Isomonodromic Deformations for Linear $q$-Difference Systems of Degree One

We construct isomonodromy transformations for linear $q$-difference systems of the form $Y(qz)=A(z)Y(z)$, where $A(z)=A_0+z{A}_1$ has diagonal leading coefficient. These transformations shift eigenvalues of the leading coefficient $A_1$ together with roots of $\det A(z)$. They lift compatibility to the right eigenpairs of $A(z)$, yielding a discrete local tau function. The resulting deformation equations preserve the Birkhoff connection matrix, and reduce in their $q\to 1$ limit to the isomonodromic deformation of a meromorphic connection on ${\mathbb{P}}^1$ with an irregular singularity of Poincaré rank one at $\mathrm{\infty }$.