While reading the book 'Langlands correspondence for loop groups', I came across the definition of the Weil group $W_F$ and the Weil-Deligne group $W'_F = W_F \ltimes \mathbb{C}$ with action $$\sigma x\sigma^-1 = ||\sigma||x, \sigma \in W_F,x\in \mathbb{C}.$$ In it, they give the definition of an $n$-dimensional complex representation of $W'_F$:
"An $n$-dimensional complex representation of $W'_F$ is by definition a homomorphism $\rho': W'_F \rightarrow GL_n(\mathbb{C})$, which may be described as a pair $(\rho,u)$, where $\rho$ is an $n$-dimensional representation of $W_F$, $u \in \mathfrak{gl}_n (\mathbb{C})$, and we have $\rho(\sigma)u\rho(\sigma) = ||\sigma||u$."
Now, I understand what $\rho$ means and why you define that action of $\rho(\sigma)$ on $\rho'(\mathbb{C})$, but I don't understand the meaning of $u \in \mathfrak{gl}_n (\mathbb{C})$, shouldn't that be $u \in GL_n(\mathbb{C})$ ?
For a reference of the book: http://goo.gl/pq5XZ, page 4.