Let $A, B\in M\_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a cyclic pair.


Is there only a finite number of flags that are stabilized both by $A$ and $B$?


Note, that if we already know that $A$ or $B$ is a cyclic endomorphism, then it is the case.



Commuting matrices and cyclic modules
Let $A=\left(\begin{smallmatrix}1&0&0\\1&1&0\\0&0&1\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}1&0&0\\0&1&0\\1&0&1\end{smallmatrix}\right)$. Then your hypotheses are satisfied, with $v=\left(\begin{smallmatrix}1\\0\\0\end{smallmatrix}\right)$, and there are infinitely many flags stabilised by $A$ and $B$ as long as your field $\mathbb C$ is infinite. The point is that you can start with the span of some linear combination of the second and third basis elements as your one dimensional subspace, stabilised by $A$ and $B$, and then the rest of the flag is determined. Presuming that $\mathbb C$ denotes the complex numbers, this should answer your question.


P.S. I've assumed that your $\mathbb{C}[A.B]$ is a misprint for $\mathbb{C}[A,B]$, the algebra generated over $\mathbb C$ by the commuting elements $A$ and $B$.


