Local submodules and the multiplicity of irreducible subquotients in category \({\mathcal O}\) (Q1087625)

The author studies the structure of the set of submodules of the module M in the category \({\mathcal O}\) over a Kac-Moody Lie algebra. The set CL(\(\lambda)\) of the so-called ''local'' submodules of the module M with given quotient \(\simeq L(\lambda)\) is defined and it is proved that the cardinality of a basis of CL(\(\lambda)\) (in the sense defined in the paper) is equal to the multiplicity of L(\(\lambda)\) as irreducible subquotient of M.