用户: 香蕉三千/Sh(G)

Let $G$ be a connected reductive group over a $p$-adic field $F$.

Let $\breve{F}$ be the completion of the maximal unramified extension of $F$, $\Gamma$ be the absolute Galois group of $F$, and $\sigma \in \Gamma$ be the Frobenius of $F$.

Let $G^{qc}$ be its quasi-split inner form over $F$ and fix an inner twisting $G_{\breve{F}}\cong G_{\breve{F}}^{qc}$.

Choose a maximal split torus $A$ of $G^{qc}$, we have $A\subseteq T\subseteq B=TU\subseteq G^{qc}$.

Absolute root datum $(X^{\ast }(T),\Phi ,X_{\ast }(T),{\Phi }^{\vee })$.

Relative root datum $(X^{\ast }(A),{\Phi }_0,X_{\ast }(A),{\Phi }_0^{\vee })$.

${\Phi }^{+}$ the positive roots, $\Delta$ the simple roots of $T$ with respect to $B$.

${\rho }_G=1/2{\sum }_{a\in {\Phi }^{+}}a.$

${\pi }_1(G)={\pi }_1(G^{qc})=X_{\ast }(T)/{\Phi }^{\vee }$.

$B(G)=G(\breve{F})/h\sim gh\sigma (g{)}^{-1}$.

Newton map $\nu :B(G)\to X_{\ast }(A{)}_{\mathbb{Q},dom}$.

Kottwitz map $\kappa :B(G)\to {\pi }_1(G{)}_{\Gamma }$.

$X_F$ the FF curve over $F$.

Start with $b\in B(G)$. We have a $G$-bundle ${\mathcal{E}}_b$ on $X_F$ and $J_b$

We assume

定义 0.1. $$\kappa (b)={\mu }^{♭},\nu (b)\le \tilde{\mu }.$$

Here $\mu :{\mathbb{G}}_m\to G$ over $\bar{F}$ a dominant cocharacter, with image ${\mu }^{♭}\in {\pi }_1(G{)}_{\Gamma }$.

Denote by $E$ the reflex field of the conjugacy class of $\mu$. Then $\tilde{\mu }\in X_{\ast }(A{)}_{\mathbb{Q},dom}$ is the $\mathrm{Gal}(E/F)$-average of $\mu$.

定义 0.2. $\mathrm{Sh}(G,b)\to \mathrm{Spd}\breve{E}$ is the moduli space parametrizing modifications of $G$-bundles ${\mathcal{E}}_b⇢{\mathcal{E}}_1$ on $X_F$ meromorphy bounded by $\mu$.

We have a Satake sheaf ${\mathcal{S}}_{\mu }$ on $\mathrm{Sh}(G,b)$.

命题 0.3. $\dim \mathrm{Sh}(G,b)=⟨2{\rho }_G,\mu ⟩.$

证明.

$\square$

命题 0.4. $$j_b^{\ast }{T}_{{\mu }^{-1}}{j}_{1!}{\mathcal{F}}_{\pi }=R{\Gamma }_c(\mathrm{Sh}(G,b))[\pi ].$$