23 problems

Let $(T,U)$ be a minimal elliptic pair, and let $\chi\colon \mathcal{T}^{+}(\mathcal{O}_k)\to\overline{\mathbb{Q}}_\ell^{\times}$ be a smooth character. Let $s_\chi\geq 0$ denote t…

Assume the regular-element setting of the source: ${\mathbf L}={\mathbf T}=C_{\mathbf G}(D)$ is a torus, ${\mathcal O}Gb$ is the principal block, $\ell\nmid(q-1)$, $F$ acts trivial…

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…

Assume $\mathbf{G}'$ is a twisted Levi subgroup of $\mathbf{G}$ containing $\mathbf{T}$. Let $\alpha$ be any representation of $\mathbb{G}'_r(\mathbb{F}_q)$ of depth $<r$, and let…

Assume that $p\neq 2$ is not bad for $\mathbf{G}$, and let $(\mathbf{T},\theta)$ be an unramified Howe-factorizable pair. Let $\vec\phi=(\phi_{-1},\ldots,\phi_d)$ be a Howe factori…

Splendid Morita equivalence conjecture. The endoscopic equivalence of Theorem $$ induces a splendid Morita equivalence in the sense of Rickard.

$t$-exactness conjecture. The endoscopic equivalence of Theorem $$ is $t$-exact.

Endoscopic equivalence conjecture. There exists an endoscopic equivalence as in Theorem $$ without any hypothesis on $\ell$.

The scalar product conjecture. For all $(\theta',\mathbb{T}'_r,\mathbb{B}'_r)$, one has

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…

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…

Disjunction conjecture for Deligne–Lusztig cohomology.

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 the reductive group, $W$ its Weyl group, $b$ the relevant element, and $X_w(b)$ the associated $p$-adic Deligne–Lusztig space for $w obreakrelaxin W$. An element $w$ is…

Let $r$ divide $n'$ and let $\theta \colon \mathbb T_h(\mathbb F_q) \cong \mathbb W_h^\times(\mathbb F_{q^n}) \to \overline{\mathbb Q}_\ell^\times$ be a character with trivial…

Let $r$ divide $n'$ and let $\theta \colon \mathbb T_h(\mathbb F_q) \cong \mathbb W_h^\times(\mathbb F_{q^n}) \to \overline{\mathbb Q}_\ell^\times$ be a character with trivial…

Let $\ell$ be a prime and let $G=\mathbb{G}(q)$ be a finite group of Lie type, where $p\ne\ell$ and $q$ is a power of $p$. Let $D\in\operatorname{Syl}_{\ell}(G)$, let…

Let $(K,\mathcal{O},k)$ be an $\ell$-modular system, let $B$ be a unipotent block with defect group $D$, and let $d$ be the order of $q$ modulo $\ell$. Let $C$ be the reduction mod…

Let $G$ be a finite reductive group with Weyl group $W$, let $F$ be the relevant Frobenius endomorphism, and let $\boldsymbol{T}$ be an $F$-stable torus. For $w otin W$, let $Qw$ d…

Macdonald's conjecture. For every such regular character $\theta$, there exists an irreducible representation $\pi_\theta$ of $G({\mathbb F}_q)$ of dimension

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…

Let $X_h$ be the closed subscheme of $U^{n,q}_h$ defined over ${\mathbb F}_{q^n}$, with commuting left action of $U^1_L/U^h_L$ and right action of $U^{n,q}_h({\mathbb F}_{q^n})$. F…