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By Stanislaw Saks

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In particular all roots have the same length and all dβ are 1. 2. Let β be an arbitrary positive root. Then for any λ ∈ X + for which λ − β is also in X + we have ExtiCA (∇A (λ), ∇A (λ − β)) A/([ λ + ρ, β ∨ − 1])A 0 if i = 1, otherwise. 2. Adjacent Weyl modules. Let R again be arbitrary. Denote by A(v−q) the localization of A at the maximal ideal generated by v − q. 1. 3. Suppose l ≥ h. Let (λ, λ ) be a pair of adjacent l-regular dominant weights separated by the hyperplane Hβ,n and with λ > λ .

Given partitions λ, µ and ν, there exists a polynomial fλµ ν ν in one variable with coefficients in Q[ξ], such that fλµ (n) = kλµ (n) for all n. 3 (1). Similarly, we conjecture that there exists polynomials gλµ (x) with coefficients in Q[ξ], such that gλµ (n) = bλµ (n) for all n. 1 is a result in [FH]. 1 holds. Set B to be the ring of polynomials in Q[x] which take integer values at integers. We can define a B[ξ]-algebra K with a basis given by the symbols Γλ , where λ runs over all partitions, and the multiplication given by ν fλµ Γν .

N)Γµ (n) = ν The main result of this Note is the following theorem on these structure conν stants kλµ (n). 1. (1) For any n, kλµ ν negative integral coefficients. , odd). ν (2) We have kλµ (n) = 0 unless |ν| ≤ |λ| + |µ|. ν (n) is independent of n. (3) If |ν| = |λ| + |µ|, then kλµ It follows from (2) and (3) that the center Z(Hn ) is naturally a filtered algebra and the structure constants of the associated graded algebra are independent of n. 1 which ν simply states that kλµ (n) are polynomials in n, and further implications on the algebra generators of Z(Hn ).

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