274 problems

Let $G$ be a reductive group and let a local system on a smooth curve be irreducible and rigid, with quasi-unipotent local monodromies and finite-order abelianization. Simpson's co…

Full-faithfulness conjecture. The functor associated with the non-formally-completed prestack,

Let $G$ be a gauge group and let ${}^LG$ denote its Langlands dual group. The associated Yang–Mills theories are compared at the quantum level. Montonen–Olive conjecture. The Yang–…

Let $G$ be a reductive group, let $\chi\in\pi_1(G)$ and let $w\in Z_G^{\vee}$, where $Z_G$ is the center of $G$. Let $\mathrm{Higgs}_G(\chi)$ denote the connected component of the…

Let $G=\operatorname{GL}_r$. Let $\mathcal{B}\subset(\mathrm{B}_{\operatorname{GL}_r}^{\mathrm{red}})^{\mathrm{cl}}$ be an open subset, and set … Let…

Let $X$ be a smooth projective curve over $\mathbb C$, let $G$ be a reductive group, and let $\mathrm{Higgs}_G(X)$ denote the moduli stack of $G$-Higgs bundles on $X$. Let…

Let $X$ be a smooth complete curve, let $G$ be a reductive group, and let $\Bun_G$ be the moduli stack of principal $G$-bundles on $X$. Write $\Shv_{\Nilp}(\Bun_G)$ for the subcate…

Let $E$ be a field of characteristic zero. Let $G$ be a reductive group with Langlands dual group $G^{\vee}$, let $X$ be the underlying curve, and let…

Let $G$ be a reductive group, let $P\subset G$ be a standard parabolic with Levi subgroup $M$, and let $\bP\subset \bG$ be the corresponding parabolic in the Langlands dual group w…

Let $G$ be a connected reductive group, let $cG$ be its Langlands dual, and let $cBun_G$ denote the moduli stack of $G$-bundles on $X$. Let $cLocSys_\cG$ be the derived stack of…

Laumon's compatibility conjecture. The restriction of $\Eis^n_{n_1,\ldots,n_k}({\mathcal S}')$ to $\Bun_{GL(n)}\subset\operatorname{Coh}_n$ is canonically isomorphic to…

Irreducibility conjecture. Moreover, if ${\mathcal S}$ is irreducible, then so is $\Eis({\mathcal S})$.

Let $C$ be the underlying smooth projective curve, let $G$ be a reductive group, and let $\mathcal F_{\mathcal P^\vee}$ be the holonomic $\mathcal D_{\mathcal Bun_G}$-module associ…

Let $G$ be a reductive group, let $G^\vee$ be its Langlands dual, and let … be the indicated rational powers of the canonical line bundles. For $\kappa\in\mathbb C$ with…

Coincidence conjecture. The isomorphisms supplied by Proposition 'IC calculation' and Theorem 'FFKM thm' (1) coincide. This is a compatibility assertion internal to the proof of th…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing its universal Borel…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing deformations of…

Let $G$ be a reductive group, $X$ a smooth complete curve, $\cG$ the Langlands dual group, and $E_\cG$ a $\cG$-local system on $X$. A Hecke eigensheaf is an object…

The geometric Langlands correspondence. There exists a canonical equivalence of categories

The geometric Langlands correspondence. There exists a canonical equivalence of categories

Let $\hg$ be an affine Kac–Moody algebra at the critical level, let $\hg_\crit\operatorname{mod}_\nilp^{I,m}$ be the full subcategory of $I$-monodromic $\hg_\crit$-modules supporte…

Full Hecke-property conjecture. Then ${\mathcal S}_E$ satisfies the full Hecke property.

Geometric Langlands conjecture. If $E$ is an absolutely irreducible $n$-dimensional local system on $X$, then there exists a perverse sheaf ${\mathcal S}_E\in P(\Bun_n)$ that is a…

Let $G$ be a connected reductive group, let $\check G$ be its Langlands dual group, and let $\Lambda_G$ be the union of conormal bundles to the $G({\mathbf O})$-orbits in the affin…

Let $G$ be the group in the paper, let $P$ be a standard proper parabolic with Levi subgroup $M$, and let $W$ be an irreducible local system on $X$ of rank $r=\dim V^\gamma$. For…