21 problems

Let $w=sw's$ be an element of the Weyl group with $ll(w)=ll(w')+2$, and let $Z=X(w)\cup X(sw')$. For $i\geq 0$, write … where … and … Let…

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…

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 $X_h$ 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…

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…

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 $G=\b G^F$ be a finite group of Lie type, let $B$ be a unipotent $\ell$-block with abelian defect group, let $b$ be its Brauer correspondent, and let $Y$ be the associated Deli…

Let $G=G(q)$ be a group of Lie type, let $\bar{\Q}_\ell$ be an algebraic closure of the $\ell$-adic numbers, and let $\ell$ be a prime dividing $\Phi_d(q)$. For a unipotent charact…

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…