97 problems

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. The homomorphism $c$ is always an isomorphism.

Let $\bG$ be a reductive algebraic group over a field of characteristic $p$, let $F$ be a Frobenius endomorphism, and let $\chi\in\operatorname{Irr}(\bG^F)$. Assume that $p$ is a g…

Let $G$ be a split reductive group over a number field $F$, with fixed Borel subgroup $B$, and let $P$ run over the standard parabolic subgroups of $G$. Denote by $m_{F;G}$ and…

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…

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

In a reductive group, Deligne–Lusztig varieties are associated with elements of the braid monoid of the Weil group, and the varieties appearing in Broué's conjecture correspond to…

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

Let $G$ be the algebraic group in the paper, let $\bG^1$ be its component, and let $\tSS$ be the simple perverse sheaf on $\bG^1$ whose restriction to $G^1$ is the distinguished si…

Let $G$ be a reductive algebraic group, and let ${\cal F}$ be a constructible sheaf on $G$ that is equivariant under the adjoint action. The preceding result asserts nonnegativity…

Braverman's conjecture. The Euler characteristic satisfies

Let ${\mathbf G}$ be a reductive group over a finite field, let $\rho$ be a good representation of ${\mathbf G}$, and let $\psi$ be a non-trivial additive character of the finite f…

Let ${\mathbf G}$ be a reductive group over a finite field, let $\rho$ be a good representation, and let $\psi$ be a non-trivial additive character of the finite field. Let…

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…

Refined $B_e(G)_{\mathrm{basic}}$-local Langlands conjecture. There is a commutative diagram

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

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 $\widehat{(\hat{\Delta}_{\mathrm{red}})}=(L,H_L,\iota_L,\rho_{H_L})$ be the quadruple associated to the reduced dual construction. For a Levi subgroup $M$ of a reductive group…

Let $G$ be an arbitrary connected reductive group, let $k$ be a field, and let $\Delta(G,k)$ be its spherical building. Let ${\mathcal{OD}}(\Delta(G,k))$ denote the ordered decompo…

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…

Spaltenstein's conjecture. Every stabiliser sheet of $\mathfrak{g}$ contains exactly one nilpotent orbit.

Let $G$ be a reductive group, let $\Gamma$ be a closed subset of its roots, and let $H_{\Gamma}$ be a maximal-rank subgroup of $G$ whose roots form $\Gamma$. Let $G^{\mathbb{L}}$ b…

Let $G$ be a reductive algebraic group with positive roots $R^G_+$, let $U^G_+$ be the analogous unipotent subgroup, and let $\bigl(M_\alpha\bigr)_{\alpha\in R^G_+}$ be coordinates…