The SL1(D)-distinction problem

Abstract

We use the local theta correspondences between the quaternionic Hermitian groups and the quaternionic skew-Hermitian groups to understand the distinction problem for the symmetric pair SL 2 (E)/SL 1 (D), where E is a quadratic field extension of a nonarchimedean local field extension F and D is a 4-dimensional division quaternion algebra over F .