ScalingStacks

Let $I\subset S$ and let $W_{I}$ be the subgroup of $W$ generated by $I$. Let $V_{I}=\bigoplus_{s\in I}ke_{s}$ and $P_{I}=k[V_{I}]$. We have $V=V_{I}\oplus V^{I}$, hence $P=P_{I}\otimes k[V^{I}]$. We deduce also that $V^{*}=(V^{I})^{\perp}\oplus(V_{I})^{\perp}$, hence the composition of canonical maps $(V^{I})^{\perp}\hookrightarrow V^{*}\twoheadrightarrow(V_{I})^{*}$ is an isomorphism. We identify $(V^{I})^{\perp}=\bigoplus_{s\in I}k\alpha_{s}$ and $(V_{I})^{*}$ via this isomorphism.

The compositions of canonical maps $\bigcap_{s{\not\in}I}\ker\alpha_{s}\to V\to V/V^{I}$ and $V_{I}\to V\to V/V^{I}$ are isomorphisms: this provides an isomorphism $V_{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigcap_{s{\not\in}I}\ker\alpha_{s}$. We denote by $\rho_{I}:P\twoheadrightarrow P_{I}$ the morphism given by the composition $V_{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigcap_{s{\not\in}I}\ker\alpha_{s}\hookrightarrow V$.

We have a functor $\gamma_{I}:P_{I}^{\mathrm{en}}\operatorname{\!-Mod}\nolimits\to P^{\mathrm{en}}\operatorname{\!-Mod}\nolimits$ sending $M$ to $k[V^{I}]\otimes M$, where $k[V^{I}]$ is the regular $k[V^{I}]^{\mathrm{en}}$-module and $P^{\mathrm{en}}$ is decomposed as $P^{\mathrm{en}}=k[V^{I}]^{\mathrm{en}}\otimes P_{I}^{\mathrm{en}}$. We obtain a fully faithful monoidal functor