TITLE: Fixed subspaces of a family of representations $\rho_t: F_2\to GL(n,\mathbb C)$
QUESTION [5 upvotes]: Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\setminus 0$ such that $A_t(v_t)=B_t(v_t)=v_t$. Is it true that such a common (non-zero) vector exists for all $t\in \mathbb R$?

REPLY [4 votes]: Define the sets
$$M = \{ (A,B,l)\in GL_n(\mathbb{C})\times GL_n(\mathbb{C})\times\mathbb{P}^{n-1}(\mathbb{C}) : Aw=Bw=w \text{ for all } w\in l \},$$
$$N = \{ (A,B)\in GL_n(\mathbb{C})\times GL_n(\mathbb{C}):\exists v\in\mathbb{C}^n\setminus\{0\}, Av=Bv=v\}, $$
where a point of $\mathbb{P}^{n-1}(\mathbb{C})$ is thought of as a line in $\mathbb{C}^n$. The set $M$ is closed and analytic (actually algebraic). Let
$$\pi: GL_n(\mathbb{C})\times GL_n(\mathbb{C})\times\mathbb{P}^{n-1}(\mathbb{C})\to GL_n(\mathbb{C})\times GL_n(\mathbb{C})$$ be the projection. Then $\pi(M)=N$, and since $\pi$ is proper, the set $N$ is closed and analytic.