73 problems

Kapustin–Witten mirror symmetry conjecture. For Langlands-dual groups, certain topological and geometrical invariants of the corresponding character varieties should coincide or mi…

Let $G$ be a reductive group, let ${}^{L}G$ be its Langlands dual group, and let $Y$ be a closed $3$-manifold. The associated $G$- and ${}^{L}G$-skein modules are vector spaces who…

Let $G$ be a complex reductive group with Langlands dual group $G^L$, and let $\mathcal H_G$ and $\mathcal H_{G^L}$ be their Higgs moduli spaces. Let $\sigma_G$ be an anti-holomorp…

Let ${\cal D}$ be the positive double, let ${\cal D}^{\vee}$ be its Langlands dual, and let ${\cal D}({\mathbb Z}^t)$ be the set of integral tropical points. Let…

Spherical weight conjecture. For every $\theta\in \cowts_{A^0}\cap \CV(G/S)$, the irreducible object $'\IC^\theta\in \catp_\CH({}'Z)$ belongs to $\catq(Z)$.

Langlands duality conjecture. The Langlands dual quantum group $U_{\tilde q}(\check{\mathfrak g})$ may generally be obtained by explicit calculations of the centralizer in the appr…

Let ${^{\scriptscriptstyle L\!}}{\mathcal O}$ be a nilpotent orbit in the Lie algebra of the Langlands dual group, let $\mathcal O=d({^{\scriptscriptstyle L\!}}{\mathcal O})$ be th…

Let $G$ be a complex reductive group and let ${^{\scriptscriptstyle L\!}}G$ be its Langlands dual. For a nilpotent orbit ${^{\scriptscriptstyle L\!}}{\mathcal O}$ in the Lie algebr…

Let $G$ be a gauge group with Lie algebra ${\mathfrak g}$, and let ${{}^LG}$ be its Langlands dual group. Let $\tau$ and ${{}^L\tau}$ be the gauge couplings, and let…

Let $G$ be a gauge group and let the hypermultiplet transform in the adjoint representation of $G$. Write ${{}^LG}$ for the Langlands dual group. Langlands-dual S-duality conjectur…

Langlands-dual sectors homotopy conjecture. Corresponding sectors for Langlands-dual groups should be homotopy equivalent. The paper states that this conjecture has a counterexampl…

Safronov's root-of-unity skein conjecture. There is a line bundle $\mathcal{L}_q$ on $\mathrm{Loc}_{G^\vee}(M)$ such that

Skein Langlands conjecture. The generic $G$-skein module is stable under Langlands duality:

Let $G$ be a reductive group, let $X$ be a spherical variety for $G$, and set $M:=T^*X$. Write $X_\mathcal{K}$ for the corresponding affine Grassmannian space, and let…

Quantum deformation and Langlands duality conjecture. There exists a square-zero deformation

Let $G$ be a connected reductive group, let $\fY$ be the spherical variety above, and let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups. Let…

Let $G$ be a connected reductive group over $\mathbb C$, let $\fY$ be a smooth affine spherical $G$-variety with connected stabilizer in a Borel subgroup $B$ of a general point, an…

Let $Y$ be an integer homology sphere, let $V_q(Y)$ and $V_{\tilde q}(Y)$ be the spaces of $q$-series associated to $Y$, and let $S$ denote the predicted Langlands-duality map on s…

Let $M_3$ be a $3$-manifold, and let $\operatorname{sk}^{e,m}(M_3)$ be its skein module doubly graded by electric and magnetic fluxes $e$ and $m$. Ben-Zvi–Gunningham–Jordan–Safrono…

Let $\Sigma$ be a compact oriented $2$-manifold and let $G$ be a semisimple group, with Langlands dual group $G^{\vee}$. Let…

Let $C$ be a smooth projective curve with $g(C)\geq 2$, let $G$ be a connected semisimple group, and let $G^{\vee}$ be its Langlands dual. Let ${\mathcal L}$ be a line bundle on…

Let $G$ be a Chevalley group over $\mathbb{F}_q$ and let $\check G$ be its Langlands dual Chevalley group. Define the category of potent representations of $G(\mathbb{F}_q)$ as the…

Let $G\circlearrowright M$ be a polarized hyperspherical $G$-variety with $M=T^*X$, and let $\check G\circlearrowright\check M$ be its Langlands-dual polarized hyperspherical varie…

Let $G$ be a reductive group and let $\check G$ be its Langlands dual group. Denote by $\mathcal{H}_G^{dR}$ and $\mathcal{H}_{\check G}^{dR}$ the de Rham potent Hecke monads, and b…

Safronov's conjecture. There is a natural isomorphism