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.