On some simple geometric structure of affine Deligne–Lusztig varieties for $${{\,\textrm{GL}\,}}_n$$

Abstract

In this paper we study the geometric structure of affine Deligne–Lusztig varieties $$X_{\lambda }(b)$$ X λ ( b ) for $$\textrm{GL}_n$$ GL n and b basic. Motivated by Görtz and He (Camb J Math 3:323–353, 2015) and Chan and Ivanov (Math Ann 380:1801–1890, 2021), we consider a new condition on $$\lambda $$ λ . If this is satisfied, then $$X_{\lambda }(b)$$ X λ ( b ) is the disjoint union of classical Deligne–Lusztig varieties times finite-dimensional affine spaces.