The cohomology of the affine Deligne-Lusztig varieties in the affine flag manifold of $GL_2$

TL;DR

This paper classifies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2 and analyzes the associated sigma-stabilizer representations on their etale cohomology.

Contribution

It provides a complete classification of these varieties and describes the sigma-stabilizer representations in terms of inductions and noncuspidal representations.

Findings

All affine Deligne-Lusztig varieties in the affine flag manifold of GL_2 are classified up to isomorphism.

The representations of the sigma-stabilizer on the etale cohomology are explicitly described.

The representations are expressed as inductions from compact subgroups and related to noncuspidal representations.

Abstract

This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.