Simple characters for principal orders in Mm(D)

In this paper we give an alternative review over the subject of strata and simple characters as more generally already considered in the references [P. Broussous, M. Grabitz, Pure elements and intertwining classes of simple strata in local central simple algebras, Comm. Algebra 28 (11) (2000) 5405–5442] and [V. Sécherre, Représentations lisses de Gl(m,D), I: charactères simples, Bull. Soc. Math. France 132 (3) (2004) 327–396]. Finally, we prove the intertwining implies conjugacy property of simple characters as known in the split case [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of Gl(N) via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993].