Cohomology of quantum groups: An analog of Kostants Theorem

We prove the analog of Kostants Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostants cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $mathfrak{g} = mathfrak{sl}(n)$. We also show that Kostants formula holds when $q$ is specialized to an $ell$-th root of unity for odd $ell ge h-1$ (where $h$ is the Coxeter number of $mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.