The University of Chicago Determining Whether Certain Affine Deligne-lusztig Sets Are Empty a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics By

Let F be a non-archimedean local field, let L be the maximal unramified extension of F , and let σ be the Frobenius automorphism. Let G be a split connected reductive group over F , and let B∞ be the Bruhat-Tits building associated to G(L). Let B1 be the building associated to G(F ). Then σ acts on G(L) with fixed points G(F ). Let I be the Iwahori associated to a chamber in B1. We have the relative position map inv : G(L)/I × G(L)/I → W̃ , where W̃ is the extended affine Weyl group of G. If w ∈ W̃ and b ∈ G(L), then the affine Deligne-Lusztig set Xw(bσ) is {x ∈ G(L)/I : inv(x, bσ(x)) = w. This dissertation answers the question of which Xw(bσ) are non-empty for certain G and b.