ScalingStacks

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.