$V^H:=\{v\in V:hv=v,\,\forall h\in H\}$ is the fixed point set.
I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those given by
$$x=[x_1,\cdots,x_n]\mapsto\left(\begin{array}{cc}\cos2\pi\langle a,x\rangle & \sin 2\pi\langle a,x\rangle \\ -\sin\pi\langle a,x\rangle & \cos2\pi\langle a,x\rangle \end{array}\right);$$ with $a\in\mathbb{Z}^n$ and $\langle a,x\rangle=\sum_ja_jx_j$.
Also, every subgroup of $T$ is compact and abelian, thus its irreducible real representations are either one-dimensional and of real type or two dimensional and of complex type.
I'm also trying to approach it via decompositions in weight spaces, but I'm kinda stuck.