Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $\chi_U^\mathbb{R}:G\to\mathbb{R}$ as $\chi_U^\mathbb{R}(g)=\operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $\chi_V:G\to\mathbb{C}$ as $\chi_V(g)=\operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=\mathbb{C}\otimes_\mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $\chi_U^\mathbb{R}$ and $\chi_{e_+(U)}$ for irreducible $U$?

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    $\begingroup$ If $E$ is a real vector space then $ \mathbb{C} \otimes_{\mathbb{R}}E =1 \otimes_{\mathbb{R}}E+i\otimes_{\mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $\mathbb{R}$-linear maps $E \to \mathbb{R}$ have a natural $\mathbb{C}$-linear extension $\mathbb{C} \otimes_{\mathbb{R}}E \to \mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) \to \mathbb{R}$. $\endgroup$ – reuns Jan 15 at 0:31
  • $\begingroup$ But $End(C\otimes_\mathbb{R} U)=C\otimes_\mathbb{R} End(U)$? $\endgroup$ – Andre Gomes Jan 15 at 3:35
  • $\begingroup$ $End_k(V)$ : the $k$-linear maps $V \to V$ $\endgroup$ – reuns Jan 15 at 4:07

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