Distinction inside L-packets of SL(n)

Abstract

If E/F is a quadratic extension p-adic fields, we first prove that the SL_n(F)-distinguished representations inside a distinguished unitary L-packet of SL_n(E) are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of N(E)/N(F). Then we establish a global analogue of this result. For this, let E/F be a quadratic extension of number fields and let π be an SL_n(AF)-distinguished square integrable automorphic representation of SL_n(AE). Let (σ,d) be the unique pair associated to π, where σ is a cuspidal representation of GL_r(AE) with n=dr. Using an unfolding argument, we prove that an element of the L-packet of π is distinguished with respect to SL_n(AF) if and only if it has a degenerate Whittaker model for a degenerate character ψ of type r^r:=(r,…,r) of N_n(A_E) which is trivial on N_n(E+A_F), where Nn is the group of unipotent upper triangular matrices of SL_n. As a first application, under the assumptions that E/F splits at infinity and r is odd, we establish a local-global principle for SL_n(A_F)-distinction inside the L-packet of π. As a second application we construct examples of distinguished cuspidal automorphic representations π of SL_n(A_E) such that the period integral vanishes on some canonical copy of π, and of everywhere locally distinguished representations of SL_n(AE) such that their L-packets do not contain any distinguished representation.