Shimura Varieties and Trace Formula Seminar

We will introduce the twisted relative trace formula and prove a relative fundamental lemma in this setting using an analogue of Kottwitz's argument in the base change situation. Applications to the study of distinction on unitary groups will be given, including the following: Let pi be a cuspidal automorphic representation of the quasi-split unitary group U(2n) satisfying appropriate local conditions. Assume that pi admits a weak base change Pi to GL(2n) that is cuspidal. If L(s,\wedge^2 Pi) has a pole at s=1 and L(1/2,Pi) \neq 0, then some pi' equivalent to pi at almost all split places is U(n) \times U(n) distinguished. If pi' also admits a weak base change to GL(2n), then pi and pi' are nearly equivalent. Time and interest permitting, open problems will be discussed.