Smooth representations of \(\mathrm{GL}_m(D)\). VI: Semisimple types (Q2902002)

For parts I--III see [\textit{V. Sécherre}, Bull. Soc. Math. Fr. 132, No. 3, 327--396 (2004; Zbl 1079.22016); Compos. Math. 141, No. 6, 1531--1550 (2005; Zbl 1082.22011); Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 951--977 (2005; Zbl 1106.22014)]. For part IV see [the authors, J. Inst. Math. Jussieu 7, No. 3, 527--574 (2008; Zbl 1140.22014)] and for part V see [\textit{P. Broussous} and the authors, Doc. Math., J. DMV 17, 23--77 (2012; Zbl 1280.22018)].NEWLINENEWLINENEWLINEA complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact non-Archimedean local field is given. More precisely, for each inertial class in the Bernstein spectrum, a type is constructed and its Hecke algebra is computed. The Hecke algebras that arise are all naturally isomorphic to products of affine Hecke algebras of type A. For cuspidal classes, the simple type is unique up to conjugacy.