Supercuspidal blocks and the local Langlands correspondence

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$, and let ${\mathrm{R}\mathrm{e}\mathrm{p}}^{\mathfrak{s}}(G(F))$ be a supercuspidal Bernstein block. In this paper, we define a cohomology class $[{\mu }_{\mathfrak{s}}]$ represented by a 2-cocycle that encodes the equivalence class of ${\mathrm{R}\mathrm{e}\mathrm{p}}^{\mathfrak{s}}(G(F))$. We then formulate a conjecture describing $[{\mu }_{\mathfrak{s}}]$ in terms of the enhanced $L$-parameter $(\phi ,\rho )$ attached to an irreducible representation in ${\mathrm{R}\mathrm{e}\mathrm{p}}^{\mathfrak{s}}(G(F))$ via the conjectural local Langlands correspondence. Finally, we verify this conjecture for the local Langlands correspondence for non-singular supercuspidal representations constructed by Kaletha.