Unipotent ℓ-blocks for simply connected p-adic groups

Let $F$ be a nonarchimedean local field and $G$ the $F$-points of a connected simply connected reductive group over $F$. We study the unipotent $\ell$-blocks of $G$, for $\ell \ne p$. To that end, we introduce the notion of $(d,1)$-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and $d$-Harish-Chandra theory. The $\ell$-blocks are then constructed using these $(d,1)$-series, with $d$ the order of $q$ modulo $\ell$, and consistent systems of idempotents on the Bruhat–Tits building of $G$. We also describe the stable $\ell$-block decomposition of the depth zero category of an unramified classical group.