A \mu-ordinary Hasse invariant

TL;DR

This paper generalizes the Hasse invariant for specific unitary Shimura varieties, demonstrating the -ordinary locus is affine and applying this to relate Galois and automorphic representations.

Contribution

It introduces a new -ordinary Hasse invariant for certain Shimura varieties and proves the affineness of the -ordinary locus, enhancing understanding of Galois-automorphic correspondence.

Findings

Constructed a -ordinary Hasse invariant for unitary Shimura varieties.

Proved the -ordinary locus is affine.

Strengthened a theorem relating Galois and automorphic representations.

Abstract

We construct a generalization of the Hasse invariant for certain unitary Shimura varieties of PEL type whose vanishing locus is the complement of the so-called \mu-ordinary locus. We show that the \mu-ordinary locus of those varieties is affine. As an application, we strengthen a special case of a theorem of one of us (W.G.) on the association of Galois representations to automorphic representations of unitary groups whose archimedean component is a holomorphic limit of discrete series.