On the restriction to D* x D* of representations of p-adic GL(2,D)

Abstract

Let D be a division algebra over a nonarchimedean local field. Given an irreducible representation π of \GL₂(D), we describe its restriction to the diagonal subgroup D* x D*. The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that π is D* x D* -distinguished if and only if π admits a Shalika model. We further prove that if D is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to D* x D* in the quaternionic case.