Jacquet-Langlands correspondence and distinction: the case of cuspidal level 0 representations

摘要

Let K/F be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GL(m)(D) be an inner form of GL(n)(F) and GL(mu)(Delta) = (M-m(D)circle times(F) K)(X). Then GL(mu)(Delta) is an inner form of GL(n)(K) and the quotients GL(mu)(Delta)/GL(m)(D) and GL(n)(K)/GL(n)(F) are symmetric spaces. Using the parametrization of Silberger and Zink, we determine conditions of GL(m)(D)-distinction for level zero cuspidal representations of GL(mu)(Delta) which are the image of a level zero cuspidal representation of GL(n)(K) by the Jacquet-Langlands correspondence. We also show that a level zero cuspidal representation of GL(n)(K) is GL(n)(F)-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL(m)(D)-distinguished.