30 problems

Renormalized r-matrix correspondence conjecture. Under the equivalence of Proposition E1, the renormalized $r$-matrix corresponds to the lowest non-trivial loop degree part of the…

Let $G$ be the reductive group defining the Kottwitz–Viehmann variety, let $F$ be its base field, let $T$ be a maximal torus, and let $G(F)^{\mathrm{rs}}$ denote the regular semisi…

Let $G$ be a connected reductive algebraic group, let $\operatorname{Gr}=G(\mathcal K)/G(\mathcal O)$ be its affine Grassmannian, and let $\lambda_1^\vee,\ldots,\lambda_k^\vee$ be…

Let $K_0\backslash\mathcal{G}_0$ be the quotient under consideration, let $\lambda$ be a dominant coweight, let $\omega_i$ be the fundamental coweights for $1\leq i\leq n-1$, and s…

Let $X$ be a smooth projective curve, let $S\subseteq X$ be a nonempty set containing the bad points $\mathcal{R}$ of a parahoric Bruhat–Tits group scheme $\mathcal{G}$, and let…

Let $G$ be the reductive group, $T$ a maximal torus, and $\rho$ the half-sum of positive roots used to define the affine Grassmannian strata. For a dominant coweight…

Representability conjecture. The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}$ is represented by a normal $k$-scheme. This predicts that the indicated truncated double quot…

Orbit-stratification conjecture. There exists a finite partially ordered set $(X,<)$ describing the $L^+G$-orbits in $\widetilde{\operatorname{Gr}}_{\mu_{\bullet}}$. Equivalently,…

Let $cmathfrak g=\operatorname{Lie}(GL_n)$ or $cmathfrak g=\operatorname{Lie}(SL_n)$, and let $gammabin g(cO)$ be elliptic, regular semisimple, and topologically nilpotent. Let…

Let $A$ be a subset of the affine simple reflections and let $y$ be an element of ${}^A W_{\mathrm{ext}}^S$; denote by $\mathsf{C}_{y,A}$ the Serre subcategory of…

Let $G$ be as in the paper, let $\mathrm{Gr}^{\circ}$ be the connected component of the affine Grassmannian containing the base point, and let $I_{\mathrm{u}}$ be the preimage of t…

Let $E$ be a $p$-adic field with perfect residue field, let $C$ be a perfect non-archimedean field of characteristic $p$ with an open and bounded valuation subring $C^+$, and set…

Centralizer transitivity conjecture. The group $(LG)_{\mathfrak g}$ acts transitively on $\operatorname{Gr}_{\mathfrak g,\mathcal O}$. The paper notes that this fiber is nonempty a…

Shallow minimal reduction type conjecture. For every $[w]\in\underline W$,

Let $M_3$ be a 3-manifold, let $G$ be the relevant group, and let … be the moduli space associated with the disk $D^2$, viewed conceptually as the cotangent bundle of the affine Gr…

Let $G$ be a group, with formal disc and loop groups $G(\mathcal{O})$ and $G(\mathcal{K})$, and let $\operatorname{Gr}_G$ be the affine Grassmannian. Let $\mathcal{V}$ be any verte…

Localization equivalence conjecture. The functor $\widetilde\Psi_w$ is a graded monoidal equivalence

Let $\mathcal{W}_n^{\mathrm{triv}}$ be the dg-stack defined from chains of local systems on $\mathcal{D}^*=\operatorname{Spec}(\mathbb{C}((z)))$, with the rank-$n$ local system ide…

Let $G$ be a reductive group over the relevant field, let $x$ be an absolutely special vertex, and let $\tGr_{G,x}^{\leq \bar{\mu}}$ be the corresponding normalized Schubert variet…

Let $G$ be a reductive group, let $\Gr_G$ be its affine Grassmannian, and let $D(\Gr_G)^{\operatorname{Hecke}}$ denote the derived category of Hecke eigen-modules, namely $D$-modul…

The equivalence conjecture. The map above is an isomorphism of Picard groupoids. This predicts that factorizable line bundles on the Ran affine Grassmannian are completely paramete…

KLMW conjecture. The square is Cartesian. Equivalently, the shuffle equations cut out $\mathcal{G} r_{SL_n}$ inside $\mathcal{S}\mathcal{G} r$. The paper proves this conjecture ove…

Let $X$ be a curve, let $G$ be a constant group scheme over $X$, and let $\ell$ be an integer prime to the characteristic. Set $A=\mu_\ell$, so that…

Let $X$ be a curve and $G$ a constant group scheme over $X$. Let $(K_2)_{\operatorname{Zar}}$ be the Zariski sheafification of the prestack assigning $K_2(A)$ to an affine scheme…

Let $G$ be a quasi-split reductive group over $E$, let $\Gamma=\operatorname{Gal}(\overline E/E)$, and let $L^+G$ act on the affine Grassmannian $\operatorname{Gr}$. Write…