\[
  \nu/\mu \longrightarrow \operatorname{Aut}(\mu)\simeq(\mathbf{Z}/h\mathbf{Z})^{*}
\]

\[
  \underline{\mathfrak{L}}^{\mathrm{rss}}\simeq \mathrm{Kill}\times_S
  \underline{t}^{\mathrm{rss}}
\]

\[
  \underline{\mathfrak{g}}^{\mathrm{rss}}\longrightarrow
  \mathfrak{J}=\underline{t}^{\mathrm{rss}}/W
\]

\[
  \underline{\mathfrak{g}}'=\underline{\mathfrak{g}}\times_{\mathfrak{J}}\mathfrak{J}'
\]

\[
  \struck{\ill{}}\ T_D\simeq T\times_S D, \quad\text{donc}\quad
  \hat T_D\simeq M_D,\ \text{où}\ M=\hat T\ \text{réseau sur}\ S.
\]

\[
  W_K\simeq W\times_S K
\]

\[
  G\xrightarrow{\ f\ } I
\]

\[
  f\,|\,T_0 = \text{morph.\ can.}\ T_0\to T_0/W_0\simeq I .
\]

\[
  p\colon \mathfrak{B}\longrightarrow G'
\]

\[
  \tilde f\colon\mathfrak{B}\to T,\qquad h\colon\mathfrak{B}\to G
  \qquad\text{tels que }\ \pi\tilde f=f h
\]

\[
  \begin{cases}
  h(g,B)=g\\
  \tilde f(g,B)=g\in B/B_u(-)\simeq T(-),
  \end{cases}
\]

\[
  \operatorname{Pic}(\mathfrak{B})\xrightarrow{\ \sim\ }
  \operatorname{Pic}(\mathfrak{B}^{\mathrm{reg}})\xleftarrow{\ \sim\ }
  \operatorname{Pic}(G'^{\mathrm{reg}})
  \qquad
  \operatorname{Pic}(D)\xrightarrow{\ \wr\ }\operatorname{Pic}(\mathfrak{B})
\]

\[
  \underline{\mathrm{Pic}}_{\mathfrak{B}/S}\xrightarrow{\ \sim\ }
  \underline{\mathrm{Pic}}_{\mathfrak{B}^{\mathrm{reg}}/S}\xleftarrow{\ \sim\ }
  \underline{\mathrm{Pic}}_{G'^{\mathrm{reg}}/S}
  \qquad
  M\to P\simeq\underline{\mathrm{Pic}}_{D/S}\xrightarrow{\ \wr\ }
  \underline{\mathrm{Pic}}_{\mathfrak{B}/S}
\]

\[
  \begin{aligned}
  &G^{\mathrm{rs}}\subset G^{\mathrm{reg}} && \text{ouvert des pts de $G$ qui
  sont réguliers semi-simples}\\
  &G'^{\mathrm{rs}}\subset G'^{\mathrm{reg}},\ \
  \mathfrak{B}^{\mathrm{rs}}\subset\mathfrak{B}^{\mathrm{reg}} && \text{images
  inverses,}
  \end{aligned}
\]

\[
  \mathfrak{B}^{\mathrm{rs}}\xrightarrow{\ \sim\ }G'^{\mathrm{rs}}
\]

\[
  \struck{\underline{\mathrm{Iso}}(T)\simeq\underline{\mathrm{Iso}}(T')}
  \ \text{donc}
\]

\[
  0\to H^1(\Gamma,\underbrace{H^0(T',\underline{O}^{*}_{T'})}_{M\times k'^{*}})
  \to\operatorname{Pic}(T)\to
  H^0(\Gamma,\underbrace{H^1(T',\underline{O}^{*}_{T'})}_{=0})
\]

\[
  \operatorname{Pic}(T)=H^1(\Gamma,M)\times
  \underbrace{H^1(\Gamma,k'^{*})}_{=0\ \text{H.~90}}
  =H^1(\Gamma,M)=H^1(k,\underline{M})
\]

\[
  \operatorname{Pic}(D)\longrightarrow\operatorname{Pic}(\mathfrak{B})
\]

\[
  0\to\operatorname{Pic}(D)\to\operatorname{Pic}(\mathfrak{B})\to
  H^1(\Gamma,M_{\bar\xi}),
  \qquad
  \operatorname{Pic}(\mathfrak{B}_\eta)\simeq\operatorname{Pic}(T_\eta)
  \simeq\operatorname{Pic}(T_\xi)
\]

\[
  \operatorname{Pic}(S)\xrightarrow{\ \sim\ }\operatorname{Pic}(G')
\]

\[
  \operatorname{Pic}(G')\hookrightarrow
  \operatorname{Pic}(G'^{\mathrm{reg}})\,[\simeq
  \operatorname{Pic}(\mathfrak{B}^{\mathrm{reg}})\leftleftarrows
  \operatorname{Pic}(\mathfrak{B})]
\]

\[
  \cdots\to\operatorname{Pic}(S)\to\operatorname{Pic}(G')\longrightarrow
  \underbrace{H^1(\xi,M)}_{\text{groupe fini}}
\]

\[
  0\to\operatorname{Pic}(S)\to\operatorname{Pic}(G)\to\operatorname{Pic}(G_\xi)
\]

\[
  \operatorname{Coker}\bigl(\operatorname{Pic}(S)\to
  \operatorname{Pic}(G^{\mathrm{reg}})\bigr)\ \text{est fini (et $=$ nul si
  $G_\xi$ est simplement connexe et déployé)}
\]

\[
  \operatorname{Lie}P=\sum_{k\geqslant 0}\mathfrak{g}(k),\qquad
  \operatorname{Lie}U=\sum_{k\geqslant 1}\mathfrak{g}(k),\qquad
  \operatorname{Lie}L=\mathfrak{g}(0).
\]

\[
  g\longmapsto\operatorname{ad}(g).x
\]

\[
  g\longmapsto\operatorname{ad}(g).e
\]

\[
  \mathrm{ST}_c\to\underline{\varnothing}_c,\qquad
  (\mathrm{CL})_{c'}\to\mathrm{Par}_{c'},
\]

\[
\begin{cases}
  N \cap N(D) = \mathfrak{z}(\mu) \cap N(D) \\
  N \cap T' = \mathfrak{z}(\mu) \cap T' \\
  N \cap N' = N(\mu) \cap N'
\end{cases}
\]

\[
  \det \begin{pmatrix}
    n_1 + 1 & n_2 & n_3 & \cdots & n_r \\
    n_1 & n_2 + 1 & n_3 & \cdots & n_r \\
    \vdots & & & & \\
    n_1 & n_2 & & \cdots & n_r + 1
  \end{pmatrix} = \textstyle\sum n_i + 1
\]

\[
  N(T) \cap N(D) = \struck{T \cap \mathrm{Norm}(\mu)}\ N(\mu) \cap N(D)
  \qquad (\mu \subset T \text{ par } \gamma), \quad
  \mathbb{Z}/h\mathbb{Z} \simeq \mu_h .
\]

\[
  \mathfrak{g}_\psi \simeq \Delta \quad (\text{ou } \mathfrak{g}_\psi \simeq
  \Delta^{-1}, \text{ car } \Delta^{\otimes 2} \simeq \mathbf{1}),
\]

\[
  (T, T', B, B') \quad \text{avec} \quad
  \begin{cases}
    T, T' \text{ tores maximaux en apposition} \\
    B, B' \text{ des Borels contenant } T, T' \text{ respectivement}
  \end{cases}
  \qquad [\,(h,p) = 1\,]
\]

\[
  \begin{cases}
    W \text{ ordre du groupe de Weyl} \\
    h \text{ ordre de l'élt de Coxeter} \\
    \varphi \text{ l'indicatrice d'Euler} \\
    z \text{ ordre du centre}
  \end{cases}
\]

\[
  e_{\alpha_1} + \cdots + e_{\alpha_r} + e_{-\psi} = e
\]

\[
  (Q \times Q')/\Phi .
\]

\[
  \sum_{1}^{r} \lambda_{i}\, e_{\alpha_{i}} + \mu\, e_{-\psi},
\]

\[
  \sum_{\substack{\alpha \in R\\ o(\alpha) = j}} \mathfrak{g}_{\alpha}
  \;+\; \sum_{\substack{\alpha \in R\\ o(\alpha) = j-h}} \mathfrak{g}_{\alpha}.
\]

\[
  K : \mu_{h} \longrightarrow T
\]