Semicontinuity

Abstract

In this chapter, we study how, conjecturally, the Kazhdan-Lusztig cells change whenever the weight function changes.

Appendices

Notes

The semicontinuity conjecture was stated by the author in [Bon2].

Exercises

Exercise 7.1.

Check Conjecture 7.2.1 whenever W is of type $B_2$ or $G_2$.

Exercise 7.2.

We aim to justify the terminology “Semicontinuity conjectures”. If ${\mathscr {X}}$ is a topological space and if $({\mathscr {E}},{\leqslant })$ is a poset, then a map $f : {\mathscr {X}}\rightarrow {\mathscr {E}}$ is called lower semi-continuous if, for all $e \in {\mathscr {E}}$, the set

$$ \{x \in {\mathscr {X}}~|~f(x) {\leqslant }e\} $$

is closed in ${\mathscr {X}}$. We define analogously the notion of an upper semi-continuous map.

Let $? \in \{L,R, LR\}$ and assume that Conjecture 7.2.1 holds. We fix $w \in W$ and we denote by $\gamma _w^? : V \rightarrow {\mathscr {P}}(W)$ the map that sends $\varphi \in V$ to the Kazhdan–Lusztig left, right or two-sided (according to the value of ?) $\varphi$-cell of w. Show that $\gamma _w^?$ is lower semi-continuous (for the order $\subset$ on ${\mathscr {P}}(W)$).

Exercise 7.3.

Let $s \in S$ and x, $y \in W$ be such that $sx {\varvec{<}}x {\varvec{<}}y {\varvec{<}}sy$ and $\ell (y)-\ell (x)=1$. Show that $\{\varphi \in V~|~h_{s,y, x}^\varphi \ne 0\}$ is closed in V.