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Register a new userDimensions of Newton strata in the adjoint quotient of reductive groups
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Authors
Robert E. Kottwitz
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This paper generalizes the classical theory of Newton polygons from the case of general linear groups to the case of split reductive groups. It also gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive groups.
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Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strata have a natural reduction of structure to a (not necessarily split) maximal torus in LG, and the authors use this property to compute the corresponding moduli spaces. This theory generalizes a natural condition on algebraic connections (the GL_n case), which plays an important role in the global analysis of meromorphic connections and isomonodromic deformations.
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.
We study the relation of the notion of weak admissibility in families of filtered phi-modules, as considered in a companion paper, with the adjoint quotient. We show that the weakly admissible subset is an open subvariety in the fibers over the adjoint quotient. Further we determine the image of the weakly admissible set in the adjoint quotient generalizing earlier work of Breuil and Schneider.
Let G be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of G. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple G, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev-Manins spaces of weighted pointed curves and with Kauszs compactification of GL(n).
We construct the Frobenius structure on a rigid connection $mathrm{Be}_{check{G}}$ on $mathbb{G}_m$ for a split reductive group $check{G}$ introduced by Frenkel-Gross. These data form a $check{G}$-valued overconvergent $F$-isocrystal $mathrm{Be}_{check{G}}^{dagger}$ on $mathbb{G}_{m,mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $check{G}$-local system $mathrm{Kl}_{check{G}}$ constructed by Heinloth-Ng^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $mathrm{Be}_{check{G}}^{dagger}$ when $check{G}$ is almost simple (which recovers the calculation of monodromy group of $mathrm{Kl}_{check{G}}$ due to Katz and Heinloth-Ng^o-Yun), and establish functoriality between different Kloosterman $check{G}$-local systems as conjectured by Heinloth-Ng^o-Yun. We show that the Frobenius Newton polygons of $mathrm{Kl}_{check{G}}$ are generically ordinary for every $check{G}$ and are everywhere ordinary on $|mathbb{G}_{m,mathbb{F}_p}|$ when $check{G}$ is classical or $G_2$.
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