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Register a new userOn the Hodge-Newton decomposition for split groups
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Authors
Robert E. Kottwitz
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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 Mazurs 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.
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A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katzs result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.
We establish a categorical version of Vogan duality for quasi-split real groups. This proves a conjecture of Soergel in the quasi-split case.
In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.
This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the ${ mu }$-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Tits {e}chelonnage root system $Sigma_0$, the Knop root system $widetilde{Sigma}_0$, and the Macdonald root system $Sigma_1$, in terms of Galois actions on the absolute roots $Phi$; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis.
Recently, Lu and Wang formulated a Drinfeld type presentation for $imath$quantum group $widetilde{{mathbf U}}^imath$ arising from quantum symmetric pairs of split affine ADE type. In this paper, we generalize their results by establishing a current presentation for $widetilde{{mathbf U}}^imath$ of arbitrary split affine type.
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