Semisimplicity of rational representations descends along field extensions

Given: An algebraic group $G$ over $k$, a finite-dimensional rational representation $(V,r)$, a field extension $k^{\mathrm{\prime }}\supseteq k$, and the base changes $G_{k^{\mathrm{\prime }}}$ and $(V_{k^{\mathrm{\prime }}},r_{k^{\mathrm{\prime }}})$. Put $A=\mathrm{\Gamma }(G,{\mathcal{O}}_G)$; finite representation matrices have entries in $A$, whether or not $G$ is affine.

Base change of matrix coefficients. Base change of the morphism $r:G\to {\mathrm{GL}}_V$ defines the representation on $V_{k^{\mathrm{\prime }}}$, and preserves invariant subspaces. A finite-dimensional subspace $C\subseteq A$ remains linearly independent after base change: the restriction maps from $C$ to the rings of affine opens jointly detect zero, and finitely many suffice. Indeed choose a finite intersection of their kernels of minimal dimension; if it were nonzero, another restriction would lower its dimension. Thus $C$ injects into a finite direct sum of affine-open coordinate rings. Tensoring with $k^{\mathrm{\prime }}$ preserves this injection by finite coefficient comparison, and these rings become the coordinate rings of the base-changed affine opens by Affine fibre products are spectra of tensor products. Consequently $C{\otimes }_k{k}^{\mathrm{\prime }}\to \mathrm{\Gamma }(G_{k^{\mathrm{\prime }}},{\mathcal{O}}_{G_{k^{\mathrm{\prime }}}})$ is injective. In the affine case these are the usual comodule coefficient calculations of Rational representations and comodules of an affine group scheme; the argument does not require global affineness.

Hom representation and scalar extension. For finite-dimensional $W$, the space $M={\mathrm{Hom}}_k(V,W)$ with $(g\cdot f)(v)=g\cdot f(g^{-1}v)$ is a finite-dimensional rational representation of $G$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational). For nonaffine $G$ its matrices are still regular: they are the products of the representation matrix on $W$ and the inverse representation matrix on $V$, so the displayed action gives a morphism into ${\mathrm{GL}}_M$ directly. Choose a finite basis $(e_i)$ of $V$ with dual basis $(e_i^{\ast })$, and a finite basis $(w_j)$ of $W$. The rank-one maps $E_{ij}:v\mapsto e_i^{\ast }(v)w_j$ form a $k$-basis of $M$; after scalar extension, their corresponding maps on $V_{k^{\mathrm{\prime }}}$ with values in $W_{k^{\mathrm{\prime }}}$ form a $k^{\mathrm{\prime }}$-basis of ${\mathrm{Hom}}_{k^{\mathrm{\prime }}}(V_{k^{\mathrm{\prime }}},W_{k^{\mathrm{\prime }}})$. Thus the canonical map $M{\otimes }_k{k}^{\mathrm{\prime }}\to {\mathrm{Hom}}_{k^{\mathrm{\prime }}}(V_{k^{\mathrm{\prime }}},W_{k^{\mathrm{\prime }}})$ sending $f\otimes a$ to $a{f}_{k^{\mathrm{\prime }}}$ is an isomorphism (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$).

Equivariance is a finite linear system. In a finite basis $m_1,\dots ,m_N$ of $M$, write $g\cdot m_j={\sum }_i{a}_{ij}(g)m_i$ with $a_{ij}\in A$. For $f={\sum }_j{z}_j{m}_j$, fixedness is exactly ${\sum }_j{a}_{ij}{z}_j=z_i$ as regular functions on $G$, for every $i$. Let $C\subseteq A$ be the span of $1$ and all $a_{ij}$. Comparing coefficients in a finite basis of $C$ turns these identities into finitely many linear equations over $k$. By [F1] that coefficient basis stays independent on $G_{k^{\mathrm{\prime }}}$, so the base-changed equations express exactly $G_{k^{\mathrm{\prime }}}$-fixedness. Fixedness in the Hom action means equivariance of $f$. Adding the finite coordinate equations $f{\mathrm{\mid }}_W={\mathrm{id}}_W$ defines the affine solution set $S=\{f\in M:f\text{ is }G\text{-equivariant},f{\mathrm{\mid }}_W={\mathrm{id}}_W\}\mathrm{.}$ This uses the finite matrix interpretation of rationality; in the affine case it is the usual comodule condition (Rational representations and comodules of an affine group scheme, [F2]).

Linear systems over a field. Every finite matrix over a field is row equivalent to a matrix in reduced row echelon form, obtained by Gauss-Jordan elimination (Gauss–Jordan elimination reduces every finite matrix over a field to reduced row echelon form). Row operations are invertible and preserve solution sets over every extension field; a system in reduced row echelon form is solvable if and only if it has no row $(0\cdots 0\mid c)$ with $c\mathrm{\ne }0$, and when it is solvable, setting the free variables equal to $0$ and solving the pivot equations gives a solution with coordinates in the field generated by the coefficients, hence in $k$ when the coefficients lie in $k$.

Splitting implies semisimplicity. If every subrepresentation of a finite-dimensional rational representation $U$ is a direct summand, then $U$ is semisimple: choose a nonzero subrepresentation of minimal dimension, which is simple, split it off, and iterate on the complement of smaller dimension (Simple and semisimple rational representations).