56 problems

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

Finkelberg–Mirković conjecture. There exists an equivalence

Good-characteristic parity conjecture. If $p$ is a good prime for ${\mathbf G}$, then the standard sheaves with coefficients in a field of characteristic $p$ are $*$-parity.

Freeness conjecture. The stalks of $\mathcal I_!(\lambda,\mathbb Z)$ are free.

Assume that $G$ is of type A. Let $\widetilde{\mathrm{KP}}_0$ be the full subcategory of the Koszul perverse coherent heart generated under extensions by the objects in…

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\mathrm{KP}_{\mathrm{coh}}^{\widehat{G}_{\mathcal{O}}}(\mathcal{R}_{G,\mathfrak{g}})$ be the Kos…

Let $G$ be a complex reductive group, let $X$ be a smooth affine spherical $G$-variety with connected stabilizers in the open Borel orbit, and set $M=T^*(2)(X)$. Let…

Assume that $G$ extends to a reductive group over ${\mathcal O}$, and let $\Loc_{\widehat{G}}^{\ur}$ denote the stack of unramified local Langlands parameters. Let…

Let $G_4=\operatorname{Res}_{\mathbf C[[t^{1/4}]]/\mathbf C[[t]]}\mathrm{PGL}_2$. Let $\overline{\mathcal O}$ be the minimal nilpotent coadjoint orbit closure of $\mathfrak{so}_8$,…

Let $(\mathrm{std}\otimes\mathfrak{sl}_2^*)(4,2,0,-2)$ be the graded vector space of pairs of binary quadratic forms with the weights specified by the coefficients in the source, a…

Let $G$ be the group appearing in the geometric Satake setting, let $G_n=\operatorname{Res}_{\mathbf{C}[[t^{1/n}]]/\mathbf{C}[[t]]}G$, and let $\check G$, $\check\gamma$, and…

Let the total stack of the multiplicative Hitchin fibration be the stack under consideration, and let its singularities be studied via its intersection complex. A Satake sheaf deno…

Let $G$ be a reductive group with Langlands dual $G^\vee$, let $\mathfrak{g}$ be the adjoint representation of $G$, and let $P$ and $P^\vee$ be the weight lattices of $G$ and…

Equivariant geometric Satake conjecture. The codomain of the natural map

Fixed-point conjecture. The fixed-point set

Let $e\in\mathfrak{sp}(6)$ be a nilpotent element of Jordan type $(3,3)$, let $\operatorname{SL}(2)$ be a maximal reductive subgroup of its centralizer, and choose an…

Let $0<K<N-1$ be positive integers, let $c\in\mathbb{C}^{\times}\setminus\mathbb{Q}^{\times}$, and set … Let…

Let $n$ and $k$ be nonnegative integers, let $c\in\mathbb{C}^{\times}\setminus\mathbb{Q}^{\times}$, and set … Let…

Generalized geometric Satake conjecture. The vector space

Braverman–Finkelberg–Nakajima conjecture. The fixed-point locus satisfies

Let $\mathcal M^\nu(\lambda,\mu)$ be the deformation associated with a decomposition $W=W^1\oplus W^2$, let $\lambda^1$ and $\lambda^2$ be the corresponding dominant coweights, and…

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi$ be the one-parameter subgroup defined by a character of the gauge group, and l…

Let $F=\mathbb F_q((t))$, let $L$ and $L^+$ denote formal loop and arc spaces, and let $X$ and $G$ be defined over $\mathbb F_q$, with $X$ smooth. Let…

Anderson's fusion conjecture. This fusion product should match the multiplication in the coordinate ring $\mathbb{C}[N]$ of $N$.

Let $G^\vee$ be a semisimple complex algebraic group, and let $G$ denote its complex Langlands dual group. Let $\lambda$ and $\mu$ be dominant weights for a maximal torus of…