38 problems

Homological-dimension-one conjecture. The quotient ${\mathsf V}/\!\!/G$ is either an affine space or a hypersurface; equivalently,

Mumford's conjecture. For some $d\in \mathbb{Z}_{+}$, the submodule

Let $G$ be a reductive group over a field $k$ of good characteristic, with Lie algebra ${\mathfrak g}$. Let ${\cal N}_1$ be the restricted nilpotent cone, let $e\in{\cal N}_1$, and…

Lusztig's cohomological realization conjecture. Every irreducible representation of $G_r^F$ appears in the virtual representation

Reductive-group triangulation uniqueness conjecture. Then $D$ has at most one triangulation of parameter $\delta$.

Let $G$ be a connected, quasi-split reductive group over $F$, let $(G',\xi,z)$ be a rigid inner twist of $G$, and let $\Phi_\mathrm{e}(G;[z])$ denote the set of $\widehat{G}$-conju…

Ben-Zvi–Sakellaridis–Venkatesh duality conjecture. The dual of $\Delta$ is

Let $S$ be a Noetherian scheme and let $G$ be an isotrivial reductive group over $S$, meaning that $G$ becomes split after a finite étale cover of $S$. A $G$-equivariant $S$-fiberw…

Chevalley restriction conjecture for commuting schemes. The morphism $\vartheta$ is an isomorphism. This asserts that invariant functions on commuting $n$-tuples are completely det…

Kobayashi's conjecture. The reductive homogeneous space $G/H$ admits compact quotients if and only if it admits standard ones.

Abelianness and strict invariance conjecture. For every reductive group $G$ over $k$, the sheaf $\pi_0^{{\mathbb A}^1}(G)$ is a sheaf of abelian groups. Moreover,…

Let $G$ be a reductive group over a $p$-adic field $F$, let $x$ be a point in its Bruhat–Tits building, and let $K_\ell:=K_{G,x,\ell}$ be the $\ell$th Moy–Prasad subgroup at $x$. W…

Suppose that $(\widetilde G_{k^{\mathrm a}},\Gamma_{k^{\mathrm a}})$ is quasisemisimple. Write $G^{\mathrm{red}}$ for the maximal pseudo-reductive quotient of $G$, which is reducti…

Let $G$ be a connected real linear reductive Lie group, let $H$ be a reductive subgroup of $G$, and let $\Gamma$ be a discrete subgroup of $G$. Let the Zariski closure of $\Gamma$…

Let $bbalpha:bbG\tobbG'$ be an isomorphism of connected reductive groups, let $bbpi\inbbPi_{bbC}(bbG)$, and let ${}^{bbalpha}bbpi$ be the irreducible representation of $bbG'$ obtai…

Let $G$ be a reductive group with cover $G(F)_{x_G}$, let $(\dot s,\mathcal H)$ be a refined endoscopic datum for $G$, and let $H(F)_{x_H}$ be the corresponding endoscopic cover. L…

Characteristic-two conjecture. If $H$ is non-$G$-dh, then $p=2$.

Let $F$ be a non-archimedean local field, let $G^*$ be a quasi-split connected reductive group over $F$, and let $C$ be an algebraically closed field of characteristic $0$. For a t…

Let $p \in \mathbb{P} \cup \{0\}$, and let $(\mathcal{G},\mathcal{H},\theta)$ be a $p$-embedding of abstract indices. The abstract-index admissibility conjecture. Then…

The cofree representation characterization conjecture. begin{enumerate} item If $V$ is cofree, then it is cnpure. item If $G$ is semisimple and $V$ is irreducible, then purity of…

Let $(G,H)$ be a real symmetric pair, where $G$ is a complex connected reductive group. Let $\theta$ be the involution defining $H$, so that $H=G^{\theta}$, and let…

Pappas–Rapoport–Smithling conjecture. The set of maximal elements in $\{\bar{\mu}\}$ with respect to the Bruhat order is precisely the set $\Lambda_{\{\mu\}}$.

Let $G$ be any connected reductive group and suppose that $\boldsymbol{k}$ is big enough for $G$. For a fixed nilpotent orbit $\mathscr{O}$, consider the cuspidal pairs…

Absolute convergence conjecture. For every $f\in C_c^\infty(G(\mathbb A)^1)$, one has

Let $G$ be a reductive group, let $r$ be the rank of the free group, and write $\mathfrak{X}_r(G)^{\mathrm{red}}$ and $\mathfrak{X}_r(G)^{\mathrm{sing}}$ for the reducible and sing…