Decompositions in categories of highest weight modules

Abstract

Let g be a semisimple finite-dimensional subalgebra of a Lie algebra a defined over C. To analyze an a-module X we often restrict the action from a to g and decompose X as a g-module. For example, if X is U(g)-locally finite then X is completely reducible as a g-module. In this article we focus on this decomposition problem not in cases when X is U(g)-locally finite but instead when X admits a nondegenerate g-invariant bilinear form. Irreducible a-modules often admit invariant bilinear or Hermitian forms which by irreducibility are nondegenerate (or zero). This leads us to study a category 9 of g-modules X that are highest weight modules and admit nondegenerate invariant bilinear forms. Our first main result, Theorem 1.8, gives an essentially unique decomposition of X into indecomposable selfdual submodules and classifies these indecomposable self-dual modules. The choice of category studied here is further suggested by the role of the Zuckerman derived functors ri [12]. These functors are fundamental to representation theory and the theory of Harish-Chandra modules. Let A be a real semisimple Lie group with Lie algebra a,, and complex Lie algebra a Let G be a maximal compact subgroup of A with complex Lie algebra g. Let p = m@u be a parabolic subalgebra of g with reductive component m and nilradical u and set s = dim(u). The functors r’ are the right derived functors of the g-finite submodule functor defined on the category of U(m)-locally finite g-modules. Theorem 1.8, recast using derived functors, would assert that any X in 9 has an essentially unique decomposition into special indecomposable submodules A4 which have cohomology PA4 = 0 for all i except possibly i = s.