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1 we have: a(i,α)⊗1 a(j,β)⊗1 = σi σj a1⊗α¯ a1⊗β¯ = k,δ δ Nijk Nαβ σk a1⊗δ¯ . δ = Note by the characterization of exp (cf. p. 4 of [FKW], if Nijk Nαβ 0, then (k, δ) ∈ exp. Using (1) again we have: a(i,α)⊗1 a(j,β)⊗1 = (k,δ)∈exp δ Nijk Nαβ a(k,δ)⊗1 . (3) now follows from (1) of Prop. 2 and (2). 14) of [KW], and using the notations there we have b(i, α) = b( , ; ) = N a( )a( , b(i, α) is given by )a( ). ) Note that d( ) = a( a(0) , and we have similar identities when and , 0 . The proof of (4) is now complete by definitions.

Equal to a function continuous from [0, T ] into L2 (R) and f (t), g(t) − f (s), g(s) = for all s, t ∈ [0, T ]. t s d f (τ ) , g(τ ) dτ + dτ t s d g(τ ) , f (τ ) dτ dτ 48 A. Constantin, L. Molinet Throughout this paper, we will denote by {ρn }n≥1 the mollifiers ρn (x) := ρ(ξ ) dξ R −1 n ρ(nx), x ∈ R, n ≥ 1, where ρ ∈ Cc∞ (R) is defined by e1/(x 0 ρ(x) := 2 −1) for |x| < 1, for |x| ≥ 1. The next two approximation results will be also very useful. Lemma 3. Let f : R → R be uniformly continuous and bounded.

This is also known as diagram automorphisms since they correspond to the automorphisms of Dynkin diagrams. 2) of [Wal] and the formula for statistical dimensions above. Also note that this ZN action preserves exp and therefore induces a ZN action on exp. For each (i, α) ∈ exp, we will denote by [i, α] its orbit in exp under the ZN action. 6. Let H ⊂ GL be as in the previous paragraph. , if σ (i) = i, σ (α) = α for some (i, α) ∈ exp, then σ = id. Then the covariant representations πi,α are irreducible and πi,α is unitarily equivalent to πj,β as covariant representations iff σ (i) = j, σ (α) = β for some σ ∈ ZN ; (3) Suppose the conditions of (2) hold.