COMPOSITION FACTORS OF COHOMOLOGY GROUPS FOR QUANTUM GROUPS

Let U be a quantum group over A=Z Ω with a symmetric Cartan matrix, where Ω is the ideal of Z generated v-1, k be an algebraically closed of characteristic zero. Consider a ring homomorphism A→k(v|→ξ), let U k=U? 瑼 k. Suppose that ξ is a primitive p th root of unity. Let χ∈X + be pregular weihts and let λ is maximal anong weights strongly likned to χ. We give the neceessary and sufficient condition for L k(λ) (or L k(χ)) to be a composition factor of H i k(w·χ). If the condition is satisfied, then it occurs with multiplicity 1.