Let $G$ be a finite group and $H_1$ and $H_2$ are two proper subgroups of $G$. Also, let $\rho:G \rightarrow \mathbb{C}^m \times \mathbb{C}^m$ be an irreducible non-trivial representation of $G$. Let $V_1$ and $V_2$ be subspaces of $\mathbb{C}^m$ such that $V_1 \cap V_2=0$. Further, $V_1$ is the maximal subspace such that for every element $h_1 \in H_1$ and $v \in V_1$, $\rho(h_1) v = v$. Similarly, $V_2$ is the maximal subspace such that for every element $h_2 \in H_2$ and $v \in V_2$, $\rho(h_2) v = v$.

Is it true that $V_1$ and $V_2$ are orthogonal to each other? If not, can you provide a counterexample?

Let $G$ be a finite group and $H_1$ and $H_2$ are two proper subgroups of $G$. Also, let $\rho:G \rightarrow \mathbb{C}^m \times \mathbb{C}^m$ be an irreducible non-trivial representation of $G$. Let $V_1$ and $V_2$ be subspaces of $\mathbb{C}^m$ such that $V_1 \cap V_2=0$. Further, $V_1$ is the maximal subspace such that for every element $h_1 \in H_1$ and $v \in V_1$, $\rho(h_1) v = v$. Similarly, $V_2$ is the maximal subspace such that for every element $h_2 \in H_2$ and $v \in V_2$, $\rho(h_2) v = v$.

Is it true that $V_1$ and $V_2$ are orthogonal to each other (w.r.t the standard inner product on $\mathbb{C}^m$) ? If not, can you provide a counterexample?