59 problems

Mirković–Vilonen's conjecture. The cohomology modules of the stalks of ${\mathcal{I}}_!(\lambda,\mathbb{Z})$ are free.

Let $X_{\la}$ be the affine Grassmannian Schubert variety associated with a dominant coweight $\la$, let $\mathcal A(\la)$ be the union of Schubert varieties in the affine flag var…

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 $k$ be a positive integer, let $q$ denote the parameter used in the $k$-Schur functions, and let $H_*(\operatorname{Gr}_{\mathrm{aff}})$ be the homology ring of the affine Gras…

Malkin–Ostrik–Vybornov conjecture. The pairs

The affine Grassmannian singularities of types $a_2$, $ac_2$, $ag_2$, $c_2$, and $cg_2$ all have codimension $4$. Their intersection cohomology agrees within the pairs of types…

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 $C$ be a smooth projective curve, let $S$ be a finite set of points containing all bad points of the group scheme $\mathcal{G}$, and let $q^*$ be the pullback map from the Pica…

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 $G$ be a reductive algebraic group in characteristic $p$, let $\operatorname{Rep}_0^{\operatorname{ext}}(G)$ denote the relevant extended principal block, and let $\Phi_H$ be t…

Let $Y_\mu(\gamma)$ be the affine Lusztig variety in the affine Grassmannian associated with an element $\gamma$ and a dominant coweight $\mu$, and let the Langlands dual group hav…

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

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…

Frobenius–Mirković equality conjecture. One has

Frobenius–Mirković reducedness conjecture. If ${\mathbf k}$ has characteristic $0$, then $S_\mu^{{\mathrm{FM}}}$ is reduced, and hence

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…

Fix integers $0<l<k<n$. Let $A(l,n)$ denote the space of affine $l$-planes in $\boldsymbol{R}^n$, let $G(k,n)$ be the Grassmannian of $k$-dimensional linear subspaces, and let…

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…

Peterson's conjecture. The small-torus equivariant homology of the affine Grassmannian is isomorphic to the equivariant quantum cohomology of the flag variety:

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

Let $G$ be quasisimple of adjoint type, let $cmathfrak g=\operatorname{Lie}(G)$, and let $\operatorname{cox}\bin W$ be a Coxeter element of order $n$. For $m$ coprime to $n$, let…

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 $G=PGL_n$, let $cmathfrak g=\operatorname{Lie}(PGL_n)$, and let $gammabin g(cO)$ be elliptic, regular semisimple, and topologically nilpotent. Let $[operatorname{Sp}gamma]$ den…

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…