18 problems

Let $G$ be one of the groups $SL_2$, $SL_3$, or $Sp_4$. Let $b=\epsilon^\nu$, and let $\ell(b)$ denote the length of the translation in $\widetilde W$ determined by $\nu$. Eventual…

Let $G$ be a connected reductive group over $F$, let $b\in G(L)$, and let $\mu$ be a dominant coweight such that the affine Deligne–Lusztig variety $X_\mu(b)$ is non-empty. Write…

Let $x\in\widetilde W$ and let $[b]\in B(G)$ be integral, meaning that $\operatorname{defect}(b)=0$. For each $v\in\operatorname{LP}(x)$ and $u\in W$, define the multiset … Set…

Let $G$ be the reductive group in the setup, let $x=w\varepsilon^\mu\in\widetilde W$ lie in a shrunken Weyl chamber, so that $\operatorname{LP}(x)=\{v\}$ for a uniquely determined…

Let $G=Sp_4(L)$, where $L$ is as in the paper. For $bin G(L)$, let … be the $\sigma$-centralizer of $b$, and let $\operatorname{def}(b)$ be the difference between the $F$-rank of…

Let $G$ be the reductive group, let $b\in G(\breve{F})$ and let $\mu$ be a cocharacter, with associated affine Deligne–Lusztig variety $X^{\mathcal{K}_p}_\mu(b)$. Write…

Let $X_{c,b}^{\caG_{\bfx}^{\rm ad}}$ be the affine Deligne–Lusztig variety associated with the Coxeter element $c$, the element $b$, and the adjoint parahoric group scheme…

Zhou's conjecture. If $(\lambda,b)$ is Hodge–Newton irreducible, then $\eta_G$ induces a bijection

Local model conjecture. For every closed point $x\in |({X_{\mathscr{G}}^{\leq\mu}(b)})_{{k_F}}|$ there exists a closed point $y\in |({{\mathcal{A}}_{\mathscr{G},\mu}})_{k_F}|$ such…

Chen–Zhu conjecture. There exists a natural bijection

Let $G$ be the reductive group in the local Shimura-variety setting, let $mu$ be the relevant dominant cocharacter, and let $b\in B(G)$ be a $\sigma$-conjugacy class. Write…

Let $G$ be the reductive group, $L$ its relevant local field extension, $P$ the parahoric subgroup, $\lambda$ a coweight, and $b\in G(L)$. Let $X(\lambda,b)_P$ be the associated cl…

Let $G=\operatorname{GL}_n$ and let $b=1$, so that $J_1(F)=G(F)$. For a suitable pair $(f,\underline{w}_f)$, write … for the affine Deligne–Lusztig induction map, and suppose that…

Let $G$ be a reductive group, let $mu$ be a Hodge cocharacter, and let $K$ be a parahoric subgroup. Assume that $(G,\mu,K)$ is of Coxeter type, and let the corresponding basic affi…

Basic-class comparison conjecture. There exists $N_b\in\mathbb Z_{\ge 0}$ such that, for every $x\in\widetilde W$ with $\ell(x)\ge N_b$,

Reuman's conjecture. If $x$ lies in the shrunken Weyl chambers, then $X_x(b)\ne\emptyset$ if and only if

P-alcove conjecture. $X_x(b)\ne\emptyset$ if and only if, for every semistandard $P=MN$ for which $x\mathbf a$ is a $P$-alcove, $b$ is $\sigma$-conjugate to an element $b'\in M(L)$…

Let $G$ be the group under consideration, let $x\in\widetilde{W}$ be an element of its affine Weyl group, and let $b\in G$. Let $I$ be an Iwahori subgroup, let $X_x(b)$ be the asso…