The cohomology of semisimple Lie algebras with coefficients in a Verma module

Abstract:

The structure of the cohomology of a complex semisimple Lie algebra with coefficients in an arbitrary Verma module is completely determined. Because the Verma modules are infinite-dimensional, the cohomology need not vanish (as it does for nontrivial finite-dimensional modules). The methods presented exploit the homological machinery of Cartan-Eilenberg [3]. The results of [3], when applied to the universal enveloping algebra of a semisimple Lie algebra and when coupled with key results of Kostant [12], Hochschild-Serre [9], yield the basic structure theorem-Theorem 4.19. Our results show, incidently, that an assertion of H. Kimura, Theorem 2 of [13] is false. A counterexample is presented in §6.