33 problems

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

Finkelberg–Mirković conjecture. There exists an equivalence

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…

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 $G$ 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…

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

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…

Let $\mathbold{G}$ be a complex reductive group, let $\lambda$ be a dominant coweight lying in the coroot lattice, and let $W$ be the Weyl group of $\mathbold{G}$, canonically iden…

The Satake-action conjecture. Under the Satake isomorphism, the action of $\mathbf J$ coincides with the usual Hecke algebra action on

Chen–Zhu conjecture. There exists a natural bijection

Let $\boldsymbol{J}$ be the algebra of regular functions on the stack $[\hat G\sigma_p/\hat G]$, acting on $\operatorname{H}_c(\operatorname{Sh}_{\mu,\overline{k}_v})$, and identif…

Let $\boldsymbol{J}$ be the algebra acting on the compactly supported cohomology of the Shimura variety, and let…

Let $G$ be the reductive group in the local Shimura-variety setting, let $mu$ be the relevant dominant cocharacter, and let $b\in B(G)$ be a $\sigma$-conjugacy class. Write…

Generalized Ginzburg–Kazhdan conjecture. The variety $W^{g,b}$ is isomorphic to the Coulomb branch of the gauge theory associated with the quiver for a Riemann surface with $b$ leg…

Let $G_{\Bbbk}$ be the semisimple group in characteristic $p$, let $\Repe$ be its extended principal block, let ${\EuScript Gr^\vee}$ be the affine Grassmannian of the Langlands du…

Let $G$ be a complex reductive group, let $X$ be a smooth complex projective curve, and let $Gr_G=G(\mathcal K)/G(\mathcal O)$ be the affine Grassmannian, where…

Let $Q$ be an affine quiver, let $\mathcal M_C(\lambda,\mu)$ be its Coulomb branch, and let $\mathfrak A_\chi(\lambda,\mu)$ be the attracting set for the chosen cocharacter $\chi$.…

Let $boldsymbol{G}^ atur$ be a complex connected reductive group Langlands-dual to $boldsymbol{G}$, let $\beta$ be a field of coefficients, and let…

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…

Let $G$ be the group under consideration, $G^\vee$ its Langlands dual group, $KConv^{\widetilde{G^\vee}}(Gr)$ and $KConv^{\widetilde{G^\vee}\times\mathbb C^\times}(Gr)$ the equivar…