  Kazhdan--Lusztig polynomials arise in the context of Hecke algebras
associated to Coxeter groups. The computation of these polynomials is very
difficult for examples of even moderate rank. In type $A$ it is known that the
leading coefficient, $\mu(x,w)$ of a Kazhdan--Lusztig polynomial $P_{x,w}$ is
either 0 or 1 when $x$ is fully commutative and $w$ is arbitrary. In type $D$
Coxeter groups there are certain "bad" elements that make $\mu$-value
computation difficult.
  The Robinson--Schensted correspondence between the symmetric group and pairs
of standard Young tableaux gives rise to a way to compute cells of Coxeter
groups of type $A$. A lesser known correspondence exists for signed
permutations and pairs of so-called domino tableaux, which allows us to compute
cells in Coxeter groups of types $B$ and $D$. I will use this correspondence in
type $D$ to compute $\mu$-values involving bad elements. I will conclude by
showing that $\mu(x,w)$ is 0 or 1 when $x$ is fully commutative in type $D$.
