On the Hodge-Newton decomposition for split groups

By R. Kottwitz. 2003

55 citations recorded by the source corpus.

The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazur's inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f087debf0b43b81561585e0e50c435a233764b4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Dimensions of some affine Deligne–Lusztig varieties (2005, background): “As in [16] (see also [17,15]), there is a simple criterion for Xμ(b) to be non-empty (see Proposition 5.”
Connected components of affine Deligne–Lusztig varieties in mixed characteristic (2013, result): “sumption on M, which is incorrect. The present assertion is the corrected statement and follows from the proof of [MV], which in its turn is nothing but a variant of the original proof of Kottwitz in [Ko3]. It remains to show that if g∈XG µ(b)(R) where R is a frame for a smooth connected ¯k-algebra ”
Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties (2014, background): “Another proof is given by Kottwitz in [16].”
On a conjecture of Kottwitz and Rapoport (2008, background): “One implication, namely that X μ (b) being non-empty implies κM (b) P μ, is the group-theoretic version of Mazur’s Inequality and a proof of this fact can be found in [10], Theorem 1.”
Affine Deligne–Lusztig varieties in affine flag varieties (2008, methodology): “This proves (2). Finally, (3) follows from (1) and (2). Remark 12.1.3. For aﬃne Deligne-Lusztig varieties in the aﬃne Grassmannian, it isknown that XG µ (b) 6= ∅ if and only if [b] ∈ B(G,µ) (cf. [KR],[K3],[Lu],[Ga],[W]). The condition [b] ∈ B(G,µ) means that ηG(b) = µ in ΛG and νb ≤ µ (“Mazur’s ineq”