We study the geometric structure of affine Deligne-Lusztig varieties $X_{\lambda}(b)$ for $\mathrm{GL}_n$ and $b$ basic. Motivated by the work of Görtz-He-Nie and Chan-Ivanov, we consider a new condition on a cocharacter $\lambda$. If this is satisfied, then $X_{\lambda}(b)$ is the disjoint union of classical Deligne-Lusztig varieties times finite-dimensional affine spaces.

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