Hecke algebras and the Schlichting completion for discrete quantum groups

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions.


Introduction
Hecke algebras, originally studied in the analysis of Hecke operators for elliptic modular forms, play a prominent rôle in representation theory and harmonic analysis. In applications to number theory one is typically interested in Hecke operators associated with arithmetic groups. Abstractly, the relevant operators can be described starting from a discrete group Γ together with a commensurated subgroup, i.e. a subgroup Λ ⊂ Γ such that Λ ∩ Λ g has finite index in Λ for all g ∈ Γ, where Λ g = gΛg −1 . At this level of generality, Hecke operators can be viewed as Γ-equivariant bounded operators on ℓ 2 (Γ/Λ) and can be described using the Hecke algebra H(Γ, Λ), which is nothing but the space of functions on double cosets c c (Λ\Γ/Λ) equipped with a suitable convolution product.
In their seminal paper [BC95], Bost and Connes exhibited an intriguing connection between Hecke algebras, number theory and noncommutative geometry. The Hecke algebra underlying the Bost-Connes system is part of a quantum statistical mechanical system whose equilibrium states are intimately related to class field theory, and the time evolution of the system can be explicitly described by means of the modular function ∇ : g → [Λ : Λ ∩ Λ g ]/[Λ g : Λ ∩ Λ g ], comparing the number of left and right cosets in a double coset.
Hecke operators and Hecke algebras can also be defined for locally compact groups G together with compact open subgroups H ⊂ G. Moreover, both situations are related by the Schlichting completion construction which associates to each discrete Hecke pair (Γ, Λ) in a canonical way a pair (G, H) consisting of a totally disconnected locally compact group G and a compact open subgroup H ⊂ G such that H(Γ, Λ) ∼ = H(G, H) [Sch80]. Analytical properties of the algebra of Hecke operators are often easier to analyze at the level of the Schlichting completion, see [Tza03], [AD14] and [KLQ08].
The aim of this article is to extend some of this theory to the case of locally compact quantum groups, both in the discrete and compact open settings. This exhibits new combinatorial behavior which is invisible in the classical case, related to the "relative dimension" constants naturally associated with quantum subgroups of discrete quantum groups. A major motivation for the passage to the quantum framework, apart from producing interesting examples of von Neumann algebras, is to create new locally compact quantum groups out of known discrete quantum groups. For this purpose we develop a generalization of the Schlichting completion procedure, which provides a slightly new perspective even for classical groups, and leads to a deeper understanding of the quantum quotient spaces Γ/Λ for discrete quantum groups. The Schlichting completion yields algebraic quantum groups in the sense of Van Daele [VD98], and we obtain concrete examples by considering pairs (Γ, Λ) of discrete quantum groups arising from HNN extensions.
Let us describe the main results obtained in the article. After collecting some preliminaries in Section 2, we begin our analysis in the setting of a subgroup Λ in a discrete quantum group Γ in Section 3. We give a detailed description of the noncommutative quotient space Γ/Λ and of the associated module category, see Proposition 3.5 and Theorem 3.10. This allows us to obtain an explicit formula for the quantum analogue µ of the counting measure on Γ/Λ, in terms of the equivalence relation induced by Λ on irreducible corepresentations of Γ, see Definition 3.13. We also give in Proposition 3.17 a categorical interpretation of the constants κ that appear in this formula. These constants are trivial in the classical case.
With this in place it is easy to write down the definition of the convolution product of the Hecke algebra H(Γ, Λ), see Definitions 3.19 and 3.24. We prove in Theorem 3.29 that this algebra is canonically isomorphic to the algebra of Hecke operators, i.e. Γ-equivariant linear maps on c c (Γ/Λ). Moreover, in Theorem 3.32 we give a combinatorial characterization of the boundedness of the Hecke operators on ℓ 2 (Γ/Λ), in terms of the constants κ.
Next we investigate the modular properties of the canonical state on H(Γ, Λ) and give an explicit formula for the corresponding modular operator in Proposition 3.42 and Theorem 3.36. This formula involves the number of left and right cosets in double cosets, as in the classical case, but in general also the modular structure of the discrete quantum group Γ, as well as the constants κ. In Paragraph 3.3 we consider an explicit class of examples arising from HNN extensions.
In Section 4 we switch to the setting of a locally compact quantum group G with a compact open quantum subgroup H, working in the framework of algebraic quantum groups. In this case it is easier to define the Hecke algebra H(G, H) since compactly supported functions on H\G/H are also compactly supported on G, and one can directly use the convolution product of the quantum algebra of functions O c (G). Again, we establish a canonical isomorphism of H(G, H) with the algebra of G-equivariant maps on c c (G/H) in Proposition 4.3. As an example, we discuss the case of the quantum doubles G = H ⊲⊳Ĥ of a compact quantum group H: in this case H(G, H) identifies with the algebra of characters of H.
In Paragraph 4.3 we associate a pair (G, H) to each discrete Hecke pair (Γ, Λ) by means of a quantum analogue of the Schlichting completion. More precisely, we construct O c (G) directly as a subalgebra of ℓ ∞ (Γ) using the Hecke convolution product between c c (Γ/Λ) and c c (Λ\Γ), see Definition 4.5. This seems to be a new point of view even in the classical case. Moreover we establish in Proposition 4.14 a canonical identification between H(G, H) and H(Γ, Λ). It follows in particular that Hecke operators are bounded on ℓ 2 (G/H) as well as on ℓ 2 (Γ/Λ), and this yields an analytic proof of the combinatorial property of the constants κ mentioned above, see Definition 3.30 and Theorem 3.32.
Finally, in Paragraph 4.4 we study the notion of reduced pair both in the discrete setting and in the compact open one, with the property of being reduced corresponding to faithfulness of the Γ-action on Γ/Λ, resp. of the G-action on G/H. We construct a reduced pair associated to an arbitrary Hecke pair in Propositions 4.19 and 4.23. Moreover we prove that the Schlichting completion G is non-discrete whenever the Hecke pair (Γ, Λ) is reduced and Λ is infinite, see Lemma 4.21. It follows that the Schlichting completions of the Hecke pairs constructed in Section 3.3 via HNN extensions yield non-discrete locally compact quantum groups, whose modular automorphisms can be computed by the explicit formulas of Paragraph 3.2.
We would like to thank the anonymous referee for their careful reading of our original manuscript and a number of valuable suggestions and comments.

(Quantum group) preliminaries
In this short section we introduce general conventions, fix our notation and offer a brief review of some definitions and facts from the theory of quantum groups. For more details we refer the reader to the following sources: [KS97], [Wor98], [VD98], [KV00], [NT13], [VY20].
Tensor products of algebras, minimal/spatial tensor products of C * -algebras and Hilbert space tensor product of spaces/operators will be usually denoted by ⊗; if we want to stress that we are dealing with the algebraic tensor product we will use the symbol ⊙. If ϕ is a linear form on an algebra A and a ∈ A we denote aϕ, ϕa the forms given by aϕ(b) = ϕ(ba) and ϕa(b) = ϕ(ab), for all b ∈ A.
2.1. Algebraic quantum groups. By definition, an algebraic quantum group G is given by a multiplier Hopf * -algebra O c (G) together with positive invariant functionals [VD98]. Recall from [KVD97] that one can associate to G in a canonical way a locally compact quantum group, i.e. a (reduced) Hopf C * -algebra C 0 (G) satisfying the axioms of Kustermans and Vaes [KV00] and containing O c (G) as a dense * -subalgebra. We denote by ϕ, ψ the left and right Haar weights of G, which are defined on O c (G). We denote the dual multiplier Hopf algebra by D(G).
Not all locally compact quantum groups arise in this way. In particular classical locally compact groups which fit into the algebraic quantum group framework are precisely the ones admitting a compact open subgroup [LVD08,Section 3], and this is precisely the class of groups which naturally appears in the study of Hecke algebras.
More specifically, if G is a locally compact group with a compact open subgroup H ⊂ G, then we get an algebraic quantum group by considering where O(H) is the usual space of representative functions on H, which embeds canonically in C 0 (G) via extension by 0, and (g · f )(x) = f (xg) for g ∈ G and f ∈ C 0 (G) is the action by right translation. The resulting multiplier Hopf * -algebra is independent of the choice of H, and in fact uniquely determined [LVD08].
A morphism between algebraic quantum groups from G 1 to G 2 is given by a nondegenerate * -homomorphism π : O c (G 2 ) → M(O c (G 1 )), compatible with the comultiplications. Observe that the algebraic multiplier algebra M(O c (G 1 )) typically contains operators which are unbounded at the Hilbert space level. However, if W G 2 denotes the multiplicative unitary associated with G 2 , then (π⊗id)(W G 2 ) is a unitary element of M(O c (G 1 )⊙D(G 2 )), and hence it induces a bounded (unitary) operator on L 2 (G 1 ) ⊗ L 2 (G 2 ). From the properties of the multiplicative unitary at the algebraic level it follows that this yields in fact a bicharacter at the C * -level, and from [MRW12] we conclude that π extends to a nondegenerate * -homomorphism C u 0 (G 2 ) → M(C u 0 (G 1 )) between the universal completions constructed in [Kus03]. That is, π determines a morphism between the associated locally compact quantum groups. For simplicity we will abbreviate C b (G 1 ) = M(C u 0 (G 1 )). If G is an algebraic quantum group, then an algebraic compact open quantum subgroup H ⊂ G is given by a non-zero central projection [KKS16,Theorem 4.3, Proposition 4.4, Corollary 3.8]. Then C(H) = p H C 0 (G) is a Woronowicz-C * -algebra and the canonical morphism from H to G corresponds to the Hopf * -homomorphism π H : C 0 (G) → C(H), f → p H f . We note that it seems unclear whether central projections of C 0 (G) satisfying the above condition (which a priori describe all open compact quantum subgroups of G) automatically lie in O c (G).
2.2. Discrete quantum groups. An algebraic quantum group Γ is called discrete if the corresponding algebra O c (Γ) is a direct sum of matrix algebras. The existence of invariant functionals is then automatic and we have C 0 (Γ) = C u 0 (Γ). In this case we use a different notation to emphasize the analogy with the classical situation: we put O c (Γ) = c c (Γ), . The left, resp. right Haar weights of Γ are denoted h L , h R and their modular groups σ L , σ R ; the scaling automorphism group will be denoted τ . We will also use the antipode S, the unitary antipode R and the co-unit ǫ.
We choose a set I(Γ) of representatives of irreducible objects up to equivalence. We can then identify ℓ ∞ (Γ) = ℓ ∞ -α∈I(Γ) B(H α ) and we have c c (Γ) = B(H α ) and c(Γ) = B(H α ). We denote by p α ∈ c c (Γ), α ∈ I(Γ) the minimal central projections and write a α = p α a for a ∈ c(Γ). The coproduct is determined by the formula (p β ⊗ p γ )∆(a)v = va α for any v ∈ Hom(α, β ⊗γ) and a ∈ c c (Γ). Let us record the following fact which is certainly well-known to experts.
which yields the result.

Discrete quantum Hecke pairs
In this section we define the Hecke algebra associated to a discrete quantum group Γ and a commensurated quantum subgroup Λ. We start by studying the structure of the quotient spaces Γ/Λ, Λ\Γ, first at the level of the set of irreducible corepresentations I(Γ), and then at the finer level of the quantum algebras of functions ℓ ∞ (Γ/Λ), ℓ ∞ (Λ\Γ). We obtain in particular in Theorem 3.10 a description of ℓ ∞ (Γ/Λ) using the classical quotient space I(Γ)/Λ.
We can then define the Hecke algebra H(Γ, Λ) and its convolution product, and prove that it is represented by Hecke operators on c c (Γ/Λ). We give a combinatorial characterization of the ℓ 2 -boundedness of Hecke operators and describe the modular properties of the canonical state of H(Γ, Λ). Finally we investigate examples arising from quantum HNN extensions.
Recall that the quantum quotient spaces are given by the algebras One can use the same conditions to define c(Γ/Λ), c(Λ\Γ) but one has to be a bit more careful for the spaces of finitely supported functions. One can check that p σ ∈ c(Γ/Λ) for any σ ∈ I(Γ)/Λ and one defines c c (Γ/Λ) = Span{p σ c(Γ/Λ), σ ∈ I(Γ)/Λ} (and similarly for the left versions). One can show that p σ c(Γ/Λ) = p σ ℓ ∞ (Γ/Λ) for any σ ∈ I(Γ)/Λ [VV13, Lemma 3.3]. It is easy to check that the coproduct ∆ restricts to von Neumann, resp. algebraic left coactions ∆ : , and similarly to right coactions ∆ : The next Lemma works in general for open quantum subgroups of locally compact quantum groups, see [KKS16, Lemma 3.1, Corollary 3.9]. The fact that the unitary antipode exchanges the left and right von Neumann algebraic homogeneous spaces for any closed quantum subgroup can be found for example in [KS20]. We include a simple proof for the discrete case.
The  Let us apply Definition 3.4 to the space B(H α , H β ), viewed as a corepresentation of Γ via the identification with Hᾱ ⊗ H β given by S → (id ⊗ S)t α . By Frobenius reciprocity we then have Recall that for a ∈ ℓ ∞ (Γ) and α ∈ I(Γ) we denote by a α ∈ B(H α ) the component p α a of a.
Further, recall that [DCKSS18, Theorem 5.2, Theorem 5.6] show that each [α] corresponds to a certain equivalence class of minimal central projections in ℓ ∞ (Γ/Λ), determined by the left adjoint action of Γ, and p [α] is equal to the sum of the projections in the aforementioned equivalence class. Thus the fact that each p α ℓ ∞ (Γ/Λ) is simple is equivalent to the equivalence relation above being trivial. This need not be the case, as already classical Clifford theory shows.
We will now represent invariant functions using ∆(p Λ ), see Theorem 3.10, which is essentially a consequence of Proposition 3.5 and "strong invariance" of the Haar weights ([KV00, Proposition 5.24]). The quantities κ below will play a fundamental role in the sequel. Recall that for v ∈ Corep(Γ) we denote by v Λ the sum of all irreducible subobjects of v equivalent to an element of I(Λ).
We first make the connection between the constants κ α,β and the Hopf algebra structure of ℓ ∞ (Γ). Recall that we denote aϕ = ϕ( · a), ϕa = ϕ(a · ) if ϕ is a linear form on an algebra and a an element of this algebra.
Lemma 3.9. For every α, β ∈ I(Γ) we have Proof. We first show that for any α, β, This scalar can be computed by evaluating both sides against h R . We obtain, after dividing both sides by dim q (β): . Note that an analogous formula appears already in work of Izumi, see the remark before Corollary 3.7 in [Izu02].
Theorem 3.10 below is a very useful tool for the study of Hecke algebras associated to discrete quantum groups. It is merely a materialization of Proposition 3.5, which says that one can recover a ∈ p [α] c(Γ/Λ) from a α , and it is a simple consequence of the so-called strong invariance properties of the Haar weights. It is well-known in the case when Λ = {e} -then p Λ is the support of the co-unit and κ α = 1 for every α. It has several corollaries important for what follows.
Theorem 3.10. For any a ∈ c c (Γ/Λ), b ∈ c c (Λ\Γ) and any choices of representatives Proof. We start with a ∈ c c (Λ\Γ), α ∈ I(Γ) and λ ∈ I(Λ). By strong right invariance we . Summing over λ ∈ I(Λ) and applying Lemma 3.9 we obtain . Now we observe that the right-hand side also lies in Consequently we can apply Proposition 3.5 which yields )∆(p Λ ) and summing over [α] we obtain the first equality in (3.1). The missing identities in (3.1) and (3.2) follow by replacing a, b with a * , b * and taking adjoints on both sides. Alternatively one can use the fact that as we are in the context of discrete quantum groups we have h L (ab) = h L (bS 2 (a)), h R (ab) = h R (bS −2 (a)).
Let us finally introduce the quantum analogues µ, ν of the counting measures on Γ/Λ resp. Λ\Γ. We will see at Proposition 3.21 that µ is the (necessarily unique, up to scalar multiplication) Γ-invariant weight on c c (Γ/Λ). Observe that, applying (3.1) to a * we obtain This explains the last identity in the following definition. By Corollary 3.11 we also see that µ, ν and (a | b) as defined below do not depend on the choices of γ ∈ [γ].
Finally for a ∈ c c (Γ/Λ) we write a Γ/Λ := (a | a) 1/2 . Remark 3.14. This definition agrees via the Fourier transform with the scalar product on the "dual algebra" C[Γ/Λ] introduced in [VV13]. Recall that this algebra is defined as the relative tensor product [Ver04]. Following [VV13] we consider the linear bijection F −1 , which corresponds to (the inverse of) the Fourier transform when Λ = {e}. Using Theorem 3.10 one can check the following explicit formula for the inverse map On the other hand the space C[Γ/Λ] has a natural prehilbertian structure given by (x | y) = ǫE(x * y). One can then easily check that F Λ is an isometry with respect to this scalar product on C[Γ/Λ] and the one of Definition 3.13 on c c (Γ/Λ).
3.1.3. The constants κ. In contrast to the classical case, where they trivialize, the constants κ of Definition 3.8 play an important role in the theory of quantum group Hecke algebras (or in the quantum version of the Clifford theory). In this subsection we give more details about them, which will however not be used in the rest of the article (except in Proposition 3.22).
Let us first note that related constants already appeared in Lemma 3.5 of [KKSV], which states the existence, for every α ∈ I(Γ), of a constant η α > 0 such that where [α] ∈ Λ\I(Γ). As we have h L pᾱ = dim q (α) qTrᾱ, comparing the formulas shows that we have in fact η α = κᾱ/ dim q (α).
The next example shows that we do not have κᾱ = κ α in general.
Example 3.15. We consider the quantum subgroup Λ ⊂ Γ = Z * Λ, where Λ is any discrete quantum group. Denote by a the generating corepresentation of Z and take v ∈ I(Λ).
and if Λ is finite, we are in the setting of Hecke pairs. For Λ finite and non classical we can further If one allows non commensurated quantum subgroups, one can take for Λ the dual of SU(2) and we see again that the ratios κᾱ/κ α are not bounded when α varies. Notice also that v = α ⊗ā, κ v = dim q (v) 2 and κ α = 1. This shows that there is no control on the ratios κ γ /κ α either when γ ⊂ α ⊗ β with β ∈ I(Γ) fixed.
We shall now give another interpretation of the constants κ α,β in terms of the Corep(Γ)module category Corep(Γ/Λ). Let us first introduce some more relevant notation. For π ∈ Corep(Γ) and α ∈ I(Γ) we denote M α,π = Hom(α, π), so that H π ≃ α∈I(Γ) H α ⊗M α,π canonically. When π = β ⊗ γ we denote C β,γ α = M α,π , so that the dimensions c β,γ α = dim(C β,γ α ) are the structure constants of the based Grothendieck ring of Corep(Γ). We can extend this notation to objects of Corep(Γ/Λ). For Recall that the modular group of the left Haar weight is implemented by the (possibly In general F does not belong to c c (Γ/Λ) but we still have an induced modular structure on this subalgebra. More precisely, take α ∈ I(Γ) and decompose We summarize the conclusions of this discussion in the next proposition.
The element F i does not depend on the choice of α ∈ I(Γ) such that M i,α = 0. We introduce the quantum dimensions, quantum multiplicities and quantum structure coefficients as follows: Proof. The element F i does not depend on α because Ad(F it i ) corresponds for every t ∈ R to the restriction of σ L t to ℓ ∞ (Γ/Λ)q i , where q i denotes the relevant minimal central projection of ℓ ∞ (Γ/Λ).
By considering the case of a quantum subgroup Λ ⊂ Γ = Λ×Λ ′ with Λ, Λ ′ non unimodular one sees that the elements F i , L i,α , D β,j i can be non trivial, and that their traces are in general different from the classical dimensions dim(i) = dim(H i ), the classical multiplicity mult(i, α) = dim(Hom(i, α)) and the classical coefficients c β,j i = dim(Hom(i, β ⊗ j)).
We denote by C[Γ], C[Γ/Λ] the free vector spaces generated by I(Γ), I(Γ/Λ), endowed with the scalar products such that I(Γ), I(Γ/Λ) are orthonormal bases. To any representation π ∈ Corep(Γ/Λ) corresponds naturally an element When π comes from Corep(Γ) one can use quantum multiplicities and we denotẽ We have then the following result generalizing (and resulting from) Corollary 3.12: Proof. We have to show that for i ∈ I(Γ/Λ), α ∈ I(Γ), the number qmult(i, α)/ dim q (α) depends only on i and the class [α] ∈ I(Γ)/Λ. Denote by q i ∈ c c (Γ/Λ) ⊂ ℓ ∞ (Γ) the minimal central projection of c c (Γ/Λ) corresponding to i ∈ I(Γ/Λ). We have (q i ) α = q i p α and so by Corollary 3.11 the number κ −1 h L (q i p α ) only depends on [α]. On the other hand we can compute it using the canonical identification H α ≃ i H i ⊗ M i,α as follows: The assertion follows since dim q (α) 2 /κ α only depends on [α] by Corollary 3.12.
3.2.1. The convolution product. Now we introduce the Hecke convolution product. Note that, thanks to Theorem 3.10, the different expressions for a * b given in the next definition are indeed equal. By comparing (3.5) and (3.7) one sees that they do not depend on the choices of α ∈ [α], β ∈ [β].
In the next proposition we give a description of the convolution product between c c (Γ/Λ) and c c (Λ\Γ) = S(c c (Γ/Λ)) using only the structure of the Γ-invariant subalgebra ℓ ∞ (Γ/Λ) ⊂ ℓ ∞ (Γ), without explicit reference to the quantum subgroup Λ. Note that by Proposition 3.18 the fractions appearing in the expression (3.8) only depend on i (and not on the choice of β i ).
Proposition 3.22. Choose for each i ∈ I(Γ/Λ) an element β i ∈ I(Γ) such that qmult(i, β i ) = 0 and consider the linear forms ϕ i : Proof. We start from the last expression in equation (3.7). We shall compute the linear form [β] κ −1 β S(b β )h L on the matrix blocks of c c (Γ/Λ) using the canonical identification Puttingβ = β i for each i ∈ I(Γ/Λ) we obtain (3.8).
It follows from (3.5) and Theorem 3.10 that p Λ is a unit for the convolution product.
By uniqueness and the invariance result of Proposition 3.21, the functional µ, ν of Definition 3.13 are the left, resp. right Haar weights of Γ/Λ, normalized by the condition µ(p Λ ) = ν(p Λ ) = 1. Moreover, from the formula a * b = (id ⊗ S(b)µ)∆(a) we recognize the usual convolution product of the discrete quantum group algebra ℓ ∞ (Γ/Λ) (or its opposite, depending on conventions), transported from the product of C[Γ/Λ] via the Fourier transform. Restricting the action of C[Γ] to C[Λ] we obtain the associated space of fixed points Proposition 3.27. Let Λ be a quantum subgroup of Γ. The map ev Λ : F → f := F (p Λ ) defines an antimultiplicative isomorphism from End Γ (c c (Γ/Λ)) to c c (Γ/Λ) Λ , with inverse bijection T given by The same equivariance formula can also be written F ((a α h R ⊗ id)∆(p Λ )) = (a α h R ⊗ id)∆(f ) for any a ∈ c(Γ) and α ∈ I(Γ).
Moreover, take a ∈ p [α] c c (Γ/Λ). Using Theorem 3.10 and the previous equivariance formula we can write This shows that T is a left inverse of ev Λ .
Conversely starting from f ∈ c c (Γ/Λ) Λ we can consider T (f ) : a → a * f . We already noticed at Proposition 3.23 that T (f )(a) ∈ c c (Γ/Λ), and after Definition 3.19 that T (f ) is equivariant with respect to the left Γ-action induced by ∆. Since p Λ is the unit of the convolution product, T is a right inverse of ev Λ . Finally T is antimultiplicative by associativity of the convolution product.
We now use the prehilbertian structure on c c (Γ/Λ) obtained from the functional µ and consider the corresponding subspace of adjointable operators in End Γ (c c (Γ/Λ)). Note that the formula (3.3) for (a | b) given in Definition 3.13 also makes sense for a ∈ c c (Γ/Λ), b ∈ c(Γ/Λ), or for a ∈ c(Γ/Λ), b ∈ c c (Γ/Λ). We use this in the following proposition. Note also that p Λ Γ/Λ = 1. Proof. We compute, using (3.3) and (3.5): We also used the identities S −1 (a α )h R • S −1 = S 2 (a * α )h L = h L a * α which are easy to check.
In the next theorem, which offers an alternative description of the Hecke algebra, we write End ′ (c c (Γ/Λ)) ⊂ End(c c (Γ/Λ)) for the subspace of adjointable maps, i.e. of maps T for which there exists S ∈ End(c c (Γ/Λ)) satisfying (a | T b) = (Sa | b) for all a, b ∈ c c (Γ/Λ). Now we investigate the question whether Hecke operators extend to bounded operators on ℓ 2 (Γ/Λ), the completion of c c (Γ/Λ) with respect to the norm · Γ/Λ arising from Definition 3.13. In the setting of discrete quantum groups we prove in Theorem 3.32 that this is equivalent to a combinatorial property of the constants κ α introduced in Definition 3.30 below. Surprisingly we could not prove directly that this property always hold for Hecke pairs. However we will show later in Section 4.3, using the Schlichting completion, that this is indeed the case, by showing that Hecke operators are always bounded.
Property (RT) is of course always verified in the classical case since then κ α = 1 for all α ∈ I(Γ). In general it does not hold for all corepresentations, as shown by the (noncommensurated) Example 3.15.
Proof. Assume first that T (b) is bounded with respect to · Γ/Λ for all b ∈ H(Γ, Λ). In particular T (p τ ) is bounded for every τ ∈ Λ\I(Γ ′ )/Λ. Consider ξ [α] = κ 1/2 α p [α] / dim q (α), which has norm 1 with respect to · Γ/Λ . By definition of the scalar product, and writing τ as a disjoint union of finitely many left classes [β] we have Using (3.9) in the preceding lemma we can further compute: Finally we decompose (γ ⊗β) into irreducible subobjects α ′ and select the ones in [α]. Using Corollary 3.12 this yields As a result we have, for any α ′ , γ ∈ I(Γ) and β ∈ I(Γ ′ ) such that α ′ ⊂ γ ⊗β: This shows the existence of the constants C β and the direct implication. Conversely, assume that (RT) holds. Taking b ∈ c c (Λ\Γ ′ /Λ), we can assume that b ∈ p τ c c (Λ\Γ ′ /Λ) with τ ∈ Λ\I(Γ ′ )/Λ, and we then have a decomposition b = The sum is in fact finite since its terms vanish unless α ′ ⊂ γ ⊗β. Now we use the case when Λ is the trivial subgroup, denoting · Γ the hermitian norm on c c (Γ) and T Γ the homomorphism from c c (Γ) to End(c c (Γ)). We know that T Γ (d) extends to a bounded operator on ℓ 2 (Γ) for any d ∈ c c (Γ), because it is an operator from the right regular representation of Γ. Fix a choice of representatives β for the classes [β] and put M = max [β]⊂τ T Γ (b β ) (so that M might depend on the choice made). We then have We have moreover c γ Γ = √ κ γ c Γ/Λ , a α ′ Γ = √ κ α ′ a Γ/Λ by definition.
As a result we can write:  Finally we can use Cauchy-Schwarz to write, for any a, b ∈ c c (Γ/Λ):

Modular structure.
Definition 3.33. The canonical state on H(Γ, Λ) is given by the formula Since the sesquilinear form ( · | · ) is positive-definite, ω is faithful, i.e. ω(f ♯ * f ) = 0 implies f = 0. To investigate the modular properties of ω we first construct a quantum analogue of the classical modular function. We consider the restrictions of the functionals µ, ν introduced in Definition 3.13 to c c (Λ\Γ ′ /Λ). These forms are faithful by Proposition 3.5, and so we can make the following definition.
The next result shows that the operator ∇ indeed plays the role of the modular operator for the canonical state on the Hecke algebra (or rather its relevant von Neumann algebraic completion).
We also have the following variant of the last step of the computation: since h R is σ R -invariant and σ R commutes with S, it is easy to check that From these properties we first deduce: 1 (a  *  b)), hence by Proposition 3.23 we have (∇ −1 a) * (∇ −1 b) = ∇ −1 (a * b). This implies (∇ k a) * (∇ k b) = ∇ k (a * b) for all k ∈ Z and, by the usual argument, (∇ z a) * (∇ z b) = ∇ z (a * b) for all z ∈ C. It follows that the maps θ t are multiplicative for the convolution product. They are also compatible with the involution since σ R t S = Sσ R t and (∇ it a) ♯ = ∇ it♯ a ♯ = ∇ it a ♯ for real t, using the property S(∇ −1 ) = ∇.
Proof. We use the property (∇a) * (∇b) = ∇(a * b) for a, b ∈ c c (Λ\Γ ′ /Λ), established in the proof of the previous proposition. Since S(∇) = ∇ −1 we can write We shall now give an explicit formula for the modular function ∇ in terms of the structure of the inclusion Λ ⊂ Γ. This will involve quantum analoguesL α ,R α of the counting functions L, R which arise from the interplay between the modular structure of the Haar weight h R and the structure of the quantum quotient space Λ\Γ/Λ.
For every α ∈ I(Γ), we have a unique h L -preserving (resp. h R -preserving) conditional expectation from p α c(Γ) = B(H α ) onto the subalgebra p α c(Λ\Γ/Λ). We consider the following related maps: In the classical case, E L α (f ) is the constant function on α , equal to the value f (α). Let us record the following property of these maps in connection with Woronowicz' modular element. F 2 a) for all a ∈ c c (Γ). In particular we have, for a ∈ c c (Λ\Γ/Λ) and α ∈ I(Γ): As h R is faithful on p α c c (Λ\Γ/Λ), we can infer that p α E R α (F −2 ) = (p α E L α (F 2 )) −1 and we conclude by Proposition 3.5.
Definition 3.40. Fix α ∈ I(Γ ′ ) and choose elements δ i ∼ α (resp. ǫ j ∽ α) such that α is the disjoint union of the classes [δ i ] ∈ Λ\I(Γ) (resp. [ǫ j ] ∈ I(Γ)/Λ). We define the following elements of p α c c (Λ\Γ/Λ): Remark 3.41. Note that we have F ∈ c(Λ\Γ/Λ) iff Λ is unimodular (i.e. F | Λ = I), since ∆(F ) = F ⊗ F . In this case we have E L α (F t ) = E R α (F t ) = p α F t for all t ∈ R, and hence, writing F α = p α F : Since dim q (δ i ) 2 /κδ i only depends on the class [δ i ] ∈ Λ\I(Γ) we can drop the constraint δ i ∼ α in the definition ofL α and we see in particular thatL α ,R α only depend on α in this case. If we have moreover κ δ = κδ for all δ ∈ I(Γ), the terms dim q (δ i ) 2 /κδ i only depend on δ i = α and hence we havẽ On the other hand let us consider the case when p α c c (Λ\Γ/Λ) = Cp α . Then we have with the same simplification as above if κ δ = κδ for all δ.
By Corollary 3.11 we have . Recall moreover that we have κǭ/κᾱ = (dim q (ǫ)/ dim q (α)) 2 when ǫ ∽ α. Hence we can write . We proceed similarly on the other side with classes [δ i ] ∈ Λ\I(Γ) and δ i ∼ α : This yields the result by definition of ∇.

Examples: HNN extensions.
Let Γ 0 be a discrete quantum group with two quantum subgroups Λ ǫ ⊂ Γ 0 (ǫ = ±1). Following [Fim13], we start with an isomorphism between the two quantum subgroups, described via a Hopf * -algebra isomorphism θ : C[Λ 1 ] → C[Λ −1 ] and we form Γ = HNN(Γ 0 , θ). Recall that C[Γ] is generated by C[Γ 0 ] and a group- the canonical conditional expectations. The algebra C[Γ] is the direct sum of the subspaces The subspaces C[Γ] n , n ≥ 1 span the kernel of the canonical conditional expectation [Fim13]. It follows in particular that I(Γ) is partitioned into the subsets Proposition 3.43. Assume that the quantum subgroups Λ ǫ have finite index in Γ 0 and at least one of them is distinct from Γ 0 . Then Γ 0 is commensurated in Γ, not normal, and of infinite index.
Example 3.47. One can construct quantum examples as follows. Take two finite quantum groups Σ ±1 , for instance duals of classical finite groups. Form the restricted product Γ 0 = ′ k∈Z * Σ sgn(k) , which is the dual of a profinite group if Σ ±1 is the dual of a finite classical group. If one group Σ ǫ is not classical, Γ 0 is a unimodular non-classical discrete quantum group. Consider the finite index subgroups Λ ǫ = ′ k∈Z * ,k =ǫ Σ sgn(k) . We have evident isomorphisms Λ ǫ ≃ Γ obtained by shifting the copies of Σ ǫ towards k = 0 in the restricted product. We denote by θ : C[Λ 1 ] → C[Λ −1 ] the corresponding isomorphism. Denoting #Σ −1 and similarlyL w = #Σ 1 , where we denote #Σ = dim(c(Σ)). As a result the modular function ∇ of the Hecke pair (Γ, Γ 0 ) is non trivial as soon as Σ 1 , Σ −1 have different dimensions/cardinals. If one of Σ ±1 is non classical (e.g. the dual of a non abelian finite group), the HNN extension Γ is neither classical, nor co-classical (but it is unimodular).
Example 3.48. One can also construct quantum examples by taking for Γ 0 the dual of a compact group G, and using quantum subgroups Λ ǫ associated with quotients H ǫ = G/K ǫ . The index of Λ ǫ in Γ is finite iff K ǫ is finite. If G is connected, the subgroups K ǫ must then be central, and we have #I(Γ 0 )/Λ ǫ = #K ǫ .
Assume that G is a connected compact Lie group. Then the fundamental group of H ǫ remembers the cardinality of the kernel K ǫ , and since we assume H 1 and H −1 to be isomorphic we will always have L( w ) = #K 1 = #K −1 = R( w ) in this case. Similarly, subobjects ofγ ⊗ γ factor through the center Z(G) for any γ ∈ I(Γ 0 ) ⊂ Rep(G), hence always belong to I(Λ ǫ ) so that we have dim q (γ ⊗ γ) Λǫ = dim q (γ ⊗ γ) andL w =R w = (#K 1 )p w . Moreover, in most simple Lie groups the center is cyclic so that #K ǫ determines K ǫ and Λ 1 ≃ Λ −1 implies in fact Λ 1 = Λ −1 . This does not mean that the Hecke algebra will be completely trivial. One can also take for θ a non-inner automorphism of H to make the construction more interesting, so that the resulting Γ looks like a variant of the partial crossed-product construction.
A typical case is given by G = SU(n) ։ H 1 = H −1 = P SU(n), to be compared with the "classical case" of the Baumslag-Solitar group BS(n, n) = HNN(Z, id : nZ → nZ). On the other hand Z(Spin(4k)) = (Z/2Z) 2 , so that the dual of SO(4k) can be realized in two different ways as a subgroup of the dual of Spin(4k). Of course one can also look at SO(3) × SU(2) which is a quotient of SU(2) × SU(2) in two different ways. In all these cases we have ∇ = 1 becauseL w =R w .
Note that since K ǫ is contained in any maximal torus of G, the above construction is compatible with q-deformations -K ǫ remains a quantum subgroup of the compact quantum group G q corresponding to G. However, we still get ∇ = 1, by essentially the same reasoning since the fusion ring of G q is the same as the one of G.

Compact open quantum Hecke pairs
In this section we introduce Hecke algebras in the setting of locally compact (algebraic) quantum groups with compact open quantum subgroups. We then describe a generalized Schlichting completion, which allows us to subsume the Hecke algebras from section 3 in this setting, and deduce some analytic consequences. Finally, we describe how to pass from arbitrary Hecke pairs to their reduced versions, and exhibit some new examples of algebraic quantum groups. The definition of compactly supported functions on the quantum homogeneous space G/H is easier than in the discrete case, since the relevant invariant functions on G are also compactly supported. Namely, we define It is shown in [LVD] that this algebra is a direct sum of matrix algebras, i.e. it corresponds to a "discrete" quantum space. We denote by c 0 (G/H) the closure of c c (G/H) in C 0 (G) and ℓ 2 (G/H) its closure in L 2 (G). It can be shown that Together with the * -structure f ♯ := F −1 (F (f ) * ), not to be confused with the given * - The left regular representation of the dual algebra λ : D(G) → B(L 2 (G)) is then given by λ(F (f ))(Λ(g)) = Λ(f * g), see [VY20, Section 4.2.2]. We also have the right regular representation ρ : D(G) → B(L 2 (G)) given by ρ(F (f )) =Ĵ λ(F (f )) * Ĵ . HereĴ is the modular conjugation operator forφ, the dual left Haar weight also given by the formulâ ϕ(F (f )) = ǫ(f ).
Explicitly we have ρ(F (f ))(Λ(g)) = Λ(g * σ −i/2 (f )) for all f, g ∈ O c (G). Here, by slight abuse of notation, we writeσ −i/2 (f ) instead of F −1 (σ −i/2 (F (f )), where (σ t ) t∈R is the modular group ofφ. Note that the map f →σ −i/2 (f ) is an algebra isomorphism from (O c (G), * ) to D(G). We have δp H = p H because H is compact andσ t (p H ) = p H for all t ∈ R because the restriction ofφ : D(G) → C to D(H) is the left Haar weight ofĤ. Proof. Let f ∈ O c (G). Using the definition of the convolution product we calculate This allows us to give the following definition.
Definition 4.2. The Hecke algebra of (G, H) is the * -algebra H(G, H) = c c (H\G/H) with the convolution product and * -structure ♯ as above.
We obtain a nondegenerate * -representation of H(G, H) on ℓ 2 (G/H) by considering the restriction of the right regular representation. Here we use that for f ∈ c c (H\G/H) and g ∈ c c (G/H) the element g * f is again contained in c c (G/H), so that ρ(F (f )) indeed maps ℓ 2 (G/H) to itself. We also observe that ρ • F : H(G, H) → B(ℓ 2 (G/H)) is antimultiplicative for the convolution product, that is, ρ(F (f * g)) = ρ(F (g))ρ(F (f )) for all f, g ∈ H(G, H).
In the same way as in the discrete case we view c c (G/H) as a D(G)-module, see the discussion before Proposition 3.27, and obtain the space End G (c c (G/H)) of D(G)-module maps. Proof.
Since T (f ) for f ∈ H(G, H) commutes with the left convolution action of D(G) it is clear that T is well-defined, and it is obvious from the definition that T is antimultiplicative. Let us verify that T is an isomorphism by verifying that the above formula for T −1 yields indeed its inverse. Well-definedness is again easy to check, and using left H-invariance of f ∈ H(G, H) = c c (H\G/H) we get where ad(W ) is conjugation with the multiplicative unitary W ∈ L ∞ (H)⊗L(H). This is a special case of the generalized quantum doubles studied in [BV05].
In fact, the quantum double of a compact quantum group is naturally an algebraic quantum group in the sense of Van Daele, which allows us to give algebraic descriptions of almost all the data involved [DVD04], [VY20].
with the product and * -structure induced from O(H). This is precisely the algebra of characters inside O(H).
We note that, with suitable adjustments, similar computations go through for generalized quantum doubles built out of compact and discrete quantum groups.
4.3. The Schlichting completion. Let Γ be a discrete quantum group and Λ ⊂ Γ be a quantum subgroup. If Λ ⊂ Γ is almost normal (see Definition 3.3) we shall construct a pair (G, H) consisting of an algebraic quantum group G and a compact open quantum subgroup H ⊂ G, playing the role of the Schlichting completion of the Hecke pair (Γ, Λ) [Sch80].
More precisely, our strategy is as follows. We first define the algebra O c (G) as a subalgebra of ℓ ∞ (Γ) using the "discrete" Hecke convolution product, see Definition 4.5. The key point of the construction consists then in proving that this algebra is a multiplier Hopf * -algebra, and more specifically, that the coproduct takes its values in the appropriate subspace of M(O c (G) ⊙ O c (G)), see Proposition 4.6.
It is then easy to see that the projection p Λ corresponds to a CQG algebra O c (H) ⊂ O c (G), and that c c (G/H) = c c (Γ/Λ) as subspaces of ℓ ∞ (Γ), see Propositions 4.10 and 4.11. Using the Haar functional of O c (H) and the Γ-invariant functional µ on c c (Γ/Λ) one can then construct the integrals of O c (G), so that G is in fact a locally compact quantum group by [KVD97]. We end the section by making the connection between the "discrete" and "compact open" Hecke algebras H(Γ, Λ) and H(G, H).
Similarly one can define the support Supp(ϕ) of a linear functional ϕ ∈ c c (Γ) * as the set of elements γ ∈ I(Γ) such that p γ ϕ = 0. If ϕ has finite support, it extends uniquely to a normal functional ϕ ∈ ℓ ∞ (Γ) * .
In the following lemmas we will always assume that O c (G) and C 0 (G) arise in the above way from a Hecke pair. Note that by definition of the convolution product, e.g. (3.5), a * b is a finite sum of elements of ℓ ∞ (Γ) so that it is indeed in ℓ ∞ (Γ). It is easy to check, using write this as product of all a γ * f γ , where a γ = δ γΛ ∈ c c (Γ/Λ), f γ = ev γΛ ∈ c c (Γ/Λ) * for γ ∈ F . This yields the equality O c (G) = O c (G).
From the fact that both C 0 (G) and C 0 (G) are completions of O c (G) = O c (G) inside B(ℓ 2 (Γ)) we get C 0 (G) = C 0 (G). Finally, the claim about the canonical subgroups follows We now return to the general setup of quantum Hecke pairs. Proposition 4.10. Equipped with the restriction ∆ H of (p H ⊗ p H )∆, O(H) (resp. C(H)) is a CQG algebra (resp. a Woronowicz C * -algebra).
Proof. Recall that the comultiplication on O c (G) is implemented on the Hilbert space level by conjugation with the multiplicative unitary for Γ, i.e. for f ∈ ℓ ∞ (Γ) we have ∆(f ) = W * (1 ⊗ f )W in B(ℓ 2 (Γ) ⊗ ℓ 2 (Γ)). In particular, the comultiplication of O(H) extends continuously to a unital * -homomorphism ∆ H : C(H) → C(H) ⊗ C(H). Since we already know that O(H) is a Hopf * -algebra the cancellation conditions for C(H) are satisfied. Hence C(H) is a Woronowicz C * -algebra.
This implies that there is a Haar functional on O(H), obtained by restricting from the Haar state of C(H). We conclude that O(H) is a CQG algebra, compare [KS97].
So the corresponding compact quantum group H is an "algebraic" compact open quantum subgroup of G, with restriction map induced by the projection p H = p Λ . We denote by h its Haar functional. We can identify the corresponding homogeneous space Proof. Take a ∈ c c (Γ/Λ). Then we have a = a * p Λ hence a ∈ O c (G). By definition of c(Γ/Λ) we have (1 ⊗ p Λ )∆(a) = a ⊗ p Λ hence a ∈ c c (G/H). For the converse inclusion, take x ∈ c c (G/H). In particular we have (1 ⊗ p Λ )∆(x) = x ⊗ p Λ so x ∈ c(Γ/Λ). It remains to prove that x has finite support in this algebra, i.e. p τ x = 0 for all but a finite number of classes τ ∈ I(Γ)/Λ. It is clearly sufficient to prove this for x = a * b with a ∈ c c (Γ/Λ), b ∈ c c (Λ\Γ), since taking products of such elements reduces the support. But this results from Lemma 4.4 and the Hecke condition: if γ ∈ Supp(a * b) then γ ⊂ α ⊗ µ with α ∈ Supp(a) and µ ∈ β , β ∈ Supp(b); writing β as a finite union of right classes [γ] and decomposing α ⊗ γ into a finite number of irreducibles δ we see that Supp(a * b) in included in the union of the finite number of right classes [δ].
by Proposition 4.6, since x ∈ O c (G). It remains to show that z = (yϕ ⊗ id)∆(x) ∈ c c (G/H) for any ϕ ∈ c c (Γ) * with finite support. But we can write where all terms belong to the corresponding algebraic tensor products, and since (1 ⊗ p Λ ) ∆(x) = x ⊗ p Λ we recognize (1 ⊗ p Λ )∆(z) = z ⊗ p Λ . The last assertion is trivial because we can check the equality (id ⊗ µ)((y ⊗ 1)∆(x)) = µ(x)y by multiplying by an arbitrary central projection p α .
Theorem 4.13. Suppose that (Γ, Λ) is a quantum Hecke pair. Then G introduced in Definition 4.5 is an algebraic quantum group. We call the pair (G, H) the Schlichting completion of (Γ, Λ).
Proof. We apply Proposition 4.12 to the functional µ on c c (Γ/Λ) from Definition 3.13, which is invariant by Proposition 3.21, and defines at the same time an invariant functional on c c (G/H) by Proposition 4.11. We have ϕ(p H ) = µ(p Λ ) = 1, hence ϕ does not vanish. Finally it is positive because µ ⊗ h is positive; indeed µ and h have both C * -algebraic realizations -recall that µ is a sum of positive forms on each matrix factor of c c (G/H) = c c (Γ/Λ). Now we compare the Hecke algebras of a Hecke pair and of its Schlichting completion. This will provide an analytic proof that the "discrete" Hecke operators on ℓ 2 (Γ/Λ) are bounded. Hence it suffices to observe that the latter is obtained by restriction of the right regular representation of D(G) on L 2 (G), which acts by bounded operators.
Note that we use in the fourth equality the fact that π H (σ −i/2 (f )) is H-invariant, hence constant. 4.4. Reduction procedure. Starting from a discrete Hecke pair (Γ, Λ), it can well happen that the Schlichting completion G is in fact discrete or even trivial. This is connected to the faithfulness of the action of Γ on Γ/Λ and to the reduction procedure that we describe now.
Suppose that Γ is a discrete quantum group with a quantum subgroup Λ and the corresponding projection p Λ ∈ ℓ ∞ (Γ). By c c (Γ/Λ) ∨ we denote finitely supported functionals on Γ/Λ as in the previous subsection.
Here we are concerned with the case C 0 (X) = c 0 (Γ/Λ), with α being the appropriate restriction of ∆.
Proof. Denote byα the action ofΓ on ℓ ∞ (Γ/Λ). Note that this is again given by the (suitable restriction of) the coproduct of ℓ ∞ (Γ). Thus to see that (Γ,Λ) is reduced it suffices to show that N(Γ Γ/Λ) = N(Γ Γ /Λ); this in turn will follow once we establish the latter part of the proposition.
Note that the inclusion ℓ ∞ (Γ/Λ) ⊂ N(Γ Γ/Λ) can be informally understood as the classically obvious fact that the kernel of the action of Γ on Γ/Λ is contained in Λ.
Proposition 4.20. Let (Γ, Λ) be as above and let (Γ,Λ) be its reduction. Then (Γ, Λ) satisfies the Hecke condition if and only if (Γ,Λ) does, and if this is the case, the corresponding Hecke algebras are isomorphic.
Proof. The first statement follows from Proposition 3.7, as the proposition above implies that the actions of Λ on ℓ ∞ (Γ/Λ) and ofΛ on ℓ ∞ (Γ/Λ) are given by the same von Neumann algebraic morphism and the notion of finite orbits does not formally involve any quantum group structure. The second statement follows now from the identification of Hecke algebras as certain commutants with respect to these actions.
We can now characterize the Hecke pairs that give rise to non-discrete Schlichting completions as follows.
Proof. Recall that we have by construction the strictly dense (equivalently, so-dense) inclusion C 0 (G) ⊂ ℓ ∞ (Γ). Multiplying by p Λ = pΛ = p H we see that C(H) is strictly dense (equivalently, so-dense) in ℓ ∞ (Λ), so thatΛ is finite if and only if H is finite. Now if H is finite, [KKS16,Proposition 4.5] implies that G is discrete. On the other hand if G is discrete, H, being an open (hence also closed by [KKS16, Theorem 3.6]) quantum subgroup of G, must be discrete by [DKSS12, Theorem 6.2]. Finally a discrete and compact quantum group must be finite.
In particular if (Γ, Λ) is a reduced Hecke pair, the associated Schlichting completion is discrete if and only if Λ is finite. This shows, with the help of Lemma 3.44, that the examples of quantum Hecke pairs discussed in the previous section lead to non-discrete Schlichting completions.
Corollary 4.22. The Schlichting completions associated with the HNN Hecke pairs of Example 3.47 are non-discrete locally compact quantum groups, with non trivial modular group as soon as #Σ 1 = #Σ −1 .
Note that the scaling constant of these quantum groups equals 1, since they arise from algebraic quantum groups.
We address now the reduction procedure for compact open Hecke pairs. Suppose that G is an algebraic quantum group and that H is an algebraic compact open quantum subgroup of G, given by a projection p H ∈ O c (G). We shall check that the procedure described above again yields a reduction of the pair (G, H).
Consider the * -algebra generated as follows: Observe that the Schlichting completion is constructed specifically so that the resulting pair (G, H) is reduced. Further we will call (G, H) a Schlichting pair whenever G is an algebraic quantum group, H is an algebraic compact open quantum subgroup of G and the pair (G, H) is reduced.
Suppose that we have two locally compact quantum groups G 1 , G 2 with respective open quantum subgroups H 1 , H 2 corresponding to projections P 1 ∈ C b (G 1 ), P 2 ∈ C b (G 2 ). We say that a morphism from G 1 to G 2 , described via a Hopf-C * -algebra morphism π : C 0 (G 2 ) → C b (G 1 ), maps H 1 to H 2 if π(P 2 ) ≥ P 1 . One may check, using [KKS16, Corollary 3.8] that indeed one obtains then (by restriction and multiplying by P 1 ) a quantum group morphism from H 1 to H 2 .
The following abstract characterization of the Schlichting completion for classical groups appears in [Tza03, Proposition 4.1]. The injectivity of the map ι ′ corresponds in the classical case to the density of the image of Γ in G ′ , and the identity ι ′ (p H ′ ) = p Λ , to the fact that Λ is the preimage of H ′ .
Proposition 4.26. Let (Γ, Λ) be a Hecke pair and (G, H) its Schlichting completion, with the canonical embedding ι : O c (G) → ℓ ∞ (Γ) defining the morphism from Γ to G. Then for any other Schlichting pair (G ′ , H ′ ) and any morphism from Γ to G ′ mapping Λ to H ′ and given by an injective map ι ′ : O c (G ′ ) → ℓ ∞ (Γ), there exists a unique morphism from G to G ′ , described by a map σ : O c (G ′ ) → O c (G), such that ι • σ = ι ′ . If in addition we assume that ι ′ (p H ′ ) = p Λ then the morphism from G to G ′ is an isomorphism. Proof. A moment of thought shows that it suffices to show that ι ′ (O c (G ′ )) ⊂ ι(O c (G)). Ignoring the injective embedding maps, and using the fact that both (G, H) and (G ′ , H ′ ) are Schlichting pairs, it suffices to note that c c (G ′ /H ′ ) ⊂ c c (G/H). But this follows as we have P ≤ P ′ (again viewing both as projections in ℓ ∞ (Γ)). The second part follows similarly.