On the cuspidal representations of ${\mathrm{GL}}_2(F)$ of level 1 or 1/2 in the cohomology of the Lubin-Tate space $\mathcal{X}({\pi }^2)$

Abstract

In this paper, we compute irreducible components which appear in the stable reduction of the Lubin-Tate curve of level two, in the mixed characteristic case. We also compute the action of the central division algebra of invariant 1/2, the action of ${\mathrm{GL}}_2$, and the inertia action explicitly. As a result, in a sense, we observe that, in the cohomology group of the stable reduction of the Lubin-Tate curve for ${\mathrm{GL}}_2$, the local Langlands correspondence and the local Jacquet-Langlands correspondence for ${\mathrm{GL}}_2$ are realized for the cuspidal representations of level 1 or 1/2.

arXiv comment53 pages