Rank-one local systems on the elliptic curve

Let

Define ${\mathcal{L}}_k$ as the quotient of $\mathbb{C}\times \mathbb{C}$ under the action

The trivial connection on the cover descends. With this convention, parallel transport around the first basic cycle has monodromy ${\omega }^k$, and the second basic cycle has monodromy one.

Equivalently, these local systems arise from the cyclic covering

This is an actual local-system construction on a fixed curve, not a category of $D$-modules on ${\mathrm{Bun}}_{{\mathbb{G}}_m}(E)$.

Let ${\mathsf{Loc}}_d(E)$ consist of the finite-rank unitary local systems whose second monodromy is identity and whose first monodromy $U$ satisfies $U^d=I$. Morphisms are parallel complex-linear maps. Adjoints use the parallel Hermitian structures.