Uniqueness of Branching through regular unipotent elements

Let $\mathrm{G}$ be a complex simple algebraic group and let ${\mathrm{G}}_0\subset \mathrm{G}$ be a closed connected subgroup containing a regular unipotent element of $\mathrm{G}$, with semisimple rank at least $2$. Using Dynkin’s classification, we prove that the restriction of an irreducible finite-dimensional representation of $\mathrm{G}$ to ${\mathrm{G}}_0$ determines the representation up to an outer automorphism of $\mathrm{G}$ preserving ${\mathrm{G}}_0$. We extend this method to the diagonal embedding ${\mathrm{G}}_0\hookrightarrow \mathrm{G}\times \mathrm{G}$ for the specific pairs $({\mathrm{S}\mathrm{O}}_{2k}(\mathbb{C})\times {\mathrm{S}\mathrm{O}}_{2k}(\mathbb{C}),{\mathrm{S}\mathrm{O}}_{2k-1}(\mathbb{C}))$, $(E_6\times E_6,F_4)$ and $(Spi{n}_8(\mathbb{C})\times Spi{n}_8(\mathbb{C}),G_2)$ and show that uniqueness continues to hold. Finally, we give examples showing that, in the diagonal setting, restriction to the principal ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ alone is not sufficient to establish uniqueness.