Gamma factors of intertwining periods and distinction for inner forms of GL(n)

Abstract

Let F be a p-adic field, E be a quadratic extension of F, D be an F-central division algebra of odd index and let θ be the Galois involution attached to E/F. Set H=GL(m,D), G=GL(m,D⊗FE), and let P=MU be a standard parabolic subgroup of G. Let w be a Weyl involution stabilizing M and Mθw be the subgroup of M fixed by the involution θw:m↦θ(wmw). We denote by X(M)w,− the complex torus of w-anti-invariant unramified characters of M. Following the global methods of [30], we associate to a finite length representation σ of M and to a linear form L∈HomMθw(σ,C) a family of H-invariant linear forms called intertwining periods on IndPG(χσ) for χ∈X(M)w,−, which is meromorphic in the variable χ. Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied in [13], to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of G, extending respectively the results of [42] and [26] for D=F, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of [12] which in the case of the group G asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of G.