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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.

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In this formulation (which is completely standard), the image of $1\in \mathbb{C}$ under \rho' is not, as you might expect, $u$, but rather $\text{exp}(u)$, which is indeed in $GL_n(\mathbb{C})$. $u$ itself is nilpotent, and is usually called the monodromy operator attached to the Weil-Deligne representation \rho'. You should check that the claimed relation $\rho(\sigma)u\rho(\sigma)^{-1}=||\sigma||u$ then holds (you forgot the inverse).

A very nice reference for this is Tate's article "Number theoretic background" in "Automorphic Forms, Representations, and L-functions", Proceedings of Symposia in Pure Mathematics Volume 33, Part 2.

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    Ok, thank you very much for your ans$w$er.2011-10-17