On the Hodge-Newton decomposition for split groups. (English) Zbl 1074.14016

Kottwitz, Robert E.

Int. Math. Res. Not. 2003, No. 26, 1433-1447 (2003).

Summary: The main purpose of this paper is to prove a group-theoretic generalization of a theorem of N. M. Katz on isocrystals [Astérisque 63, 113–164 (1979; Zbl 0426.14007)]. Along the way, we re-prove the group-theoretic generalization of Mazur’s inequality for isocrystals due to M. Rapoport and M. Richartz [Compos. Math. 103, 153–181 (1996; Zbl 0874.14008)], and generalize, from split groups to unramified groups, a result from the author and M. Rapoport [Comment. Math. Helv. 78, No.1, 153–184 (2003; Zbl 1126.14023)] which determines when the affine Deligne-Lusztig subset \(X^G_{\mu(b)}\) of \(G(L)/G({\mathcal O}_L)\) is nonempty.

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
22E50	Representations of Lie and linear algebraic groups over local fields
20G25	Linear algebraic groups over local fields and their integers

Citations:

Zbl 0426.14007; Zbl 0874.14008; Zbl 1126.14023