On the geometry of certain non-basic affine Deligne-Lusztig varieties

Abstract

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O_F$, uniformizer $t$, residue field $\kappa=\mathcal O_F/(t)$, and let $L=\breve F$ be the completion of the maximal unramified extension of $F$, with ring of integers $\mathcal O_L$ and Frobenius automorphism $\sigma = \sigma_{L/F}$. Let $G$ be an unramified connected reductive group over $F$, fix a Borel subgroup $B\subset G$ and a maximal split torus $T\subset B$, and write $K=G(\mathcal O_L)$ for the hyperspecial subgroup. For $b\in G(L)$ and a dominant cocharacter $\mu\in X_\bullet(T)$, the (affine Grassmannian) affine Deligne--Lusztig variety is the locally closed subfunctor[ %X^G_{\mu}(b);\subset; G(L)/K,\qquad X^G_{\mu}(b)(\bar\kappa) ;=;\bigl{,x\in G(L)/K:;x^{-1}b\sigma(x)\in Kt^{\mu}K,\bigr}, ] viewed as a (perfect) $\kappa$-scheme locally of finite type. For a Levi subgroup $M\subset G$ and a basic element $b\in M(L)$, there is a natural ``reduction to Levi'' morphism (see \cite[Prop.~5.6.1]{GHKR06}) [ \beta:\ X^G_{\mu}(b);\longrightarrow; \bigsqcup_{\mu_M\in S_M(\mu,v_b)} X^M_{\mu_M}(b), ] where the disjoint union runs over those $M$-dominant cocharacters $\mu_M$ that are maximal with the properties $(\mu_M){\mathrm{dom}}\preceq\mu$ and $X^M{\mu_M}(b)\neq\varnothing$. \medskipWe already have a good understanding of the geometric structure of the affine Deligne--Lusztig variety when $b$ is basic: Fox and Imai studied the irreducible components of the affine Deligne-Lusztig variety $X^G_{\mu}(b)$ over $G = \text{GU}(2,n-2)$ when $b$ is a basic element in $G$ (see \cite{FI22}). This thesis focuses primarily on the case $G=\GL_n$ when $b$ is not basic in $G$ but basic in $M$, which is a \emph{standard} (block diagonal) Levi, to study $X^G_{\mu}(b)$, using the reduction to Levi map $\beta$. A further emphasis is placed on the \emph{Drinfeld case}, where the base $X^M_{\mu_M}(b)$ has dimension zero, and in this case we prove that $\beta$ is a (globally) trivial algebraic bundle with constant affine space fiber. Moreover, when $\mu$ is the $G$-dominant representative of an $M$-minuscule cocharacter $\mu_M$, we show that $\beta$ is a Zariski locally trivial vector bundle over $X^M_{\mu_M}(b)$ under natural non-degeneracy conditions on $(M,\mu,b)$. In this setting we compute the dimension of every geometric fiber of $\beta$ and, under explicit slope-compatibility hypotheses, establish that $\beta$ is a Zariski locally trivial algebraic vector bundle over $X^M_{\mu_M}(b)$. Our approach exploits an Iwasawa-type factorization relative to $P=MN$ together with an explicit block-matrix/lattice analysis that converts the fiber condition into linear-algebraic data governed by valuations of matrix entries.