The induced coordinate module E(lambda)

Let $(G,T)$ be a split reductive group over $k$, and let $B^0=B^{-}$ be a provided opposite Borel subgroup. For a provided character of $B^0$ whose restriction to $T$ is $\lambda \in X(T)$, define $E(\lambda )$ to be the $k$-subspace of $O(G)$ (The coordinate Hopf algebra of an affine group scheme) satisfying $f(gb)=f(g)\lambda (b^{-1})$ for every $k$-algebra $R$, $g\in G(R)$ and $b\in B^0(R)$. Here $\lambda$ on $B^0$ denotes that provided extension. The subspace is stable under the left regular action $(gf)(x)=f(g^{-1}x)$ and hence is a rational $G$-module (Rational representations and comodules of an affine group scheme). Its underlying subspace and this action are choice-free. In the source convention this is ${\mathrm{Ind}}_{B^0}^G(k_{\lambda })$.

Assuming AC (The Axiom of Choice) for the cited split-Borel structure, $B^0=U^{-}⋊T$ by the dimension-and-exact-image argument in the Remarks of Primitive vectors for a Borel pair, applied to the opposite Borel. The projection to $T$ extends every $\lambda \in X(T)$ uniquely to a character of $B^0$ trivial on $U^{-}$. Thus the construction applies to every weight $\lambda$ of the given split torus. The same structural input gives $B^{-}\cap B=T$ and the open big cells $U^{-}B$ and $U{B}^0$ (Root subgroups of a split reductive group, Bruhat decomposition for a split reductive group, Borel subgroups, maximal tori and Borel pairs).

• Well-definedness. The defining condition is checked on $R$-points for every $k$-algebra $R$; since $f$ is a regular function on the affine group scheme $G$, the condition is an identity of morphisms and the set $E(\lambda )$ is a $k$-subspace of $O(G)$ stable under the left regular action: if $f$ satisfies the condition and $g_0\in G(R)$, then $(g_{0}f)(xb)=f(g_0^{-1}xb)=f(g_0^{-1}x)\lambda (b^{-1})=(g_{0}f)(x)\lambda (b^{-1})$ for $x\in G(R)$, $b\in B^0(R)$, so $g_{0}f\in E(\lambda )$.
• The big cell. The opposite Borel $B^0=B^{-}$ is the one appearing in the definition, so an element of $E(\lambda )$ is a function on $G$ whose restriction to each right $B^0$-coset transforms by ${\lambda }^{-1}$; the big cell $U{B}^0$ is the open cell of the Bruhat decomposition associated with $B$ and $B^0$.
• Induced module. The identification with ${\mathrm{Ind}}_{B^0}^G(k_{\lambda })=\{f\in \mathrm{Mor}(G,{\mathbb{A}}^1):f(gb)=b^{-1}f(g)\}$ is the source's definition of the induced module; the translation convention $\lambda (b^{-1})$ matches the left regular action used here.
• Choice scope. For a provided subgroup and character, the equivariance subspace and its left action use no choice principle. AC is inherited only for the supplemental split-Borel projection and big-cell facts above. The comultiplication alone describes right translation, so it is not the coaction label for the left action used here.