  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.
