The local Ginzburg-Rallis model for generic representations
This paper presents a continuation of the author's works on the local Ginzburg-Rallis model for generic representations in the \(p\)-adic and the real case. Let \(\chi\) denote a character of \(F^{\times}\), where \(F\) is either a \(p\)-adic field or the field of the real numbers. In that model, to an irreducible admissible representation \(\pi\) of \(\mathrm{GL}(6, F)\) with central character \(\chi^{2}\), one associates a certain Hom space, the dimension of which is denoted by \(m(\pi)\). Furthermore, to an irreducible admissible representation \(\pi_D\) of \(\mathrm{GL}(3, D)\), where \(D\) is the unique quaternion algebra over \(F\), with central character \(\chi^{2}\), one can also associate a certain Hom space, the dimension of which is denoted by \(m(\pi_D)\). It is known that \(0 \leq m(\pi), m(\pi_D) \leq 1\). In the paper under the review, the author proves that for an irreducible generic representation \(\pi\) with central character \(\chi^{2}\) and the local Jacquet-Langlands correspondence \(\pi_D\) of \(\pi\) to \(\mathrm{GL}(3, D)\), we have \(m(\pi) + m(\pi_D) = 1\).
scientific article; zbMATH DE number 7005762
scientific article