The essentially tame Jacquet-Langlands correspondence for inner forms of \(\text{GL}(n)\)
The objective of the paper under review is to render the Jacquet--Langlands correspondence for inner forms of \(\text{GL}(n)\) explicit -- for the class of representations that can be explicitly described through the theory of admissible pairs. This is the class of irreducible essentially tame, cuspidal representations of parametric degree \(n\). Let \(F\) be a non-archimedean local field of finite residue characteristic and \(D\) a central division algebra over \(F\) of index \(d\). For \(m\in\mathbb{N}\), let \(\mathcal{A}^\square_m(D)\) be the set of equivalence classes of irreducible smooth representations of \(\text{GL}_m(D)\) essentially square integrable modulo the centre. If \(n=md\), the Jacquet--Langlands correspondence provides a canonical bijection \(\mathcal{JL}:\mathcal{A}^\square_n(F)\longrightarrow\mathcal{A}^\square_m(D)\). The parametric degree \(\delta(\pi)\) of \(\pi\in\mathcal{A}^\square_m(D)\) is a positive integer dividing \(n\). For \(m\neq1,n\), the authors base their definition on the classification of supercuspidal representations in terms of types given in [\textit{S. Stevens} and \textit{V. Sécherre}, J. Inst. Math. Jussieu 7, 527--574 (2008; Zbl 1140.22014)]. As a matter of fact, a representation \(\pi\) of parametric degree \(n\) is cuspidal. Next, for a representation \(\pi\in\mathcal{A}^\square_m(D)\), let \(t(\pi)\) be the number of unramified characters of \(F^\times\) such that \((\chi\circ\text{Nrd})\pi\cong\pi\). Then \(t(\pi)\mid\delta(\pi)\) and \(\pi\) is called essentially tame, if the residue characteristic does not divide \(\delta(\pi)/t(\pi)\). Let \(\mathcal{A}^{\text{et}}_m(D)\subset\mathcal{A}^\square_m(D)\) denote the set of equivalence classes of irreducible essentially tame representations of \(\text{GL}_m(D)\) of parametric degree \(n\). The Jacquet--Langlands correspondence preserves both the parametric degree and the \(t\)-invariant, hence induces a canonical bijection \(\mathcal{JL}:\mathcal{A}^{\text{et}}_n(F)\longrightarrow\mathcal{A}^{\text{et}}_m(D)\). In [J. Am. Math. Soc. 18, No. 3, 685--710 (2005; Zbl 1073.11070)], the authors gave a parametrisation of \(\mathcal{A}^{\text{et}}_n(F)\) in terms of admissible pairs. An admissible pair \((E|F,\xi)\) of degree \(n\) consists of an extension \(E|F\) of degree \(n\) and a quasicharacter \(\xi\) of \(E^\times\) satisfying two conditions. Write \((E|F,\xi)\longmapsto{}_F\Pi_{\xi}\) for the canonical bijection obtained there, and \(P_n(F)\) for the set of \(F\)-isomorphism classes of admissible pairs of degree \(n\). Here, they generalise this to obtain a parametrisation of \(\mathcal{A}^{\text{et}}_m(D)\) by \(P_n(F)\) via the theory of Bushnell--Kutzko types for \(\text{GL}_m(D)\) developed by Sécherre and Sécherre--Stevens. The main results of the paper read as follows (Theorems A and D of the Introduction): \textit{Theorem A} If \((E|F,\xi)\) is an admissible pair such that \([E:F]=n\), then there is a canonically determined tamely ramified character \(\nu={}_D\nu_\xi\) of \(E^\times\) such that \((E|F,\nu\xi)\) is admissible, \(\nu^2=1\) and \({}_D\Pi_\xi=\mathcal{JL}({}_F\Pi_{\nu\xi})\). Combining this with the explicit description of the Local Langlands Correspondence \(\mathcal{L}\) given in [Proc. Lond. Math. Soc. (3) 101, No. 2, 497--553 (2010; Zbl 1198.22009)] this yields: \textit{Theorem D} If \((E|F,\xi)\) is an admissible pair such that \([E:F]=n\), then there is a canonically determined tamely ramified character \(\lambda={}_D\lambda_\xi\) of \(E^\times\) such that \({}_D\Pi_{\lambda\xi}={}_D\mathcal{L}({}_F\Sigma_\xi)\). For the case \(m=1\), the second result was conjectured by Bushnell--Fröhlich in 1983.
explicit local Jacquet-Langlands correspondence
scientific article; zbMATH DE number 5942223