Block decompositions for p-adic classical groups and their inner forms

For an inner form G of a general linear group or classical group over a non-archimedean local field of odd residue characteristic, we decompose the category of smooth representations on Z[μ_p∞,1/p]-modules by endo-parameter. We prove that parabolic induction preserves these decompositions, and hence that it preserves endo-parameters. Moreover, we show that the decomposition by endo-parameter is the Z̄[1/p]-block decomposition; and, for R an integral domain, introduce a graph whose connected components parameterize the R-blocks, in particular including the cases R=Z̄_ and R=F̄_ for ≠ p. From our description, we deduce that the Z_̄-blocks and F_̄-blocks of G are in natural bijection, as had long been expected. Our methods also apply to the trivial endo-parameter (i.e., the depth zero subcategory) of any connected reductive p-adic group, providing an alternative approach to results of Dat and Lanard in depth zero. Finally, under a technical assumption (known for inner forms of general linear groups) we reduce the R-block decomposition of G to depth zero.