On the decomposition of a representation of \(GL(3)\) restricted to \(GL(2)\) over a \(p\)-adic field (Q1210440)

Let \(F\) be a nonarchimedean local field. Consider irreducible admissible representations \(V\) of \(\text{GL}_N(F)\) and \(W\) of \(\text{GL}_{n- 1} (F)\). By restriction \(V\) is also a \(\text{GL}_{n - 1} (F)\)-module. A theorem of Bernstein asserts that the dimension of the space \(\text{Hom}_{\text{GL}_{n-1} (F)} (V,W)\) is at most one. The present paper is concerned with the question which pairs \((V,W)\) do have nonzero homomorphisms. A complete answer in terms of induced representations and, via Langlands-correspondence, in terms of Weil group representations is given for the case \(W = \mathbb{C}\) and \(n =3\). A conjecture for \(n > 3\) is stated. It further is proven that there are always homomorphisms if \(V\) and \(W\) are generic, i.e. possess unique Whittaker models.