144 problems

Mirković–Vilonen's conjecture. The cohomology modules of the stalks of ${\mathcal{I}}_!(\lambda,\mathbb{Z})$ are free.

Toda's chi-independence conjecture. For any $\gamma\in B_S(\alpha)$, the invariant $\Phi(\alpha+n\delta)$ is independent of $n\in\mathbb{Z}$.

Finkelberg–Mirković conjecture. There exists an equivalence

In generalized Springer theory, consider cuspidal simple perverse sheaves with coefficients reduced modulo a prime $p$. Mautner's cleanness conjecture. These cuspidal perverse shea…

Bobadilla–Kollár's conjecture. If $f$ is a $\mathbb Z$-homology fiber bundle, then $f$ is smooth.

Let $\Lambda_V$ be the representation variety under consideration, let $G_V$ act on it, and let $\Lambda_b$ be an irreducible component indexed by a canonical-basis element $b$. A…

Laumon's compatibility conjecture. The restriction of $\Eis^n_{n_1,\ldots,n_k}({\mathcal S}')$ to $\Bun_{GL(n)}\subset\operatorname{Coh}_n$ is canonically isomorphic to…

Irreducibility conjecture. Moreover, if ${\mathcal S}$ is irreducible, then so is $\Eis({\mathcal S})$.

Let $X$ be a complex variety, let $T^*X$ be its cotangent bundle, and let $\Lambda\subset T^*X$ be a conical Lagrangian variety, such as the conormal variety to a stratification of…

Rationality and finite-dimensionality conjecture. The pairing takes values in rational Laurent series, and the quotient

Let $f:Y'\to Y”$ be a proper map between schemes over an algebraically closed field, and let $\CF$ be an irreducible $\ell$-adic perverse sheaf on $Y'$ obtained by the six operatio…

Perverse sheaves on flag varieties form a bounded derived category with a Serre functor. Beilinson–Bezrukavnikov–Mirkovic conjecture. The Serre functor of the bounded derived categ…

Let $\overline{A}$ and $\overline{X}$ be quasi-projective varieties over $\mathbb C$, with $\overline{A}$ smooth, and let $\overline{f}:\overline{A}\to\overline{X}$ be a projective…

Derived Koszul duality conjecture. The category of perverse sheaves on the affine flag variety of a semi-simple group, smooth along the Schubert stratification, is derived Koszul e…

Let $Q$ be a Dynkin quiver obtained by orienting the Dynkin diagram of $\mathfrak g$, let $\operatorname{rep}(Q,{\mathbf V})$ be its representation space, and let $\mathcal O$ be a…

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 $X$ be the curve and $G$ the reductive group of the paper. Let $E$ be a $\check{G}$-local system on $X$, and put $W=V^{\gamma}_E$. Assume that $W$ is irreducible and that whene…

Let $f_{\pi^l}$ be the functions on $({\cal O}/\pi^l{}cal O)^{n+1}$ obtained from the spherical vector, and let $I$ be the polynomial appearing in the formula $\psi(-I/(x_0\pi^l))$…

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…

Let $X$ be a smooth projective curve over a ground field $k$, let $E$ be an $m$-dimensional local system on $X$, and let $\Bun_m$ be the moduli stack of rank $m$ vector bundles on…

Braverman's conjecture. The Euler characteristic satisfies

Iwahori–Whittaker–Iwahori-monodromic equivalence conjecture. There exists an equivalence between the mixed versions of $D_{\mathcal{IW}}$ and the category of Iwahori-monodromic she…

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$, let $\mathcal{D}:{\mathcal{P}}\to{\mathcal{P}}$ be Verdier duality, and let…

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$ constructible with respect to the stratification by the…

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…