ScalingStacks

Let ${\mathcal{C}}{ox}$ be the poset of finite Coxeter groups, viewed as a category. The objects are Coxeter groups $(W,S)$ and $\operatorname{Hom}\nolimits((W,S),(W^{\prime},S^{\prime}))$ is the set of injective maps $f:S\to S^{\prime}$ such that $m_{f(s),f(t)}=m_{st}$ for all $s,t\in S$. Given $s\in S$, we denote by $i_{s}:(W_{S\setminus s},S\setminus s)\to(W,S)$ the inclusion.

Let ${\mathcal{F}}$ be a full subposet of ${\mathcal{C}}{ox}$ closed below.

Let ${\mathcal{H}}_{(W,S)}={\mathbf{Z}}[q^{\pm 1}]B_{(W,S)}/((T_{s}-1)(T_{s}+q))_{s\in S}$ be the Hecke algebra of $(W,S)$.

Markov traces, with a possibly more general definition, have been studied by Jones and Ocneanu in type $A$ [Jo], Geck-Lambropoulou in type $B$ [GeLa], Geck in type $D$ [Ge], and Kihara in type $I_{2}(n)$ [Ki]. Gomi has provided a general construction of Markov traces for Weyl groups, using Lusztig’s Fourier transform [Go]. More recently, Lasy has studied Markov traces in relation with Gomi’s definition and Soergel bimodules [La1].

One can ask in addition that the functors $T_{S,s,\pm}$ are invertible. On the other hand, one can get a more general definition by dropping the functoriality of ${\mathcal{C}}$ and by requiring the existence of functors $D_{S,s,\pm}:{\mathcal{C}}_{S\setminus s}\to{\mathcal{C}}_{S}$ such that $M_{S}(\gamma_{S\setminus s}(?)\cdot F_{s}^{\pm 1})\simeq D_{S,s,\pm}M_{S\setminus s}(?)$.

The first “trace” condition, once formulated in the appropriate homotopical setting, leads to a universal solution (“abelianization”) that can be described explicitly [BeNa]. It would be interesting to find a suitable formulation of the second condition in this setting leading to a universal solution. It would also be interesting to study functoriality with respect to cobordism in type $A$, along the lines of [KhTh] and [ElKr].

Let ${\mathcal{S}}{oe}$ be the category of Soergel bimodules: this is the full subcategory of $P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits$ whose objects are direct summands of direct sums of objects of the form $\theta_{s_{1}}\cdots\theta_{s_{n}}\langle r\rangle$, for some $s_{1},\ldots,s_{n}\in S$ and $r\in{\mathbf{Z}}$.

There is a ${\mathbf{Z}}[q^{\pm 1}]$-algebra morphism ${\mathcal{H}}\to K_{0}({\mathcal{S}}{oe})$ given by $T_{s}\mapsto[F_{s}]$, and that morphism is actually an isomorphism (cf [Soe, Theorem 1] and [Li, Théorème 2.4]).

We consider now a Markov $2$-trace in the following setting. Assume the functor ${\mathcal{C}}$ takes values in graded triangulated categories and $M_{(W,S)}$ is the restriction of a graded triangulated functor $M_{(W,S)}:\operatorname{Ho}\nolimits^{b}({\mathcal{S}}{oe}_{(W,S)})\to{\mathcal{C}}_{(W,S)}$. In particular, it induces a ${\mathbf{Z}}[q^{\pm 1}]$-linear map ${\mathcal{H}}_{(W,S)}\to K_{0}({\mathcal{C}}_{(W,S)})$. Let $R=\operatorname{colim}\nolimits_{(W,S)\in{\mathcal{F}}}K_{0}({\mathcal{C}}_{(W,S)})$, a ${\mathbf{Z}}[q^{\pm 1}]$-module. Assume there are commuting endomorphisms $t_{\pm}$ of $R$ compatible with the action of $[T_{S,s,\pm}]$ on $K_{0}({\mathcal{C}}_{(W,S)})$, for all $(W,S)\in{\mathcal{F}}$ and $s\in S$, via the canonical maps $\iota_{(W,S)}:K_{0}({\mathcal{C}}_{(W,S)})\to R$.

Define $\tau_{(W,S)}:B_{(W,S)}\to R$ by $\tau(b)=\iota_{S}([M_{S}(F_{b})])$. We have the following immediate proposition.

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.

Passing to homology, we obtain the following result.

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].

By shifting suitably the invariants, we can get rid of the automorphisms $T_{S,s,\pm}$, but we lose functoriality (it would be interesting to see if functoriality with respect to an appropriate notion of cobordisms can be implemented). In order to do this, we need to use $\frac{1}{2}{\mathbf{Z}}$-complexes.

Given ${\mathcal{A}}$ an additive category, the category of $\frac{1}{2}$-complexes in ${\mathcal{A}}$ has objects $(C^{i},d^{i})_{i\in\frac{1}{2}{\mathbf{Z}}}$ where the differential has degree $1$, and morphisms are $\frac{1}{2}{\mathbf{Z}}$-graded maps commuting with the differential. Its homotopy category is denoted by $\operatorname{Ho}\nolimits_{\frac{1}{2}}({\mathcal{A}})$ and, when ${\mathcal{A}}$ is an abelian category, its derived category by $D_{\frac{1}{2}}({\mathcal{A}})$.