37 problems

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 defining the affine Deligne–Lusztig variety, let $\boldsymbol{\theta}$ denote the induced action on the relevant character lattices, and let…

Let $G$ be a reductive group over a local field, let $\mu\in X_*(T)_+$ be a dominant cocharacter, and let $\tau$ represent the unique basic $\sigma$-conjugacy class such that…

Let $k$) be an algebraic closure of $\a0\mathbb F_p$, let $W(k)$ be its ring of Witt vectors with Frobenius automorphism $\sigma$, and put $L=W(k)[1/p]$. Let $G$ be a connected red…

Kottwitz–Rapoport conjecture. The nonemptiness pattern of $X(\mu,b)$ is given by Mazur's inequality.

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

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 $G$ be a reductive group over the relevant local field, let $K$ be a $sigma$-stable hyperspecial subgroup containing an Iwahori subgroup $I$, and let $mu$ be a noncentral cocha…

Let $\mathcal{B}$ be a hereditary $\mathcal{O}_X$-order that is Iwahori at each $y\in Y$, and let $I_\bullet=(I)$. Write $\widetilde{\mathbf{w}}_{\operatorname{bsc}}$ for the basic…

Refined virtual dimension conjecture. For every $[b]\in B(G)_x$,

Let $w$ be an element of the affine Weyl group and let $b$ be an element of the Kottwitz set $B(G)$ such that the affine Deligne–Lusztig variety $X_w(b)$ is nonempty. Write…

Let $w=t^{\mu}y$ with $y$ in the finite Weyl group and $\mu$ dominant regular, and let $\ell(y)$ denote the length of $y$. For the associated affine Deligne–Lusztig variety…

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 $G$ 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…

Irreducible-component conjecture. One has

Let $F$) be a nonarchimedean local field, let $G$ be a quasi-split connected reductive group over $F$, and let $K$ be a special parahoric subgroup. For…

Let $\mathit{k}$ be the residue field. Consider a Rapoport–Zink space and the corresponding affine Deligne–Lusztig set, with their natural geometric structures; write the perfectio…

Fargues–Rapoport conjecture. For $p$-adic period domains, the weakly admissible locus coincides with the admissible locus if and only if the pair $(\mu,b)$ is Hodge–Newton decompos…

Let $G$ be a reductive group over $F$, let $S$ be a minisotropic torus, and let $E/F$ be a totally ramified Galois extension over which $S$ splits. Let $J\subseteq G(\breve E)$ be…

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…