Rings of differential operators on singular generalized multi-cusp algebras

Abstract

The aim of the paper is to study the ring of differential operators \(\mathcal{D}(A(m))\) on the generalized multi-cusp algebra \(A(m)\) where \(m\in \mathbb{N}^n\) (of Krull dimension \(n\)). The algebra \(A(m)\) is singular apart from the single case when \(m=(1, \ldots , 1)\). In this case, the algebra \(A(m)\) is a polynomial algebra in \(n\) variables. So, the \(n\)'th Weyl algebra \(A_n=\mathcal{D}(A(1, \ldots , 1))\) is a member of the family of algebras \(\mathcal{D}(A(m))\). We prove that the algebra \(\mathcal{D}(A(m))\) is a central, simple, \(\mathbb{Z}^n\)-graded, finitely generated Noetherian domain of Gelfand-Kirillov dimension \(2n\). Explicit finite sets of generators and defining relations is given for the algebra \(\mathcal{D}(A(m))\). We prove that the Krull dimension and the global dimension of the algebra \(\mathcal{D}(A(m))\) is \(n\). An analogue of the Inequality of Bernstein is proven. In the case when \(n = 1\), simple \(\mathcal{D}(A(m))\)-modules are classified.