Fixed-point schemes and centralizers of linearly reductive actions

Let a linearly reductive algebraic group $H$ fix $x\in X(\bar{k})$, write $A={\mathcal{O}}_{X_{\bar{k}},x}$ with maximal ideal $\mathfrak{m}$, and let $V=\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. Each jet $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^n$ is a finite-dimensional rational $H$-module. Complete reducibility [F2] makes the projection between successive jets split equivariantly. Starting with the identity on $V$, choose compatible equivariant lifts $V\to \mathfrak{m}\mathrm{/}{\mathfrak{m}}^n$ recursively; the existing AC premise permits this countable choice. Their inverse limit gives equivariant formal parameters in $\widehat{A}$. A cotangent basis identifies $\widehat{A}$ with $\bar{k}[\text{\hspace{0.17em}⁣}[V]\text{\hspace{0.17em}⁣}]$: the associated-graded polynomial isomorphism [F3] lifts degree by degree to a unique complete coordinate isomorphism. Thus the formal $H$-action is linear on these chosen parameters. This uses the full scheme action on finite jets, not merely $H(\bar{k})$, and allows nonsmooth linearly reductive $H$.

Decompose $V=V_0\oplus V_1$ into its trivial and nontrivial simple summands. The formal fixed ideal is generated by all coefficients of the differences between the action coaction and identity on parameters. These coefficients span $V_1$: their quotient is the largest trivial quotient of $V$, which is exactly $V_0$ by complete reducibility. On the quotient power-series ring in $V_0$ every parameter is fixed, so all higher differences vanish too. Hence the completed fixed local ring is $\bar{k}[\text{\hspace{0.17em}⁣}[V_0]\text{\hspace{0.17em}⁣}]$. Exactness of completion [F3] identifies it with completion of the actual fixed local ring. Its associated graded is polynomial, and the converse regularity criterion [F3] makes that local ring regular. This proves smoothness at every geometric fixed point, and therefore smoothness of $X^H$. Its tangent space is the invariant tangent space by the same linear coefficient equations. In particular it applies both to smooth linearly reductive $G$ and to arbitrary multiplicative-type $H$.

For smooth $G$ and dense $S\subseteq G(k)$, the subset is schematically dense and remains so after field extension. The closed stabilizer of every geometric point of $X^S$ contains $S$, hence contains $G$ scheme-theoretically. On each finite jet the action-coefficient equations vanish on $S$ exactly when they vanish on $G$, by the same schematic density after tensoring with the coefficient algebra. Thus the fixed ideals are equal, giving $X^S=X^G$ as schemes, not only as geometric point sets. step 2.1 proves their smoothness.

For a linearly reductive $H$ acting by automorphisms on smooth $G$, step 2.1 makes $G^H$ smooth. For conjugation this fixed functor is precisely the scheme centralizer $C_G(H)$. Now let $H$ be of multiplicative type, possibly nonsmooth. Its linear reductivity [F2] proves smoothness of $C=C_G(H)$. Put $N=N_G(H)$. Its connected identity component acts on $H$ by conjugation; rigidity [F4] makes that action the identity on every base algebra, since its value at the identity is the identity. Thus $N^{\circ }\subseteq C$, giving $N^{\circ }=C^{\circ }$ as subgroup schemes. The smoothness of $C^{\circ }$, and translation of the neutral component after algebraic closure, prove that $N$ is smooth. No statement that normalizing $H^{\circ }$ is the same as centralizing $H$ is used.

If $g$ is semisimple over algebraically closed $k$, [F5] places the reduced closure $G_0$ of its powers in a diagonal matrix group. It is a smooth diagonalizable subgroup, hence linearly reductive. Its powers are dense, so step 3.1 gives $X^g=X^{G_0}$, and step 2.1 gives smoothness. The conclusion is independent of the chosen faithful representation.

If $H$ is smooth, its geometric points are schematically dense by [F4]. Commuting with, or conjugating onto itself, those points is then exactly the corresponding geometric centralizer or normalizer condition for the subgroup variety $H$; a closed smooth subgroup with that geometric point set is unique, since smoothness gives reducedness and reduced closed subschemes with equal geometric points coincide. These are the last pointwise identifications in the Statement. For nonsmooth $H$ they are omitted: the displayed ${\mu }_p$ example has trivial geometric point set but its two distinct weight characters on the standard module give diagonal scheme centralizer. Thus all scheme-smoothness claims are preserved while the pointwise boundary is exact.