Decomposition of D-modules over a hyperplane arrangement in the plane
AbstractLet α1,α2,…,αm be linear forms defined on ℂn and 
$\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$
, where V(αi)={p∈ℂn:αi(p)=0}. The coordinate ring 
$O_{\mathcal{X}}$ 
of 
$\mathcal{X}$ 
is a holonomic An-module, where An is the nth Weyl algebra and since holonomic An-modules have finite length, 
$O_{\mathcal{X}}$ 
has finite length. We consider a “twisted” variant of this An-module which is also holonomic. Define Mαβ to be the free rank-1 ℂ[x]α-module on the generator αβ (thought of as a multivalued function), where 
$\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ 
and the multi-index β=(β1,…,βm)∈ℂm. Our main result is the computation of the number of decomposition factors of Mαβ and their description when n=2.