Critical points of master functions and mKdV hierarchy of type $C^{(1)}_{n}$

Abstract

We consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $C_n^{(1)}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb C^m$, where $m$ are positive integers. For each cell we define an injective rational map $\mathbb C^m \to \mathcal M(C_n^{(1)})$ of the cell to the space $\mathcal M(C_n^{(1)})$ of Miura opers of type $C_n^{(1)}$. We show that the image of the map is invariant with respect to all mKdV flows on $\mathcal M(C_n^{(1)})$ and the image is point-wise fixed by all mKdV flows $\frac{\partial}{\partial t_r}$ with index $r$ greater than $2m$.