In previous joint work with J.-S. Koskivirta, we introduced the notion of quasi-constant character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions minuscule and co-minuscule. In this note we characterize quasi-constant fundamental weights in terms of the Weyl group of the corresponding maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of the Weyl group of the corresponding maximal sub-root system.

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks G-Zip(mu). Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags G-ZipFlag(mu), fibered in flag varieties over G-Zip(mu). It provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by Griffiths-Schmid and the flag space for Siegel varieties studied by Ekedahl-van der Geer. Four applications are obtained: (1) Pseudo-representations are attached to the coherent cohomology of Hodge-type Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with nondegenerate limit of discrete series Archimedean component. (3) It is shown that all Ekedahl-Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre's letter to Tate onmod p modular forms is generalized to general Hodgetype Shimura varieties.

We define stacks of zip flags, which form towers above the stack of G-zips of Moonen, Pink, Wedhorn and Ziegler in [14-16]. A stratification is defined on the stack of zip flags, and principal purity is established under a mild assumption on the underlying prime p. We generalize flag spaces of Ekedahl-Van der Geer [4] and relate them to stacks of zip flags. For large p, it is shown that strata are affine. We prove that morphisms with central kernel between stacks of G-zips have discrete fibers. This allows us to prove principal purity of the zip stratification for maximal zip data. The latter provides a new proof of the existence of Hasse invariants for Ekedahl-Oort strata of good reduction Shimura varieties of Hodge-type, first proved in [8].

First the Griffiths line bundle of a Q-VHS V is generalized to a Griffiths character grif(G, mu, r) associated to any triple (G, mu, r), where G is a connected reductive group over an arbitrary field F, mu is an element of X-*(G) is a cocharacter (over (F) over bar) and r : G -> GL(V) is an F-representation; the classical bundle studied by Griffiths is recovered by taking F = Q, G the Mumford-Tate group of V, r : G -> GL(V) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to grif(G, mu, r). The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of G-Zips. When G is F-simple, we show that, up to positive multiples, the Griffiths character grif(G, mu, r) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by -mu. As an application, we show that the Griffiths line bundle of a projective G-Zip(mu)-scheme is nef.

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of G-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type An 1, C 2, and Fp-split groups of type A 2 (this includes all Hilbert{Blumenthal varieties and should also apply to Siegel modular 3-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

In [131, initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if mu is a quasi-constant cocharacter of an F-p-group G, then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G-Zip(mu), without any restrictions on p, even if the pair (G, mu) is not of Hodge type and even if mu is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit.