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Does anyone know if there exists some natural way to interpret the Kazhdan-Lusztig C-basis in a categorification of the Hecke algebra ? The category of Soergel bimodules categorifies the C'-basis but it doesn't seem clear to me whether this or some other related structure (Rouquier complexes, sheaves...) could be adapted to categorify the C-basis instead ? I am currently working on some elements of the Hecke algebra which seem to have remarkable positivity properties when written in the C-basis (---> (-1) signs occur when writing them in the C'-basis).

The $C$ and $C'$-basis are defined in Kazhdan-Lusztig 79 :

For each element $w\in W$ there is a unique selfdual $C_w\in H$ such that

$$C_w=\sum_{y\leq w} \epsilon_y \epsilon_w q_w^{1/2} q_y^{-1} \bar{P_{y,w}} T_y$$

with $P_{y, w}\in\mathbb{Z}[q^{1/2}, q^{-1/2}]$ a polynomial in $q$ having degree at most $\frac{1}{2}(\ell(w)-\ell(y)-1)$ for $y\leq w$, $P_{w,w}=1$. The set $(C_w)_{w\in W}$ forms a basis, the $C$-basis of the Hecke algebra.

For each element $w\in W$, there is a unique selfdual $C'_w\in H$ such that

$$C'_w=q_w^{-1/2} \sum_{y\leq w} P_{y,w} T_y$$

with $P_{y, w}\in\mathbb{Z}[q^{1/2}, q^{-1/2}]$ a polynomial in $q$ having degree at most $\frac{1}{2}(\ell(w)-\ell(y)-1)$ for $y\leq w$, $P_{w,w}=1$. The set $(C_w')_{w\in W}$ forms a basis, the $C$-basis of the Hecke algebra.

the problem of applying the involution $j$ from 7.9 of Humphreys to my elements is that one gets (-1) signs before some coefficients of $C'$-elements

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    $\begingroup$ Could you remind us which are the C and C' bases? When I hear "Kazhdan-Lusztig basis" I think (or rather I repeat what what K-L themselves said) intersection cohomology sheaves on the flag variety, which is already a natural categorified interpretation, but maybe you mean some variant? $\endgroup$
    – David Ben-Zvi
    Commented Sep 24, 2012 at 15:02
  • $\begingroup$ The remark at the end of Section 7.9 in Humphreys connects the C and C' bases by an automorphism of the Hecke algebra. If your elements don't already depend on the specifics of your categorification, you might be better off pushing your elements through that automorphism and studying their images. $\endgroup$
    – Alexander Woo
    Commented Sep 24, 2012 at 17:56
  • $\begingroup$ How are the $\mathbb{Z}[q,q^{-1}]$-lattices spanned by the C and C' bases related? Often in categorification one sees two lattices. One lattice is spanned by projective objects while the other is spanned by simples. The minus signs in the change of basis can interpreted as taking the Euler characteristic of a projective resolution. $\endgroup$
    – David Hill
    Commented Sep 25, 2012 at 15:58
  • $\begingroup$ I should add that even if the lattices are the same, one can still make sense of all this. For example, the C basis may correspond to simple objects and the C' basis corresponds to so-called standard objects. $\endgroup$
    – David Hill
    Commented Sep 25, 2012 at 16:11

1 Answer 1

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If you identify the Hecke algebra with the Grothendieck principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlands dual options you pick doesn't matter) such that $T_y$ is the class of a Verma module with highest weight $-y\rho-\rho$ (so $T_1$ corresponds to the anti-dominant Verma), then the $C'$-basis will match with the classes of tilting modules, and the $C$ basis with the simple modules (I tend to prefer matching the $C'$-basis with projectives, but then the $C$-basis is something horrible).

I'm not really sure where this is written down properly; philosophically, the point is that everything will be fixed in place once you decide what functor to send the bar involution to. If you send it the contragredient dual, then simples and tiltings are obvious sets of modules fixed by this duality, and they satisfy the triangularities you've written above.

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  • $\begingroup$ Thank you for this answer. In case we match the $C'$-basis with projective modules of a graded version of category $\mathcal{O}$ then the Hecke algebra is categorified as a module over itself by graded translation functors which categorify the right multiplication by elements of the $C'$-basis. In case I follow your idea is it obvious on how to categorifiy the Hecke algebra as a module over itself ? $\endgroup$
    – Thomas Gobet
    Commented Sep 26, 2012 at 9:16
  • $\begingroup$ The same way, just with slightly different conventions; you conjugate the projective story by $T_{w_0}$, since there is a derived equivalence sending projectives to tiltings lifting $T_{w_0}$. Alternatively, you get the basis of indecomposable projectives by applying indecomposable projective functors to the dominant Verma module (which is projective) and the indecomposable tiltings by applying them to the anti-dominant Verma module (which is tilting). $\endgroup$
    – Ben Webster ♦
    Commented Sep 26, 2012 at 15:41

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