Conjectural algebraic formulas for representations of \(\text{GL}_n\) (Q1585481)

Local Langlands correspondence establishes a bijection \(\phi_n\) between the set \(\Pi_n\) of \(n\)-dimensional representations of the Galois group \(\text{Gal}(\overline F/F)\) of a local non-archimedean field \(F\), and the set \(GL(n,F)\hat{\;}\) of irreducible non-degenerate representations of \(GL_n(F)\). This implies that one can associate to a pair \((E,\chi)\) of a commutative semisimple algebra \(E\) of rank \(n\), and a character \(\chi\) of \(E^*\), an irreducible representation \(\pi_\chi\) of \(GL_n(F)\). An explicit construction of \(\pi_\chi\) is not known. The authors propose in the paper an explicit construction of the representation \(\pi_\chi\) if \(n=4\). In the paper of \textit{D. Kazhdan} [Prog. Math. 92, 125-158 (1990; Zbl 0781.22013)], such a formula is proved for \(n=3\).