Convolution Is Pointwise Multiplication
Let \(G = \operatorname{Spec} A\) be an affine group scheme over \(R\) with coordinate Hopf algebra \(A\), and let \(T\) be an \(R\)-scheme. Under the affine identification \[ G(T) \;=\; \operatorname{Hom}_{R\text{-sch}}(T, G) \;\cong\; \operatorname{Hom}_{R\text{-alg}}\bigl(A,\, \Gamma(T)\bigr), \] the group law of \(G\) on \(T\)-points corresponds to the convolution product \[ \varphi * \psi \;=\; \mu_{\Gamma(T)} \circ (\varphi \otimes \psi) \circ \Delta_A , \] with unit \(\eta \circ \varepsilon_A\) and inverse \(\varphi \circ S_A\).
Specialization to characters. For an affine group scheme \(G'\), the pointwise product of characters \(\chi_1 \chi_2 : G' \to \mathbb{G}_m\) corresponds, under the coordinate-ring bridge (Homomorphism of Affine Group Schemes), to the convolution product on \(\operatorname{Hom}_{R\text{-bialg}}(R[\mathbb{Z}], O(G'))\), and under Characters Are Group-Like Elements to the ring product of group-like elements. This is the justification for carrying character groups by convolution structures in the Lean development.
Proof
The group law \(G \times_R G \to G\) corresponds contravariantly to the comultiplication \(\Delta_A : A \to A \otimes_R A\). Given \(T\)-points \(\varphi, \psi : A \to \Gamma(T)\), their product point is the composite \(A \xrightarrow{\Delta} A \otimes A \xrightarrow{\varphi \otimes \psi} \Gamma(T) \otimes \Gamma(T) \xrightarrow{\mu} \Gamma(T)\) — the convolution. The unit point corresponds to the counit and the inverse to precomposition with the antipode, by the same translation of the unit and inversion morphisms. For characters: the pointwise product \(\chi_1\chi_2\) is the composite \(G' \xrightarrow{\Delta_{G'}} G' \times G' \xrightarrow{\chi_1 \times \chi_2} \mathbb{G}_m \times \mathbb{G}_m \xrightarrow{m} \mathbb{G}_m\), whose coordinate-ring translation is exactly the convolution of the corresponding bialgebra maps. On group-like elements \(u_i = \chi_i(e^1)\), convolution evaluates to \(u_1 u_2\).