We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation

THE ORTHOGONAL FREE QUANTUM GROUPS

PIERRE FIMA AND ROLAND VERGNIOUX

Abstract. We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II1-factors and Brannan’s result showing that the orthogonal free quantum groups have Haagerup’s approximation property. We also deduce Ozawa-Popa’s property strong (HH) and give a new proof of Isono’s result about strong solidity.

1. Introduction

The orthogonal free quantum groups FO_n are given by universal unital C^∗-algebras A_o(n) = C^∗(FO_n) which were introduced by Wang [Wan95] as follows:

Ao(n) =C^∗hvij,1≤i, j≤n|vij =v_ij^∗ and (vij) unitaryi.

Equipped with the coproduct defined by the formula ∆(vij) =P

vik⊗vkj, this algebra becomes a WoronowiczC^∗-algebra [Wor98] and hence corresponds to a compact and a discrete quantum group in duality [Wor98, PW90], denoted respectivelyO⁺_n andFOn.

TheseC^∗-algebras have been extensively studied since their introduction, and it has been noticed that the discrete quantum groupsFO_n share many analytical features with the usual free groups F_n from the operator algebraic point of view. Let us just quote two such results that will be of interest for this article:

(1) the von Neumann algebrasL(FOn) associated toFOn,n≥3, are full II1factors [VV07] ; (2) the von Neumann algebrasL(FO_n) have Haagerup’s approximation property [Bra12].

On the other hand there is one result that yields an operator algebraic distinction betweenFOn

andFn: it was shown in [Ver12] that the firstL²-cohomology group ofFOn vanishes.

The purpose of this article is to present slight reinforcements and alternate proofs of the results (1), (2) above. Namely we will:

(1) show that the discrete quantum groups FOn are not inner amenable (see [MvN43, Eff75]

for non abelian free groups);

(2) construct an explicit proper cocycle witnessing Haagerup’s property [DFSW13].

Moreover, putting these constructions together we will obtain a new result, namely Property strong (HH) from [OP10b], which is a strengthening of Haagerup’s property and corresponds to the existence of a proper cocycle in a representation which is weakly contained in the regular representation. This sheds interesting light on the result from [Ver12] mentioned above, according to which such a proper cocycle cannot live in a representation strongly contained in the regular representation (or multiples of it). Note that the cocycle corresponding to the original proof [Haa79] of Haagerup’s property forF_n does live in (multiples of) the regular representation.

Finally we will mention applications to solidity: indeed the constructions of the article and Property strong (HH) allow for a new proof of the strong solidity of L(FOn) using the tools of [OP10a, OP10b]. Notice that strong solidity has already been obtained in [Iso15] using Property AO⁺ from [Ver05, VV07] and the tools of [PV14].

Acknowledgments. The second author wishes to thank Uwe Franz for interesting discussions about the interpretation of [Bra12, Theorem 4.5] in terms of conditionally negative type functions.

The first author acknowledges the support of the ANR grants NEUMANN and OSQPI.

2000Mathematics Subject Classification. 46L65 (20G42, 46L10, 20F65).

1

2. Notation

Ifζ,ξare vectors in a Hilbert spaceH, we denoteωζ,ξ = (ζ| · ξ) the associated linear form on B(H), and we putωζ =ωζ,ζ ∈B(H)⁺_∗.

We will use at some places the q-numbers [k]_q =U_k−1(q+q⁻¹) = (q^k−q^−k)/(q−q⁻¹), where (U_k)_k∈N are the dilated Chebyshev polynomials of the second kind, defined byU₀ = 1, U₁ =X andXU_k =U_k−1+U_k+1.

Following [KV00], a locally compact quantum group G is given by a Hopf-C^∗-algebra C₀(G), with coproduct denoted ∆, satisfying certain axioms including the existence of left and right invariant weightsϕ,ϕ⁰ :C₀(G)₊ →[0,+∞] :

ϕ((ω⊗id)∆(f)) =ω(1)ϕ(f) and ϕ⁰((id⊗ω)∆(f)) =ω(1)ϕ⁰(f),

for allω ∈ C₀(G)^∗₊ and f ∈ C₀(G)₊ such that ϕ(f) <∞. When there is no risk of confusion, we will denote bykfk²₂ =ϕ(f^∗f)∈[0,+∞] the hilbertian norm associated withϕ. Using a GNS construction (H,Λϕ) forϕ, one defines the fundamental multiplicative unitary V ∈ B(H ⊗H), denotedW in [KV00], by putting V^∗(Λϕ⊗Λϕ)(f ⊗g) = (Λϕ⊗Λϕ)(∆(g)(f ⊗1)). Recall thatG is called unimodular, or of Kac type, ifϕ=ϕ⁰.

We will assume for the rest of the article that G is discrete, meaning for instance that there exists a non-zero vector ξ0 ∈ H such that V(ξ0⊗ζ) = ξ0⊗ζ for all ζ ∈ H. In that case one can always identify C0(G) with its image by the GNS representation on H, moreover it can be reconstructed fromV as the norm closure of (id⊗B(H)_∗)(V), and the coproduct is given by the formula ∆(f) =V^∗(1⊗f)V.

FromV one can also construct the dual algebraC_red^∗ (G) as the norm closure of (B(H)_∗⊗id)(V), equipped with the coproduct ∆(x) =V(x⊗1)V^∗(opposite to the dual coproduct of [KV00]). Since (ω_ξ0 ⊗id)(V) = 1, this C^∗-algebra is unital, and it is in fact a Woronowicz C^∗-algebra [Wor98].

Moreover the restriction ofωξ₀ is the Haar statehofC_red^∗ (G). When there is no risk of confusion, we will denotekxk²₂=h(x^∗x) the hilbertian norm associated withh. Using Λh:x7→xξ0as a GNS construction forhwe have (Λh⊗Λh)(∆(x)(1⊗y)) = ∆(x)V(ξ0⊗yξ0) =V(xξ0⊗yξ0), so thatV coincides with the multiplicative unitary associated to (C_red^∗ (G),∆) as constructed in [BS93].

The von Neumann algebras associated to Gare denoted L^∞(G) =C0(G)⁰⁰ andM =L(G) = C_red^∗ (G)⁰⁰, they carry natural extensions of the respective coproducts and Haar weights.

Recall that a corepresentation X of V is a unitaryX ∈ B(H⊗HX) such that V12X13X23 = X23V12. There exists a so-called maximal C^∗-algebra C^∗(G) together with a corepresentation V∈M(C0(G)⊗C^∗(G)) such that any corepresentationXcorresponds to a unique∗-representation π : C^∗(G) → B(HX) via the formula X = (id⊗π)(V) [BS93, Corollaire 1.6]. We say that X, π are unitary representations of G. When X = V, we obtain the regular representation π= λ: C^∗(G) →C_red^∗ (G). Moreover C^∗(G) admits a unique Woronowicz C^∗-algebra structure given by (id⊗∆)(V) =V12V13, and λis then the GNS representation associated with the Haar statehofC^∗(G).

On the dual side, a unitary corepresentation of G, or representation of V, is a unitary Y ∈ M(K(H_Y)⊗C^∗(G)) such that (id⊗∆)(Y) =Y₁₂Y₁₃, corresponding to a∗-representation ofC₀(G) as above. We denote Corep(G) the category of finite dimensional (f.-d.) unitary corepresentations ofG, and we choose a complete set IrrGof representatives of irreducible ones. In the discrete case, it is known that any corepresentation decomposes in IrrG. We have in particular a∗-isomorphism C0(G) 'c0−L

r∈IrrGB(Hr) such that V corresponds to L

r∈IrrGr. We denote pr ∈C0(G) the minimal central projection corresponding toB(Hr), andCc(G)⊂C0(G) the algebraic direct sum of the blocksB(Hr). The∗-algebraCc(G), together with the restriction of ∆, is a multiplier Hopf algebra in the sense of [VD94], and we denote its antipode by ˆS.

On the other hand we denote C[G]_r ⊂ C^∗(G) the subspace of coefficients (ω⊗id)(r) of an irreducible corepresentationr, andC[G] the algebraic direct sum of these subspaces. The coproduct onC^∗(G) restricts to an algebraic coproduct onC[G], which becomes then a plain Hopf∗-algebra.

Recall also thatλrestricts to an injective map on C[G], hence we shall sometimes considerC[G] as a subspace ofC_red^∗ (G). We will denote byandS the co-unit and the antipode ofC[G], and we note thatextends toC^∗(G): it is indeed the∗-homomorphism corresponding toX = idH⊗id_C.

We denote (fz)z the Woronowicz characters of C^∗(G), satisfying in particular h(xy) = h(y (f1∗x∗f1)), whereφ∗x= (id⊗φ)∆(x) andx∗φ= (φ⊗id)∆(x). We put as wellFv= (id⊗f1)(v)∈ B(Hv) for any f.-d. unitary corepresentationv ∈B(Hv)⊗C^∗(G) and qdimv= TrFv. Denoting vζ,ξ= (ωζ,ξ⊗id)(v), the Schur orthogonality relations [Wor87, (5.15)] read, for v,wirreducible:

(1) h(v^∗_ζ,ξwζ⁰,ξ⁰) = δv,w

qdimv(ζ⁰|F_v⁻¹ζ)(ξ|ξ⁰).

Note that in Woronowicz’ notation we havevkl =ve_k,e_l if (ei)i is a fixed orthonormal basis (ONB) of Hv. Similarly, we have the formula ϕ(a) = qdim(v) Tr(Fva) for the left Haar weight ϕ and a∈B(Hr)⊂C0(G) [PW90, (2.13)] — note that the coproduct onC0(G) constructed fromV in [PW90] is opposite to ours, so thatϕ=h_dR with their notation.

Finally we will use the following analogue of the Fourier transform:

F:Cc(G)→C_red^∗ (G), a7→(ϕ⊗id)(V(a⊗1)).

One can show that it is isometric with respect to the scalar products (a|b) =ϕ(a^∗b), (x|y) =h(x^∗y) fora,b∈Cc(G),x,y∈C_red^∗ (G) — see also Section 3.1.

In this article we will be mainly concerned with orthogonal free quantum groups. For Q ∈ GL_n(C) such that QQ¯ ∈CI_n, we denote A_o(Q) the universal unitalC^∗-algebra generated by n² elements v_ij subject to the relations Q¯vQ⁻¹ = v and v^∗v = vv^∗ = I_n⊗1, where v = (v_ij)_ij ∈ Mn(Ao(Q)) and ¯v = (v^∗_ij)ij. Equipped with the coproduct ∆ given by ∆(vij) =Pvik⊗vkj, it is a full WoronowiczC^∗-algebra, and we denoteG=FO(Q) the associated discrete quantum group, such that C^∗(FO(Q)) = Ao(Q). In the particular case Q =In we denote Ao(n) = Ao(In) and FOn=FO(In).

It is known from [Ban96] that the irreducible corepresentations of FO(Q) can be indexedv^k, k∈Nup to equivalence, in such a way thatv⁰= id_C⊗1, v¹ =v, and the following fusion rules hold

v^k⊗v^l'v^|k−l|⊕v^|k−l|+2⊕ · · · ⊕v^k+l.

Moreover the contragredient of v^k is equivalent to v^k for allk. The classical dimensions dimv^k satisfy dimv⁰ = 1, dimv¹ = n and ndimv^k = dimv^k−1+ dimv^k+1. We denote ρ ≥ 1 the greatest root of X²−nX + 1, so that dimv^k = [k+ 1]_ρ. Similarly we have qdimv⁰ = 1 and qdimv¹qdimv^k= qdimv^k−1+ qdimv^k+1. Moreover, ifQis normalized in such a way thatQQ¯ =

±In, we have qdimv= Tr(Q^∗Q). We denoteq∈]0,1] the smallest root ofX²−(qdimv)X+ 1, so that qdimv^k = [k+ 1]q.

3. The adjoint representation ofFOn

The aim of this section is to prove that the non-trivial part ad^◦of the adjoint representation of FO_n factors through the regular representation. Equivalently, we will prove that all states of the formω_ξ◦ad^◦, withξ∈H, factor throughC_red^∗ (FO_n). For this we will use the criterion given by Lemma 3.5 below, which is analogous to Theorem 3.1 of [Haa79].

3.1. Weak containment in the regular representation. In this section we gather some useful results which are well-known in the classical case. We say that an element f ∈ L^∞(G) is a (normalized) positive type function if there exists a stateφonC^∗(G) such thatf = (id⊗φ)(V).

The associated multiplier isM = (id⊗φ)∆ :C[G]→C[G]. The next lemma shows that it extends to a completely positive (CP) map onC_red^∗ (G) characterized by the identity (id⊗M)(V) =V(f⊗1).

In the locally compact case, the lemma is covered by [Daw12, Theorem 5.2]. For the convenience of the reader we include a short and self-contained proof for the discrete case.

Lemma 3.1. Let φbe a unital linear form on C[G] and considerM = (id⊗φ)∆ :C[G]→C[G].

ThenM extends to a CP map onC_red^∗ (G)if and only if φextends to a state ofC^∗(G).

Proof. By Fell’s absorption principleV₁₂V13=V23V₁₂V^∗23, ∆ extends to a∗-homomorphism ∆⁰ : C_red^∗ (G)→ C_red^∗ (G)⊗C^∗(G). Hence if φextends to a state of C^∗(G), M extends to a CP map M = (id⊗φ)∆⁰ onC_red^∗ (G). For the reverse implication, sinceC^∗(G) is the envelopingC^∗-algebra ofC[G], it suffices to prove thatφ(xx^∗)≥0 for allx∈C[G].

We choose a corepresentation v ∈B(Hv)⊗C_red^∗ (G). Let L be a GNS space for B(Hv), with respect to an arbitrary given state. For a ∈ B(Hv), resp. ω ∈ B(Hv)^∗, we denote ˆa, ˆω the

corresponding elements of L = B(C, L) resp. L^∗ = B(L,C), and we identify B(Hv) with a subspace ofB(L) via left multiplication. Similarly we denote ˆv the element ofB(C, L)⊗C_red^∗ (G) corresponding to v. We have then ˆvˆv^∗ ∈ B(L)⊗C_red^∗ (G) and (ˆω ⊗1)ˆvˆv^∗(ˆω^∗ ⊗1) = xx^∗ if x= (ω⊗id)(v).

IfM is CP, the following element ofB(L)⊗C_red^∗ (G) is positive:

(id⊗M)(ˆvvˆ^∗) = (id⊗id⊗φ)(id⊗∆)(ˆvˆv^∗)

= (id⊗id⊗φ)(v12vˆ13vˆ₁₃^∗ v₁₂^∗ ) =v(X⊗1)v^∗,

whereX = (id⊗φ)(ˆvvˆ^∗)∈B(L). We conclude thatX is positive, henceφ(xx^∗) = ˆωXωˆ^∗≥0 for

anyx∈C[G].

We will be particularly interested in the case whenφ factors throughC_red^∗ (G), or equivalently, whenφis a weak limit of states of the formPn

i=1ω_ξi◦λwithξ_i∈H. In that case we say thatf,φ are weakly associated toλ, or weakly`². We have the following classical lemma [Fel63, Lemma 1]:

Lemma 3.2. Let π : C^∗(G) → B(K) be a ∗-representation. Assume that there exists a subset X ⊂ K such that Span π(C[G])X =K and ωξ◦π is weakly `² for all ξ ∈ X. Then π factors throughC_red^∗ (G).

Proof. As noted in the original paper of Fell, the proof for groups applies in fact to general C^∗- algebras, and in particular to discrete quantum groups. More precisely, takeA=C^∗(G), T =π

andS={λ}in [Fel63, Rk 1].

On the other hand we say that φ∈C[G]^∗ is an `²-form if it is continuous with respect to the

`<sup>2</sup>-norm on C[G], i.e. there exists C ∈ Rsuch that |φ(x)|<sup>2</sup> ≤Ch(x<sup>∗</sup>x) for all x∈C[G]. In that case we denotekφk<sub>2</sub> the corresponding norm. Clearly, ifφis`² then it is weakly`².

Although we will not need this in the remainder of this article, the following lemma shows that

`<sup>2</sup>-forms can also be characterized in terms of the associated “functions” inC0(G) by means of the left Haar weight. Note that in the unimodular case we have simplyϕ(f f<sup>∗</sup>) =ϕ(f<sup>∗</sup>f) =kfk<sup>2</sup><sub>2</sub>. Lemma 3.3. Let φ∈C[G]<sup>∗</sup> be a linear form and put f = (id⊗φ)(V). Then φ is an`²-form if and only ifϕ(f f^∗)<∞.

Proof. We put p0= (id⊗h)(V)∈Cc(G), which is also the central support of ˆ, and we note the following identity (see the Remark after the proof) in the multiplier Hopf algebraCc(G) : (2) ∀f ∈Cc(G) ( ˆS⊗ϕ)(∆(p0)(1⊗f)) =f.

From this we can deduce that the scalar product in H implements, via the Fourier transform, the natural duality between C_c(G) and C[G] which is given, for x = (ω ⊗id)(V) ∈ C[G] and f = (id⊗φ)(V) ∈ C_c(G), by hf, xi = (ω⊗φ)(V) = φ(x) = ω(f). Indeed we have, using the identities ( ˆS⊗id)(V) =V^∗ andV13V23= (∆⊗id)(V):

(xξ₀|F(f)ξ₀) =h(x^∗F(f)) = (ϕ⊗h)((1⊗x^∗)V(f⊗1))

= (¯ωSˆ⊗ϕ⊗h)(V₁₃V₂₃(1⊗f ⊗1)) = (¯ωSˆ⊗ϕ)(∆(p₀)(1⊗f)) = ¯ω(f).

This yields, forx∈C[G] andφsuch thatf = (id⊗φ)(V)∈Cc(G), the formulaφ(x) =ω(f) = (F(f^∗)ξ₀|xξ₀). We get in particularkφk₂=kF(f^∗)k₂=kf^∗k₂=p

ϕ(f f^∗).

Now for a general φand r ∈ IrrG, denote φ_r the restriction of φ to C[G]_r. We have kφk²₂ = P

r∈IrrGkφrk²₂, (id⊗φ_r)(V) =p_rf and ϕ(f f^∗) =P

r∈IrrGϕ(f f^∗p_r), so that the identitykφk²₂ =

ϕ(f f^∗) still holds in [0,+∞].

Remark 3.4. The identity (2) is equivalent to the fact that the Fourier transformFis isometric.

Indeed a simple computation yields

h(F(a)^∗F(b)) =ϕ[a^∗( ˆS⊗ϕ)(∆(p₀)(1⊗b))].

Note that (2) can be easily proved by standard computations inCc(G), and this is one convenient way of establishing the fact thatF is isometric.

Now we consider the case of the discrete quantum group FOn, whose irreducible corepresenta- tionsv^k∈B(Hk)⊗Ao(n) are labeled, up to equivalence, by integersk. Recall that we denoteC[G]k

the corresponding coefficient subspaces, and (Uk)k the series of dilated Chebyshev polynomials of the second kind. In [Bra12], Brannan shows that the multiplier Ts : C_red^∗ (FOn) → C_red^∗ (FOn) associated with the functional

τs:C[FOn]→C, x∈C[FOn]k7→ U_k(s) Uk(n)(x)

is a completely positive map as above, for everys∈]2, n] — see also Section 4.1.

Besides it is known that FOn satisfies the Property of Rapid Decay. More precisely, for any k∈Nand anyx∈C[G]k ⊂C_red^∗ (G) we havekxk ≤C(1 +k)kxk2, whereCis a constant depending only onn [Ver07]. On the other hand we denote byl:C[G]→Cthe length form onFOn, whose restriction to C[G]k coincides by definition with k, and we recall that the convolution of two linear formsφ, ψ∈C[G]^∗ is defined byφ∗ψ= (φ⊗ψ)∆. It is well-known that the convolution exponentiale^φ=P

φ^∗n/n! is well-defined onC[G], and in the case ofl(or any other central form) we have simplye^−λl(x) =e^−λk(x) ifx∈C[G]k.

Using the multipliers Tsand Property RD we can follow the proof of [Haa79, Theorem 3.1] and deduce the following lemma:

Lemma 3.5. A stateφonC^∗(FOn)factors throughC_red^∗ (FOn)if and only ife^−λl∗φis an`²-form for allλ >0.

Proof. We denote byφkthe restriction ofφto the f.-d. subspaceC[G]k. Since these subspaces are pairwise orthogonal with respect toh, we havekφk²₂ =P

kφkk²₂. Now for everys < nthere is a λ >0 such thatU_k(s)/U_k(n)≤e^−λk for allk. We can then write

kτs∗φk²₂=X

k(τs∗φ)kk²₂=XUk(s)²

Uk(n)²kφkk²₂≤X

e^−2λkkφkk²₂=ke^−λl∗φk²₂.

Now ife^−λl∗φis`²for allλ >0, we conclude that this is also the case ofτs∗φfor alls∈]2, n[.

In particularτs∗φis weakly`<sup>2</sup> for all s. We have (τs∗φ)(x)→φ(x) ass→nfor allx∈C[G], andτs∗φis a state for alls, henceτs∗φ→φweakly and it follows thatφis weakly`².

Conversely, assume that φ factors throughC_red^∗ (G). Using Property RD we can write for any x∈C[G]k ⊂C_red^∗ (G):

|φ(x)| ≤ kxk ≤C(1 +k)kxk₂,

hencekφkk2≤C(1 +k). This clearly implies thatke^−λl∗φk2 is finite for everyλ >0.

3.2. The adjoint representation. Recall that we denote by λ : C^∗(G) → B(H) the GNS representation of the Haar state h, and consider the corresponding right regular representation ρ:C^∗(G)→B(H),x7→U λ(x)U given by the unitaryU(Λ_h(x)) = Λ_h(f₁∗S(x)) = Λ_h(S(x∗f₋₁)).

We have in particularρ(v_ij)Λ_h(x) = Λ_h(x(f₁∗v_ji^∗)), ifv_ij =v_ei,ej are the coefficients of a unitary corepresentation in an ONB. Recall that [U C_red^∗ (G)U, C_red^∗ (G)] = 0.

The adjoint representation of Gis ad :C^∗(G)→B(H), x7→ P

λ(x₍₁₎)ρ(x₍₂₎). Here ∆ :x7→

Px₍₁₎ ⊗x₍₂₎ is the coproduct from C^∗(G) to C_red^∗ (G)⊗maxC_red^∗ (G). We have (id⊗ad)(v) = (id⊗λ)(v)(id⊗ρ)(v) ifv is a corepresentation ofG. Here are two explicit formulae, forx∈C[G] andvij coefficient of a unitary corepresentation in an ONB:

ad(x)Λh(y) =X

Λh(x₍₁₎y(f1∗S(x₍₂₎))), ad(vij)Λh(y) =X

Λh(viky(f1∗v_jk^∗ )).

The corepresentation of V associated to ρ is W = (1⊗U)V(1⊗U), and the one associated to ad is A = V W. Note that we have, in the notation of [BS93], W = Σ ˜VΣ. In particular [BS93, Proposition 6.8] shows that W(1⊗f)W^∗ =σ∆(f) forf ∈C0(G). This means that the multiplicative unitaryW^∗ is associated to the discrete quantum groupG^co-op.

Lemma 3.6. The canonical lineCξ₀⊂H is invariant for the adjoint representationad :C^∗(G)→ B(H)if and only if Gis unimodular. In that case,ξ₀ is a fixed vector forad.

Proof. For x ∈ C[G] one can compute ad(x)ξ0 = Px₍₁₎(f1 ∗S(x₍₂₎))ξ0, which equals in the unimodular casePx₍₁₎S(x₍₂₎)ξ₀=(x)ξ₀. So in that caseξ₀is fixed.

Furthermore, for v∈IrrGthe previous computation together with Woronowicz’ orthogonality relations lead to

(ξ0|ad(vij)ξ0) =X

k

h(vik(f1∗v_jk^∗ )) =X

k

h((v_jk^∗ ∗f₋₁)vik)

=X

k

δjiδkk

qdimv = dimv qdimv(vij).

In particular, denotingφ:x7→(ξ0|ad(x)ξ0), we see that (id⊗φ)(v) = (dimv/qdimv)id. Now if the lineCξ0 is invariant, φis a character and since v is unitary we must have dimv= qdimv for

allv∈IrrG. This happens exactly whenGis unimodular.

Definition 3.7. If G is a unimodular discrete quantum group, we denote ad^◦ the restriction of the adjoint representation ofC^∗(G) to the subspaceH^◦=ξ₀^⊥⊂H.

Our aim in the remainder of the section is to show that ad^◦ factors through λ in the case of G=FO_n, n≥3 — or more generally for unimodular orthogonal free quantum groups. We will need some estimates involving coefficients of irreducible corepresentations ofG.

In order to carry on the computations, we will need to be more precise about the irreducible corepresentations of FO(Q). We denote by v =v¹ ∈ B(Cⁿ)⊗Ao(Q) the fundamental corepre- sentation. Then we introduce recursively the irreducible corepresentationv^k ∈ B(Hk)⊗Ao(Q) as the unique subcorepresentation of v^⊗k not equivalent to any v^l, l < k. In this way we have Hk ⊂H₁^⊗k, withH0 =C andH1 =Cⁿ. We denote by Pk ∈B(H₁^⊗k) the orthogonal projection ontoHk. Recall that each irreducible corepresentation of Ao(Q) is equivalent to exactly one v^k, and that we have the equivalence of corepresentations

Hk⊗Hl'H_|k−l|⊕H_|k−l|+2⊕ · · · ⊕Hk+l−2⊕Hk+l.

It is known that dimHk = [k+ 1]ρ= (ρ^k+1−ρ^−k−1)/(ρ−ρ⁻¹). Note in particular that we have D₁ρ^k ≤dimH_k ≤D₂ρ^k with constants 0< D₁< D₂ depending only onn.

Let us also denote by Q^k_r ∈ L(H₁^⊗k) the orthogonal projection onto the sum of all subspaces equivalent toH_r, so that P_k =Q^k_k. If p_r is the minimal central projection ofC₀(FO_n) associated withv^r, thenQ^k_r also corresponds to ∆^k−1(pr) via the natural action ofC0(FOn) onH1.

Note that the subspace ImQ^a+b+c_r (Pa+b⊗Pc) (resp. ImQ^a+b+c_r (Pa⊗Pb+c)) corresponds to the unique subcorepresentation ofv^a+b⊗v^c (resp. v^a⊗v^b+c) equivalent to v^r, when it is non-zero.

Whenr=a+b+cboth spaces coincide withHa+b+c. On the other hand one can show that these subspaces ofH₁^⊗a+b+c are pairwise “far from each other” when r < a+b+c andb is big. More precisely, Lemma A.4 of [VV07] shows that

k(Pa+b⊗Pc)Q^a+b+c_r (Pa⊗Pb+c)k ≤C1q^b

for some constant C₁ > 0 depending only on q. Indeed when r varies up to a+b+c−2 the maps on the left-hand side live in pairwise orthogonal subspaces of H₁^⊗a+b+c and sum up to (P_a+b⊗P_c)(P_a⊗P_b+c)−P_a+b+c.

Since we are interested in ad^◦, we assume in the remainder of this section that FO(Q) is uni- modular : equivalently,Qis a scalar multiple of a unitary matrix, orqρ= 1.

In the following lemma we give an upper estimate for |(ωζ ⊗Trk)((Pl⊗Pk)Q^k+l_r Σlk)|, where ζ is any vector inHl, Trk is the trace on B(Hk), and Σlk :Hl⊗Hk →Hk⊗Hl is the flip map.

Note thatk(Pl⊗Pk)Q^k+l_r Σlkk ≤ 1 so that (dimHk)kζk² is a trivial upper bound, which grows exponentially withk. In the lemma we derive an upper bound which is polynomial ink(and even constant forr < k+l).

Note that it is quite natural to consider maps like (Pl⊗Pk)Q^k+l_r Σlk when studying the adjoint representation ofA_o(n). A non-trivial upper bound for the normk(P1⊗P_k)Q^k+1_k−1Σ_1kkwas given in [VV07, Lemma 7.11], but it does not imply our tracial estimate below.

We choose a fixed vectorT1=H1→H1with norm√

dimH1—T1is unique up to a phase, and we have (id1⊗T₁^∗)(T1⊗id1) =µid1 with µ=±1. We choose then fixed vectorsTm∈Hm⊗Hm

inductively by puttingTm= (Pm⊗Pm)(id_m−1⊗T1⊗id_m−1)T_m−1. If (e^m_s)s is an ONB ofHm, we haveT_m=Pe^m_s ⊗e¯^m_s where (¯e^m_s)_s is again an ONB ofH_m, andkT_mk=√

dimH_m. Finally, we denote more generally

T_ab^m:H₁^⊗a−m⊗H₁^⊗b−m→H₁^⊗a+b, ζ⊗ξ7→X

ζ⊗e^m_s ⊗¯e^m_s ⊗ξ.

Lemma 3.8. Let (e^k_p)p be an ONB of Hk. There exists a constant C ≥1, depending only on n, such that we have, for any0≤m≤l≤k,l6= 0,ζ∈Hlandr=k−l, k−l+ 2, . . . , k+l−2m−2:

X

p

(T_lk^m∗(ζ⊗e^k_p)|Q^k+l−2m_r T_kl^m∗(e^k_p⊗ζ))

≤C^lkζk²,

X

p

(T_lk^m∗(ζ⊗e^k_p)|Pk+l−2mT_kl^m∗(e^k_p⊗ζ))

≤kC^lkζk² if l6= 2m,

≤(Ck)^lkζk² ifl= 2m.

Proof. In this proof we denotedk = dimHk = qdimv^k and idk ∈B(H₁^⊗k) the identity map. Let us denote by S^k_r, S₊^k the sums in the statement as well as S^k = S₊^k +P

rS_r^k. We first prove the estimate forS_r^k: using the Lemma A.4 from [VV07] recalled above, witha=c =l−mand b=k−l, we can write

|S_r^k|=

X

p

(T_lk^m∗(ζ⊗e^k_p)|Q^k+l−2m_r T_kl^m∗(e^k_p⊗ζ))

≤X

p

|(T_lk^m∗(ζ⊗e^k_p)|(Pl−m⊗Pk−m)Q^k+l−2m_r (Pk−m⊗Pl−m)T_kl^m∗(e^k_p⊗ζ))|

≤ k(P_l−m⊗P_k−m)Q^k+l−2m_r (P_k−m⊗P_l−m)k × kTmk²×X

p

kζk²ke^k_pk²

≤C1q^k−ldkdmkζk²≤C1D₂²q^−l−mkζk².

The case ofS₊^k is more involved. First notice that|S₊^k−S^k|=|P

rS^k_r| ≤(l−m)C1D²₂q^−l−mkζk² according to the estimate above. This shows that the estimate forS₊^k is equivalent to the same estimate for S^k with a possibly different constant. In the case when 2m < l we will prove the estimate forS₊^k using an induction over k. We first perform the following transformation using scalar products inH_l−m⊗H_k−l⊗H_l−m:

S^k = X

p,q,i,j

(T_lk^m∗(ζ⊗e^k_p)|e^l−m_i ⊗e^k−l_q ⊗e^l−m_j )×(e^l−m_i ⊗e^k−l_q ⊗e^l−m_j |T_kl^m∗(e^k_p⊗ζ))

= X

p,q,i,j,s,t

(ζ⊗e^k_p|e^l−m_i ⊗e^m_s ⊗e¯^m_s ⊗e^k−l_q ⊗e^l−m_j )×

×(e^l−m_i ⊗e^k−l_q ⊗e^m_t ⊗e¯^m_t ⊗e^l−m_j |e^k_p⊗ζ)

= X

q,i,j,s,t

(ζ|e^l−m_i ⊗e^m_s )×(e^l−m_i ⊗e^k−l_q ⊗e^m_t |P_k(¯e^m_s ⊗e^k−l_q ⊗e^l−m_j ))×(¯e^m_t ⊗e^l−m_j |ζ)

=X

q,s,t

(T_l,k−l+2m^m∗ (ζ⊗¯e^m_s ⊗e^k−l_q ⊗e^m_t )|PkT_k−l+2m,l^m∗ (¯e^m_s ⊗e^k−l_q ⊗e^m_t ⊗ζ)).

One can factor (P_k−l+m⊗id_l−m) out ofPk and let it move to the right of T_k−l+2m,l^m∗ . Similarly, one can factor (id_l−m⊗P_k−l+m) out, let it move throughT_k−l+2m,l^m∗ on the left-hand side of the scalar product, and take it back to the right-hand side thank to the sum overqandt (by cyclicity of the trace). This yields:

S^k =X

q,s,t

(T_l,k−l+2m^m∗ (ζ⊗¯e^m_s ⊗e^k−l_q ⊗e^m_t )|

|P_kT_k−l+2m,l^m∗ ((P_k−l+m⊗id_m)(id_m⊗P_k−l+m)(¯e^m_s ⊗e^k−l_q ⊗e^m_t )⊗ζ)), which can be compared to:

S₊^k−l+2m=X

p

(T_l,k−l+2m^m∗ (ζ⊗e^k−l+2m_p )|PkT_k−l+2m,l^m∗ (e^k−l+2m_p ⊗ζ))

=X

q,s,t

(T_l,k−l+2m^m∗ (ζ⊗e¯^m_s ⊗e^k−l_q ⊗e^m_t )|P_kT_k−l+2m,l^m∗ (P_k−l+2m(¯e^m_s ⊗e^k−l_q ⊗e^m_t )⊗ζ)).

More precisely, using again the Lemma A.4 from [VV07] witha=c=mandb=k−l we get

|S^k−S₊^k−l+2m| ≤X

q,s,t

kTmk²k(Pk−l+m⊗id_m)(id_m⊗P_k−l+m)−P_k−l+2mkkζk²

≤X

q,s,t

kTmk² C1q^k−lkζk²≤C1d³_md_k−lq^k−lkζk²≤C1D₂⁴q^−3mkζk².

Using our previous estimate for thel−m−1 termsS_r^k we obtain a recursive inequation forS₊^k:

|S₊^k|=|S^k−X

r

S_r^k| ≤ |S^k|+C1D₂²(l−m)q^−l−mkζk²

≤ |S₊^k−l+2m|+C1(D²₂l+D⁴₂)q^−3lkζk².

Fork≤2l we use the trivial estimate |S₊^k| ≤ dkkTmk²kζk² ≤D₂²q^−3lkζk². Sincel−2m >0 an easy induction onk yields|S^k₊| ≤klC^lkζk², which implies the estimate of the statement up to a change of the constantC.

For the case 2m > lwe consider the vector ¯ζ= (ζ^∗⊗id)Tl, which satisfies (id⊗T_m^∗)(ζ⊗id) = µ^l−m( ¯ζ^∗⊗id)(id⊗Tl−m) and (T_m^∗ ⊗id)(id⊗ζ) =µ^l−m(id⊗ζ¯^∗)(Tl−m⊗id). We transform S^k as follows:

S^k = Tr (ζ^∗⊗idk)(id_l−m⊗Tm⊗id_k−m)(id_k−m⊗T_m^∗ ⊗id_l−m)(idk⊗ζ)

= Tr (idm⊗T_l−m^∗ ⊗idk−m)( ¯ζ⊗idk+l−2m)(idk+l−2m⊗ζ¯^∗)(idk−m⊗Tl−m⊗idm)

= Tr (idk+l−2m⊗ζ¯^∗)(idk−m⊗Tl−m⊗idm)(idm⊗T_l−m^∗ ⊗idk−m)( ¯ζ⊗idk+l−2m) , S¯^k = Tr ( ¯ζ^∗⊗id_k+l−2m)(idm⊗T_l−m⊗id_k−m)(id_k−m⊗T_l−m^∗ ⊗idm)(id_k+l−2m⊗ζ)¯

. As a result, ¯S^k can be obtained from S^k+l−2m by replacing ζ with ¯ζ andm with l−m, and we can apply the case 2m < lto obtain the result — observe thatk+l−2m≤k andkζk¯ =kζk.

Iflis even, we still have to deal with the case 2m=l, which is the subject of the next lemma.

Lemma 3.8bis. Let (e^k_p)p be an ONB of Hk. There exists a constantC ≥1, depending only on n, such that we have, for anyζ, ξ∈H2mand0<2m=l≤k:

X

p

((idm⊗T_m^∗ ⊗idk−m)(ζ⊗e^k_p)|(idk−m⊗T_m^∗ ⊗idm)(e^k_p⊗ξ))

≤(Ck)^lkζkkξk.

Proof. We need in fact to prove a more general statement, wheremis allowed to take two different values ¯m,m∈N^∗on the left and on the right of the scalar product, with ¯m+m≤k. Forζ∈H_{2 ¯m}, ξ∈H_2m we denote

S^k(ζ, ξ) =X

p

((idm¯ ⊗T_m^∗_¯ ⊗id_k−m¯)(ζ⊗e^k_p)|(id_k−m⊗T_m^∗ ⊗idm)(e^k_p⊗ξ)).

We first transform this quantity by writing the trace ofB(Hk) as the restriction to a corner of the trace ofB(Hk−1⊗H1):

S^k(ζ, ξ) =X

q,s

((idm¯ ⊗T_m^∗_¯ ⊗id_k−m−1¯ )(ζ⊗e^k−1_q )⊗e¹_s|(id_k−m⊗T_m^∗ ⊗idm)(Pk(e^k−1_q ⊗e¹_s)⊗ξ))

=X

q

((id_m¯ ⊗T_m^∗_¯ ⊗id_k−m−1¯ )(ζ⊗e^k−1_q )|(id_k−m⊗T_m^∗ ⊗id_m−1)(P_k⊗id_2m−1)(e^k−1_q ⊗ξ)),˜ where ˜ξ=P

(e¹_s⊗id_2m−1⊗e^1∗_s )(ξ)∈H1⊗H_2m−1 has the same norm asξ. Observe that one has (e^k−1_q ⊗ξ) = (id˜ k⊗Pm⊗id_m−1)(e^k−1_q ⊗ξ) so that one can replace˜ T_m^∗ on the right-hand side of the scalar product byT_m−1^∗ (id_m−1⊗T₁^∗⊗id_m−1) orT₁^∗(id₁⊗T_m−1^∗ ⊗id₁).

Then we use the following generalization of Wenzl’s induction formula, which can be found e.g.

in [VV07, Equation (7.4)]:

P_k = (P_k−1⊗id₁) +

k−1

X

i=1

(−µ)^k−id_i−1 dk−1

(id_i−1⊗T₁⊗id_k−i−1⊗T₁^∗)(P_k−1⊗id₁).

Substituting Pk in the formula for S^k(ζ, ξ) above, we see that most terms vanish. The terms corresponding tok−m+ 1≤i≤k−1 vanish because in that case both legs ofT1 in the factor (id_k−m⊗T_m^∗ ⊗id_m−1)(id_i−1⊗T1⊗id_k−i−1⊗T₁^∗⊗id_2m−1) hit the left leg of Tm which lies in

Hm. The terms corresponding to ¯m+ 1≤i≤k−m−1 vanish because in that case the vectorT1

goes through (id_k−m⊗T_m^∗ ⊗id_m−1) and hitse^k−1_q on the left of the scalar product. Finally, the terms corresponding to 1≤i≤m¯ −1 vanish because the vectorT1hitsζ on the left of the scalar product.

Int the term with (P_k−1⊗id1) we replaceT_m^∗ byT_m−1^∗ (id_m−1⊗T₁^∗⊗id_m−1) on the right-hand side of the scalar product and obtain

X

q

((idm¯ ⊗T_m^∗_¯ ⊗id_k−m−1¯ )(ζ⊗e^k−1_q )|(id_k−m⊗T_m−1^∗ ⊗id_m−1)(e^k−1_q ⊗ξ⁰)) whereξ⁰= (T₁^∗⊗id_2m−2)( ˜ξ) =P

(¯e^1∗_s ⊗id_2m−2⊗e^1∗_s )(ξ)∈H_2m−2. We recognizeS^k−1(ζ, ξ⁰) and we note thatkξ⁰k ≤d1kξk. We remark also that whenm= 1 we have ξ⁰ = 0: indeed in the Kac case Σ11T1 is proportional toT1, andT₁^∗(H2) ={0}.

In the term i=k−m, the right-hand side of the scalar product reads:

A= (id_k−m⊗T₁^∗⊗id_m−1)(id_k−m+1⊗T_m−1^∗ ⊗id_m)×

×(id_k−m−1⊗T₁⊗id_3m−2)(id_k−2⊗T₁^∗⊗id_2m−1)(e^k−1_q ⊗ξ)˜

=µ(id_k−m−1⊗T_m−1^∗ ⊗id_m)(id_k−2⊗T₁^∗⊗id_2m−1)(e^k−1_q ⊗ξ)˜

=µ(id_k−m−1⊗T_m^∗ ⊗id_m)(e^k−1_q ⊗ξ).˜

Now we decompose ˜ξ= ξ⁰⁰+Q^2m_2m−2( ˜ξ), where ξ⁰⁰ = P2m( ˜ξ). Summing overq, the terms corre- sponding toξ⁰⁰yield the quantityS^k−1(ζ, ξ⁰⁰) and we note thatkξ⁰⁰k ≤ kξk.

On the other hand, by Wenzl’s recursion formula, Q^2m_2m−2( ˜ξ) = ˜ξ−P_2m( ˜ξ) decomposes as a linear combination of vectors of the form (idj⊗T1⊗id_2m−2−j)(ξ⁰). Again, the contributions with j6=m−1 vanish, either becauseT1hits the right leg ofT_m^∗, or because it hitse^k−1_q on the left-hand side of the scalar product (ifk >m¯+m). The corresponding term inA, without the multiplicative factor (−1)^md_m−1/d_2m−1, is

A⁰=µ(id_k−m−1⊗T_m^∗ ⊗id_m)(id_k+m−2⊗T₁⊗id_m−1)(e^k−1_q ⊗ξ⁰)

=µ(id_k−m−1⊗T₁^∗⊗idm)(id_k−m⊗T_m−1^∗ ⊗idm+1)(id_k+m−2⊗T1⊗id_m−1)(e^k−1_q ⊗ξ⁰)

= (idk−m⊗T_m−1^∗ ⊗idm−1)(e^k−1_q ⊗ξ⁰),

and summing the scalar products overqwe recognize againS^k−1(ζ, ξ⁰).

Finally fori= ¯mwe let the vectorT1from Wenzl’s formula go to the left-hand side of the tensor product. Noting thatT_m^∗(idm⊗T₁^∗⊗idm) =T_m+1^∗ onHm+1⊗H₁^⊗m+1andT₁^∗(id1⊗T_m^∗_¯⊗id1) =T_m+1^∗_¯ onHm+1¯ ⊗Hm+1¯ we obtain

((id_m−1¯ ⊗T_m+1^∗_¯ ⊗id_k−m−2¯ )(ζ⊗e^k−1_q )|(id_k−m−1⊗T_m+1^∗ ⊗id_m−2)(e^k−1_q ⊗ξ)).˜ In absolute value, the sum overq of these terms is less than

dimH_k−1kTm+1¯ kkTm+1kkζkkξk=d_k−1p

dm+1¯ dm+1kζkkξk.

Putting everything together we have obtained

|S^k(ζ, ξ)| ≤ |S^k−1(ζ, ξ⁰)|+^dk−m−1_d

k−1 |S^k−1(ζ, ξ⁰⁰)|+^dk−m−1_d ^dm−1

k−1d2m−1 |S^k−1(ζ, ξ⁰)|+

+^d_d^m−1¯

k−1 d_k−1p

d_m+1¯ d_m+1 kζkkξk

≤ |S^k−1(ζ, ξ⁰⁰)|+ 2|S^k−1(ζ, ξ⁰)|+D²₂q^{−2(m+ ¯m)}kζkkξk.

This allows to prove, by induction over m+ ¯m, that we have |S^k(ζ, ξ)| ≤D²₂(3q⁻²k)^{m+ ¯m}kζkkξk for all ζ ∈ H_{2 ¯m}, ξ ∈ H_2m and all k ≥ m¯ +m. We initialize the induction at m+ ¯m = 2, i.e.

m = ¯m = 1. Then ξ⁰ = 0, and by an easy induction over k ≥ 2 the inequation above yields

|S^k(ζ, ξ)| ≤D²₂q⁻⁴kkζkkξk. Now if the result holds atm+ ¯m−1, and assuming thatm >1, we apply the induction hypothesis toS^k−1(ζ, ξ⁰) in the inequation above and get

|S^k(ζ, ξ)| ≤ |S^k−1(ζ, ξ⁰⁰)|+D²₂(2(3q⁻²(k−1))^{m+ ¯m−1}+q^{−2(m+ ¯m)})kζkkξk

≤ |S^k−1(ζ, ξ⁰⁰)|+D²₂(3q⁻²)^{m+ ¯m}(k−1)^{m+ ¯m−1}kζkkξk.