On the Restriction to 𝒟* × 𝒟* of Representations of p-adic GL2(𝒟)

Let be a division algebra over a nonarchimedean local field. Given an irreducible representation π of GL2(), we describe its restriction to the diagonal subgroup × . 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 × -distinguished if and only if π admits a Shalika model. We further prove that if is a quaternion division algebra then the twisted Jacquetmodule is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to × in the quaternionic case.

Kirillov Theory for a Class of Discrete Nilpotent Groups

This paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups G. G is assumed to be discrete, countable and π-radicable, with π containing the primes less than or equal to the nilpotence class of G. In addition, it is assumed that all of the characters of G have idempotent absolute value. Such groups are shown to be plentiful.