Monodromy of a generalized Lame equation of third order

Abstract

We study the monodromy of the following third order linear differential equation \[y'''(z)-(α\wp(z;τ)+B)y'(z)+β\wp'(z;τ)y(z)=0, \] where $B\in\mathbb{C}$ is a parameter, $\wp(z;τ)$ is the Weierstrass $\wp$-function with periods $1$ and $τ$, and $α,β$ are constants such that the local exponents at the singularity $0$ are three distinct integers, which can always be written as $-n-l, 1-l, n+2l+2$ after a dual transformation, where $n,l\in\mathbb{N}$. This ODE can be seen as the third order version of the well-known Lamé equation $y''(z)-(m(m+1)\wp(z;τ)+B)y(z)=0$. We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that \begin{itemize} \item[(i)] if $n, l$ are both odd, then the monodromy can not be unitary; \item[(ii)] if $n$ is odd and $l$ is even, then there exist finite values of $B$ such that the monodromy is the Klein four-group and hence unitary; \item[(iii)] if $n$ is even, then whether there exists $B$ such that the monodromy is unitary depends on the choice of the period $τ$. \end{itemize} The methods of studying the second order Lamé equation can not work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable $SU(3)$ Toda system in another work (Chen-Lin, J. Differ. Geom. to appear).