Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators In

For the algebra In of polynomial integro-differential operators over a field K of characteristic zero, a classification of simple weight and generalized weight (left and right) In-modules is given. It is proven that the category of weight In-modules is semisimple. An explicit description of generalized weight In-modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight In-modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight In-modules is given. In the wild case `natural` tame subcategories are considered with explicit description of indecomposable modules. It is proven that every generalized weight In-module is a unique sum of absolutely prime modules. For an arbitrary ring R, we introduce the concept of {\em absolutely prime} R-module (a nonzero R-module M is absolutely prime if all nonzero subfactors of M have the same annihilator). It is shown that every indecomposable generalized weight In-module is equidimensional. A criterion is given for a generalized weight In-module to be finitely generated.