\[
\begin{array}{ll}
\mathbf{P}_1 \times \mathbf{P}_1 & \mathrm{PGL}_2 \times \mathrm{PGL}_2 \\[2pt]
\mathbf{P}_2 & \mathrm{PGL}_3 \\[2pt]
F_n \quad n \geq 2 & \bigl(\mathrm{GL}(2)\cdot \Gamma(\mathcal{O}(n))\bigr)/\mu_n
\end{array}
\qquad
\begin{array}{l}
(F_0 = \mathbf{P}_1 \times \mathbf{P}_1) \\[2pt]
(F_1 = \mathbf{P}_2 \text{ éclaté})
\end{array}
\]

\[
  \mathrm{Psaut}_S(X), \quad \mathrm{Baut}_S(X)
\]

\[
  \mathrm{Psaut}_k(K) ; \qquad
  \mathrm{Cr}_{n,k} = \mathrm{Baut}_k\bigl(k[t_1,\ldots,t_n]\bigr)
\]

\[
  \mathrm{Psaut}_k(K)(k) \simeq \mathrm{Aut}_k(K)^{\circ}, \qquad
  \mathrm{Lie}\bigl(\mathrm{Baut}_k(K)\bigr) \simeq \mathrm{Der}_k(K)^{\circ}
\]

\[
  G \longrightarrow \mathrm{Psaut}_k(X) \; ?
\]

\[
  G \times X \dashrightarrow X
\]

\[
  \left.
  \begin{array}{l}
    g, h \in G(S) \\
    x \in X(S)
  \end{array}
  \right\}
  \quad
  \begin{array}{l}
    h \cdot x \text{ défini} \\
    g(h \cdot x) \text{ défini}
  \end{array}
  \;\Longrightarrow\;
  \bigl(gh \cdot x \text{ défini et égal à } g(h\,x)\bigr)
\]

\[
  T \subset G \subset \mathrm{Baut}(T)
\]

\[
  \begin{array}{c}
    T \subset G \\
    \cup \\
    H
  \end{array}
  \qquad
  \left\{
  \begin{array}{l}
    T \times H \longrightarrow G \quad \text{immersion ouverte} \\
    \struck{\ill{}} \quad \bigcap \text{ conjugués de } H = e
  \end{array}
  \right.
\]

\[
  G \dashrightarrow T, \qquad (g, t) \mapsto f(gt)
\]

\[
  \Longrightarrow \quad f(tg) = t\,f(g)
\]

\[
  \begin{array}{c}
    T \subset G \\
    \subset G'
  \end{array}
\]

\[
  \exists\,!
  \left\{
  \begin{array}{l}
    x_\alpha : \mathbf{G}_a \xrightarrow{\ \sim\ } U \\
    \rho_\alpha : \mathbf{G}_m \longrightarrow T
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{l}
    t\,x_\alpha(\lambda)\,t^{-1} = x_\alpha\bigl(\alpha(t)\lambda\bigr) \\
    f\bigl(x_\alpha(\lambda)\bigr) = \rho_\alpha(1+\lambda)
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{l}
    \langle \rho_\alpha, \alpha \rangle = 1 \\
    H = x_\alpha(1)\,T\,x_\alpha(-1) \qquad (?)
  \end{array}
  \right.
\]

\[
  \mathfrak{g} = \mathfrak{g}_0 + \sum_{\alpha \in R} \mathfrak{g}^{\alpha},
  \qquad R \subset \mathrm{Hom}(T, \mathbf{G}_m)
\]

\[
  \mathfrak{g}_0 = \mathrm{Lie}\,T \quad (T = \mathrm{Cent}_G T), \qquad
  \dim \mathfrak{g}^{\alpha} = 1
\]

\[
  T_\chi = (\mathrm{Ker}\,\chi)^{\circ} \subset T, \qquad
  Z_\chi = \mathrm{Cent}(T_\chi)
\]

\[
  Z_\chi / T_\chi \hookrightarrow \mathrm{Aut}(\mathbf{P}^1) = \mathrm{PGL}(2)
  \qquad (\text{car } \mathbf{P}^1 \text{ de dim } 1)
\]

\[
    \left\{
    \begin{array}{l}
      \dim \mathfrak{g}_\alpha = 1 \\
      \exists\,! \; x_\alpha : \mathbf{G}_a \longrightarrow Z_\chi \subset G,
        \quad x_\alpha(\mathbf{G}_a) = U_\alpha, \;
        \mathrm{Lie}(U_\alpha) = \mathfrak{g}_\alpha \\
      \phantom{\exists\,! \;} t\,x_\alpha(\lambda)\,t^{-1} =
        x_\alpha\bigl(\alpha(t)\lambda\bigr) \\
      \rho_\alpha : \mathbf{G}_m \longrightarrow T, \quad
        f\bigl(x_\alpha(\lambda)\bigr) = \rho_\alpha(1+\lambda), \quad
        \langle \rho_\alpha, \alpha \rangle = 1
    \end{array}
    \right.
  \]

\[
    \mathrm{Lie}\,Z_\chi = \mathfrak{g}_0 +
      \sum_{\beta \in \mathbf{Q}\chi \,\cap\, R} \mathfrak{g}^{\beta}
  \]

\[
  G^{\circ} = L \cdot U, \qquad L \supset T
\]

\[
  \varphi_\alpha \begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}
    = x_\alpha(\lambda), \qquad
  \varphi_\alpha \begin{pmatrix} 1 & 0 \\ \lambda & 1 \end{pmatrix}
    = x_{-\alpha}(\lambda)
\]

\[
  \varphi_\alpha \begin{pmatrix} \mu & 0 \\ 0 & 1 \end{pmatrix}
    = \rho_\alpha(\mu), \qquad
  \varphi_\alpha \begin{pmatrix} 1 & 0 \\ 0 & \mu \end{pmatrix}
    = \rho_{-\alpha}(\mu)
\]

\[
  f\left(\varphi_\alpha \begin{pmatrix} a & b \\ c & d \end{pmatrix}\right)
    = \rho_\alpha(a+b)\,\rho_{-\alpha}(c+d)
\]

\[
  \varphi_\alpha \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot t
    = t \cdot \rho_\alpha\!\left(a + \frac{b}{\alpha(t)}\right)
      \rho_{-\alpha}\bigl(c\,\alpha(t) + d\bigr)
\]

\[
  w_\alpha \cdot t = \struck{\ill{}} \; s_\alpha(t)
    = t\;\alpha^{*}\bigl(\alpha(t)\bigr)^{-1}, \qquad
  \alpha^{*} = \rho_\alpha - \rho_{-\alpha}
\]

\[
  \left\{
  \begin{array}{l}
    \text{Si } M = \mathrm{Hom}(T, \mathbf{G}_m), \; \ill{} \text{ syst. de
      racines de } L \text{ est } \{M, R_s, \; \alpha \mapsto \rho_\alpha -
      \rho_{-\alpha}\} \\
    \phantom{\text{Si } M} R_s = R \cap (-R)
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{l}
    \mathrm{Norm}_{G^{\circ}}(T) = T \cdot W^{*} \\
    W^{*} \text{ engendré par les } w_\alpha \; (\alpha \in R_s), \quad
      R_s = R \cap (-R)
  \end{array}
  \right.
\]

\[
  \rho_{s_\alpha(\beta)} = s_\alpha(\rho_\beta), \qquad
  \mathrm{int}(w_\alpha)\,x_\beta = x_{s_\alpha(\beta)}
\]

\[
  \rho_\alpha = \rho_\beta \quad \text{ou} \quad
  \langle \rho_\alpha, \beta \rangle = \langle \rho_\beta, \alpha \rangle = 0
\]

\[
  [X_\alpha, X_\beta] = \bigl(\langle \rho_\alpha, \beta \rangle -
    \langle \rho_\beta, \alpha \rangle\bigr) X_{\alpha+\beta}
  \qquad \text{si } \alpha + \beta \neq 0
\]

\[
  [X_\alpha, X_{-\alpha}] = \rho_\alpha - \rho_{-\alpha}
\]