  We give a complete description of the category of smooth complex
representations of the multiplicative group of a central simple algebra over a
locally compact nonarchimedean local field. More precisely, for each inertial
class in the Bernstein spectrum, we construct a type and compute its Hecke
algebra. The Hecke algebras that arise are all naturally isomorphic to products
of affine Hecke algebras of type A. We also prove that, for cuspidal classes,
the simple type is unique up to conjugacy.
