Geometric approach to the local Jacquet-Langlands correspondence (Q2879441)

Let \(D\) be a central division algebra over a \(p\)-adic field \(F\) with \(\dim_F D = n^2\). Then the local Jacquet-Langlands correspondence determines a natural correspondence between irreducible discrete series representations of \(\mathrm{GL}_n (F)\) and irreducible smooth representations of \(D^\times\). In this paper, the author provides a purely geometric approach to the local Jacquet-Langlands correspondence under the assumption that the invariant of the division algebra is \(1/n\). He uses the local \(\ell\)-adic étale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients and proves that this correspondence preserves irreducible representations assuming that \(n\) is prime. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case without using a global automorphic technique or the classification of supercuspidal representations of \(\mathrm{GL}(n)\).