On some simple geometric structure of affine Deligne-Lusztig varieties for $GL_n$

Abstract: In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_n$ and $b$ basic. We introduce a new condition on $\lambda$. If this is satisfied, then the corresponding affine Deligne-Lusztig variety is the disjoint union of classical Deligne-Lusztig varieties times finite-dimensional affine spaces.