Sur les $\ell$-blocs de niveau zéro des groupes $p$-adiques II

Unipotent $\ell$-blocks for simply-connected $p$-adic groups

“…is attached a system of conjugacy classes on the Bruhat-Tits building. By [Lan18b] section 4.3, if Γ(φ, σ) is of order prime to ℓ, all of these conjugacy classes are of order prime to ℓ, and if Γ(φ, σ) is not of order prime to ℓ, then none of them are. Thus we get the result.…”

“…A new method, using consistent systems of idempotents on the Bruhat-Tits building, was used in [Dat18] to construct in depth zero the ℓ-blocks for GL n . Then, this was used in [Lan18a] and [Lan18b] to obtain decompositions of the depth zero category over Z ℓ , for a group which is split over an unramified extension of F . These decompositions are constructed using Deligne-Lusztig theory.…”

“…Let Rep 1 Q ℓ (G) be the subcategory of unipotent representations. Using [Lan18b] (with the system of conjugacy classes composed of the trivial representation for every polysimplex), we also get a ℓ-unipotent category over Z ℓ : Rep 1 Z ℓ (G). The unipotent ℓ-blocks are the ℓ-blocks of Rep 1 Z ℓ (G).…”

“…The unipotent ℓ-blocks are the ℓ-blocks of Rep 1 Z ℓ (G). In [Lan18b], the idempotents are constructed using Deligne-Lusztig theory. A first difficulty for an ℓ-block decomposition is that Deligne-Lusztig theory does not produce primitives idempotents.…”

“…These categories correspond to the primitive idempotents in the stable Bernstein centre, as defined in [Hai14]. In [Lan18b], the decomposition into stable blocks of the depth zero category is given by…”