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\ldots < n_k = n f = (V_1, \ldots, v_n) $ \\ +\underline{Beachte:} $V_i = V_{i-1} \oplus y_i, y_i := < e_{n_i+1}, \ldots, e_{n_i} > $ \\ + $ v_i = n_i, v_2 = n_2 - n_1, v_3 = n_3 - n_2, \ldots, v_k = n_k - n_{k-1}, v_i = \dim_F(y_i) $ \\ +$L_f = \set{\text{Matrix mit von 0 verschiedenen Blöcken der Größen }v_i \times v_i\text{ auf der Diagonale aus } G}$ \\ +... + +$ v = (v_1, \ldots, v_k) \vDash n $ Wir schreiben $P_v = U_v \rtimes L_v$ anstatt $P_f, L_f, U_f$. ($\nu = (n) \Rightarrow P_{(n)} = G = L_{(n)}, U_{(n)} = (1)$) \\ +\underline{Sonderfall:} $v = (1^n) = (1, \ldots, 1) \vDash n, P_v = B = u \cdot T$, Borus. + +\begin{definition} +Eine $n \times m$-Matrix $A$ heißt (untere) \underline{unitriangulär}, falls folgendes gilt: $A_{ii} = 1, A_{ij} = 0 \forall i < j$ (allgemeine untere Dreiecksmatrix mit 1 auf der Diagonale), analog obere. +\end{definition} + +\begin{lemma} % 2.2.5 +Sei $V = F^n, f = (W_1, \ldots, W_n)$ mit $W_i = < e_1, \ldots, e_i >, \xi = (e_1, \ldots, e_n)$ natürliche Basis von $V$. \\ +Sei $X$ ein $B$-invarianter Unterraum von $V$, d.h. $bx \in X \forall b \in B, x \in X (\Rightarrow bX = X$. \\ +Dann ist $W = W_i$ für ein $1 \leq i \leq n$. +\end{lemma} + +\begin{bew} +Sei $1 \leq k \leq n$ minimal mit $X \subseteq W_k$. Wir zeigen: $X = W_k$. \\ +Dann existiert ein $x = \sum\limits_{i=1}^k \alpha_i e_i$ mit $\alpha_k \neq 0$ (da $k$ minimal). \\ +... $ \exists b \in B: b \cdot x = e_k \Rightarrow e_k \in X$ \\ +Nun ist $(E + e_{k-1,k}) e_k = e_k + e_{k-1} \in X \Rightarrow e_{k-1} \in X$, analog $\forall i \leq k: e_i \in X \Rightarrow W_k \subseteq X \Rightarrow W_k = X$. +\end{bew} + +\begin{satz} % 2.2.6 +Sei $B \leq H \leq G$. Dann is $H$ eine Standardparabolische Untergruppe, d.h. $ \exists \nu \vDash n, H = P_{\nu} $ +\end{satz} + +\begin{bew} +Sei $X$ ein $H$-invarianter Unterraum von $V$. Dann ist $X$ auch $B$-invariant, wil $B \subseteq H$. +Also gibt es ein $1 \leq i \leq n: X = W_i = < e_1, \ldots e_i >, \xi = (e_1, \ldots e_n)$ natürliche Basis wegen 2.2.5. \\ +Seien $W_{\alpha_1}, \ldots, W_{\alpha_r}$ mit $1 \leq \alpha_1 < \alpha_2 < \ldots < \alpha_r = n$ genau die $H$-invarianten +Unterräume von $V$. $\underline{\alpha} = (\alpha_1, \ldots, \alpha_r), W_{\alpha_i} = < e_1, \ldots, e_{\alpha_i} > $ +$ F_{\underline{\alpha}} = (W_{\alpha_1}, \ldots, W_{\alpha_r})$ ist $H$-invariante Fahne von Dimensionstyp $\underline{\alpha}$. \\ +Sei $\mu_1 = \alpha_1,_2 = \alpha_2 - \alpha_1, \ldots, \mu_r = \alpha_r - \alpha_{r-1}$ +$ \Rightarrow \mu = (\mu_1, \ldots, \mu_r) \vDash n $. \\ +$\Stab_G(F_{\underline{\alpha}}) = P_{\mu}, H \leq P_\mu$ \\ +Zu zeigen: $H = P_{\mu}$. \\ +\underline{Spezialfälle} +\begin{enumerate}[1.)] + \item $r = 1, \mu = (n), P_{\mu} = G$ \\ + Zu zeigen: $H = G$ \\ + $ < H e_1 > = H$-invarianter Unterraum $\Rightarrow V = < H e_1 >$ + $ \Rightarrow \exists h \in H: he_1 = \alpha_1 e_1 + \ldots + \alpha_n e_n$ mit $\alpha_n \neq 0$ \\ + Bruhat-Zerlegung: $h \in B w B, \exists w \in W $ \\ + 2.1.9 und 2.1.5 $\Rightarrow$ Für $g \in BwB$ hat $g$ als Matrix die Form ... \\ + d.h. hier für $h \in BwB: w(1) = n$ \\ + Beachte: Wegen $B \subseteq H$ ist $h = b_1 w b_2 \Rightarrow w = b_1^{-1} h b_2^{-1} \in H$ \\ + Sei $1 < j \leq n$ mit $w(j) = 1$ (Ohne Einschränkung $n \geq 2$) \\ + Dann ist $X_{1j} = \set{x_{1j}(\alpha) | \alpha in F} \leq B \leq H$ \\ + $X_{n1} = X_{w(1)w(j)} = wX_{1j}w^{-1} \in H$ (mit 2.1.4) \\ + Sei $1 \leq i < m < n \Rightarrow X_{im} \subseteq B \subseteq H $ \\ + Dann ist $X_{nm}(\alpha) = [x_{n1} (\alpha), x_{1m}(1) ] \in H \forall \alpha \in F$ (mit 2.1.4) \\ + $ \Rightarrow X_{nm} \subseteq H$ \\ + $ \forall 1 \leq < n: x_{i1}(\alpha) = [x_{in}(\alpha, x_{n1}(1)] \in H \Rightarrow X_{i1} \subseteq H$ \\ + $ \forall 1 < i,m \leq n, i \neq m: x_{im}(\alpha) = [ x_{i1}(\alpha), x_{1m}(1) ] \in H \Rightarrow X_{im} \subseteq H$ \\ + Wir haben gezeigt $X_{ij} \subseteq H \forall 1 \leq i, j \leq n, i \neq j$ \\ + Da $T \subseteq B \subseteq H \Rightarrow H = G$. + \item $r = 2, I_{\underline{\alpha}} = W_m \lneq V = W_n$ \\ + Wir wissen schon: $H \leq P_{\mu}, \mu = (m, n - m) \vDash n$ \\ + Klar: $U_\mu \subseteq B \subseteq H, P_\mu = U_\mu \rtimes L_\mu$, es genügt also zu zeigen: $L_\mu \subseteq H$. \\ + $L \cong \GL_m(F) \times \GL_{n-}(F)$ \\ + $\GL_m(F) = < \text{Diagonalmatrizen in } \GL_m(F) \text{ und } x_{ij}(\alpha) >$, analog $\GL_{n-m}$ \\ + Es genügt also zu zeigen: $X_{ij} \in H \forall 1 \leq i, j \leq m, i $ und $m + 1 \leq i, j \leq n$ \\ + Sei $X_1 = < H e_1 > = H$-invarianter Unterraum von $W_m \Rightarrow X_1 = W_m$ \\ + D.h $\exists h \in H, h e_1 = \alpha e_1 + \ldots \alpha_m e_m$ mit $\alpha_m \neq 0$. \\ + Sei $w \in W$ mit $h \in BWB$. Wie oben folgt aus 2.1.9 und 2.1.5 $w(1) = m$ und daher $X_{m1} \subseteq H$. \\ + Beachte: $w \in H \Rightarrow w^{-1}(1) = j \leq m$ \\ + $X_{m1} = X_{w(1)w(j)} = w X_{1j} w^{-1} \in H$. \\ + Kommutatoren wie im ersten Spezialfall $\Rightarrow X_{ij} \subseteq H \forall 1 \leq i, j \leq m \Rightarrow \GL_m(F) \subseteq H$. \\ + $$ ebenfalls $H$-invariant $\Rightarrow \exists h \in H: h e_{m+1} = \alpha_{m+1} e_{m+1} + \ldots + \alpha_n e_n$ mit $\alpha_n \neq 0$. \\ + Es folgt analog wie eben $X_{ij} \subseteq H \forall m+1 \leq i, j \leq n \Rightarrow \GL_{n-m}(F) \subseteq H$. \\ + $ \Rightarrow P_\mu \subseteq H \Rightarrow H = P_\mu$ +\end{enumerate} +Übung: Allgemeiner Fall. +\end{bew} + \end{document}