$\DeclareMathOperator{\GL}{GL}$Let $G$ be a finitely presented group with generators $g\_1,\dotsc, g\_k$. Suppose we have a family of representations $\rho\_t:G\to \GL(n,\mathbb C)$ with $t\in [0,1]$ smoothly dependent on $t$. Suppose that for any $t\in [0,1]$ and $i\in\{1,\dotsc,k\}$, $\rho\_t(g\_i)$ is diagonalisable, and for $t>0$ all representations $\rho\_t$ are conjugate.


**Question.** Is it true that $\rho\_0$ is conjugate to $\rho\_t$ for $t>0$?


(**PS** *Yves gave a counter-example in a [comment](https://mathoverflow.net/questions/376770/a-limit-of-conjugate-representations-in-mathrmgln-mathbb-c#comment955517_376770)*.)


Is the answer positive, if we additionally ask $\rho\_t(G)\subset \GL(n,\mathbb C)$ to be conjugate to a subgroup of $\operatorname U(n)\subset \GL(n,\mathbb C)$ for all $t$?



A limit of conjugate representations in $\mathrm{GL}(n,\mathbb C)$
It is proven in Lemma 1.25 of the book "Varieties of representations" by Lubotzky and Magid (Memoirs of AMS, vol. 336, 1985) that each $GL(n,{\mathbb C})$-conjugation orbit of a semisimple representation $\rho\in R\_n={\mathrm Hom}(\Gamma, GL(n, {\mathbb C}))$ is Zariski-closed in $R\_n$. If a representation is unitarizable, then it is completely reducible, i.e. [semisimple](https://en.wikipedia.org/wiki/Semisimple_representation#:%7E:text=In%20mathematics%2C%20specifically%20in%20representation%20theory%2C%20a%20semisimple,example%20of%20the%20general%20mathematical%20notion%20of%20semisimplicity.). Thus, you get the property you are after. Note that you do not need finite presentability of $\Gamma$, only finite generation.


