We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks G-Zip µ . 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 µ , fibered in flag varieties over G-Zip µ . 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 Hodgetype Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with non-degenerate 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 on mod p modular forms is generalized to general Hodge-type Shimura varieties.

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 A n 1 , C 2 , and Fp-split groups of type A 2 (this includes all Hilbert-Blumenthal varieties and should also apply to Siegel modular threefolds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected-Hodge-type.

We define stacks of zip flags, which form towers above the stack of G-zips of Moonen, Pink, Wedhorn and Ziegler in [MW04], [PWZ11] and [PWZ15]. 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 [EvdG09] 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 [GKb].(3) If B ⊂ P 0 is a Borel subgroup, then the projection G-ZipFlag Z → G-Zip Z factors through G-ZipFlag Z → G-ZipFlag (Z,P0) , with flag variety fibers P 0 /B. (4) There is a second zip datum Z 0 associated to (Z, P 0 ) and a smooth map G-ZipFlag (Z,P0) → G-Zip Z0 . It induces a stratification of G-ZipFlag (Z,P0) parameterized by I0 W , where I 0 is the type of P 0 .The stratification in (4) is termed fine, see §3.1. We also define a coarse stratification of G-ZipFlag (Z,P0) , see §2.2; it generalizes the Bruhat stratification of G-Zip Z studied by Wedhorn [Wed14], which in turn generalizes the a-number stratification studied by Oort and others (cf. introduction of loc. cit.).In general, the zip datum Z 0 is not attached to a cocharacter, unless P 0 = P . There seems to be no known algebraic counterpart of the stack G-Zip Z0 in the theory of Shimura varieties. Thus we see that the group-theoretic approach of G-Zips reveals genuinely new structure. Nevertheless, following (i) the analogy between stacks of G-Zips and Griffiths-Schmid manifolds suggested in [GKb] (see esp. §I.4) and (ii) the approach to Griffiths-Schmid manifolds in [GGK13], it appears that there may be a complex-geometric analogue of G-Zip Z0 among the objects studied in loc. cit.; see §6.4.Th. 1 generalizes the construction of the stack of zip flags G-ZipFlag Z in [GKb] to an intermediate parabolic B ⊂ P 0 ⊂ P . In particular, when P 0 = B one has G-ZipFlag (Z,P0) = G-ZipFlag Z and when P 0 = P one has G-ZipFlag (Z,P0) = G-Zip Z . In turn, the case of G-ZipFlag Z is a group-theoretical generalization of the flag space -and its stratification -associated to the moduli space of abelian varieties A g ⊗ F p by Ekedahl and van der Geer [EvdG09].

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