Leading coefficients of Kazhdan--Lusztig polynomials and fully commutative elements

Abstract

Let $W$ be a Coxeter group of type ${\tilde{A}}_{n-1}$. We show that the leading coefficient, $\mu (x,w)$, of the Kazhdan--Lusztig polynomial $P_{x,w}$ is always equal to 0 or 1 if $x$ is fully commutative (and $w$ is arbitrary).

Approximately 10 pages, AMSTeX