Orthogonality of characters of \(\text{GL}_n\) over a local field of non-zero characteristic
\textit{L.~Clozel} [Prog. Math. 101, 101-121 (1991; Zbl 0760.22023)] proved the orthogonality of characters of square integrable representations with a given central character for a general reductive group over a non-Archimedean local field of characteristic zero. The main goal of the paper under review is the proof of the analogous statement for the group \(\text{GL}_n(F)\) when \(F\) is a local field of non-zero characteristic. Using this statement, the author is also able to prove the following theorem (which extends to the non-zero characteristic case a result obtained by \textit{J.~D.~Rogawski} [in Duke Math. J. 50, 161-196 (1983; Zbl 0523.22015)]): Let \(D\) be a central division algebra over \(F\) with dimension \(n^2\). Then there exists a unique bijective correspondence between the set of equivalence classes of essentially square integrable irreducible representations of \(\text{GL}_n(F)\) and the set of equivalence classes of irreducible representations of \(D^*\) such that the characters of two representations whose equivalence classes correspond are equal, up to the sign \((-1)^{n-1}\), on elements which have the same separable characteristic polynomial. This result is the first step in the proof of the existence of a Jacquet-Langlands correspondence between \(\text{GL}_n\) and its inner forms in the case of non-zero characteristic (for the case of zero characteristic, see the results obtained by \textit{P.~Deligne, D.~A.~Kazhdan} and \textit{M.-F.~Vignéras} [in Représentations des groupes réductifs sur un corps local, 33-117 (1984; Zbl 0583.22009)] and \textit{Y.~Z.~Flicker} [in J. Ramanujan Math. Soc. 5, No. 2, 107-122 (1990; Zbl 0722.11028)]). As a consequence, the author shows that there is transfer of all orbital integrals of \(D^*\) to \(\text{GL}_n(F)\), and of all the orbital integrals of \(\text{GL}_n(F)\) which vanish on all non-elliptic regular semisimple elements to \(D^*\). The paper uses some of the results obtained by \textit{B.~Lemaire} [in Ann. Inst. Fourier 46, No. 4, 1029-1059 (1996; Zbl 0853.22012) and Compos. Math. 100, No. 1, 41-75 (1996; Zbl 0856.22024)].
scientific article; zbMATH DE number 1406357