Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis vectors which contains $C_y$ also contains $C_x$. Similarly for $x\leq_R y$ and right ideals.
Note that $x\leq_L y$ if and only if $x^{-1}\leq_R y^{-1}$, so all the information is in one of these preorders.
Now, we can define a geometric preorder satisfying similar properties, following Steinberg. Let $\mathfrak{n}$ be the usual strictly upper-triangular matrices in $\mathfrak{sl}_n$ and let $\mathfrak{n}\cap\mathfrak{n}^w$ be its intersection with itself conjugated by $w$. We let $x\leq_l y$ if $B\cdot (\mathfrak{n}\cap\mathfrak{n}^y)\subset B\cdot (\mathfrak{n}\cap\mathfrak{n}^x)$, and $x\leq_r y$ if $x^{-1}\leq_ly^{-1}$.
Do the preorders $\leq_L$ and $\leq_l$ coincide?
It's a theorem (look, for example in Appendix B of Borho and Brylinski, Differential... III) that the equivalence relations induced by them (the left cells) do; they're both distinguished by their $Q$-symbol under Robinson-Schensted.
Do these preorders have a nice description in terms of Robinson-Schensted?
EDIT: After speaking with various people, I believe this is an open question, and quite an interesting one in my opinion.
