On the geometry of Certain Non-Basic Affine Deligne-Lusztig Varieties

Abstract

Let $F$ be a non-Archimedean local field, let $L=\breve F$, and let $G=\mathrm{GL}_n$. Let $M\subset G$ be a standard Levi subgroup and let $b\in M(L)$ be basic in $M$, but not necessarily basic in $G$. For a dominant cocharacter $μ$, we study the reduction-to-Levi morphism $β:X^G_μ(b)\to \bigsqcup_{μ_M\in S_M(μ,ν_b)}X^M_{μ_M}(b)$ for affine Deligne--Lusztig varieties in the affine Grassmannian. Using an Iwasawa factorization relative to $P=MN$, we reduce the fiber condition to explicit Frobenius-twisted lattice equations in the off-block coordinates. In the Drinfeld case, where the base $X^M_{μ_M}(b)$ is zero-dimensional, we prove that $β$ is globally trivial with constant affine-space fiber in the non-basic cases considered. More generally, in the minuscule case we develop a nonzero-slope lattice-theoretic criterion which shows that the fibers are affine spaces and that $β$ is Zariski locally a trivial affine-space bundle in the non-basic cases considered. We also give examples in the non-minuscule setting where the fibers need not be affine spaces.