33 problems

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 $\catq(Z)$.

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…

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

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 $C$ be a smooth projective curve and let $G$ be a semisimple group, with Langlands dual group $G^{\vee}$. Let…

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 $G\circlearrowright M$ 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

Let $G$ be a connected, simply connected, and semisimple complex Lie group. Let $F$ be a finite subgroup of its center $Z(G)$, and let ${}^L\tilde{G}$ be the universal covering of…

Let $\mathcal{U}_q(\hat{\mathfrak{g}})$ and $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ be Langlands-dual quantum affine algebras, let $P'$ be the specified sublattice of the…

Let $\mathcal{U}_q(\hat{\mathfrak{g}})$ and $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ be quantum affine algebras of Langlands-dual types, let $P'$ be the sublattice specifi…

Let $\Gamma$ be a finite subgroup of $SU(2)$. For an Abelian central subgroup $Z\subset G$ of a connected compact simple Lie group $G$, set $H=G/Z$, with central extension … Let…

Let $\Gamma$ be a finite group, and let $G$ be a connected compact simple Lie group. Write $N(\Gamma,G)$ for the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by…

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 $\scrX$ be a hyperspherical $G$-variety and $\scrX^\vee$ its dual hyperspherical $G^\vee$-variety, with quantizations $Q\scrX$ and $Q\scrX^\vee$. Let…

Let $G$ be a complex reductive group with Lie algebra $\fg$, let $\fg^\vee$ be the Langlands-dual Lie algebra, and let $Z_\fg=T^*\CB\times_{\CN_\fg}T^*\CB$ be the Steinberg variety…

Let $G$ be the group appearing in the skein construction, let $Z$ denote the relevant central subgroup, and let $\pi_1$ denote the corresponding fundamental-group twist. For a clos…

Skein-module Langlands duality conjecture. There is a linear isomorphism

Let $G$ be a connected complex semi-simple group with real structure, let $\mathcal C$ be a compact complex Riemann surface with antiholomorphic involution $\tau$ and no real point…

Let $\alpha_{i_1},\dots,\alpha_{i_k}$ be distinct cominuscule roots. Suppose the corresponding sections $b_{i_j}$ of $L_{\alpha_{i_j}}K$ have $m$ distinct zeros $x_1,\dots,x_m$, wh…

Let $\Gamma=F_r$ be the free group of rank $r$, and let $G$ and $G^L$ be complex reductive Langlands dual groups. Write $\mathcal{X}_rG$ and $\mathcal{X}_rG^L$ for their character…