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

Abstract

For an inner form $\mathrm{G}$ 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 $\mathbb{Z}[\mu_{p^{\infty}},1/p]$-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 $\overline{\mathbb{Z}}[1/p]$-block decomposition; and, for $\mathrm{R}$ an integral domain, introduce a graph whose connected components parameterize the $\mathrm{R}$-blocks, in particular including the cases $\mathrm{R}=\overline{\mathbb{Z}}_{\ell}$ and $\mathrm{R}=\overline{\mathbb{F}}_\ell$ for $\ell\neq p$. From our description, we deduce that the $\overline{\mathbb{Z}_\ell}$-blocks and $\overline{\mathbb{F}_\ell}$-blocks of $\mathrm{G}$ 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 $\mathrm{R}$-block decomposition of $\mathrm{G}$ to depth zero.