A criterion for hypersymmetry on discrete groupoids

Given a Fell bundle $\mathcal{C}\mathrm{\Xi }$ over the discrete groupoid $\mathrm{\Xi }$, we study the symmetry of the associated Hahn algebra ${\mathrm{\ell }}^{\mathrm{\infty },1}(\mathrm{\Xi }\text{\hspace{0.17em}⁣}\mid \text{\hspace{0.17em}⁣}\mathcal{C})$ in terms of the isotropy subgroups of $\mathrm{\Xi }$. We prove that $\mathrm{\Xi }$ is symmetric (resp. hypersymmetric) if and only if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also characterize hypersymmetry using Fell bundles with constant fibers, showing that for discrete groupoids, ‘hypersymmetry’ equals ‘rigid symmetry’.