19 problems

Level-rank conjecture. The two image sets are unions of $H_{L,\lambda}^G(\zeta_m)$- and $H_{M,\mu}^G(\zeta_l)$-blocks, respectively; the bijection between them induced by…

Let $G$ be a finite reductive group, and let $(M,\mu)$ be a $\Phi_m$-cuspidal pair with relative Weyl group $W_{M,\mu}^G$. Let $H_{M,\mu}^G(x)$ be the Broué–Malle algebra, and let…

Let $W$ be a Weyl group, let $\mathcal{J}$ be its set of near involutions, and write $\dim(M_w)=P_w(q)$, where $P_w(t)\in\mathbb{Q}[t]$. For $w\in\mathcal{J}$, let…

Let $\mathbb{G}$ be a generic finite reductive group, let $e>0$, and let $(\mathbb{L},\lambda)$ be a $\Phi_e$-cuspidal pair. Let $H_{W_{\mathbb{G},\mathbb{L},\lambda}}(\vec{u})$ be…

Weak e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$,

Cabanes--Enguehard conjecture. The relation $\leq_e$ is transitive and therefore coincides with $\ll_e$. The conjecture concerns the poset of $e$-pairs and would make the directly…

The block-isomorphism character-triple conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$, there is an ${\rm Aut}_{\mathbb F}({\mathbf{G}}^F)_B$-equivariant b…

The e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$,

Unipotent-family conjecture. There exists a unique bijection

Broué–Malle–Michel conjecture. There exists such a parabolic subgroup $\mathbf P$ for which the cohomology modules in distinct degrees have no common irreducible constituent, and f…

Let $G$ be a twisted reductive group over an algebraically closed field, with Borel subgroup $B$. Let $W$ be its corresponding Weyl group, let $w\in W$, and let $s_i$ be a simple r…

Let $W$ be the Weyl group of a maximal torus contained in a Borel subgroup $B$ of a twisted reductive group over an algebraically closed field. Let $w$ be a minimal-length element…

Let $G$ be a finite reductive group with Frobenius endomorphism $F$, let $k$ be the underlying field of characteristic $p$, and let $d\in\Delta$ index the corresponding grading of…

Counting conjecture. Suppose that $p$ is good for $\mathbf{G}$. Then $\alpha_\ell$ is the number of unipotent irreducible Brauer characters of $G$.

Let $G$ be a finite reductive group with Weyl group $W$, defined over a field of characteristic $p$, and let $k$ be a field of characteristic $ell$ with $ell eq p$. Let $d$ denote…

Let ${\mathbf G}$ be a connected reductive algebraic group with Steinberg endomorphism $F$, let $G={\mathbf G}^F$, let $W$ be its Weyl group, and let $Q_w$ denote the restriction t…

Abelian defect group cohomology conjecture. The two $\mathbf G^F$-modules $H_c^{\mathrm{odd}}$ and $H_c^{\mathrm{even}}$ are disjoint, and $F^\delta$ is semisimple on…

Let $G$ be a finite reductive group, let $\ell$ satisfy the preceding assumptions (B1)–(B3), let $T_w$ be the associated torus, let $b$ be the principal $\ell$-block of $G$, and le…

Let $G$ be a finite reductive group whose principal $\ell$-block has cyclic defect, and let $h_0$ be the relevant order of the roots of unity occurring in the block. Let $q^\delta$…