17 problems

Representations of the Iwahori–Hecke algebra ${\mathcal{H}}_n$ of type $A_{n-1}$ can be constructed from the action of the braid group $B_n$ on the homology of configuration spaces…

Let $F$ be a finite extension of $\mathbb{Q}_p$ or of $\mathbb{F}_p((t))$, let $\mathbb{G}$ be a split reductive group over $F$, and set $G=\mathbb{G}(F)$. Let $C$ be a characteris…

Let $\mathcal{H}(B\setminus G/B)$ be the spherical Hecke algebra, let $\mathcal{F}$ be the functor and $\Lambda_B$ the idempotent used in the paper, and let $M$ be a simple module…

Let $W$ be a finite Coxeter group with simple generators $S$, let $\mathcal{H}_K$ be the corresponding Hecke algebra over $K$, and let $E_v^\lambda$ be a simple $\mathcal{H}_K$-mod…

Let $W$ be a finite Coxeter group, let $\mathcal{H}$ be its Iwahori--Hecke algebra, and let a $W$-graph representation mean a representation afforded by a $W$-graph. A representati…

Brubaker–Bump–Friedberg conjecture. The map $\mathcal{L}$ is an $\mathcal{H}$-map from $M$ to $V_{\varepsilon}$ for some choice of linear character $\varepsilon$ of $\mathcal{H}_0$…

Let $G$ be split and let $\mathfrak{i}=[T,1]_G$ be its Iwahori-spherical Bernstein component, with $T^{\vee}$ a maximal torus in the complex dual group and $W_{\mathfrak{i}}$ the c…

Let $G^+$ be the semigroup of a $p$-adic loop group on which the Cartan decomposition holds, let $I$ be an Iwahori subgroup, and let $\mathcal{H}(G^+,I)$ be the algebra of complex-…

Polynomiality conjecture. Each $a_{\mathbf w,\mathbf v}^{\mathbf u}$ is a polynomial in the parameters in ${\mathcal Q}$ with integral coefficients depending only on the geometry o…

Unique nonnegative constituent conjecture. For each left cell ${\mathfrak{C}}$ of $W$, there is a unique $E\in{\mathcal{S}}_L(W)$ such that $m({\mathfrak{C}},E)>0$; moreover, for t…

Let $W$ be a finite Coxeter group with weight function $L$, let ${\operatorname{Irr}}(W)$ denote its irreducible representations, and let $c_{w,E}$ and $f_E$ be the leading coeffic…

Let $\hat{\Phi}$ be a simply-laced root system, let $u\leqslant v$ in its Bruhat order, and let $S'(u,v)=\{\alpha\in\hat{\Phi}^+\mid u\leqslant ur_\alpha<v\}$. Let $\tilde m(u,v)$…

Let $\hat{\Phi}$ be a simply-laced root system, let $u\leqslant v$ in its Bruhat order, and let $S(u,v)=\{\alpha\in\hat{\Phi}^+\mid u\leqslant vr_\alpha<v\}$. Let $M_v$ be the rele…

FM-partition conjecture. Suppose $\operatorname{char}(F)=infty$ and that $la$ is neither $2$-regular nor $2$-restricted. Then the Specht module

Generation conjecture. The $\mathbb B[\xi]$-algebra $\mathcal K$ is generated by $E_r$, $r=1,2,\ldots$.

Polynomiality conjecture. There exists a polynomial $f_{\lambda \mu}^\nu$ in one variable with coefficients in $\mathbb Q[\xi]$ such that

Let $W$ be a finite group with generating set $S$, let $k=\mathbb{F}_\ell$ be the field of $ll$ elements, and let $\theta_\ell^\zeta:R\to k$ be the ring homomorphism sending…