Subsection 4.2.1. (0P2P) · ScalingStacks

4.2.1. Main Theorem

We put $\operatorname{HH}\nolimits_{i}=\operatorname{HH}\nolimits_{i}^{(W,S)}=\operatorname{Tor}\nolimits_{i}^{P^{{\mathrm{en}}}}(P,-):P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits\to P\operatorname{\!-modgr}\nolimits$ . This gives rise to functors $\operatorname{HH}\nolimits_{i}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)$ and to a functor $\operatorname{HH}\nolimits_{*}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)^{{\mathbf{Z}}}$ .

Given $I\subset S$ , we have a functor $\rho_{I}^{*}:D^{b}(P_{I}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}\to D^{b}(P\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ . This defines a functor from ${\mathcal{C}}{ox}$ to graded triangulated categories $(W,S)\mapsto D^{b}(P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ . Our grading here is the one coming from $P_{S}\operatorname{\!-modgr}\nolimits$ .

The following theorem is a consequence of Theorem 5.1 below.

0P20

Theorem 4.5. The functors $\operatorname{HH}\nolimits_{*}^{S}$ define a Markov $2$ -trace on finite Coxeter groups with value in ${\mathcal{C}}_{S}=D^{b}(P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ and with $T_{S,s,+}=[-1]\{1\}$ and $T_{S,s,-}=\operatorname{Id}\nolimits$ .

Passing to homology, we obtain the following result.

0P21

Corollary 4.6. The functors $H^{*}\operatorname{HH}\nolimits_{*}^{S}$ define a Markov $2$ -trace on finite Coxeter groups with value in ${\mathcal{C}}_{S}=\bigl((P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}\bigr)^{{\mathbf{Z}}}$ and with $T_{S,s,+}=[-1]\{1\}$ and $T_{S,s,-}=\operatorname{Id}\nolimits$ .

The construction of §4.1.3 provides a Markov trace, recovering a result of Lasy [La1, La2]. Note that Webster and Williamson [WeWi] have shown that for finite Weyl groups, this is actually Gomi’s trace, as conjectured by J. Michel. That has been shown to hold also in type $I_{2}(n)$ by Lasy [La1].

0P22

Corollary 4.7 (Lasy). The following defines a Markov trace on finite Coxeter groups:

{\mathcal{B}}_{(W,S)}\ni b\mapsto\sum_{d,i,j}(-1)^{j}\dim H^{j}(\operatorname{HH}\nolimits_{i}^{S}(F_{b}))_{d}q^{-d}t^{-i}\in{\mathbf{Z}}[q^{\pm 1},t^{\pm 1}]

corresponding to $t_{+}=-t$ and $t_{-}=1$ .

Raphaël Rouquier

Original source: arXiv:1203.5065v1