Uniqueness of Branching through regular unipotent elements

Abstract

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 $({\mathrm{SO}}_{2k}(\mathbb{C})\times {\mathrm{SO}}_{2k}(\mathbb{C}),{\mathrm{SO}}_{2k-1}(\mathbb{C}))$, $(E_6\times E_6,F_4)$ 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 \(\mathrm{SL}_2(\mathbb C)\) alone is not sufficient to establish uniqueness.