Kazhdan-Lusztig polynomials for the longest element in finite Coxeter groups

Question

Suppose $W$ is a finite Coxeter group and that $w_\circ$ is the longest element. I want to prove that $P_{x,w_\circ}=1$ for all $x \in W$.

I know from Corollary 7.14 in the book Humphreys p. 167 that when $x < w_\circ$, $P_{x,w_\circ}=P_{sx,w_\circ}$ for some $s \in S$ such that $sw_\circ<w_\circ$ and $sx > x$, but I can't figure out how to utilize this, since the polynomials $P_{x,w_\circ}$ are not quite clear to me.

Answer 1

Notice that $sw_°<w_°$ for all $s\in S$. We can deduce by induction that $P_{x,w_°}=P_{w_°,w_°}$ for all $x$. Then we just need to know that $P_{w_°,w_°}=1$ , which is often part of the definition.

It's not always necessary to understand something completely to prove something about it.

Answer 2

Okay, here is a possible answer. Suppose $x \in W$ and not equal to $w_\circ$. Since we have that $sw_\circ<w_\circ$ for all $s \in S$ corollary 7.14 in the above mentioned book, gives that for $s\in S$ s.t. $sx>x$ the following holds:

$$P_{x,w_\circ}=P_{sx,w_\circ}$$

If $sx \neq w_\circ$, then $sx<w_\circ$ and reapply the corollary for $s'$ and $sx$ s.t. $s'sx>sx$. Eventually this means this gives that $s_r\dots s'sx=w_\circ$ and the following equality holds:

$$ P_{x,w_\circ}=P_{w_\circ,w_\circ}$$

And since $P_{w,w}=1$ by definition of the Kazhdan Lusztig polynomials, the result follows.