The $\hat{I}¼$-Trace System

Abstract

We study a simple 1-parameter perturbation of the regular holonomic Trace System satisfied by a complex power of the root of the universal polynomial of degree k as a holomorphic function of the coefficients. We prove that these systems have many analogous properties than the Trace System studied in [4] and we prove that they are, in general, minimal extensions of a simple pole meromorphic connection on a rank $k$ trivial bundle on ${\mathbb{C}}^k$. We also examine the structure of these $D$-modules for the special values of the parameters. This explicites many examples of perverse sheaves associated to representations of the $\ddot{I}{€}_1$ of the complement of the hyper-surface $\{\ddot{I}{ƒ}_k\hat{I}”(\ddot{I}ƒ)=0\}$ in the affine space with coordinates $\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_k$, where $\hat{I}”(\ddot{I}ƒ)$ is the discriminant of the universal monic polynomial of degree $k$, $P_{\ddot{I}}ƒ(z):=z^k+{\sum }_{h=1}^k(-1{)}^h\ddot{I}{ƒ}_h{z}^{k-h}$.