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...< aiv, biv ∗ > if JV, ∗ V (λ) = ∑ ai ⊗bi. Let ρ ∈ h ∗ be the half sum of positive roots. Lemma 29. For A = U(g ) and any V, W ∈ O0, we have JV,W(tρ) → 1 when t ∈ C and t tends to infinity. Proof : In =-=[ES1]-=-, the intertwining operator Φ v (λ) was computed in terms of the Shapovalov form (formula (3-5) in [ES1]). From formula (3-5) in [ES1] it is easy to obtain the following asymptotic expansion of Φ v (λ...

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... has the form (1.3) (RV (Y1)ψ)(h) = δ(h) −1 ( ∆h − ∑ eαfα 2 sinh 2 ) − 〈ρ, ρ〉 (ψ(h)δ(h)), h ∈ t (〈α, h〉/2) α∈R + where ∆h is the Laplace operator on h, and ψ(h) = ˜ ψ(e h ), ˜ ψ ∈ F V (K). (iii) ([E],=-=[ES]-=-) Set (1.4) Ψ(h) = Tr|Nλ (Φeh ∑ 〈λ,h〉 ) = e α∈Q + Tr| Nλ[λ−α](Φ)e −〈α,h〉 . This series absolutely converges in the region {h ∈ h : Re〈α, h〉 > 0, α ∈ R + }, and its sum takes values in V ∗ [0] and is a...

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...k (λ)ea−k 21 e k 31e b−k 32 vλ ⊗ e b−m 32 e m 31e a−m 21 vλ. Finaly, given v ∈ V , there is a unique singular vector, sing(vλ ⊗v), in Mλ ⊗V of the form sing(vλ ⊗ v) = vλ ⊗ v + {lower order terms}. In =-=[ESt]-=- the vector sing(vλ ⊗ v) is given in terms of the inverse of the Shapovalov form. As a corollary we get sing(vλ ⊗ v) = = ∞∑ a,b=0 ∞∑ a,b=0 ∑ min(a,b) m,k=0 ∑ min(a,b) m,k=0 (−1) a+b B a,b m,k (λ)eb−m ...

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...sion / 1 (; x) = 2ffl \Delta / (0) 1 (; x) +O(ffl 2 ) where / (0) 1 (; x) = e x (e x \Gamma e \Gammax ) 2 `s\Gamma e x + e \Gammax e x \Gamma e \Gammax ' is the /-function for the classical case (cf. =-=[ES]-=-). The difference operators become M 1 = 2 + ffl 2 (D 2 + 4) +O(ffl 3 ) M 0 = \Gamma2 + ffl 2 (3D 2 + 8) + 4ffl 3 (D 3 + 1) +O(ffl 4 ); where commuting differential operators D 2 ; D 3 are equal D 2 =...

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...k ⎦ . (54) Also, the discritization [7,8,29,30] of the BN-CSC model and the BCN-SS model can be constructed. 4 The integrability of this model (with λ ′ 1 = 0) has been studied by Etingof and Styrkas =-=[27]-=- in terms of the representation theory of the Lie algebras. See also [18,28]. 5 Using the identity sin 2x = 2sin xcos x, we rewrote the term 1/sin 2 (π/L)2qj. 12The models which were introduced in th...

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...(r,s)∈Z2>0, 1≤rs≤n [ lim h→hr,s(t) Nr,s(t, h) h− hr,s(t) ]−1 fn−rs(t, hr,s(t) + rs) h− hr,s(t) . (1.25) Actually, one can obtain this formula by considering the Jantzen filtration (see [27], [14] and =-=[10]-=-) on M(c, h) and by some calculation on Kn. Thus from the comparison of (1.24) and (1.25), the verification of the conjecture (1.23) is reduced to the conjecture (1.20). However, it seems that the jus...