On involutive homogeneous varieties and representations of Weyl algebras
If \(M\) is a finitely generated module over a complex Weyl algebra \(A_n\) then \(M\) is provided with a filtration associated with the Bernstein filtration on \(A_n\). The corresponding graded module \(\text{gr}(M)\) over \(P_n=\mathbb{C}[X_1,\dots,X_{2n}]\) is finitely generated. Let \(\text{Ch}(M)\) be the homogeneous algebraic variety in \(\mathbb{C}^{2n}\) defined by the annihilator of \(\text{gr}(M)\) in \(P_n\). This variety is involutive in the sense that the tangent space \(L=T_p(\text{Ch}(M))\) of each point \(p\in\text{Ch}(M)\) is co-isotropic, \(L^\perp\subseteq L\), with respect to the standard symplectic form on \(\mathbb{C}^{2n}\). The author presents a construction of a series of minimal involutive homogeneous subvarieties in \(\mathbb{C}^{2d}\) for sufficiently large \(d\). Based on this construction he shows that if \(n\geq 4\), \(k\geq 4\), and \(d\in A_n\) has degree \(k\) with generic principal symbol in \(P_n\) then \(A_n/A_nd\) is a left irreducible \(A_n\)-module. Denote by \({\mathcal M}_n^k\) the full category of finitely generated \(A_n\)-modules \(M\) such that \(\text{Ch}(M)\) has all irreducible components minimal of codimension \(k\). It is shown that if \(n\geq 2\), \(1\leq k\leq n\), then \({\mathcal M}_n^k\) has uncountably many non-isomorphic irreducible modules.
scientific article; zbMATH DE number 1448643
scientific article