Cocharacter limit subgroups

The affine graded concentrator construction [F1] represents $P=P_G(\lambda )$ and the identity concentrator $U=U_G(\lambda )$ as closed subschemes of $G$. Conjugation acts by group automorphisms, so the limit of a product or inverse is the product or inverse of the limits, on every base algebra. Thus $P$ is a subgroup and its limit map $p:P\to G$ is a homomorphism. Its image lies in the fixed subgroup $Z=Z_G(\lambda )$, since a limit at zero is fixed; $p{\mathrm{\mid }}_Z=\mathrm{id}$ and $U=\ker p$ is normal. Limits in both directions force every nonzero graded coefficient to vanish, giving $P_G(\lambda )\cap P_G(-\lambda )=Z$ scheme-theoretically. Fixed smoothness [F3] gives smooth $Z$, while [F1] applied to smooth $G$ with targets $G$ and $\{e\}$ gives smooth $P,U$. Their smooth geometric-point models are unique by [F1].

Embed $G$ in $\mathrm{G}\mathrm{L}(V)$ by [F2] and choose a weight basis in which $\lambda (t)=\mathrm{diag}(t^{m_1},\dots ,t^{m_n})$, $m_1\ge \cdots \ge m_n$. In $\mathrm{G}\mathrm{L}(V)$, conjugation multiplies entry $x_{ij}$ by $t^{m_i-m_j}$. Thus $P_{\mathrm{G}\mathrm{L}(V)}$ has zero entries when $m_i-m_j<0$, $Z_{\mathrm{G}\mathrm{L}(V)}$ has zero entries for nonzero differences, and $U_{\mathrm{G}\mathrm{L}(V)}$ has identity diagonal blocks and zero entries unless $m_i-m_j>0$. These descriptions hold over every algebra. The corresponding groups for $G$ are their scheme intersections with $G$, since the orbit limit in the ambient group lies in closed $G$. Applying the intersection formula for Lie and the dual-number entry calculation gives $\mathrm{Lie}P={⨁}_{r\ge 0}{\mathfrak{g}}_r$, $\mathrm{Lie}Z={\mathfrak{g}}_0$, and $\mathrm{Lie}U={⨁}_{r>0}{\mathfrak{g}}_r$.

Multiplication $U⋊Z\to P$ has the explicit inverse $g\mapsto (gp(g{)}^{-1},p(g))$. The first component has limit identity and the second is fixed; both maps are scheme morphisms and satisfy the inverse identities on every algebra. This proves the semidirect product as a group-scheme isomorphism, without inferring it merely from tangents and geometric points.

Put $U^{-}=U_G(-\lambda )$. The weight-block equations give $U^{-}\cap P=1$ on every algebra. Let the smooth group $U^{-}\times P^{\mathrm{o}\mathrm{p}}$ act on $G$ by $(u,p)\cdot g=ugp$; its scheme stabilizer at identity is therefore trivial. After algebraic closure, [F4] identifies its orbit map $\mu :U^{-}\times P\to G$ with an isomorphism onto a locally closed orbit. Its differential at identity is addition ${⨁}_{r<0}{\mathfrak{g}}_r\oplus {⨁}_{r\ge 0}{\mathfrak{g}}_r\to \mathfrak{g}$, an isomorphism by step 1.2. Between these smooth schemes this is the étale criterion at identity (the invertible Jacobian calculation in Milne13.33's proof). Translations by the acting group carry this calculation to every point of the domain; hence the locally closed orbit immersion is étale and therefore open. Open immersion descends along the faithfully flat field extension, so $\mu$ is the asserted open immersion over $k$. The graded limit action extends to ${\mathbf{A}}^1\times U\to U$ and sends $\{0\}\times U$ to identity. Over the algebraic closure every point is connected to identity by that affine-line morphism, so $U$ is geometrically connected, even for disconnected $G$. Its weight-block matrices lie in an upper-unitriangular group, making it unipotent by [F5].

If $G$ is reductive, [F3] makes $Z=C_G(\lambda ({\mathbf{G}}_m))$ reductive. By step 2.1 the quotient $P\mathrm{/}U$ is $Z$, and $U$ is smooth connected normal unipotent by step 2.2. Thus $U\subseteq R_u(P)$; the image of $R_u(P)$ in reductive $Z$ is a smooth connected normal unipotent subgroup and is trivial. Consequently $R_u(P)=U$, proving the full reductive clause. All preceding assertions allow disconnected smooth affine $G$.