$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}\_n$ be the symmetric group, $\SYT\_n$ be the set of standard young tableaux of size $n$.


For $u\in \mathfrak{S}\_n$, let $\RSK:\mathfrak{S}\_n\to \SYT\_n^2$ denote the Robinson-Schensted-Knuth correspondance.


Let $P\_{u,w}(q)$ be the Kazhdan-Lusztig polynomial and $\mu\_{u,w}=[q^{{(l(w)-l(u)-1)}/{2} }]P\_{u,w}(q)$.


Let $\evac:\SYT\_n\to \SYT\_n$ be Schutzenberger's involution.


**Question:** I am looking for a proof of the following proposition that Schutzenberger's involution preserves the $\mu$-coefficient:



> 
> Take $u,w\in \mathfrak{S}\_n$, and let $(P\_1,Q\_1)=\RSK(u)$, $(P\_2,Q\_2)=\RSK(w)$. If we set
>  \begin{gather\*}
> u'=\RSK^{-1}(\evac(P\_1),\evac(Q\_1)) \\
> w'=\RSK^{-1}(\evac(P\_2),\evac(Q\_2)),
> \end{gather\*}
>  then we have $$\mu(u,w)=\mu(u',w').$$
> 
> 
> 



Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials
Let $P^\* = evac(P)$. As noted above, if $RSK(u)=(P,Q)$, then $RSK(w\_0uw\_0) = (P^\*,Q^\*)$. Conjugation by $w\_0$ induces an automorphism of the Hecke algebra sending $T\_x \mapsto T\_{w\_0 x w\_0}$ and $c\_x \mapsto c\_{w\_0 x w\_0}$, from which the result you want follows. However you might be interested that something stronger is true: if $u=RSK^{-1}(P,Q)$ and $\sigma(u) = RSK^{-1}(P^\*,Q)$ then $\mu(u,v)=\mu(\sigma(u),\sigma(v))$. Similarly if $\rho(u) = RSK^{-1}(P,Q^\*)$ then $\mu(u,v)=\mu(\rho(u),\rho(v))$. 


This follows from a theorem of Mathas on the action of $T\_{w\_0}$ on cell representations. The relevant paper is [Mathas, On the Left Cell Representations of Iwahori‐Hecke Algebras of Finite Coxeter Groups](https://doi.org/10.1112/jlms/54.3.475). Proposition 3.17 is the relevant result. 


