On representations of GL(n) distinguished by GL(n − 1) over a p-adic field

Abstract

For a nonarchimedean local field F, let GL(n):= GL(n, F) and GL(n−1) be embedded in GL(n) via g ↦ ( 0 1 g 0 ). Let π be an irreducible admissible representation of GL(n) for n ≥ 3. We prove that π is GL(n − 1)-distinguished if and only if the Langlands parameter L(π) associated to π by the Local Langlands Correspondence has a subrepresentation L(11 n−2) of dimension n−2 corresponding to the trivial representation of GL(n−2) such that the two-dimensional quotient L(π)/L(11 n−2) corresponds either to an infinite-dimensional representation or the one-dimensional representations \(\nu ^{ \pm (\tfrac{{n - 2}}{2})} \) of GL(2). We also prove that, for a parabolic subgroup P of GL(n) and an irreducible admissible representation ρ of the Levi subgroup of P, \(\dim _\mathbb{C} (Hom_{GL(n - 1)} [ind_P^{GL(n)} (\rho ),\mathbb{I}_{n - 1} ]) \leqslant 2\). For the standard Borel subgroup B n of GL(n) and characters x i of GL(1), we classify all representations ξ of the form \(ind_{B_n }^{GL(n)} (\chi _1 \otimes \cdots \otimes \chi _n )\) for which \(\dim _\mathbb{C} (Hom_{GL(n - 1)} [\xi ,\mathbb{I}_{n - 1} ]) = 2\).