38 problems

Let $U_L^1/U_L^h$ be a quotient of unit groups, let $\chi\colon U_L^1/U_L^h\to \overline{\mathbb{Q}}_\ell^\times$ be a character, and let $\mathbb{U}_{h,k}(\mathbb{F}_{q^n})$ be th…

Let $\ell\neq p$ be primes, let $q$ be a power of $p$, let $d$ be the multiplicative order of $q$ modulo $\ell$, and let $G=G(q)$ be a finite group of Lie type with abelian Sylow…

Let $\Gamma^\bullet$ denote the graph obtained from the Brauer tree of the principal $\ell$-block by removing the exceptional node and all edges incident to it. For each root of un…

In a reductive group, Deligne–Lusztig varieties are associated with elements of the braid monoid of the Weil group, and the varieties appearing in Broué's conjecture correspond to…

Let $W$ be the Weyl group, let $F$ be the Frobenius automorphism, and let ${\boldsymbol\pi}$ be the positive generator of the center of the pure braid group. Let…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$ and let ${\bf X}({\bf w})$ be the associated Deligne–Lusztig variety. The compactly supported cohomology groups…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let ${\mathcal H}_q(w)$ be the specialization at $x^{1/a}\mapsto q^{1/a}$ of the cyclotomic Hecke algebra ${\mathcal H}(w)$, and…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let $w=\beta({\bf w})$, let $B(w)$ be the braid group of $C_W(wF)$, and let ${\mathcal H}(w)$ be a cyclotomic Hecke algebra for…

Let $B$ be the braid group, let $B^+$ be its positive braid monoid, let $F$ act on $B$ by the Frobenius diagram automorphism, and let ${\boldsymbol\pi}$ be the positive generator o…

Let $W$ be the Weyl group of ${\bf G}$, let $B$ be its braid group, let $B^+$ be the positive braid monoid, and let $\beta:B^+\to W$ be the canonical morphism. Let $F$ be the Frobe…

Assume that ${\bf G}$ is split of type $A_2$. Let $\varphi:B^+\to\mathbb Z$ be the class function defined in the source by its values on the listed representatives and by…

Let $N=l(w_0)$, let $\rho\in{\mathcal E}({\bf G}^F,1)$, and let $A_\rho$ and $a_\rho$ be respectively the degree and valuation of the polynomial in $q$ giving $\rho(1)$. Let $E$ be…

Duality periodicity conjecture. If ${\bf w}\in B^+$ satisfies ${\bf w}\preccurlyeq\boldsymbol\pi^n$, then

Let $N=l(w_0)$, let $\rho\in{\mathcal E}({\bf G}^F,1)$, and let $A_\rho$ and $a_\rho$ be respectively the degree and valuation of the polynomial in $q$ giving $\rho(1)$. Let…

Let ${\bf t}\in B^+$, and let $F$ act on $B^+$ as in the definition of $F$-conjugacy. Two elements are $F$-conjugate if ${\bf t}'={\bf w}^{-1}{\bf t}F({\bf w})$ for some…

Let $\mathbf G$ be an algebraic group over $\bar{\mathbb F}_q$, let $F$ be a Frobenius map, let $G=\mathbf G^F$, and let $W$ be the $F$-split Weyl group. Let $w\in W$ be a good ele…

Let $\mathbb{T}_r$ and $\mathbb{U}_r$ be the groups appearing in the cohomology varieties $Y_{\mathbb{T}_r,\mathbb{U}_r}$, and let…

Let $X_{c,b}^{\caG_{\bfx}^{\rm ad}}$ be the affine Deligne–Lusztig variety associated with the Coxeter element $c$, the element $b$, and the adjoint parahoric group scheme…

Let $B$ be a unipotent block of a finite reductive group, and let $T$ denote a complex of modules satisfying Rickard's criterion for inducing a derived equivalence. Broué's Deligne…

Let $X_h$ be a finite-type variety over a finite field with geometric Frobenius $\Fr_{q^n}$ acting on its compactly supported cohomology. A variety is maximal in the sense of Boyar…

Let $G$ be a reductive group, let $mu$ be a Hodge cocharacter, and let $K$ be a parahoric subgroup. Assume that $(G,\mu,K)$ is of Coxeter type, and let the corresponding basic affi…

Conjecture A'. The restriction morphism induces an isomorphism

Let $X(c)$ be a Deligne–Lusztig variety, let $\pi:Y(\dot{c})\rightarrow X(c)$ be the covering with group containing $U_I$, and let $j:X(c)\hookrightarrow\omega X(c)$ be the open im…

Let $\chi$ be a unipotent ordinary character in the principal block, and suppose $d=1$ or $d=2$ with $\kappa=1$. Write $\mathrm{Deg}(\chi)$ for its generic degree and…

Extended Poincaré duality conjecture. Poincaré duality holds for $D_{\mathbf v(\mathbf w\varphi)^n}$: there is a perfect pairing of $D_{\mathbf v(\mathbf w\varphi)^n}$-modules