Représentations lisses de GLm(D), III : types simples

Let F be a non Archimedean local field, D a division algebra over F and G=GLm(D), with m⩾1. This paper is the continuation of [V. Sécherre, Représentations lisses de GL(m,D), I : caractères simples, Bull. Soc. Math. France 132 (3) (2004) 327-396] and [V. Sécherre, Représentations lisses de GL(m,D), II : β-extensions, Compositio Math. 141 (2005) 1531-1550], whose purpose is the generalization to G of Bushnell-Kutzko's work [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Princeton University Press, Princeton, NJ, 1993] concerning the split group GLn(F). For any r⩾1 dividing m, and for certain smooth irreducible supercuspidal representations Π0 of G0=GLm/r(D), we construct a type for the inertial class [G0r,Π0⊗r]G. We give the structure of its Hecke algebra which, as in the split case, is an affine Hecke algebra of type Ar−1.