Abstract: Let E/F be a quadratic extension of number fields. Jacquet developed the relative trace formula to show that a cuspidal representation pi of GL_n(A_E) is distinguished by the quasi-split unitary group wrt E/F if and only if pi is Galois invariant. I will discuss the local analogue of this question: what can be said about the space of functionals invariant under a unitary group.
(Joint work with Brooke Feigon and Omer Offen.)