Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M\_n(F)×M\_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.


How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?


Thank you.



algebraic closure of commuting pairs of matrices
Note that $M$ is invariant under the $GL\_n(F)$-action given by $a \cdot (x,y):=(axa^{-1},aya^{-1})$. It follows that its closure $N$ is also invariant under $GL\_n(F)$. Since $F$ is infinite and $GL\_n$ is reductive, the rational points $GL\_n(F)$ are Zariski-dense in $GL\_n(K)$ by Borel, Linear Algebraic Groups, Corr. 18.3. (This is probably the piece of information you were missing). 


Then it follows that $N$ is $GL\_n(K)$-invariant, as claimed. 


