2} ph(E_{1})
≡ ∑_{i} M_{l}(f_{i}) ph(E_{1}) ph(E_{2})^{2} ph(E_{1})M_{l}(g_{i})

mod 𝕁_{1}(L^{2}(E_{0}), L^{2}(E_{2ρ})) + 𝕁_{2}(L^{2}(E_{0}), L^{2}(E_{2ρ})),

where the operators in the last expression are the intertwiners of Lemma 10.1. By a similar calculation, we obtain

ph(E_{2}) ph(E_{1})^{2} ph(E_{2}) ≡ ∑_{i} M_{l}(f_{i}) ph(E_{2}) ph(E_{1})^{2} ph(E_{2})M_{l}(g_{i})

mod 𝕁_{1}(L^{2}(E_{0}), L^{2}(E_{2ρ})) + 𝕁_{2}(L^{2}(E_{0}), L^{2}(E_{2ρ})), and the result then follows from Lemma 10.1.

Lemma 10.3. Let μ ∈ P, i ∈ Σ and n ∈ ℕ. Then ph(F_{i})^{n} ph(E_{i})^{n} − Id ∈ 𝕁_{i}(L^{2}(E_{μ})) and ph(E_{i})^{n} ph(F_{i})^{n} − Id ∈ 𝕁_{i}(L^{2}(E_{μ})).

Proof. Let μ_{α_{i}} ∈ \frac{1}{2}ℕ be the restriction of μ to a weight of S_{q}^{i} ≅ SU_{q}(2). The operator ph(E_{i})^{n} : L^{2}(E_{μ}) → L^{2}(E_{μ+nα_{i}}) is a partial isometry, and its kernel is the span of those S_{q}^{i}-isotypical subspaces whose highest weight l ∈ \frac{1}{2}ℕ satisfies l < μ_{α_{i}} +n. Therefore ph(F_{i})^{n} ph(E_{i})^{n}−Id is a projection onto a finite number of S_{q}^{i}-types, and hence K_{q}^{i}-types, in L^{2}(E_{μ}). A similar statement can be made for ph(E_{i})^{n} ph(F_{i})^{n} − Id. The result then follows from Theorem 5.1 b).

We now augment the diagram of Lemma 10.2 by adding two more operators:

\xymatrix{ & L^{2}(E_{α_{1},0}) \ar[ddr]^{ph(E_{2})^{2}} \ar[r]^{−A_{1}} & L^{2}(E_{2α_{1}+α_{2},0}) \ar@{-->}[dr]^{ph(E_{2})} \\ L^{2}(E_{0,0}) \ar@{-->}[ur]^{ph(E_{1})} \ar@{-->}[dr]_{ph(E_{2})} &  &  & L^{2}(E_{2ρ,0}) \\  & L^{2}(E_{α_{2},0}) \ar[uur]_{ph(E_{1})^{2}} \ar[r]_{−A_{2}} & L^{2}(E_{α_{1}+2α_{2},0}) \ar@{-->}[ur]_{ph(E_{1})}}  (10.2)

where

A_{1} = ph(E_{1})^{2} ph(E_{2}) ph(F_{1})
A_{2} = ph(E_{2})^{2} ph(E_{1}) ph(F_{2}).

By Lemma 10.3, A_{1}^{∗}A_{1} ≡ Id modulo 𝕁_{1}(L^{2}(E_{α_{1}})) + 𝕁_{2}(L^{2}(E_{α_{1}})) and A_{1}A_{1}^{∗} ≡ Id modulo 𝕁_{1}(L^{2}(E_{2α_{1}+α_{2}})) + 𝕁_{2}(L^{2}(E_{2α_{1}+α_{2}})). Similar statements hold for A_{2}. The inclusion of the minus signs before A_{1} and A_{2} in the diagram (10.2) ensures that all squares in the diagram anti-commute modulo 𝕁_{1} + 𝕁_{2}. Thus (10.2) is a complex modulo 𝕁_{1} + 𝕁_{2}. The combinatorial structure underlying the diagram (10.2) is the Bruhat graph of the group G = SL(3, ℂ). Rather than detail this in generality, let us simply introduce some convenient notation.

Definition 10.4. Let Γ be the set of arrows in the diagram (10.2) and Γ^{(0)} the set of six vertices. Denote by T_{γ} the operator corresponding to γ ∈ Γ in (10.2). Also, to each arrow γ we associate a set of simple roots, denoted supp(γ) and called the support of γ, according to the Weyl reflection underlying it as follows:

\xymatrix{ & · \ar[ddr]^{\{α_{2}\}} \ar[r]^{\{α_{1},α_{2}\}} & · \ar[dr]^{\{α_{2}\}} \\ · \ar[ur]^{\{α_{1}\}} \ar[dr]_{\{α_{2}\}} &  &  & · \\  & · \ar[uur]_{\{α_{1}\}} \ar[r]_{\{α_{1},α_{2}\}} & · \ar[ur]_{\{α_{1}\}}}

Documenta Mathematica 20 (2015) 433–490

10.2 Construction of the Fredholm module

Let H_{BGG} be the ℤ/2ℤ-graded Hilbert space which is the direct sum of the six section spaces in the BGG diagram (10.2) graded by even and odd Bruhat length, namely L^{2}(E_{0}), L^{2}(E_{2α_{1}+α_{2}}) and L^{2}(E_{α_{1}+2α_{2}}) have degree 0 and the other three summands have degree 1. The sum T = ∑ _{γ}(T_{γ} + T_{γ}^{∗}) is an odd SU_{q}(3)-equivariant operator on H_{BGG}. It verifies all the axioms of an equivariant Kasparov K-homology cycle, but modulo 𝕁_{1}(H_{BGG}) + 𝕁_{2}(H_{BGG}) instead of modulo 𝕂(H_{BGG}). To refine this into a genuine Kasparov cycle we use the operator partition of unity constructed in [Yun11a], which is described in the following lemma.

Lemma 10.5. Let K_{q} = SU_{q}(3). There exist mutually commuting operators N_{γ} ∈ 𝕃(H_{BGG}), indexed by the arrows γ ∈ Γ above, with the following properties:

a) For each γ, N_{γ}𝕁_{i}(H_{BGG}) ⊆ 𝕂(H_{BGG}) for all α_{i} ∈ supp(γ).

b) For any vertex v ∈ Γ^{(0)}, ∑ _{γ∋v} N_{γ}^{2} = Id_{H_{BGG}}, where the sum is over all arrows entering or leaving v.

c) Whenever vertices v, v′ ∈ Γ^{(0)} are at distance two in the graph we have N_{γ_{1}}N_{γ_{2}} = N_{γ′_{1}}N_{γ′_{2}} where (γ_{1}, γ_{2}) and (γ′_{1}, γ′_{2}) are the unique two (undirected) paths of length two joining v, v′.

d) Each N_{γ} is K_{q}-equivariant.

e) Each N_{γ} commutes modulo compact operators with the left multiplication action of C(X_{q}), the Yetter-Drinfeld action of O(K_{q}) and all of the normalized BGG operators T_{γ′}.

Proof. Firstly, one applies the Kasparov technical theorem of Baaj-Skandalis [BS89, Theorem 4.3] with respect to the Drinfeld double G_{q} = D(K_{q}). This yields an almost G_{q}-invariant operator M_{0} ∈ L^{2}(H_{BGG}), in the sense that (C_{0}(G_{q}) ⊗ 1)(δ(M_{0}) − 1 ⊗ M_{0}) ∈ C_{0}(G) ⊗ 𝕂(H_{BGG}), which satisfies M_{0}𝕁_{1}(H_{BGG}), (1 − M_{0})𝕁_{2}(H_{BGG}) ⊆ 𝕂(H_{BGG}) and such that M_{0} commutes modulo compacts with the left multiplication action of C(X_{q}) and the normalized BGG operators. Here δ denotes the adjoint action of G_{q} induced from H_{BGG}. From almost invariance it follows that M_{0} commutes modulo compacts with the Yetter-Drinfeld action of O(K_{q}). Moreover, if M denotes the operator obtained by averaging the K_{q}-action on M_{0} using the Haar functional of C(K_{q}), then M −M_{0} ∈ 𝕂(H_{BGG}). Thus M satisfies the same properties as M_{0} but is additionally K_{q}-invariant. Now, putting N_{1} = M^{\frac{1}{2}} , N_{2} = (1 − M)^{\frac{1}{2}} , the remainder of the construction is carried out as in [Yun11a, Lemma 4.14].

Theorem 10.6. The operator F = ∑ _{γ} N_{γ}(T_{γ} + T_{γ}^{∗}) is an odd Fredholm operator on H_{BGG} which defines an equivariant K-homology class [F] ∈ KK^{SL_{q}(3,ℂ)}(C(X_{q}), ℂ). The SU_{q}(3)-equivariant index of this class in KK^{SU_{q}(3)}(ℂ, ℂ) = R(SU_{q}(3)) is the class of the trivial representation.

Proof. The fact that F defines a K-homology class in KK^{SL_{q}(3,ℂ)}(C(X_{q}), ℂ) can be proven as for Theorem 4.19 of [Yun11a]. Note that in order to prove SL_{q}(3, ℂ)-equivariance it suffices to check that the action of O(SU_{q}(3)) corresponding to the Yetter-Drinfeld structure commutes with F up to compact operators. This in turn follows using Theorem 9.3. Finally, the SU_{q}(3)-types occur in the spaces L^{2}(E_{μ}) with the same multiplicities as their classical counterparts, so the index computation follows from the fact that the classical BGG complex is a resolution of the trivial representation.

Remark 10.7. The same general construction can be used to make a Fredholm module in KK^{SL_{q}(3,ℂ)}(C(X_{q}), ℂ) with any desired SU_{q}(3)-equivariant index. A BGG complex can be formed starting with an arbitrary weight μ (in the notation we have used here it should be an anti-dominant weight), where the weights appearing in the equivalent of the diagram (10.2) are those in the ρshifted Weyl orbit of μ. The procedure above applies, and the equivariant index of the resulting KK-cycle is the class of the irreducible representation with lowest weight μ.

11 Applications to Poincaré duality and the Baum-Connes conjecture

In this section we explain how our previous constructions imply Poincaré duality in equivariant KK-theory for the flag manifold X_{q} = SU_{q}(3)/T, and a certain analogue of the Baum-Connes conjecture for the dual of SU_{q}(3). Some of the arguments will only be sketched, and for more information and background we refer to [MN06], [MN10], [Mey08], [NV10], [Voi11]. Equivariant Poincaré duality in KK-theory with respect to quantum group actions was introduced in [NV10], where it was also shown that the standard Podleś sphere is equivariantly Poincaré dual to itself with respect to the natural action of SU_{q}(2). An important ingredient in the study of Poincaré duality with respect to quantum group actions is the use of braided tensor products, and we refer to [NV10] for definitions and more details. Our aim here is to exhibit another example of equivariant Poincaré duality in the sense of [NV10], namely for the quantum flag manifold X_{q} = SU_{q}(3)/T. The key ingredient for this is the class [F] ∈ KK^{SL_{q}(3,ℂ)}(C(X_{q}), ℂ) obtained in Theorem 10.6. It yields a class in KK^{SL_{q}(3,ℂ)}(C(X_{q}) ⊠ C(X_{q}), ℂ) by precomposing the representation of C(X_{q}) with the ∗-homomorphism C(X_{q})⊠C(X_{q}) → C(X_{q}) induced by multiplication. Here ⊠ denotes the braided tensor product over SU_{q}(3), and we write D(SU_{q}(3)) = SL_{q}(3, ℂ) for the quantum double of SU_{q}(3).

Theorem 11.1. The quantum flag manifold X_{q} is SU_{q}(3)-equivariantly Poincaré dual to itself. That is, there is a natural isomorphism

KK_{∗}^{D(SU_{q}(3))}(C(X_{q}) ⊠ A, B) ≅ KK_{∗}^{D(SU_{q}(3))}(A, C(X_{q}) ⊠ B)

for all D(SU_{q}(3))-C^{∗}-algebras A and B.

Proof. With the class [F] ∈ KK^{SL_{q}(3)}(C(X_{q})⊠C(X_{q}), ℂ) at hand, the argument is completely analogous to the proof of Theorem 6.5 in [NV10], reducing it to Poincaré duality for the classical flag manifold X_{1}. We shall therefore not go into the details. Let us remark that we do not need an explicit description of the element η_{q} ∈ KK_{∗}^{D(SU_{q}(3))}(ℂ, C(X_{q}) ⊠ C(X_{q})) corresponding to the unit of the adjunction, since this element is uniquely determined from the unit η_{1} ∈ KK_{∗}^{SU(3)}(ℂ, C(X_{1}) ⊗ C(X_{1})) of the classical SU(3)-equivariant Poincaré duality thanks to the continuous field structure of q-deformations, see [NT12], [Yam13]. We recall that η_{1} is the image of the class of the trivial line bundle 1_{X_{1}} ∈ KK_{∗}^{SU(3)}(ℂ, C(X_{1})) under the wrong way map induced by the diagonal embedding X_{1} ↪ X_{1} × X_{1}. We refer the reader to [EM10b, EM10a] for a full discussion of equivariant KK-duality for manifolds, and [RS86] for more information on the case of flag varieties.

Let us now come to the Baum-Connes conjecture. We continue to write K_{q} = SU_{q}(3) and denote by K̂_{q} the discrete quantum group dual to K_{q}. The starting point of the approach in [MN06] is to view equivariant Kasparov theory as a triangulated category. More precisely, if Γ is a discrete quantum group we consider the category KK^{Γ} which has as objects all separable Γ-C^{∗}-algebras, and KK^{Γ}(A, B) as the set of morphisms between two objects A and B. Composition of morphisms is given by the Kasparov product. For a description of the structure of KK^{Γ} as a triangulated category we refer to [NV10], [Voi11]. Suffice it to say that this extra structure allows one to do homological algebra in the context of Kasparov theory. In fact, there is one further ingredient needed in the definition of the Baum-Connes assembly map. Namely, one has to identify the category CI_{Γ} of compactly induced actions within KK^{Γ}. Classically, the objects of CI_{Γ} are the C^{∗}-algebras induced from finite subgroups of the discrete group Γ. If Γ is torsion-free the situation is particularly simple, in the sense that only the trivial subgroup has to be taken into account in this case. It turns out that the dual of K_{q} behaves like a torsion-free group. More precisely, the quantum group K̂_{q} is torsion-free in the sense that any ergodic action of K_{q} on a finite dimensional C^{∗}-algebra is K_{q}-equivariantly Morita equivalent to the trivial action on ℂ, see [Mey08], [Gof12]. For a torsion-free quantum group Γ we define the full subcategory CI_{Γ} of KK^{Γ} by

CI_{Γ} = {C_{0}(Γ) ⊗ A|A ∈ KK},

where the coaction on C_{0}(Γ)⊗A is given by comultiplication on the first tensor factor. Similarly, we let CC_{Γ} ⊂ KK^{Γ} be the full subcategory of all objects which become isomorphic to 0 in KK under the obvious forgetful functor. The subcategory CC_{Γ} is localising, and we denote by ⟨CI_{Γ}⟩ the localising subcategory generated by CI_{Γ}. Moreover, the pair of localising subcategories (⟨CI_{Γ}⟩, CC_{Γ}) in KK^{Γ} is complementary, compare [Mey08]. That is, KK^{Γ}(P, N) = 0 for all P ∈ ⟨CI_{Γ}⟩ and N ∈ CC_{Γ}, and every object A ∈ KK^{Γ} fits into an exact triangle

\xymatrix{ΣN \ar[r] & Ã \ar[r] & A \ar[r] & N}

with Ã ∈ ⟨CI_{Γ}⟩ and N ∈ CC_{Γ}. Such a triangle is called a Dirac triangle for A, it is uniquely determined up to isomorphism in KK^{Γ} and depends functorially on A.

Definition 11.2. Let Γ be a torsion-free discrete quantum group and let A be a Γ-C^{∗}-algebra. The Baum-Connes assembly map for Γ with coefficients in A is the map

μ_{A} : K_{∗}(Γ ⋉_{r} Ã) → K_{∗}(Γ ⋉_{r} A)

induced from a Dirac triangle for A. If μ_{A} is an isomorphism we shall say that Γ satisfies the Baum-Connes conjecture with coefficients in A.

By the work of Meyer and Nest [MN06], this terminology is consistent with the usual definitions in the case that Γ is a torsion-free discrete group. Using the Fredholm module for the quantum flag manifold SU_{q}(3)/T in Theorem 10.6 we obtain the following result.

Theorem 11.3. The dual of SU_{q}(3) for q ∈ (0, 1] satisfies the Baum-Connes conjecture with trivial coefficients ℂ.

Proof. We shall follow the arguments in [MN07]and show ℂ ∈ ⟨CIK̂_{q}⟩. This clearly implies that μ_{ℂ} is an isomorphism. Using Baaj-Skandalis duality, it is enough to prove C(K_{q}) ∈ ⟨T_{K_{q}}⟩, where T_{K_{q}} ⊂ KK^{K_{q}} denotes the category of all trivial K_{q}-C^{∗}-algebras. We have C(T) ⊂ ⟨T_{T} ⟩ by the Baum-Connes conjecture for the abelian group T̂, where T_{T} ⊂ KK^{T} is the category of trivial T -C^{∗}-algebras. This implies C(K_{q}) = ind_{T}^{K_{q}}(C(T)) ∈ ⟨C(K_{q}/T)⟩. Hence it suffices to show C(K_{q}/T) ∈ ⟨ℂ⟩. In the case q = 1 one obtains inverse isomorphisms α_{1} : C(K_{1}/T) → ℂ^{|W |} and β_{1} : ℂ^{|W |} → C(K_{1}/T) in KK^{K_{1}} using Poincaré duality, where |W | = 6 is the order of the Weyl group of K_{1} = SU(3), see [RS86], [MN07]. For general q we could argue in a similar way by invoking Theorem 11.1. Alternatively we may proceed as follows, avoiding the use of braided tensor products. The element β_{1} is given by induced vector bundles over the flag manifold, and one obtains a corresponding class β_{q} ∈ KK^{K_{q}}(ℂ^{|W |}, C(K_{q}/T)) for any q ∈ (0, 1] using the induction isomorphism KK^{K_{q}}(ℂ, C(K_{q}/T)) ≅ KK^{T} (ℂ, ℂ). Similarly, the element α_{1} is given by twisted Dolbeault operators. Using theorem 10.6 we obtain a corresponding class α_{q} in KK^{K_{q}}(C(K_{q}/T), ℂ^{|W |}). From K_{q}-equivariance it is immediate that we have β_{q} ∘ α_{q} = Id in KK^{K_{q}}(ℂ^{|W |}, ℂ^{|W |}) for the classes thus obtained. To check α_{q} ∘ β_{q} = Id in KK^{K_{q}}(C(K_{q}/T), C(K_{q}/T)) we may use the canonical isomorphism KK^{K_{q}}(C(K_{q}/T), C(K_{q}/T)) ≅ KK^{T} (C(K_{q}/T), ℂ) and the fact that the C^{∗}- algebras C(K_{q}/T) form a T -equivariant continuous field, implementing a KK^{T} - equivalence between C(K_{q}/T) and C(K/T), see [NT12], [Yam13]. It therefore suffices to consider the effect of α_{q} ∘ β_{q} on K_{∗}^{T} (C(K_{q}/T)) ≅ R(K) ⊗_{R(T)} R(K), which is the same for all q ∈ (0, 1].

We remark that Theorem 11.3 is of rather theoretical value. In particular, it does not lead to K-theory computations similar to the ones for free orthogonal quantum groups in [Voi11].

A Some results from q-calculus

A.1 Proof of the change of basis formula of Proposition 6.2

The vectors

|ξ_{j}⟩ = | \begin{matrix}m & & 0 & −m \\ & j & & −j \\ & & 0 \end{matrix} ^{↑} , |η_{k}⟩ = | \begin{matrix}m & & 0 & −m \\ & k & & −k \\ & & 0 \end{matrix} ^{↓} ,

are the basis vectors for the 0-weight space of μ = (m, 0, −m) in the upper and lower Gelfand-Tsetlin bases, respectively. Our calculation of the change-of-basis coefficients avoids the use of raising and lowering operators from [MSK95], instead using a recurrence relation which arises by considering the bracket

⟨η_{k}|E_{1}^{∗}E_{1}|ξ_{j}⟩.  (A.1)

Letting E_{1}^{∗}E_{1} act on |ξ_{j}⟩ first, the Gelfand-Tsetlin formulae (6.3), (6.4) give

⟨η_{k}|E_{1}^{∗}E_{1}|ξ_{j}⟩ = [j][j + 1]⟨η_{k}|ξ_{j}⟩.  (A.2)

On the other hand, in the lower Gelfand-Tsetlin basis E_{1} acts according to the formula (6.3) for Ψ(E_{1}) = E_{2} (see the definition of the lower basis in Section 6.2). We get

E_{1}^{∗}E_{1}|η_{k}⟩ = \frac{[m + k + 2][m − k][k + 1]^{2}}{[2k + 1]^{\frac{1}{2}} [2k + 2][2k + 3]^{\frac{1}{2}}} |η_{k+1}⟩

+ \frac{[m + k + 2][m − k][k + 1]^{2}}{[2k + 1][2k + 2]} + \frac{[m + k + 1][m − k + 1][k]^{2}}{[2k][2k + 1]} |η_{k}⟩

+\frac{[m + k + 1][m − k + 1][k]^{2}}{[2k − 1]^{\frac{1}{2}} [2k][2k + 1]^{\frac{1}{2}}} |η_{k}⟩

Taking the inner product of this with |ξ_{j}⟩ and equating with (A.2) yields a three-term recurrence relation for ⟨η_{k}|ξ_{j}⟩. The result is simplified if we introduce the non-unit vectors

|x_{j}⟩ = [2j + 1]^{− \frac{1}{2}} |ξ_{j}⟩,  |y_{k}⟩ = [2k + 1]^{− \frac{1}{2}} |η_{k}⟩.  (A.3)

One obtains

[j][j + 1]⟨y_{k}|x_{j}⟩ = a(k)⟨y_{k+1}|x_{j}⟩ + (a(k) + c(k))⟨y_{k}|x_{j}⟩ + c(k)⟨y_{k−1}|x_{j}⟩, (A.4)

where

a(k) = \frac{[m + k + 2][m − k][k + 1]^{2}}{[2k + 1][2k + 2]} ,
c(k) = \frac{[m + k + 1][m − k + 1][k]^{2}}{[2k][2k + 1]} .

We claim that the solution of Equation (A.4) is given by q-Racah coefficients. Unfortunately, the q-Racah polynomials are typically written in terms of the non-symmetric q-numbers [[n]] = \frac{1−q^{n}}{1−q}, so we must rewrite the recurrence relation as

(1 − q^{2j})(1 − q^{2(j+1)})⟨y_{k}|x_{j}⟩ = A(k)⟨y_{k+1}|x_{j}⟩ + (A(k) + C(k))⟨y_{k}|x_{j}⟩ + A(k)⟨y_{k−1}|x_{j}⟩, (A.5)

where

A(k) = \frac{(1 − q^{2(m+k+2)})(1 − q^{2(k−m)})(1 − q^{2(k+1)})^{2}}{(1 − q^{2(2k+1)})(1 − q^{2(2k+2)})} ,
C(k) = \frac{q^{2}(1 − q^{2(k+m+1)})(1 − q^{2(k−m−1)})(1 − q^{2(k)})^{2}}{(1 − q^{2(2k)})(1 − q^{2(2k+1)})} .

These are precisely the coefficients in the recurrence relation for the q^{2}-Racah polynomials described in Equation (14.2.3) of [KLS10], with parameters α = q^{2(m+1)}, β = q^{−2(m+1)}, γ = δ = 1 and N = m. The initial condition for the recurrence relation is fixed by Equation (6.10), which gives

⟨y_{0}|x_{j}⟩ = \frac{(−1)^{j+m}}{[m + 1]} ,

and formula of Proposition 6.2 follows.

A.2 A q-integral identity for little q-Legendre polynomials

We recall the definitions of the standard q-differentiation and q-integration operators:

D_{q}f(x) = \frac{f(qx) − f(x)}{(qx − x)},
∫_{0}^{x} f(y)d_{q}y = x(1 − q) ∑_{j=0}^{∞} q^{j}f(q^{j}x).

We also recall the following basic q-derivatives, where [[n]] = \frac{1−q^{n}}{1−q}:

D_{q}x^{α} = [[α]]_{q}x^{α−1},  for all α ∈ ℝ,  (A.6)
D_{q}(x; q^{−1})_{n} = −[[n]]_{q^{−1}}(x; q^{−1})_{n−1}  (A.7)
= −q^{−(n−1)}[[n]]_{q}(x; q^{−1})_{n−1},  for all n ∈ ℕ.

Proposition A.1.

∫_{0}^{1} x^{− \frac{1}{2}} p_{k}(x|q^{2}) d_{q^{2}}x = q^{\frac{1}{2}} \frac{2k + 1}{2} _{q}^{−1} .

Proof. Let us put r = q^{2}. The little q-Legendre polynomials satisfy the following Rodrigues-type Formula (see [KLS10]):

p_{k}(x|r) = \frac{1}{[[k]]_{r}!} D_{r}^{k} [x^{k}(x; r^{−1})_{k}].

From Equations (A.6) and (A.7) one has that for all 0 ≤ i < k, D_{q}^{i}x^{k} = 0 at x = 0 and D_{q}^{i}(x; r^{−1})_{k} = 0 at x = 1. Thus, by k applications of q-integration by parts, we get

∫_{0}^{1} x^{− \frac{1}{2}} p_{k}(x|r) d_{r}x

= \frac{(−1)^{k}}{[[k]]_{r}!} [ ; [−\frac{1}{2}] ; ]_{r} r^{\frac{1}{2}} [ ; [−\frac{3}{2}] ; ]_{r} r^{\frac{3}{2}} ⋯ ; −\frac{(2k−1)}{2} ; _{r} r^{\frac{(2k−1)}{2}}∫_{0}^{1}x^{− \frac{(2k+1)}{2}} x^{k}(x; r^{−1})_{k}d_{r}x

= \frac{1}{[[k]]_{r}!}\begin{aligned}[ \\ [ & \frac{1}{2} & ] \\ & & ] \end{aligned}_{r} \begin{aligned}[ \\ [ & \frac{3}{2} & ] \\ & & ] \end{aligned}_{r} ⋯ \begin{aligned} \\ & \frac{(2k−1)}{2} & \\ & & \end{aligned}_{r} ∫_{0}^{1} x^{− \frac{1}{2}} (x; r^{−1})_{k}d_{r}x

where in the last equality we have used [[α]]_{r} = −r^{α}[[−α]]_{r}. The last q-integral can be computed by q-integrating by parts k more times, giving

\frac{(−1)^{k}}{[[k]]_{r}!} r^{\frac{1}{2}} r^{\frac{3}{2}} ⋯ r^{\frac{2k−1}{2}} r^{k}[[−k]]_{r}[[−k + 1]]_{r} ⋯ [[−1]]_{r} ∫_{0}^{1} x^{\frac{(2k−1)}{2}} d_{r}x

= r^{\frac{k}{2}} ∫_{0}^{1} x^{\frac{(2k−1)}{2}} d_{r}x

= q^{\frac{1}{2}} \frac{2k + 1}{2} _{q}^{−1} .

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Christian Voigt School of Mathematics and Statistics University of Glasgow 15 University Gardens Glasgow G12 8QW United Kingdom Robert Yuncken

(1) Université Clermont Auvergne Université Blaise Pascal Laboratoire de Mathématiques BP 10448 F-63000 Clermont-Ferrand France

(2) CNRS, UMR 6620, LM F-63178 Aubiere France Robert.Yuncken @math.univ-bpclermont.fr