83 problems

Formality conjecture. The dg-algebra $A_X$ is formal: it is quasi-isomorphic to the graded algebra

Let $G/H$ be a spherical homogeneous space, let $Y\subseteq G/H$ be a subvariety, and let $\operatorname{sLog}_{\Gamma,t}(Y)$ be its spherical amoeba and $\operatorname{strop}(Y)$…

Let $F$ be a non-archimedean local field, let $\mathbf{G}$ be a reductive group over $F$, and let $\mathbf{X}$ be a $\mathbf{G}$-spherical $F$-variety. Write $G$ for the group of…

A graded Satake diagram is a decorated Dynkin diagram arising in the classification above, with the specified type I or type II $\mathbb{Z}_2$-graded Satake-diagram structure. Non-…

Multiplicity conjecture. The unique distinguished element belongs to $\Pi_\phi(G)$ if and only if

Let $X$ be an affine spherical variety over $k$ with trivial arithmetic multiplicity. Let $\mathcal S(X(\mathbb A_k))$ be a Schwartz space in the sense described in the source, 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 $\catq(Z)$.

Let $\mathbf{G}$ be a reductive group over the nonarchimedean field $k$, let $\mathbf{X}$ be a spherical $\mathbf{G}$-variety, let $K$ be a maximal compact subgroup of…

Let $G$ be a reductive group with Langlands dual group $G^\vee$, and let $W$ be the Weyl group. Let $X$ be a smooth spherical $G$-variety such that $T^*X$ is hyperspherical and has…

Let $\sT\subset B$ be a Cartan subgroup, let $\lambda$ be a regular coweight of $G$, and let $\sT_{\lambda}$ be the subgroup of $\sT$ generated by $\exp(2\pi i\lambda)$. Let…

Let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups, and let $D^{\mathbb Z/2}$ denote the $\mathbb Z/2$-graded version of a graded category. $\mathbb Z/2$-graded Kos…

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 $F$ be a number field and let $G=\mathbf{GL}_4\times\mathbf{GL}_2$. Embed $\mathbf{GL}_2\times\mathbf{GL}_2$ into $G$ by $(g_1,g_2)\mapsto(\operatorname{diag}(g_1,g_2),g_2)$, a…

Distinguished-representation moment map conjecture. If $Z(\pi)^\vee$ is $X$-distinguished, then

Flexibility conjecture. $X$ is flexible.

Let $G$ be a complex reductive group, let $X$ be a smooth affine spherical $G$-variety with connected stabilizers in the open Borel orbit, and set $M=T^*(2)(X)$. Let…

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 $\mathcal R=(\Delta,S^p,\Sigma,\Gamma)$ be a complete spherical skeleton, where $R$ is the underlying root system, $R^+$ is its set of positive roots, and $R^+_{S^p}$ is the se…

Let $X$ be a smooth Fano variety. Let $\rho_X$ denote its Picard number and let $\iota_X$ denote its pseudo-index, defined by … Here $-K_X$ is the anticanonical divisor of $X$. The…

Let $X$ be a strongly tempered spherical variety, so that its dual group satisfies $\check{G}_X=\check{G}$. Let $\mathbf{A}=\mathbf{A}_X$, let $\delta$ denote the relevant modular…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots. Let $J_A$ be the associated group scheme and let…

Let $X=G/H$ be an affine homogeneous spherical variety without type $N$ roots, and let $P(X)$ be the associated parabolic subgroup. Abelian-centralizer equivalence conjecture. The…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots, and let $S_X$ be a polarizable symplectic dual representation of…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots. Let…