15 problems

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…

Gaitsgory's conjecture. There is an equivalence of monoidal categories

Let $G=\mathrm{SL}(2)$, let ${\mathcal H}_s$ be the Hilbert space of the deformed quantum theory, and let $\Lambda$ range over laminations. Denote by $|\Lambda\rangle$ the states w…

Let $G$ be a gauge group, $C$ a Riemann surface, and let the $H_3^+$-WZNW model provide a quantum theory whose states are represented by solutions of the KZB equations. Let the Ver…

Let $G$) be a reductive group whose derived subgroup is simply connected, with Cartan subgroup $T$, and let $\kappa$ be a non-degenerate quantum parameter for $G$. Let $\check{G}$…

Let $G$ be a non-simply laced Lie group with lacing number $r^\vee$, let $\mathcal W_{q,t}(\mathfrak{g})$ be the associated deformed $W$-algebra, and let…

Let ${\mc C}$ be a smooth projective algebraic curve, let $p_i$ be marked points, let $L_{\lambda_i,{}^Lk}$ be the relevant modules for the affine algebra at level ${}^Lk$, and let…

Let $G$ be the group associated with $\mathfrak{g}$, let $\operatorname{Gr}_G$ be its affine Grassmannian, and let $\operatorname{Whit}^c(\operatorname{Gr}_G)$ denote the Whittaker…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let ${}^L\mathfrak{g}$ be its Langlands dual, and let $\widehat{{}^L\mathfrak{g}}_{{}^Lk}\text{-mod}^0$ and…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let ${}^L\mathfrak{g}$ be its Langlands dual, and let $\widehat{{}^L\mathfrak{g}}_{{}^Lk}\text{-mod}^0$ be the semisi…

Compatibility conjecture. The resulting actions of

Let $G$ be a reductive group, let $\cG$ be its usual Langlands dual group, and let $\kappa$ be a non-degenerate twisting. Let $\CG$ and $\check\CG$ be the gerbes on $\Gr_G$ and…

Whittaker and theta-lifting conjecture. (i) $\mathcal K_E$ is a perverse sheaf, each of whose $\mathbb Z/2\mathbb Z$-parity pieces is irreducible, and

Parabolic compatibility conjecture. There is an automorphism $\delta:\operatorname{Bun}_n\widetilde\to\operatorname{Bun}_n$ such that the diagram

Quantum Langlands conjecture. There is an equivalence