81 problems

Symplectic duality conjecture. There exist many pairs of symplectic resolutions for which various dualities hold, including Koszul duality between their categories $\mathcal O$.

In the 1990s, Crane and Frenkel considered three-dimensional topological field theories of Witten, Reshetikhin and Turaev and the categorical language emerging in geometric represe…

Let $G$ be a connected complex semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Let $(e,h,f)\in\mathfrak{g}^{\times 3}$ be a principal $\mathfrak{sl}_2$-triple, a…

A generalized affine Springer fiber is the variety associated with a reductive group, a finite-dimensional representation, and an element of the corresponding loop-space representa…

Kazhdan–Lusztig irreducibility conjecture. In type A, the characteristic cycle of the IC-sheaf $\mathcal{L}_Q$ is irreducible.

Let $\Catgeomdk$ be the geometric category defined in the surrounding text, let $\Pervsmkb$ be the stated target category, and let $\ConvHecke$ be the functor constructed there. Co…

Let $\Catgeom$ be the abelian category of objects equipped with compatible Hecke isomorphisms, and call an object finitely generated if it admits a surjection from an object of the…

Let $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ be the category of $G(\CO_x)$-integrable critical-level $\hat\fg$-modules that are set-theoretically supported on…

Let $\eta\in Y^+$ and suppose that $\eta-2\check\rho\in Y^+$. Let $K^\bullet_\eta$ be the graded multiplicity space of $\IC(\ol\fM_\eta)$ in $\bigoplus_{\xi\in Y}\bp_*\IC(\fQ^\xi)$…

Let $\fM$ be the moduli space of $\bG$-torsors on $\BP^1$ with formal trivialization at infinity, stratified by $\fM_\eta$ for $\eta\in Y^+$. Write … where $K^\bullet_\eta$ is the…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. The natural projection … has target identified with … The algebra $\CA=U(\fn^L_+)$ acts on th…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. For each $\beta\in Y$, let $T_{-\beta}$ denote the corresponding semiinfinite orbit, and cons…

Quasimap coefficient conjecture. There is an isomorphism of $\fg^L$-modules

Let $\CN^L\subset\fg^L$ be the nilpotent cone, and let $\fn_-^L$ be the negative nilpotent subalgebra. The geometric action of $\fg^L$ is defined on … The local cohomology module i…

Tilting conjecture. In the $\bG[[z]]$-equivariant case, this resulting $\fg^L$-module is tilting. This predicts that the geometric $\fg^L$-action on the global cohomology of quasim…

Let $B^{\vee,+}$ and $B^{\vee,-}$ be opposite Borel subgroups of $G^{\vee}$ with $B^{\vee,+}\cap B^{\vee,-}=T^{\vee}$. Set $\mathcal{W}=B^{\vee,-}B^{\vee,+}\subset G^{\vee}$, let…

Let $M\subset G$ be a Levi subgroup contained in a parabolic subgroup, and let … be the proposed Grothendieck–Springer functor. The existing intertwiner isomorphisms are constructe…

Spectral-automorphic intertwiner conjecture. Under the relative Langlands equivalences, $I_{\mathrm{aut}}$ is equivalent to the spectral push-pull functor $I_{\mathrm{spec}}$ along

Let ${\mathfrak{f}}=\overline{\mathbb{F}}_p((t))$ and ${\mathfrak{o}}=\overline{\mathbb{F}}_p[[t]]$. Let $G$, $X$, and $Y$ be the group and the two $G$-spaces described above, and…

Let $G$ be a reductive group over the relevant local field, let $K$ be a $sigma$-stable hyperspecial subgroup containing an Iwahori subgroup $I$, and let $mu$ be a noncentral cocha…

Suppose $\mathfrak{g}$ is simply-laced, let $\lambda\in P_+$, let $\mu\in \operatorname{wt}(V(\lambda))$, and choose an integral set of parameters $\mathbf{R}$ of level $\lambda$.…

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 $G$ be a reductive group, let $\overline{\mathrm{Bun}}_{N^-}$ be the compactification used in the paper, and let $\mathrm{IC}_{\overline{\mathrm{Bun}}_{N^-}}$ denote its inters…

Let $T$ be a torus, let $\mathfrak{t}$ be its Lie algebra, and let $T^{\vee}_{\mathbb{C}}$ be the complex dual torus. Let $\mathrm{BFM}(T^{\vee})$ denote the associated BFN space.…

For a proper smooth variety $\mathscr X$ equipped with a $G[\![z]\!]$-action and a $G[\![z]\!]$-equivariant map … assume that … where…