Kazhdan-Lusztig conjecture

introduction

• 1979 conjectures
  • KL character formula
  • KL positivity conjecture

Hecke algebra

• new basis of Hecke algebra $\{\underline{H}_{x}| x\in W\}$

$$ \underline{H}_{x}=H_{x}+\sum_{y\in W, \ell(y)<\ell(x)} h_{y,x}H_{y} $$ where $h_{y,x}\in v\mathbb{Z}[v]$ is so called the Kazhdan-Lusztig polynomial

• positivity conjecture : $h_{x,y}\in \mathbb{Z}_{\geq 0}[v]$

Hodge theory

• Poincare duality
• hard Lefshetz theorem
• Hodge-Riemann bilinear relation

exposition

• Williamson- Kazhdan-Lusztig conjecture and shadows of Hodge theory