Nonemptiness of Affine Deligne–Lusztig Varieties

Definitions and mathematical statement

Let $k=\overline{{\mathbb{F}}_q}$, let $L=k((\varepsilon))$, and let $G$ be split, semisimple and simply connected over ${\mathbb{F}}_q$. Let $\sigma$ act on coefficients by $q$-power Frobenius and fix $\varepsilon$. Choose the source's base alcove $\mathbf a$, standard Iwahori subgroup $I$, split torus and dominant chamber. Let $W_a$ be the affine Weyl group, with finite Weyl group $W$ and finite simple-reflection set $S$. For a representative of $x\in W_a$, define \[ X_x(b)=\{gI\in G(L)/I:g^{-1}b\sigma(g)\in IxI\}, \] with its usual affine Deligne--Lusztig variety structure. Let $\eta_1:W_a\to W$ be the finite-part projection, and let $\eta_2(x)$ be the unique $w\in W$ such that $w^{-1}x\mathbf a$ lies in the dominant Weyl chamber. Put \[ h_x=\eta_2(x)^{-1}\eta_1(x)\eta_2(x),\qquad W_T=\langle T\rangle\quad(T\subseteq S). \] Use affine length for $x$ and the finite Weyl-group length for $h_x$.

The source defines "shrunken" by the written root-group comparison \[ U_\alpha\cap xI\ne U_\alpha\cap I \qquad\text{for every finite root }\alpha. \] There is a notation issue: $xI$ literally denotes a left coset, whereas root-group filtration comparisons normally involve the conjugate Iwahori $xIx^{-1}$ (often written ${}^xI$). The extract does not resolve whether the conjugation notation was lost. This draft retains the written test and flags it, rather than silently changing the shrunken class.

For every shrunken $x$ in the intended sense, the first proposed assertions are \[ X_x(1)\ne\varnothing\iff h_x\notin\bigcup_{T\subsetneq S}W_T, \] and, when nonempty, \[ \dim X_x(1)=\frac12\left(\ell(x)+\ell(h_x)\right). \] Membership in the complement of that union means full support in the finite simple reflections, not merely avoidance of one chosen proper parabolic subgroup.

The second assertion is retained with its exact broad quantifiers: for every $\sigma$-conjugacy class $[b]$, where $b\sim h^{-1}b\sigma(h)$, there exists $n_0=n_0([b])$ such that \[ \forall x\in W_a\text{ with }\ell(x)\geq n_0,\qquad X_x(b)\ne\varnothing\iff X_x(1)\ne\varnothing. \] The source does not restrict this second quantifier to shrunken alcoves or replace large length by distance from chamber walls. No such restriction or additional Newton-point condition is inserted without source confirmation.

Short English statement

For the intended shrunken alcoves, ask for the full-support nonemptiness criterion and exact dimension formula. Separately, ask whether every fixed sigma-conjugacy class eventually has the same nonempty loci as the identity at all sufficiently large affine lengths. The shrunken-Iwahori notation needs clarification.

Sources

• Supplied report: Buildings: Interactions with Algebra and Geometry (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/03, pp. 119--172; proposed year 2008. https://doi.org/10.4171/owr/2008/03.
• Supplied Gortz--Haines--Kottwitz--Reuman reference: https://math.umd.edu/~tjh/flagADL29.pdf.
• Local UnsolvedMath registry, permanent ID 30000910, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note reports comparison with the official report. The supplied 21 August 2026 triage labels the record partially solved, reporting one emptiness direction and many cases, while retaining a general predicted nonemptiness direction. It does not separately establish every dimension and eventual-equivalence assertion. No PDF access or independently updated status assessment is asserted here.