29 problems

Lusztig's character formula conjecture. If $p \geq h$ and $w \in W_{\mathrm{aff}}^\varnothing$ satisfies

Let $\mathbb F$ be an algebraically closed field of positive characteristic, let $\mathbb A$ be the coefficient ring used for reduction modulo the characteristic, and let…

Let $W$ be the Weyl group, let $w\in W$, and let $Y_x(\dot w)$ be the stratum defined by … For a $G$-regular linear character $\psi:U_0\to\mathcal{O}^{\times}$, let…

Let $W$ be the Weyl group, let $w\in W$, let $\dot w$ be a representative of $w$, and let $\psi:U_0\to\mathcal{O}^{\times}$ be a regular linear character. Torus restriction conject…

Let $V$ be a crystalline representation of ${\rm G}_{{\mathbb Q}_p}$, let $T$ be a Galois-invariant lattice, and write $V\bmod p$ for the semisimplification of $T/pT$. Let $\Pi(V)$…

The p-modular TQFT structure conjecture.

Let $p_+$ and $p_-$ be the parameters of the logarithmic $(p_+,p_-)$ model. Let ${\bar{\mathscr{C}}}$ be the space of generalized characters, let $\mathscr{G}$ be the subspace desc…

Let $H$ be a factorizable ribbon quantum group, with ribbon element having multiplicative Jordan decomposition. The decomposition gives $SL(2,\mathbb{Z})$-representations $\bar\pi$…

Let $k$ be a global field, let $\ell$ be a prime different from the characteristic of $k$, and let $\iota:\mathbb{C}\to\overline{\mathbb{Q}}_\ell$ be a field isomorphism. Let…

Let $M,N\in\mathbb{N}$, let $d\in\mathbb{N}_0$, let $\mathbb{F}$ be the underlying field, and let $E$ be the natural $2$-dimensional $\mathbb{F}$-representation of…

Let $\ell$ be a prime and let $q$ be a prime power. The naive converse theorem for mod $\ell$ representations of $\operatorname{GL}_2(\mathbb{F}_q)$ asks whether the relevant mod…

Let $V$ be an odd-dimensional indecomposable representation of a finite $2$-group $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$. Assuming the relevant tensor…

Let $\overline{r}:{\operatorname{Gal}}(\overline F/F)\rightarrow {\operatorname{GL}}_n(\mathbb{F})$ be a continuous homomorphism satisfying conditions (i)–(iii) in the paper, and l…

Torsion analogue of the weight-monodromy conjecture. For every integer $i$, there exists a non-zero monic polynomial $P_i(T)\in\mathbb Z[T]$ such that its roots have complex absolu…

Let $q=p^f$ be a power of the prime $p$, let $G$ be a finite Chevalley group defined over $\mathbb F_q$ with $p$ a bad prime for $G$, and let $U$ be a Sylow $p$-subgroup of $G$. Fo…

Equivalence conjecture. For sufficiently large $p$, the functor $F$ is an equivalence.

Let $F$ be a totally real number field, and let $\bar\rho: \operatorname{Gal}(\bar{\mathbb Q}/F)\rightarrow\operatorname{GL}_n(\bar{\mathbb F}_p)$ be a semisimple modular represent…

Let $\operatorname{Rep}_0$ be the principal block of finite-dimensional algebraic representations, let $\mathcal H$ be the Hecke category, and let $\Theta_s$ be the wall-crossing f…

Let $G_{\Bbbk}$ be the semisimple group in characteristic $p$, let $\Repe$ be its extended principal block, let ${\EuScript Gr^\vee}$ be the affine Grassmannian of the Langlands du…

Let $\mu$ be $p$-regular, let $x\in\Wa$ satisfy $x\pdot\mu$ dominant, and let $X_1^p$ denote the restricted region in the weight lattice. Let $\operatorname{LCF}$ denote Lusztig's…

Let $G_{\Bbbk}$ be a semisimple algebraic group with weight lattice $X$, positive roots $\Phi_+$, affine Weyl group $\Wa$, Coxeter number $h$, and $p$-dilated dot action $\pdot$. L…

Let $X$ be a smooth projective curve over ${\mathbb F}_q$, let $\ell$ be a prime different from the characteristic of ${\mathbb F}_q$, and let $T(X,2,q^n,\ell)$ and…

Let $G=({\bf Z}/p)^r$ and let $V$ be a faithful three-dimensional modular representation of $G$ over a field of characteristic $p$. Its invariant ring is $\mathbf{k}[V]^G$. Faithfu…

Let $G=({\bf Z}/p)^r=\langle e_1,\ldots,e_r\rangle$ and let $V$ be the generic three-dimensional representation over $\mathbf{k}=\mathbf{F}_p(x_{1j},x_{2j}\mid j=1,\ldots,r)$. Defi…

Let $G$ be a finite group of Lie type with Weyl group ${\mathbf{W}}$ and Frobenius-induced automorphism $\gamma$, defined over a field with parameter $q$, and let $k$ be a field of…