Determining whether certain affine Deligne-Lusztig sets are empty
  Let F be a non-archimedean local field, let L be the maximal unramified extension of F, and let fr be the Frobenius automorphism. Let G be a split connected reductive group over F, and let B(1) be the Bruhat-Tits building associated to G(F). We know that fr acts on G(L) with fixed points G(F). Let I be the Iwahori associated to a chamber in B(1). We have the relative position map, inv, from G(L)/I x G(L)/I to the extended affine Weyl group, W_e of G. If w is in W_e and b is in G(L), then the affine Deligne-Lusztig set Xw(b fr) is {x in G(L)/I : inv(x,b fr(x)) = w}. This paper answers the question of which Xw(b fr) are non-empty for certain G and b. 
Daniel C. Reuman