Subsection 8.3.1. (0PIF) · ScalingStacks

8.3.1. Isomorphism Theorem

Let $Z^{\prime}={\mathbf{R}}$ be the smooth curve with $Z^{\prime}_{o}=(-\frac{1}{2},\frac{1}{2})$ with its standard orientation. Fix an increasing homeomorphism $\alpha:{\mathbf{R}}_{>0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{>\frac{1}{2}}$ fixing the positive integers and define $\alpha^{\prime}:{\mathbf{R}}_{<0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{<-\frac{1}{2}}$ by $\alpha^{\prime}(t)=-\alpha(-t)$ .

Assume $Z(\xi_{1}^{+})\neq Z(\xi_{2}^{-})$ and assume there is a morphism $\tilde{\xi}_{1}:Z^{\prime}\to Z$ with image $Z(\xi_{1}^{+})$ and such that $\xi_{1}^{+}=\tilde{\xi}_{1}\circ\alpha$ . Put $\xi_{1}^{-}=\tilde{\xi}_{1}\circ\alpha^{\prime}:{\mathbf{R}}_{<0}\to Z$ and denote by $\xi^{-}$ the composition ${\mathbf{R}}_{<0}\xrightarrow{\xi_{1}^{-}}Z\hookrightarrow Z_{\xi_{1}}$ .

Proposition 8.1.15 gives an isomorphism of differential pointed bimodules $\hat{\kappa}_{1}:L_{\xi_{1}^{+}}(-_{2},-_{1})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R_{\xi_{1}^{-}}(-_{1},-_{2})^{\vee}$ .

Since there is no admissible path from $\xi_{2}^{-}(-1)$ to $\xi_{1}^{+}(1)$ in $Z$ , we have $A_{n}=D_{n}=G_{n}$ (with the notations of §8.2.2), hence we have an isomorphism (Lemma 8.2.5)

\nu_{n}:R_{\xi_{2}^{-}}^{\bullet}(T,-,e^{n})\wedge L_{\xi_{1}^{+}}^{\bullet}(-,S,e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(-n),\ldots,\xi_{2}^{-}(-1)\},T\sqcup\{\xi_{1}^{+}(1),\ldots,\xi_{1}^{+}(n)\}).

Consider

	$\displaystyle\lambda:L_{\xi_{1}^{+}}^{\bullet}(T,-)R_{\xi_{2}^{-}}^{\bullet}(-,S)$	$\displaystyle\to R_{\xi_{2}^{-}}^{\bullet}(T,-)L_{\xi_{1}^{+}}^{\bullet}(-,S)$
	$\displaystyle\alpha\wedge\beta$	$\displaystyle\mapsto\nu_{1}^{-1}(\alpha\cdot\beta)=((\alpha\cdot\beta)_{\xi_{2}^{-}(-1)}\boxtimes\operatorname{id}\nolimits_{T\setminus\{\chi(\alpha\circ\beta)(\xi_{2}^{-}(-1))\}})\wedge(\alpha\cdot\beta)_{\|S}.$

Since $\nu_{n}$ is an isomorphism, the morphisms (5.2.1) are isomorphisms (cf proof of Theorem 8.2.1) and we obtain from Remark 5.4.1 an isomorphism of differential pointed categories

\Delta_{E}{\mathcal{S}}_{M}^{\bullet}(Z)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}{\mathcal{S}}_{M}^{\bullet}(Z).

Composing its inverse with $\Xi$ , we deduce from Theorem 8.2.1 an isomorphism of differential pointed categories

\Xi^{\prime}:\Delta_{\lambda}{\mathcal{S}}_{M}^{\bullet}(Z)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}_{M}^{\bullet}(Z_{\xi}).

0PDT

Theorem 8.3.1. The isomorphism $\Xi^{\prime}$ provides an isomorphism of $2$ -representations, where $\Delta_{\lambda}{\mathcal{S}}_{M}^{\bullet}(Z)$ is equipped with the diagonal action and ${\mathcal{S}}_{M}^{\bullet}(Z_{\xi})$ with the action of $R_{\xi^{-}}$ .

The remainder of §8.3 is devoted to the proof of Theorem 8.3.1.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2