# Is there any relation between real and complex character functions of irreducible representations of compact lie groups?

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 $$U$$?

• 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 $Tr : End(U) \to \mathbb{R}$. – reuns Jan 15 at 0:31
• But $End(C\otimes_\mathbb{R} U)=C\otimes_\mathbb{R} End(U)$? – Andre Gomes Jan 15 at 3:35
• $End_k(V)$ : the $k$-linear maps $V \to V$ – reuns Jan 15 at 4:07