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

Abstract

Let F be a non-archimedean local field and G the F -points of a connected simply-connected reductive group over F . In this paper, we study the unipotent ℓ-blocks of G. 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 ℓ-blocks are then constructed using these (d, 1)-series, with d the order of q modulo ℓ, and consistent systems of idempotents on the Bruhat-Tits building of G. We also describe the stable ℓ-block decomposition of the depth zero category of an unramified classical group.

International Congress of Mathematicians

“…[130] and [38]). The most definitive result in this direction was then obtained by Loïc Merel [135], who showed that the torsion subgroups of elliptic curves defined over a number field K are bounded by a constant B K depending only on K, and indeed, only on the degree of K over Q.…”

“…where T is a finite group, called the torsion subgroup of E over Q. Mazur's celebrated theorem [36] lists all the possibilities for the groups T that can arise in this way: This striking result was apparently anticipated by the Italian geometer Beppo Levi [142] in 1908. It became more widely known as a precise conjecture formulated by Andrew Ogg [139] and provides the backdrop for an active area of investigation to which mathematicians like Kamienny [130], Merel [135], and many others, have made important subsequent contributions. Indeed, the study of rational points on modular curves remains a lively terrain of investigation to which a variety of approaches grounded in the pioneering insights of [36] have been applied (cf.…”

“…The somewhat larger quotient of J 0 .N / with finite Mordell-Weil group that emerges from Kolyvagin's theorem is called the winding quotient (a terminology that can be traced back to Mazur's "winding element" [42]). The winding quotient was later exploited to great effect by Merel in his extension of Theorem 4.1 to number fields of arbitrary degree [135].…”