Communications In Mathematical Physics - Volume 275 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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147) (148) L2 Denote B := O p w (b−1 ). By Theorem 8 one a0 + O p w (δ1 ), b BB = I + O p w (δ2 ), AB = O p w (149) (150) with δ1,2 estimated by (142)–(144). Since ab0 is bounded together with its derivatives, it follows that 1 := O p w (δ1 ) is bounded. Similarly 2 := O p w (δ2 ) is bounded. Thus, using Neumann’s is formula one gets that the operator I + 2 is invertible provided is small enough. So, from (150) one has (I + 2) −1 BB = I B −1 = (I + ⇐⇒ 2) −1 B. (151) Finally one has Aψ L2 = ABB −1 ψ ≤ C (I + ≤ AB L(L 2 ,L 2 ) B −1 ψ L2 −1 2) Bψ L2 ≤ C Bψ L2 L2 (152) .

It forms an explanation why there are so many more nontrivial identities on the hyperbolic, trigonometric and rational level when compared to the elliptic level. Returning to the precise description of the relevant symmetry groups, we will mainly encounter stabilizer subgroups of the isotropy subgroup W−δ . Observe that W−δ is a standard parabolic subgroup of W with respect to both bases j since −δ ∈ V + ( j ) + }, respectively ( j = 1, 2), with associated simple reflections sα , α ∈ 1 := 1 \ {α18 sα , α ∈ 2 := 2 \ {γ18 }.

Note that this subsection τi j = τi j the q-difference operators τi j are already well defined on {t ∈ C8 | 8j=1 t j = p 2 q 2 }. 6), we have θ (q −1 t8 t7±1 ; p) θ (t6 t7±1 ; p) Ie (τ68 t; z) + (t6 ↔ t7 ) = Ie (t; z), where (t6 ↔ t7 ) means the same term with t6 and t7 interchanged. For generic t ∈ C8 with 8j=1 t j = p 2 q 2 we integrate this equality over z ∈ C, with C a deformation of T which separates the upward and downward pole sequences of all three integrands at the same time. 7) as meromorphic functions in t ∈ H pq .

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