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The dependence is on the Lie subalgera of the stabilizer subgroup. So, if an orbit M is good, then any orbit isomorphic to M as a homogeneous manifold will be good. In Sect. 2). We show that for almost all f these cohomologies coincide with the usual de Rham cohomologies of the manifold M and then use this fact in Sect. 5 to prove the existence of an invariant quantization. In the proof we use methods of [DS1] and [DS2]. 3). For the case An some additional arguments are required. ) In conclusion we make two remarks.
6). It will be convenient to include the conditions c(α¯ i ) = 0. 4. a) Given a k-tuple of positive numbers (c1 , . . , ck ) := (c(α¯ 1 ), . . , ¯ for all c(α¯ k )) satisfying the Assumption, Eq. 6) uniquely defines numbers c(α) ¯ α ∧ E−α satisfies positive quasiroots α¯ = α¯ i such that the bivector v = c(α)E the condition [[v, v]] = K 2 ϕM . 8) for all quasiroots α. ¯ Proof. a) Since by assumption the denominator is never zero, Eq. 5) for all positive β. 4) becomes c(α¯ + β)(c( α) ¯ + c(β)) ¯ β) Eq.
However if ¯ β) ¯ − Kλ(γ¯ ), c(β¯ + γ¯ )λ(β¯ + γ¯ ) = c(β)λ( we have a contradiction, since either the sign in front of Kλ(α¯ + β¯ + γ¯ ) is positive and ¯ = +Kλ(α¯ + β), ¯ so 0 = λ(α¯ + β), ¯ or the sign in front of Kλ(α¯ + β¯ + γ¯ ) −Kλ(α¯ + β) is negative, implying 0 = λ(γ¯ ). We conclude that the sign in front of Kλ(γ¯ ) must be positive. In the situation of the lemma we can express c(γ¯ ) in terms of c(α) ¯ as ¯ + λ(β¯ + γ¯ )). 13) Now, we consider the pair (M, λ) as an orbit in g∗ passing through λ with the KKS Poisson bracket v = vλ = α∈ + 1 Eα ∧ E−α .
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