# Comparing the Kazhdan-Lusztig and Steinberg pre-orders

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.

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Note that your link leads to the Steinberg paper, while your slightly earlier post links to the BB paper. Anyway, Appendix B of the latter refers to the never-published 1977 Warwick preprint by Spaltenstein (then Lusztig's student) titled Remarques sur certaines relations d'equivalence dans les groupes de Weyl. In type A, where all nilpotent orbits are special, it seems plausible that the pre-orders and not just the equivalence relations should coincide. An interesting question apparently not addressed in Spaltenstein's later Springer LN volume. –  Jim Humphreys Jan 27 '11 at 21:20
Ah, thanks, I fixed that (had both of them open in long DigiSchrift links, and copied the wrong one). I saw the Spaltenstein reference, but have found no evidence that preprint ever saw the light of day. It's not on MathSciNet, nor does Googling the title turn up more than a couple other papers which cited it. –  Ben Webster Jan 27 '11 at 21:43
Spaltenstein's preprint of course preceded the KL work, but their treatment in type A probably isn't so closely linked to Hecke algebra formalism. Still, it's hard to put the pieces together. –  Jim Humphreys Jan 27 '11 at 22:40