Let $(W,S)$ be a Coxeter system. One can have the Kazhdan-Lusztig polynomial $P_{x,\ y}(q)$.

Does $P_{x,\ y}(q)=P_{x^{-1},\ y^{-1}}(q)$ for all $x,y\in W$?

Yes, this holds. It follows for example from the standard recursive construction via R-polynomials. My preferred way is the recursion given by Lusztig that constructs $\mu_{x,y}$ and $p_{x,y}$ without the detour over $R$-polynomials. In any case, it is a direct consequence of the recursion constructions.

On a more abstract level, this is an expression of the fact that $T_w \mapsto T_{w^{-1}}$ is an antiautomorphism of the Hecke algebra and $w\mapsto w^{-1}$ is an automorphism of the Bruhat-order.

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