Reading off the irreducible real representations from complex ones Let $G$ be a finite group. For any $\mathbb{C}G$-module $W$ we define its $\textbf{Frobenius-Schur -ndicator}$ to be
$$
\iota_W=\frac{1}{|G|}\sum_{g\in G}\chi_W(g^2),
$$
where $\chi_W$ is the character of $W$.
My question is whether the following statement is true:
Statement
If $U$ is an irreducible $\mathbb{R}G$-module, then its character has one of the following forms:


*

*$\chi_U$, and $U$ is an irreducible $\mathbb{C}G$-module,

*$2\operatorname{Re}\chi_W$, for some irreducible $\mathbb{C}G$-module $W$.


Moreover, for any irreducible $\mathbb{C}G$ module $V$, one of the following is true:


*

*$\iota(V)=1$, then $V$ is an irreducible $\mathbb{R}G$-module,

*$\iota(V)\neq 1$, then $2\operatorname{Re}\chi_V$ is a character of some irreducible $\mathbb{R}G$-module.

 A: We can define the inner product
$$\langle \chi_1, \chi_2 \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi_1}(g) \chi_2(g)$$
of characters for representations over $\mathbb{R}$ in the same way as over $\mathbb{C}$ (of course here we have $\overline{\chi} = \chi$). It is still true that
$$\langle \chi_V, \chi_W \rangle = \dim \text{Hom}_G(V, W)$$
but the computation of $\text{Hom}_G(V, W)$ is slightly more complicated. We still have that nonisomorphic irreducibles are orthogonal, but if $V$ is irreducible then
$$\langle \chi_V, \chi_V \rangle = \dim \text{End}_G(V)$$
can take one of three possible values, corresponding to the three finite-dimensional real division algebras over $\mathbb{R}$ (Schur's lemma guarantees that $\text{End}_G(V)$ is a division algebra): 


*

*If $\text{End}_G(V) \cong \mathbb{R}$ then $\langle \chi_V, \chi_V \rangle = 1$.

*If $\text{End}_G(V) \cong \mathbb{C}$ then $\langle \chi_V, \chi_V \rangle = 2$.

*If $\text{End}_G(V) \cong \mathbb{H}$ then $\langle \chi_V, \chi_V \rangle = 4$.


Now, if $V$ is an irreducible real representation, consider the complexification $V \otimes \mathbb{C}$. It has the same character as $V$, and hence $\langle \chi_V, \chi_V \rangle$ takes the same values. 


*

*If $\langle \chi_V, \chi_V \rangle = 1$ then $V \otimes \mathbb{C}$ is irreducible.

*If $\langle \chi_V, \chi_V \rangle = 2$ then $V \otimes \mathbb{C}$ is the direct sum of two nonisomorphic irreducibles, whose characters must be complex conjugates of each other. These are given by $V$ together with the action of $\text{End}_G(V) \cong \mathbb{C}$, where there are two ways of identifying $\text{End}_G(V)$ with $\mathbb{C}$ related by complex conjugation.

*If $\langle \chi_V, \chi_V \rangle = 4$ then $V \otimes \mathbb{C}$ is the direct sum of two isomorphic irreducibles, whose character must be real. These are given by $V$ together with the action of any copy of $\mathbb{C}$ inside $\text{End}_G(V) \cong \mathbb{H}$. 


(This does not quite follow from a dimension count but it does follow from the identity $\text{End}_G(V \otimes \mathbb{C}) \cong \text{End}_G(V) \otimes \mathbb{C}$.)
This gives your first statement. The second statement is also true (each of the three cases above corresponds to a possible value of the Frobenius-Schur indicator) but the proof escapes me for the moment. 
