ScalingStacks

Proof. [02SN]

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Proof.

From the description of Dψ|V⁡(Λ)D_{\psi}|_{V(\Lambda)} and Proposition 4.37, we know that

degDψ⁡(V⁡(Λ))=k!​volM~​(Λ)⁡(Δψ,Λ).\deg_{D_{\psi}}(V(\Lambda))=k!\operatorname{vol}_{{\widetilde{M}}(\Lambda)}(\Delta_{\psi,\Lambda}).

Since

volM~​(Λ)(Δψ,Λ)=1[M(Λ):M~(Λ)]volM⁡(Λ)(Λ∗),\operatorname{vol}_{{\widetilde{M}}(\Lambda)}(\Delta_{\psi,\Lambda})=\frac{1}{[M(\Lambda):{\widetilde{M}}(\Lambda)]}\operatorname{vol}_{M(\Lambda)}(\Lambda^{\ast}),

the result follows from the definition of the multiplicity. ∎

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