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

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