Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators \(\mathbb{I}_n\) (Q2152632)

The authors study (generalized) simple weight left modules over the algebra of polynomial integro-differential operators \[ \mathbb I_n =K\langle x_1,\dots ,x_n,\partial_1,\dots ,\partial_n,\int _1,\dots ,\int_n\rangle, \] where \(K\) is a field of characteristics zero. In Section 2, the classification of this type of modules is given (Theorem 2.3). Moreover, it is shown that the category of weight \(\mathbb I_n\)-modules is a semisimple category (Theorem 2.5). In Section 3, the indecomposable generalized weight \(\mathbb I_n\)-modules are considered. For each orbit \(\mathcal{O} = \mathcal{M}_n/G\), where \(\mathcal{M}_n\) is the maximal spectrum of the polynomial algebra \(D_n= K[H_1,\ldots H_n]\) where \(H_i:=\partial_ix_i\) and \(G =\langle\sigma_1,\ldots \sigma_n\rangle\subseteq Aut_K(D_n)\) with \(\sigma_i(H_j)=H_j-\delta_{ij}\) where \(\delta_{ij}\) is the Kronecker delta, the subcategory \(GW(\mathbb I_n,\mathcal{O})\) of generalized \(\mathbb I_n\)-modules with \(\mathrm{Supp}\subseteq\mathcal{O}\) is a direct sums of subcategories generated by a special single weight \(\mathbb I_n\)-module (Theorem 3.2). A criterion for this categories to be finite representation type, tame or wild is presented. From this result an explicit description of indecomposable generalized weight \(\mathbb I_n\)-modules is given (Theorem 3.6). Section 4 is devoted to the case of right \(\mathbb I_n\)-modules.