On symmetric partial differential operators

Abstract

Let \(\sigma _{1}, \dots , \sigma _{k}\) be the elementary symmetric functions of the complex variables \(x_{1}, \dots , x_{k}\). We say that \(F \in {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\) is a trace function if their exists \(f \in {\mathbb {C}}[z]\) such that \(F(\sigma _{1}, \dots , \sigma _{k}) = \sum _{j=1}^{k} f(x_{j})\) for all \(\sigma \in {\mathbb {C}}^{k}\). We give an explicit finite family of second order differential operators in the Weyl algebra \(W_{2}:= {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\langle \frac{\partial }{\partial \sigma _{1}}, \dots , \frac{\partial }{\partial \sigma _{k}}\rangle \) which generates the left ideal in \(W_{2}\) of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As an application, we obtain for each integer k a holonomic system which is a quotient of \(W_{2}\) by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: \(z^{k} + \sum _{h=1}^{k} (-1)^{h}\sigma _{h}z^{k-h} = 0\).