the analogy with the case of an isolated hypersurface singularity. We have seen that the theorem of Malgrange can be adapted for reductive linear free divisors using the logarithmic Brieskorn lattice from above. The regularity of $(G_0(h),\nabla)$ at $\theta=0$ suggest to study the spectrum in the classical sense of Varchenko (i.e., at $\theta=0$) of this lattice. We recall the definition and calculate two examples, in order to show that this spectrum contains additional information not present in roots of the Bernstein polynomial, similarly to the case of isolated singularities.

Let $(\mathbb{E},\nabla)$ be a vector bundle on ${{\mathds C}}=\operatorname{\textup{Spec}\,}{{\mathds C}}[\theta]$ equipped with a connection with a pole at zero of order two at most, which is regular singular. The localization $\mathbb{M}:=E\otimes_{{{\mathds C}}[\theta]}{{\mathds C}}[\theta,\theta^{-1}]$ has the structure of a holonomic ${{\mathds C}}[\theta]\langle\partial_\theta\rangle$-module with a regular singularity at $\theta=0$. We suppose that the monodromy of its de Rham complex is quasi-unipotent. Denote by $V^\bullet \mathbb{M}$ the canonical V-filtration on $\mathbb{M}$ at $\theta=0$, indexed by ${{\mathds Q}}$. As this is a filtration by free ${{\mathds C}}[\theta]$-modules (and not by free ${{\mathds C}}[\theta^{-1}]$-modules as the $V$-filtration at $\theta=\infty$), we write it as a decreasing filtration. Define the spectrum of $(E,\nabla)$ to be $$\operatorname{\textup{Sp}_{\theta=0}}(E,\nabla):=\sum_{\alpha \in {{\mathds 