42 problems

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 $B^{\vee,+}$ 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…

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…

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 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 $G$ 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…

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…

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 $\alpha\geq1$, and let $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}$ be the graded algebra associated with the preprojective construction for the quiver of type $A_{2}$ with multiplicit…

Let $K\subset G$ be the symmetric subgroup of a real form $G_{\mathbb R}\subset G$. Write $X({\mathcal K})$ for the corresponding symmetric variety, $G({\mathcal O})$ for the posit…

Let $\G$ be a connected semisimple complex Lie group with a distinguished automorphism $\sigma$ fixing a pinning, let $\G_\sigma=((\G^\vee)^\sigma_0)^\vee$ be the associated endosc…

Assume $G$ is split, semisimple, and simply connected over $F=\mathbb{F}_q((t))$. Let $LG$ be the loop group, $\mathfrak A^{\operatorname{Fr}}(LG)$ the Frobenius-equivariant Grothe…

Locality conjecture. There exists a $G(\sF)\times G(\sF)$-equivariant sheaf ${\mathbf S}_{\rho}$ of complex vector spaces on $\overline{G}(\sF)$ such that…

Semi-infinite extension conjecture. If $\CF$ is the constant sheaf on $G_{\sO}$, then $f_{i_{!*}\CF}$ corresponds under the Satake isomorphism to…

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

Let $c\in C$ and let $\mathrm G$ be a reductive group. For $\mu\in X_+(\mathrm G^\vee)$, let $\rho_\mu$ be the corresponding irreducible representation of the Langlands dual group,…

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…

Frenkel–Gaitsgory's main conjecture. The induced functor

Let $M$ and $N$ be the parameters of the orthogonal setup, let $\bF$ be the Laurent series field, and let $U_{M,N}(\bF)$ be the unipotent group equipped with the character…

Let $V_0$ and $V_1$ be respectively orthogonal and symplectic vector spaces, let $\Ffl_1$ be the variety of complete self-orthogonal flags in $V_1$, and let $\Ffl_0$ be a chosen co…

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 $f$ be the homogeneous polynomial defining the graded matrix-factorization category $\operatorname{HMF}(f)$, and let $h$ be its degree. Let $Z_G$ be the central charge symbolic…

Finite-type extension conjecture. The category $\mathcal{P}_{coh}^{G(\mathcal{O}) \rtimes \mathbb{G}_m}(\mathrm{Gr}_{G_{Ad}})$ is a monoidal categorification of the quantum cluster…