134 problems

Luna's conjecture. Mapping ${\mathcal H}\subseteq{\mathcal G}$ to the homogeneous spherical datum of ${\mathcal G}/{\mathcal H}$ induces a bijection between the set of conjugacy cl…

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

Let $G$ be a reductive group and let $H$ be a sober spherical subgroup with $H=N_G(H)$. Let $\bar X$ be the unique simple complete toroidal embedding of $G/H$, and let $X_D$ be the…

Knop's conjecture. The weight monoid $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Let $K$ be a connected compact Lie group, let $M$ be a multiplicity-free compact Hamiltonian $K$-manifold with moment map $\mu:M\rightarrow\mathfrak k$, and fix a maximal torus…

Let $G$ be a connected reductive complex algebraic group, let $K$ be a maximal compact subgroup of $G$, and let $X$ be a spherical homogeneous algebraic variety. Let…

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…

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 $G/H$ be a spherical homogeneous space, and let $\mathfrak B(G/H)$ denote the set of $B$-orbits on $G/H$. A partial order on $\mathfrak B(G/H)$ is required to satisfy propertie…

Let $G$ be a group of type $A_n$, and consider toric limits arising from $(G/B,{\mathcal O}(-K))$. Gorenstein toric limits conjecture. The limits of $(G/B,\mathcal O(-K))$ are Gore…

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 $G$ be a reductive group acting on a smooth spherical variety $X$, and let $W$ be the Weyl group of $G$. Let $M^\vee$ be the dual Hamiltonian space attached to $X$ by relative…

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…

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 $W$ be the Weyl group of $G$, equivalently of $G^{\vee}$. Set $\fX=T^*_{\psi}\fY$, let $\CB=G/B$ be the flag variety, and let … denote the zero fiber of the moment map for the…

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…

For each $i=1,\ldots,p$, let $H_i$ be an output of Algorithm D applied to $(H,\Psi_i)$, and let $(G_i,K_i)$ be the pair obtained from $(G,H_i)$ by reduction of the ambient group. U…

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…

Uncorrected Borel–Moore homology conjecture. The conjecture attributed to the cited work proposes an isomorphism of modules

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

Let $X=G/H$ be the symmetric space under consideration, let $M=T^*X$, let $M^\vee$ be its hyperspherical dual, and let … Write $sgn$ for the sign representation of the Weyl group…

Flexibility conjecture. $X$ is flexible.