38 problems

Higher Deligne–Lusztig representations are representations associated with parahoric subgroups of $p$-adic groups, and compact induction produces representations of the ambient $p$…

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 ${\mathbf{G}}$ be a connected reductive algebraic group with Frobenius endomorphism $F$, let ${\mathbf{L}}$ be an $F$-stable Levi subgroup, and let ${\mathbf{U}}$ be the unipot…

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 ${\mathbf{G}}$ be a finite reductive group with Frobenius endomorphism $F$, let ${\mathbf{P}}$ be a parabolic subgroup with an $F$-stable Levi complement ${\mathbf{L}}$, and le…

Letellier's conjecture. The Fourier transform should satisfy

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 $(\mathbb{L},\lambda)=(\mathbb{T},1)$ for a maximal torus $\mathbb{T}$, and consider the associated Deligne–Lusztig cohomologies with their…

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…

Let $\mathbb{G}$ be a generic finite reductive group, let $e>0$ be an integer, and let $(\mathbb{L},\lambda)$ be a $\Phi_e$-cuspidal pair. The associated Deligne–Lusztig representa…

Deligne–Lusztig parabolic-independence conjecture. The maps ${\mathbf{R}}_{{\mathbf{L}}\leq{\mathbf{P}}}^{\mathbf{G}}$ and…

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…

Lusztig's disjunction conjecture. The disjunction property of the Deligne–Lusztig cohomology conjecture should hold for $w$.

Craven's conjecture on cohomological degrees. If $\rho$ occurs in $H_c^i(X(w),\overline{\mathbf Q}_\ell)$, then it occurs in degree

Disjunction conjecture for Deligne–Lusztig cohomology.

Refined Broué conjecture for torus centralizers. There is a representative $C$ in the stated quotient of homotopy categories such that the right action factors through an action of…

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…