Parabolic subgroups and Levi decomposition

Choose an integral cocharacter $\lambda$ whose simple-root pairings are zero on $I$ and strictly positive on $\mathrm{\Delta }\setminus I$: take the sum of rational dual vectors there and clear denominators as in the standard Levi lemma. Every root has simple coefficients of one sign, so $⟨\beta ,\lambda ⟩=0$ exactly for $\beta \in {\mathrm{\Phi }}_I$, and the positive pairings occur precisely on ${\mathrm{\Phi }}^{+}\setminus {\mathrm{\Phi }}_I$. By [F2], the fixed group $Z_G(\lambda )$ and $L_I=C_G(T_I)$ are smooth connected reductive with the same root spaces and torus. The image of $\lambda$ lies in $T_I$, so $L_I\subseteq Z_G(\lambda )$; equal dimension makes them equal. The cocharacter kernel $U_G(\lambda )$ is smooth connected unipotent with just the positive-pairing Lie weights. Its Lie algebra is contained in $\mathrm{Lie}U$, hence [F4] puts it in $U$. The root-coordinate theorem [F3] identifies it with the ordered product $U_I$ of the root groups in ${\mathrm{\Phi }}^{+}\setminus {\mathrm{\Phi }}_I$, in any order. Therefore $P_G(\lambda )=U_I⋊L_I$ and its unipotent radical is exactly $U_I$, since $L_I$ is reductive. It contains $B$ by the nonnegative pairings of all positive roots, and the geometric Borel/properness criterion [F3] makes it parabolic. Define this smooth group to be $P_I$.

A normalizer representative $n_w$ belongs to $P_G(\lambda )$ exactly when $w\lambda =\lambda$. Its conjugation orbit is $(\lambda -w\lambda )(t)n_w$, lying in the closed translate $T{n}_w$; a nontrivial cocharacter of a torus has no limit at zero inside that torus. The stabilizer of this dominant $\lambda$ in $W$ is $W_I$: if nontrivial $w$ fixes it, choose a left descent $s_{\alpha }$ so that $w^{-1}\alpha$ is negative by [F4]. Then $0\le ⟨\alpha ,\lambda ⟩=⟨w^{-1}\alpha ,\lambda ⟩\le 0,$ so $\alpha \in I$. The reflection $s_{\alpha }$ fixes $\lambda$ and shortens $w$; induction gives $w\in W_I$. Conversely every generator of $W_I$ fixes $\lambda$. Since $P_I$ contains $B$, a Bruhat cell is wholly contained in it exactly when its representative is, so [F1] gives $P_I={⨆}_{w\in W_I}C(w)$. The same argument shows that any dominant cocharacter with these zero simple pairings gives this $P_I$.

Let $P\supseteq B$ be any smooth closed subgroup. Over an algebraic closure put $W_P=\{w:n_w\in P\}$. It is a subgroup, because representatives multiply modulo $T\subseteq B$, and [F1] makes $P$ the union of its cells. Set $J=\{\alpha \in \mathrm{\Delta }:s_{\alpha }\in W_P\}$. For $w\in W_P$, both $B$ and $n_{w}B{n}_w^{-1}$ lie in $P$. Thus for each positive inversion root $\beta \in {\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}$, both $U_{\beta }$ and $U_{-\beta }$ lie in $P$. Their rank-one elementary products give a representative of $s_{\beta }$ in $P$. For a reduced expression of $w$, the inversion roots ${\beta }_j$ in [F4] therefore give $s_{{\beta }_j}\in W_P$. The first is $s_{{\alpha }_1}$; inductively, conjugating $s_{{\beta }_j}$ by the earlier simple factors puts $s_{{\alpha }_j}$ in $W_P$. Hence all letters lie in $J$, so $W_P=W_J$. The smooth closed groups $P$ and $P_J$ have the same geometric cells and are reduced, hence are equal; descent gives equality over $k$. In particular this classifies every smooth parabolic containing $B$, with no unsupported assumption that a general $P$ was already a limit group.

The subset $I$ is recovered by the negative simple root groups in $P_I$, or equivalently the simple reflections in $W_I$. Thus the classification is unique. Cocharacters with all positive-root pairings nonnegative give the same classification by step 2.1. Step 1.1 proves the complete ordered unipotent-radical and Levi decomposition in(b). For $I=\mathrm{\varnothing }$, the fixed group is $T$ and all positive root groups occur, giving $P_I=B$; for $I=\mathrm{\Delta }$, there are no positive pairing weights and the limit group is $G$. The torus case $\mathrm{\Delta }=\mathrm{\varnothing }$ satisfies both endpoints with $B=G$.

Over an algebraically closed field every smooth parabolic contains a Borel by [F3]. Conjugate that Borel to $B$ and apply step 3.1 to obtain a standard $P_I$. If $g{P}_I{g}^{-1}=P_J$, the Borels $gB{g}^{-1}$ and $B$ of the smooth connected affine group $P_J$ are conjugate by some $p\in P_J(k)$, by [F3]. Hence $pg$ normalizes $B$, so $pg\in B$ by its self-normality in [F3]. Therefore $g\in P_J$, forcing $P_I=P_J$ and $I=J$. This proves the full conjugacy and uniqueness in(c). Nonsmooth parabolics remain in the general conditional Definition: the Frobenius example has a nilpotent lower-entry condition $c^p=0$ and the stated proper quotient, and is excluded only from this smooth classification.