Let ${\mathrm{E}}_n({\mathbb{C}})$ denote the connected complex Lie group of type ${\mathrm{E}}_n$ for $n = 6, 7$. These two groups contain the following reductive pairs:
$\begin{align} T_1({\mathbb{C}}) \times {\mathrm{Spin}}(10,{\mathbb{C}}) & \subset {\mathrm{E}}_6({\mathbb{C}}), \cr T_2({\mathbb{C}}) \times {\mathrm{Spin}}(8,{\mathbb{C}}) & \subset {\mathrm{E}}_6({\mathbb{C}}), \cr T_1({\mathbb{C}}) \times {\mathrm{E}}_6({\mathbb{C}}) & \subset {\mathrm{E}}_7({\mathbb{C}}), \end{align}$
where $T_1({\mathbb{C}})$ and $T_2({\mathbb{C}})$ are complex tori of dimensions 1 and 2 respectively. In this talk, I will describe the dual pair correspondences arising from the minimal representations of ${\mathrm{E}}_6({\mathbb{C}})$ and ${\mathrm{E}}_7({\mathbb{C}})$. These are joint projects with Edmund Karasiewicz and Gordan Savin.
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