Lemma 6.1.3 (0P79) · ScalingStacks

0P79

Lemma 6.1.3. The maps $(f_{m^{\prime}m})$ define a twisted object $V=[\bigoplus V_{m},(f_{m^{\prime}m})]$ . There is an isomorphism of differential $(H_{r}\otimes H_{n})$ -modules

V\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{+}(r,n),\ (h\otimes h^{\prime})b_{E}\mapsto hf_{r}(\iota_{n}(h^{\prime}))T_{w_{E}}\text{ for }h\in H_{r}\text{ and }h^{\prime}\in H_{n}.

0P7A

Proof. The length property of the bijection $\beta$ above shows that the map of the lemma is an isomorphism of $(H_{r}\otimes H_{n})$ -modules. Since

d(T_{w_{E}})=\sum_{i\in E,\ j\in\{1,\ldots,r+n\}\setminus E,\ j<i}T_{w_{E}s_{i,j}},

it follows that the map of the lemma intertwines $d_{V}$ and the differential of $L^{+}(r,n)$ . The lemma follows. ∎