On the Hodge-newton Decomposition for Split Groups

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 from [KR] which determines when the affine Deligne-Lusztig subset X μ (b) of G(L)/G(oL) is non-empty. Let F be a finite extension of Qp with uniformizing element ̟. We write L for the completion of the maximal unramified extension of F in some algebraic closure F̄ of F . We write σ for the Frobenius automorphism of L over F , and we write o (respectively, oL) for the valuation ring of F (respectively, L). Let G be a split connected reductive group over o and let A be a split maximal torus of G over o. Fix a Borel subgroup B = AU containing A with unipotent radical U , as well as a parabolic subgroup P of G containing B. Write P = MN , where M is the unique Levi subgroup of P containing A and N is the unipotent radical of P . We writeXG for the quotient ofX∗(A) by the coroot lattice forG, and we define a homomorphismwG : G(L) → XG as follows. For g ∈ G(L) we define rB(g) ∈ X∗(A) to be the unique element μ ∈ X∗(A) such that g ∈ G(oL) · μ(̟) · U(L), and we define wG(g) to be the image of rB(g) under the canonical surjection from X∗(A) to XG. (This definition of wG suffices for the purposes of this paper; see §7 of [Kot97] for a definition that applies to groups G that are not split over L.) Applying the construction above to M rather than G, we obtain XM , the quotient of X∗(A) by the coroot lattice for M , and a homomorphism wM : M(L) → XM .