On Unitary Monodromy of Second-Order ODEs

Abstract: Given a second-order, holomorphic, linear differential equation $Lf=0$ on a Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is \emph{unitary} if it preserves a non-degenerate (though not necessarily positive) Hermitian form $H$ on $\mathbb{C}^2$ under the action $g\circ H\doteq g^\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a differential operator $L$ to have a unitary monodromy group, and we construct the form $H$ explicitly. First, in the case that the natural representation of $G$ on $\mathbb{C}^2$ is irreducible, we show that unitarity is equivalent to a set of easily-verified trace conditions on local monodromy matrices; in the case that it is reducible, we show that $G$ is unitary if and only if it is conjugate to a subgroup of one of two model subgroups of $\operatorname{GL}(2,\mathbb{C})$. Second, we show that unitarity of $G$ is equivalent to a criterion on the real dimension of the algebra $A$ generated by a rescaled group $G'\subset\operatorname{SL}(2,\mathbb{C})$: that $\dim(A)=1$ if $G\subset S^1$ is scalar, $\dim(A)=2$ if $G$ is abelian, $\dim(A)=3$ if $G$ is non-abelian but its action on $\mathbb{C}^2$ is reducible, and $\dim(A)=4$ otherwise. We leverage these results to extend a conjecture of Frits Beukers on the spectrum of Lam\'e operators -- namely, we give asymptotic and numerical evidence that his conjecture should apply similarly to a wider class of Heun operator. Our work makes progress towards characterizing the spectra of second-order operators on Riemann surfaces, and in particular, towards answering the accessory parameter problem for Heun equations.