On representations of \(\mathrm{GL}(n)\) distinguished by \(\mathrm{GL}(n - 1)\) over a \(p\)-adic field
The present article is concerned with admissible representations of the linear group \(\mathrm{GL}_{n}(F)\) over a non-Archimedean local field \(F\) (\(n\geq 3\)) which are distinguished by \(\mathrm{GL}_{n-1}(F)\), i.e., which admit a nonzero \(\mathrm{GL}_{n-1}(F)\)-invariant linear form. More precisely, it contains a proof of the characterization of these representations in terms of their Langlands parameters (Theorem 1.1), conjectured and proved in the case \(n=3\) by \textit{D. Prasad} [Duke Math. J. 69, No. 1, 167--177 (1993; Zbl 0827.22006)], as well as the calculation of the dimension of the space of \(\mathrm{GL}_{n-1}(F)\)-invariant forms for all parabolically induced representations from irreducible representation of a Levi subgroup. The author uses the same methods as D. Prasad: a precise study of (normalized) induced representations from maximal parabolic subgroups via Mackey theory and segments theory of A. Zelevinsky which gives a list of \(\mathrm{GL}_{n-1}(F)\)-distinguished representations, then an argument by induction on \(n\) to prove that this list is complete (Corollary 6.15). These methods allow the author to show that the dimension of the space of \(\mathrm{GL}_{n-1}(F)\)-invariant forms for a parabolically induced representation from irreducible representation of a Levi subgroup is always smaller than 2 and equal to 2 for a precise list of representations which appears in Theorem 7.17.
scientific article; zbMATH DE number 6176535
scientific article