Weight subgroups of a torus action

Decompose $\mathfrak{m}={⨁}_{\chi }{\mathfrak{m}}_{\chi }$ by [F1] and let $J_A$ be the ideal generated by ${\mathfrak{m}}_{\chi }$ for $\chi \mathrm{\notin }A$. It is a Hopf ideal. For $f\in {\mathfrak{m}}_{\chi }$, the reduced coproduct $\mathrm{\Delta }f-f\otimes 1-1\otimes f$ belongs to $\mathfrak{m}\otimes \mathfrak{m}$ and decomposes into terms of weights $\beta ,\gamma$ with $\beta +\gamma =\chi$. If $\chi \mathrm{\notin }A$, at least one of $\beta ,\gamma$ is outside $A$, so every term vanishes modulo $J_A$ in one factor. The antipode preserves weights and the augmentation ideal, and the counit kills it. Thus $C_A=\mathrm{Spec}(R\mathrm{/}{J}_A)$ is a closed $T$-stable subgroup, and its construction commutes with field extension. Using ${\mathfrak{m}}_{\chi }$, rather than all $R_{\chi }$, ensures that $1$ is never killed when $0\mathrm{\notin }A$.

Let $K$ be the closed subgroup generated by $H_1,H_2$. It can be constructed by taking the joint kernel in $O(G)$ of pullbacks under all finite word-product maps with factors $H_i$ (including inverses, which remain in $H_i$). The resulting quotient is jointly injected into their coordinate rings. Tensor joint injectivity follows by restricting a finite tensor expression to finite-dimensional coefficient spans; concatenating words and reversing them show the kernel is a Hopf ideal. After any field extension the word-source rings are reduced, so their jointly injected subalgebra is reduced; thus $K$ is geometrically reduced and smooth. Every word source is geometrically connected. Pullbacks of an idempotent of $O(K)$ are constants equal to its augmentation; joint injectivity therefore makes that idempotent constant. Thus $K$ is connected. The construction is $T$-stable.

Choose homogeneous lifts $f_1,\dots ,f_d\in \mathfrak{m}$ of a weight basis of $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. At identity, [F2] identifies the completed local ring of $G$ with $k[\text{\hspace{0.17em}⁣}[f_1,\dots ,f_d]\text{\hspace{0.17em}⁣}]$. A homogeneous element of augmentation weight $\chi$ has only nonconstant monomials of that same weight in its formal expansion: compare coefficients in each finite quotient by powers of the maximal ideal, using the equivariant parameter isomorphism. If $\chi \mathrm{\notin }A$, every such monomial involves a parameter whose weight is outside $A$, since $A$ is closed under addition. Conversely those outside-weight parameters themselves generate part of $J_A$. Hence the completed ideal of $J_A$ is exactly the ideal generated by the outside-weight parameters, and the completed local ring of $C_A$ is the power-series ring in the remaining parameters. Ideals in this Noetherian formal power-series ring are closed, so passage from the expansions to this ideal identity is legitimate. The quotient of the smooth local ring by these independent outside-weight parameters is smooth at identity by the coordinate Jacobian criterion in [F2]. Every generator of $J_A$ is zero in its completion, and completion injectivity [F2] makes it zero already in that quotient. Thus the localized $J_A$ is exactly this parameter ideal. Repeat after algebraic closure to conclude that $C_A$ is smooth at identity; translations and [F3] make it smooth everywhere. Its identity component $H_A=C_A^{\circ }$ is smooth connected and has the displayed Lie algebra.

Let $H$ be a $T$-stable smooth connected subgroup with Lie algebra contained in the selected weights. Its homogeneous formal parameters have weights in $A$. Restrict $f\in {\mathfrak{m}}_{\chi }$ with $\chi \mathrm{\notin }A$ to $H$; its expansion contains no nonconstant monomial, since all such monomial weights belong to $A$, and its constant is zero. Thus it vanishes in the completion at identity. Completion injectivity [F2] and geometric integrality [F3] show that it vanishes globally on $H$: localization of its integral coordinate ring at identity is injective. Therefore $H\subseteq C_A$, and connectedness gives $H\subseteq H_A$. If $H$ has exactly the selected Lie algebra, its dimension equals that of $H_A$ by smoothness; a closed proper subgroup of a geometrically irreducible smooth connected group has smaller dimension. Hence $H=H_A$, proving the full uniqueness and containment in(a), including empty and nonsaturated semigroups.

The fixed subgroup $G_{\alpha }=G^{T_{\alpha }}$ is smooth connected by [F4], with Lie weights precisely the rational multiples of $\alpha$, including zero. Apply the cocharacter theorem in $G_{\alpha }⋊T$ to $\lambda$. Its identity concentrator lies in the kernel $G_{\alpha }$ of projection to $T$, since conjugation leaves that projection unchanged. It is smooth connected unipotent and has exactly the strictly positive rational-$\alpha$ Lie weights by [F4]. It is $T$-stable, since the torus action commutes with $\lambda$, and equals $H_{(\alpha )}$ by(a). Now take a smooth $T$-stable $H$ with the asserted Lie containment, possibly disconnected. Its $T_{\alpha }$-fixed subgroup is smooth, and the identity concentrator there is a smooth connected subgroup of $U_{G_{\alpha }}(\lambda )$, with the same positive Lie space. Smoothness and dimension therefore make it the whole $U_{G_{\alpha }}(\lambda )$, proving containment in $H$. For $H={\alpha }_p\subseteq {\mathbf{G}}_a$ the smoothness premise fails and the stated counterexample confirms its necessity.

For any smooth connected affine $T$-group $L$, the weights occurring in its augmentation ideal are exactly the semigroup generated by its Lie weights. Indeed, a nonzero homogeneous function has a nonzero formal expansion by the injectivity in [F2] and integrality in [F3]; one of its nonconstant monomials expresses its weight as a sum of parameter weights. Conversely, any nonempty finite sum is realized by the product of the corresponding nonzero homogeneous parameter lifts, which is nonzero in the integral ring $O(L)$. Apply this to $K,H_1,H_2$. Any nonzero homogeneous augmentation function on $K$ has a nonzero pullback under some word map by joint injectivity. In the tensor product of the word-source coordinate rings, its weight is a sum of augmentation weights of those factors, with at least one nonconstant factor; it therefore belongs to the semigroup generated by the Lie weights of $H_1,H_2$. Conversely their Lie weights occur in $\mathrm{Lie}K$, since their Lie inclusions are injective, and therefore their entire semigroup lies in that of $K$. This proves(c). No equality of the generated Lie algebra with $\mathrm{Lie}K$ is asserted, and no strictly definite weight assumption or finite generating set of $A$ was used.