Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $\mathbb{I}_n$

For the algebra $\mathbb{I}_n = K {\langle x_1, \dotsc, x_n, \partial_1, \dotsc, \partial_n, \int_1, \dotsc, \int_n \rangle}$ of polynomial integrodifferential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $\mathbb{I}_n$‑modules is given. It is proven that the category of weight $\mathbb{I}_n$‑modules is semisimple. An explicit description of generalized weight $\mathbb{I}_n$‑modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $\mathbb{I}_n$‑modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $\mathbb{I}_n$‑modules is given. In the wild case ‘natural‘ tame subcategories are considered with explicit description of indecomposable modules. For an arbitrary ring $R$, we introduce the concept of absolutely prime $R$‑module (a nonzero $R$‑module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is proven that every generalized weight $\mathbb{I}_n$‑module is a unique sum of absolutely prime modules. It is also shown that every indecomposable generalized weight $\mathbb{I}_n$‑module is equidimensional. A criterion is given for a generalized weight $\mathbb{I}_n$‑module to be finitely generated.