Higher Lamé equations and critical points of master functions

Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power $U(\mathfrak n_{-})^{\otimes(n-1)}$, where $n$ is the number of singular points in $\mathbb C$ and $U(\mathfrak n_{-})$ is the enveloping algebra of the nilpotent subalgebra of $\mathfrak{gl}_{r+1}$.