12 problems

Let $R$ be a reduced, irreducible, finite root system, let $\widehat{\mathcal W}$ be its affine Weyl group, and let $k$ be a field. Form the associated moment graph…

Choose a generic line and the associated graded algebra ${\mathbb R}[T]$, with $T$ of degree $2$, and let ${\mathscr B}(x)^{\delta y}_T$ be the corresponding module. The Hard Lefsc…

Choose a generic line in $V^\ast$ and let $\rho:S\to{\mathbb R}[T]$ be the corresponding graded surjection, with $T$ of degree $2$. For an $S$-module $M$, set…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^{[y]}\hookrightarrow {\mathscr B}(x)^y$ be the inclusion of the costalk int…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^y$ be the stalk of the sheaf ${\mathscr B}(x)$ at $y\in{\mathcal W}$. For a…

Let $\mathcal W$ be the Coxeter group, let $\mathcal H$ be its Hecke algebra, and let $B(x)$ be the indecomposable projective object associated with $x\in\mathcal W$. For a graded…

Let $\mathcal W$ be the Coxeter group with Bruhat order $<$ and length function $l$. For $x,y\in\mathcal W$ with $y<x$, let $B(x)^{[y]}$ denote the corresponding graded $S$-module…

Moment-graph formula conjecture. One has

Let $\widehat{W}^{\circ}$ be the finite subset of an affine Weyl group used in the moment-graph construction, let $\mathcal{B}(w)$ be the indecomposable object associated with…

Let $w\in{\widehat{\mathcal W}}$ and suppose that $\operatorname{char} K>N(w)$. For every $x\le w$, let … be the indicated map between the costalk and stalk of the reduced Braden–M…

Let ${\widehat{\mathcal G}}$ be the affine moment graph, let $w\in\mathcal V={\widehat{\mathcal W}}$, and let $(K,w)$ be a GKM-pair. For the Braden–MacPherson sheaf…

Let ${\widehat{\mathcal G}}_k$ be the affine moment graph over the field $k$, let $\mathscr B_k(w)$ be the intersection sheaf attached to $w\in{\widehat{\mathcal W}}$, and let…