Generic distinction of induced representations: the case of symmetric spaces

Let H be the fixed-point subgroup of an involution of a reductive group G over a local field F of characteristic zero. We give a necessary condition for a parabolically induced representation of G(F) to admit a non-zero family of H(F)-equivariant linear forms depending holomorphically on the induction parameter. By works of Blanc-Delorme and Brylinski-Delorme, this condition is essentially sufficient. As a global consequence, we obtain vanishing results for regularized periods of Eisenstein series. We illustrate this on symplectic periods, where we extend Offen’s computation of truncated periods of certain cuspidal Eisenstein series to inner forms of GL(2n).