3 problems

Lusztig's conjecture. $J_u$ is isomorphic to $K_\mathcal{G}(Y\times Y)$ for some complex group $\mathcal{G}$ and finite $\mathcal{G}$-set $Y$.

Let $W$ be a bounded Coxeter group, let $J_W$ be its asymptotic Hecke algebra with basis $\{j_x\mid x\in W\}$, and let $D$ be the set of Duflo involutions in $W$. Local-unit conjec…

Let $\tilde{\mathcal{S}}_n$ be the extended affine symmetric group, let $\lambda \vdash n$ be an integer partition, and let ${\underline{c}}_\lambda$ be the corresponding two-sided…